Relativistic Astrophysics and compact objects - Dipartimento di ...

Relativistic Astrophysics and compact objects 

Luca Del Zanna 

Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze 

luca.delzanna@unifi.it 

Part of the course Astrofisica delle alte energie, 

Laurea Magistrale in Scienze Fisiche e Astrofisiche, Università di Firenze

High Energy Astrophysics 

The theory of relativity 

Applications to compact objects 

What is High Energy Astrophysics? 

This basic question does not have an easy answer. Surely the subject covers astrophysical 

environments where high energies are involved, for instance sources of X-ray or γ-ray 

emission. But even radio emission may be produced by ultra-relativistic electrons. In 

general, let us say that it deals with either violent phenomena (e.g. supernova explosions, 

relativistic fireballs) or extreme conditions (e.g. neutron stars matter, supermassive black 

holes, ultra-relativistic jets, relativistically hot plasmas, magnetar-type magnetic fields). 

Typical subjects of High Energy Astrophysics include the following: 

1 Special and General Relativity (high velocities, strong gravitational fields). 

2 Compact objects (neutron stars and black holes). 

3 Non-thermal emission (synchrotron, inverse Compton, hadronic processes). 

4 Non-thermal particles (acceleration of particles and origin of cosmic rays). 

In this course we will try to cover all the above subjects, alternating phenomenological 

issues and basic theory. Notions required: hydrodynamics, electrodynamics, stellar 

structure, basic dynamics and emission processes in Astrophysics, basic plasma physics 

and MHD. It is meant to be followed at the second year of the Laurea Magistrale. 

L. Del Zanna Relativistic Astrophysics and compact objects 2 / 181

Plan of the course 




This is the tentative plan of the corse Astrofisica delle alte energie, 6 CFU (48h), divided 

officially in 3 CFU (24h) for UNIFI (L. Del Zanna) and 3 CFU (24h) for INAF (P. Blasi): 

1 Introduction to High Energy Astrophysics (2h: L. Del Zanna). 

2 Special and General Relativity (12h: L. Del Zanna). 

3 Applications to compact objects (10h: L. Del Zanna). 

4 Phenomenology of compact objects (4h: N. Bucciantini - INAF). 

5 Non-thermal emission (8h: E. Amato - INAF). 

6 Non-thermal particles and cosmic rays (12h: P. Blasi - INAF). 

Suggested books include the following: 

M. Vietri, Astrofisica delle alte energie, Boringhieri. 

S. Rosswog & M. Brüggen, Introduction to High-Energy Astrophysics, Cambridge. 

M. Longair, High-Energy Astrophysics, Cambridge. 

G. Rybicki & A. Lightman, Radiative Processes in Astrophysics, Wiley. 

S. Shapiro & S. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars, Wiley. 

S. Weinberg, Gravitation and Cosmology, Wiley. 

B. Shutz, A first course in general relativity, Cambridge. 


Outline 

1 High Energy Astrophysics 




2 The theory of relativity 

Special Relativity 

General Relativity: the Principle of Equivalence 

Physics in an external gravitational field 

The Einstein field equations 

3 Applications to compact objects 

Explosive events: SNe and GRBs 

Relativistic stars 

Gravitational collapse and Black Holes 

Electrodynamics of compact objects 





Part I: High Energy Astrophysics 





History of High Energy Astrophysics 

1911: First balloon flights by Hess, discovery of cosmic rays. 

1934: Baade and Zwicky predict neutron stars from core-collapse in supernovae. 

1943: Seyfert catalogued the first class of AGNs. 

1956: Radio synchrotron radiation first recognized in astronomical sources (M87). 

1962: First extra-solar X-ray source (Scorpius X-1) in rocket experiments by Giacconi. 

1963: Discovery of quasars (QSOs). 

1966: Rees predicts superluminal motions in AGN jets. 

1967: Bell and Hewish discover pulsars. VLBI program starts. 

1970: First detection of QPOs. Launch of Uhuru (first X-ray dedicated satellite). 

1973: Publication of GRB observations (top secret from 1967). 

1974: Indirect proof of gravitational waves from a binary pulsar system. 

1982: Discovery of the first millisecond (recycled) pulsar in radio. 

1992: Discovery of microquasars (radio jets from compact binaries). 

1997: BeppoSAX discovers the first GRB afterglow. 

1998: First gamma-ray giant flare from a SGR. Magnetars. 

1999: Launch of Chandra and XMM-Newton X-ray missions. 

2003: Modern Cherenkov telescopes (MAGIC, HESS). Double pulsar system found. 

2008: Launch of Fermi gamma-ray satellite. 

2009: Swift (first dedicated GRB misison) discovers a GRB with z=9.4 

2010: AGILE discovers gamma flares from the Crab Nebula, confirmed by Fermi. 


The high-energy sky 




Figure: Brightest sources in the gamma-ray sky are: compact binaries, Pulsar Wind Nebulae, blazars. 





The electromagnetic spectrum 

Useful physical constants (cgs units): 

c = 3.0 10 10 cm s −1 

G = 6.7 10 −8 dyne cm 2 g −2 

h = 6.6 10 −27 erg s 

e = 4.8 10 −10 statcoulomb 

me = 9.1 10 −28 g 

mp = 1.7 10 −24 g 

kB = 1.4 10 −16 erg K −1 

σ = 5.7 10 −5 erg s −1 cm −2 K −4 

M⊙ = 2.0 10 33 g 

R⊙ = 7.0 10 10 cm 

L⊙ = 3.8 10 33 erg s −1 

1 pc = 3.1 10 18 cm 

1 AU = 1.5 10 13 cm 

1 yr = 3.2 10 7 s 

1 eV = 1.6 10 −12 erg 

Figure: The electromagnetic spectrum in different 

photon energy units. The relations are 

E = hν = hc/λ = kB T. 





Supernovae and supernova remnants 

Supernovae (SNe) are catastrophic explosions of 

stars, where the energies involved (Ekin ∼ 10 51 erg) 

are about the whole energy radiated by the 

progenitor in its entire life. They are certainly the 

high energy events studied over the longest time. 

Historical (galactic) events were recorded in 1006, 

1054, 1181, 1572, 1604, and 1680 (1987 in LMC). 

According to SN lightcurves and spectra, there are 

several types of events. Very important is the class 

of type Ia SNe, produced by explosions of white 

dwarves in binary systems, where the dimming 

rate is related to the peak luminosity, thus can be 

employed as cosmological standard candles. 

The other classes are characterized by core 

collapse: for stars with M > 8 M⊙, once the iron 

core has reached the Chandrasekhar mass of 

1.44 M⊙, a rapid collapse sets in until a proto 

neutron star with ρnuc = 2.6 10 14 g cm −3 and 

T ∼ 10 11 K is formed. Most of the energy is carried 

away by neutrinos, then material starts to bounce 

back leading to the explosion of the outer layers. 

Figure: Different lightcurves for type Ia 

and II SNe and the cartoon of 

core-collapse for type II SNe. 





A supernova event creates a remnant (SNR) that 

may shine for up to 10 5 years, while diluting stellar 

metal-enriched material in the ISM. The forward 

shock of a SNR is also a site for particle 

acceleration. In 1934, Baade and Zwicky, other 

than predicting the existence of NSs, also 

estabilished the supernova paradigm: the galactic 

cosmic rays (up to 10 15 eV) can be produced using 

a small fraction of SN energy. 

Galactic cosmic rays have an energy density 

uCR ∼ 1 eV cm −3 in a volume Vgal ∼ 200 kpc 3 and 

have a corresponding escape time of tesc ∼ 10 7 yr. 

The rate of production by SNe must be 

uCR Vgal/tesc ηESN/tSN, 

and for ESN ∼ 10 53 erg and tSN ∼ 100 yr, an 

efficiency of η = 0.05 seems to be enough. 

Observations of gamma rays from SNRs due to 

hadronic decay would provide the final proof of the 

paradigm. 

Figure: Composite images of the 

Tycho and Kepler SNRs. 





A special class of SNRs is provided by plerions, or 

Pulsar Wind Nebulae (PWNe). Their prototype is the 

Crab Nebula (M1, in the Taurus), the SNR of the 1054 

supernova observed by the Chinese astronomers. 

Other than the usual outer shell, due to the blast wave 

interacting with ISM (threaded by optical filaments due 

to Rayleigh-Taylor instabilities), the central part is filled 

with a hot bubble produced at the termination shock by 

the conversion of kinetic energy of the ultrarelativistic 

pulsar wind into heat and particles. In 2000 the X-ray 

Chandra satellite discovered jets in many PWNe. 

PWNe emit non-thermal emission at all wavelengths, 

synchrotron radiation from radio to gamma 

(ultrarelativistic electrons spiralling along magnetic 

fields) and Inverse Compton (same electrons 

interacting with background photons) from GeV to PeV 

photon energies. Typical PWN luminosities are 

L ∼ 10 38 erg s −1 . Just before the discovery of pulsars, 

Franco Pacini predicted that a young fast rotating 

magnetized NS could release via dipole radiation the 

right amount of energy to power the Crab Nebula. 

Figure: Optical and X-ray image 

of the Crab Nebula and its 

multiwavelength spectrum. 


Pulsars and magnetars 




Pulsars and magnetars are the most spectacular 

manifestations of NSs, the relic of SN explosions. 

Pulsars were discovered in 1967 by Jocelyn Bell, a 

PhD student, as sources of extremely periodic 

radio signals. Now we know about 2000 pulsars, 

all of them in our Galaxy. If pulsars are collapsed 

NSs, we expect a typical radius 

R ∼ (M⊙/ρnuc) 1/3 ∼ 10 km, 

and from the conservation of angular momentum 

and magnetic flux 

ΩR 2 = const, BR 2 = const, 

we derive that a newly born pulsar has P ∼ 1 ms 

and B ∼ 10 12 G. The periodic signal is due to 

some sort of (non-thermal) emission from the 

poles of the rotating magnetosphere, making an 

angle α with the rotation axis and pointing 

periodically towards the Earth. 

Figure: Cartoon of a pulsar. The 

pulsed radio emission is produced at 

the poles of the magnetosphere. 





Let us now describe the Pacini-Gold model of the 

rotating magnetic dipole in vacuum. If z is the 

rotation axis, the magnetic moment is 

m = (BpR 3 /2)(sin α cos ωt, sin α sin ωt, cos α), 

and the radiated EM energy in time is 

˙E = −(2/3c 3 )| ¨ m| 2 = −B 2 p R6 ω 4 sin 2 α/6c 3 , 

that must be compensated by rotational energy 

losses 

˙Erot = Iω ˙ω ⇒ ˙ω = −Kω 3 . 

Immediate observable quantities are the period 

P = 2π/ω and its slowdown rate ˙P. Thanks to this 

simple model, a pulsar in the P − ˙P diagram has a 

ready estimate for the following quantities 

˙E ∼ ˙P/P 3 

, Bp sin α ∼ P ˙P, τ ∼ P/ ˙P. 

Millisecond pulsars with low magnetic fields are 

believed to be recycled old objects, spun up by 

accretion in a binary system. 

Figure: The P − ˙P diagram for pulsars. 

Only the blue region allows the radio 

emission to operate. 





We know that NSs with a B ∼ 10 12 G magnetic 

field are born quite naturally in core-collapse 

supernovae. What happens if in the hot proto NS 

dynamo sets in and B is further amplified? For an 

initial period of 1 ms we expect B ∼ 10 15 G, so that 

the corona of such magnetar would contain an 

energy 

Emag ∼ (B 2 /8π)(4π/3)R 3 ∼ 10 48 erg. 

Giant flares associated to magnetic field 

reconnection events may be observed as 

gamma-ray bursts. 

We believe that Soft Gamma Repeaters (SGRs, 

and the lower energy class of Anomalous X-ray 

Pulsars, AXPs) may be the proof: sudden releases 

of gamma rays of 10 46 erg in a second or less 

(more than for SNe!), followed by oscillations 

related to the NS rotation. Such high magnetic 

fields are confirmed by the measure of rapid 

spin-down rates, leading to rather long periods 

(P ∼ 5 − 8 s). 

Figure: A cartoon of a magnetar and 

the lightcurve of a SGR giant flare. 


Compact binary systems 




About half of all stars are found in binary (or 

multiple) systems, when at least one of the 

members is a compact object (NS or BH) we call 

them compact binary systems of X-ray binaries. 

When a young, O/B-type, luminous (L > L⊙) and 

massive star (M > 10M⊙) is the donor companion, 

a strong stellar wind is present and the accretion is 

due to the transonic Bondi-Hoyle mechanism, 

modified by relative rotation (HMXB systems). 

These systems often show rather regular X-ray 

pulsations. 

When an evolved, low-mass star that has 

expanded to fill its Roche lobe is the donor star, 

accretion occurs via an accretion disk (LMXB 

systems). From the emitted radiation we can infer 

the presence of coronal disk winds, transient hot 

spots, and of quasi-periodic oscillations (QPOs), 

related to the dynamics of the disk and of the 

accretion. These systems are a necessary 

evolution step towards millisecond pulsars (spun 

up by accretion) and binary pulsars. 

Figure: Doppler broadened lines due 

to winds from an accreting torus and 

high-mass and low-mass X-ray binary 

systems. 





Binary NS-NS systems (at least one of them must 

be a pulsar in order to be detected!) are unique 

laboratories for testing General Relativity (GR). 

The famous Hulse-Taylor system shows an orbital 

decay due to emission of gravitational waves 

(GWs), to date only this type of indirect proof is 

available. Moreover, post-Newtonian parameters 

such as periastron advance, gravitational redshift 

and time dilation, change in orbital period and 

Shapiro delay can be measured and used to infer 

the single NS masses. 

In 2003/4 the first double pulsar system was 

discovered, with an orbital period of just 2.4 h! GR 

effects are enhanced and the relative motions 

allow us to even probe the one pulsar’s 

magnetosphere: in this fantastic astrophysical 

laboratory it has been proved that plasma is 

trapped in the closed magnetic field dipole within 

the light cylinder. 

Figure: Emission of GWs in the 

Hulse-Taylor binary and General 

Relativity effects measured in the 

double pulsar system. 


Gamma Ray Bursts 




Gamma Ray Bursts (GRBs) were discovered in 

1967 by the USA Vela satellites, looking for 

gamma emission due to nuclear experiments. 

However, only in 1973 the discovery was made 

public: a cold war gift to Astronomy! 

Lightcurves of GRBs are very irregular, duration 

varies from 10 −2 to 10 3 seconds. Variability occurs 

on τ ∼ 1 ms scales, then the typical size of the 

source must be, to preserve coherency of the 

signal 

R < c τ 300 km, 

comparable to NS or stellar mass BH radii. Until 

the 80s there was a general consensus that the 

bursters were NSs in our Galaxy. 

From the count distribution in duration a clear 

distinction between short GRBs (less than 2 s) and 

long GRBs (more than 2 s) can be made. Spectra 

are also different, those of short GRBs are harder 

(larger fraction of high-energy photons). 

Figure: Lightcurves from several 

GRBs and distribution in duration. 





During the 90s we had the proof that GRBs are 

actually distant events originated in other galaxies. 

BATSE found a perfectly isotropic distribution in 

the sky, and in 1997 the Italian-Dutch BeppoSAX 

mission measured the first X-ray afterglow 

associated with a GRB, apparently related to a 

faint galaxy. Later also radio and optical afterglows 

were found, and nowadays Swift currently 

measures spectral lines cosmological redshifts 

(the record is z = 9.4, observed in 2010). The rate 

is one GRB every 10 7 years per galaxy. 

Of course this created a problem: some GRBs 

have been observed with astonishing isotropized 

energies of Eiso = 4.5 10 54 erg, even more than 

M⊙c 2 = 1.8 10 54 erg. Only if we assume that the 

observed radiation comes from a jet of half 

aperture θjet pointing towards us we can save the 

situation, since 

Etrue 

Eiso 

= ∆Ω 

4π 

= 2 · 2π 

4π 

θjet 

0 

sin θdθ θ2 

jet 

2 , 

leading to reduction factors of about 100. 

Figure: Proofs of the extragalactic 

origin of GRBs: BATSE and the first 

afterglow by BeppoSAX. 





What is the origin of GRBs? Soon after their discovery it 

was predicted a connection with the death of massive stars 

and supernova events. The smoking gun was in 2003, 

when a supernova occurred 6 days later in the same 

position of a bright GRB, confirming a similar event 

observed by AGILE in 1998. The most plausible model for 

long GRBs is that of a collapsar: a massive iron core 

(M > 10M⊙) of a rapidly rotating Wolf-Rayet star may 

collapse into a Kerr BH accreting material from a torus. 

Polar relativistic jets are believed to be collimated by 

magnetic fields and later escape the stellar progenitor. 

Short GRBs have been observed by Swift from 2004 to be 

associated to elliptical galaxies (no young massive stars, 

many stellar relics). In a close binary NS-NS system, after 

a long inspiral phase, the final merging produces a Kerr BH 

surrounded by a T = 10 10 K hot disk, and jets are 

expected to form as in the previous case. 

In both cases, the available gravitational binding energy 

seems to be more than enough: 

E GM2 

R 

3 1053 

M 

M⊙ 

2 −1 R 

erg. 

10 km 

Figure: Mainstream models for 

GRBs: collapsar for long GRBs 

and binary NS-NS merger for 

short GRBs. 





We have seen that Γ > 100 relativistic jets must 

form in small polar angles θjet < 10 ◦ . We shall see 

that radiation from a moving source is beamed 

within an angle θ ∼ 1/Γ, this leads to observable 

change of slopes in lightcurves as the jet slows 

down (achromatic breaks). 

The model for a sudden release of relativistic 

energy with 

η = E/Mc 2 = e/ρc 2 ≫ 1, 

and of its propagation is said fireball. The GRB 

gamma prompt emission is assumed to be 

non-thermal radiation produced in colliding shock 

fronts within the jet. The X-ray and optical 

afterglow occurs after interaction with ISM. 

Viewing angle effects may also explain lower 

energy events (X-Ray Rich GRBs and X-Ray 

Flashes), since relativistic beaming and Doppler 

effects (enhancing E and ν, respectively) are 

reduced for jets observed off-axis. We shall 

discuss these effects later in the course. 

Figure: The fireball model and 

relativistic effects for lower energy 

events. 


Active Galactic Nuclei 




Active Galactic Nuclei (AGNs) are the central part 

(less than 10 ly) of active galaxies, about 3% of the 

known galaxies. They were discovered and 

recognized as extragalactic sources between the 

50s and 60s. Huge zoology: radio galaxies, 

quasars (QSOs), Seyfert, BL Lac, Markarian, 

blazars... 

The main characteristics (not necessarily all 

present in all classes) of active galaxies hosting 

AGNs are: 

compact size, luminous center, 

spectra with strong emission lines, 

strong Doppler-broadened emission lines, 

strong ultraviolet emission from center, 

strong non-thermal emission, 

jets and double radio lobes, 

short timescale variability at all wavelengths. 

Figure: Composite image of 

Centaurus A. 





Typical bolometric luminosities for quasars are 

L ∼ 10 46 erg s −1 , 

at least 100 times standard galaxies, 

concentrated in a very compact region. Time 

variability is observed in some AGNs to be as 

small as τ ∼ 1h, thus the source size must be, 

to preserve coherence of the signal 

R < cτ 3.5 10 −5 pc, 

which is about the size of our Solar System! 

Figure: VLBA observations of 

Keplerian motions in an AGN disk. 

Accretion from a disk onto a supermassive black hole (SMBH) is the answer. Consider 

matter ∆M = ˙M∆t falling radially from infinity and converting its kinetic energy into heat 

and radiation when it stops at a radius R. The available power is 

L = 1 ˙Mv 2 

2 ff = GM ˙M/R = ξ ˙Mc 2 , 

where ξ is the efficiency. For a SMBH with M ∼ 10 8 M⊙ the Schwarzschild radius is 

Rs = 2GM/c 2 10 −5 pc, 

and ξ = 1/2. For comparison, nuclear fusion (direct conversion of rest mass to energy) in 

the Sun leads to ξ = 0.007. The observed luminosity can be reached with ˙M ∼ 1 M⊙ yr −1 . 





Another clear indication that accretion onto a 

supermassive central object is what powers AGNs 

comes from the observation that AGN luminosities are 

always smaller than the so-called Eddington limit 

LEdd = 4πGMmpc 

10 

σT 

46 

 

M 

108 

erg s 

M⊙ 

−1 , 

where σT = 6.65 10 −25 cm 2 is the Thomson cross 

section for electrons and where the mass of the SMBH 

is inferred either from the bulge luminosity or from its 

virial velocity dispersion. 

The Eddington limit is due to the back reaction of the 

radiation produced by the extremely hot accreting gas. 

In a fully ionized plasma, protons and electrons are 

bound to move together, then the limit is found by 

equating the gravitational and radiation forces 

F = GMmp 

r 2 

L/c 

= σT , 

4πr2 where gravity on electrons and radiation pressure on 

protons (σ ∝ m −2 ) have been neglected. 

Figure: Hubble image of a torus 

in a galactic center; distribution of 

AGNs in the L − MBH plane. 





We have seen that a SMBH accreting mass from a 

torus is a good candidate to explain AGN 

energetics. Other observational features like 

emission lines and can be explained within a 

unified model by changing some parameters (most 

notably the viewing angle). 

One important open question is, how are the polar 

jets accelerated to ultrarelativistic speeds? Two of 

the most successful mechanisms involve the 

presence of the magnetic field: 

Magnetocentrifugal due to disk’s field, 

Blandford-Znajek due to extraction of energy 

from the rotating BH’s ergosphere. 

Jets must be relativistic because we observe 

apparent superluminal velocities in the plane of the 

sky from blobs in jets with Vjet c almost pointing 

towards us (blazars). We are going to see that 

vapp ≤ Γjetc, ⇒ Γjet ≥ vapp/c, 

and, for instance, in 3C 373 we find 

vapp 10c ⇒ Γjet ≥ 10 ⇒ Vjet/c = (1−1/Γ 2 

jet )1/2 ≥ 99%! 

Figure: The jet of M87 with features in 

apparent superluminal motion. 





The demonstration does not involve relativity, but 

only geometry under the assumption of a finite 

light propagation speed. The apparent velocity in 

the plane of the sky is 

vapp = 

Vτ sin θ 

τ − Vτ cos θ/c = 

V sin θ 

1 − (V/c) cos θ , 

where the first denominator is the arrival time of 

photons emitted in an interval τ. For a given speed 

V of the jet, vapp is a function of the observing 

angle θ. This function has a maximum for 

where 

cos θc = V/c, sin θc = Γ −1 , 

vapp(θc) = 

V/Γ 

= ΓV, 

1 − (V/c) 2 

thus for V c the maximum observed speed is 

basically vapp ≤ Γc, as used in the previous slide. 

Thus, special relativity is not violated. On the 

contrary, we have a clear demonstration of the 

presence of relativistic velocities. 

Figure: Explanation of superluminal 

motions of relativistic jets. Below: the 

function vapp(θ)/c for different values 

of V (or Γ). 









Part II: the theory of relativity 














Special Relativity: introduction 





The laws of Newtonian mechanics are invariant in any inertial system (where a body not 

subject to forces moves with uniform velocity), this is known as the Galilean principle of 

relativity. Given a transformation from an inertial system (t, x) to another one (t ′ , x ′ ) 

moving at constant speed v 

t ′ = t, x ′ = x − vt, (1.1) 

velocities and accelerations transform as u ′ = u − v ⇒ a ′ = a, so there is no way to detect 

a difference in any force obeying F = ma. 

This is not true for Maxwell’s equations, containing the speed of light c. For example, the 

equation for electromagnetic waves 

[−(1/c 2 )∂ 2 t + ∇2 ]f(t, x) = 0, (1.2) 

is not invariant, since substituting ∇ = ∇ ′ , ∂t = ∂t ′ − v · ∇′ we get something quite different. 

Thus, at the end of the 19 th century people were convinced that the light propagates in a 

privileged system (that of ether). In 1887 Michelson and Morley demonstrated that, within 

5 km s −1 , the velocity of light did not change during Earth’s orbital motion. However, in 1892 

Lorentz proposed that bodies contract like l = l0(1 − v 2 /c 2 ) 1/2 in the direction of motion 

respect to ether, explaining the negative result, and later, with Poincaré, proposed the 

famous Lorentz transformations for distribution of moving charges, that leave Maxwell’s 

equations invariant and predict the contraction. The discovery of the electron seemed to 

support these ad hoc hypotheses for a while, until experiments of light refraction in moving 

liquids led to a complete rejection of the ether option. 









The Principle of Relativity and Lorentz transformations 

In 1905 Albert Einstein (1879 - 1955) proposed a radical solution: by extending Galilean 

relativity and assuming the validity of Maxwell’s equations, he modified Newton’s laws and 

the concepts of absolute time and space. The postulates of Special Relativity are: 

1 Principle of Relativity. The laws of nature are the same in any inertial system. 

2 The speed of light is also the same in any inertial system, independently of the relative 

motion between the source and the observer. 

Here special means that we are assuming the existence of global inertial frames, where 

bodies not subject to forces maintain their status of motion with uniform velocity. In General 

Relativity this will be true just locally. From now on c = 1 and t → ct will be a distance. 

Galilean transformations are replaced by Lorentz transformations 

where 

t ′ = γ(t − v · x), x ′ = γ(x − vt), x ′ ⊥ = x⊥, (1.3) 

γ := (1 − v 2 ) −1/2 , (1.4) 

is the Lorentz factor. Notice that time and space are now mixed up (in the direction parallel 

to v), this is the only way to have a constant speed of light and invariance of Maxwell’s 

equations. The inverse relations are found by swapping (t, x) with (t ′ , x ′ ) and the sign of v. 

These transformations can be further generalized to translations in time and to rotations 

(the Poincaré group). 









Basic kinematic consequences of Lorentz transformations 

The immediate consequences of Lorentz transformations are radical changes in the 

concepts of simultaneity of events, measures of lengths and time intervals. 

Loss of simultaneity. Consider light emitted isotropically from a point B along x ′ , 

reaching points A and C along the same axis with x ′ A < x′ B < x′ C and 

x ′ C − x′ B = x′ B − x′ A = ∆l0, where for simplicity we assume v x x ′ . For the observer 

in B in the primed system the photons will reach A and C simultaneously, since the 

speed of light is unchanged by the motion. For an observer still in B at the moment of 

emission but steady in the unprimed system the photons will reach A first, thus the 

concept of simultaneity is relative. 

Contraction of length. In the same example, the distance observed in the steady frame 

is, according to (1.3), 

xB − xA = v(tB − tA ) + γ −1 (x ′ B − x′ A ), tA = tB ⇒ ∆l = γ −1 ∆l0 ≤ ∆l0 , (1.5) 

where he index 0 indicates the proper length in the rest frame. 

Time dilation. Similarly, consider an event occurring at the same place in the primed 

system, for instance the creation (1) and decay (2) of a particle. We find 

t2 − t1 = γ[t ′ 2 − t′ 1 + v(x′ 2 − x′ 1 )], x′ 2 = x′ 1 ⇒ ∆t = γ∆t0 ≥ ∆t0 , (1.6) 

where the index 0 indicates the proper time in the rest frame. Particles moving at high 

Lorentz factors are actually seen to live much longer than steady particles. 





Transformation of velocities and aberration 





Consider a point with velocity u ′ := dx ′ /dt ′ in the moving frame and u := dx/dt in the 

observer frame. From the differentials of the inverse of the relations (1.3) we find 

dt = γ(dt ′ + vdx ′ ), dx = γ(dx ′ + vdt ′ ), dy = dy ′ , dz = dz ′ , (1.7) 

hence the velocity measured in the laboratory frame is 

u = 

u ′ + v 

1 + v · u ′ , u⊥ 

u 

= 

′ ⊥ 

γ(1 + v · u ′ ) 

For parallel velocities u1 (= v) and u2 (= u ′ ) the composition rule is 

. (1.8) 

u = u1 + u2 

, (1.9) 

1 + u1u2 

and from (1 − u1)(1 − u2) > 0 it is easy to see that u < 1, as expected. 

When instead u ′ makes an angle θ ′ 0 with v, we find the formula for relativistic aberration 

tan θ = |u⊥| 

|u| = 

u ′ sin θ ′ 

γ(u ′ cos θ ′ + v) 

, (1.10) 

where u ′ = |u ′ |, and the azimuthal angle φ remains unchanged. If light is emitted in the 

moving system, we have u ′ = 1 in the above formula. 





Relativistic beaming and relativistic Doppler effect 





Consider then a photon with θ ′ = π/2, in the laboratory frame we observe 

tan θ = 1/(γv) ⇒ sin θ = 1/γ , (1.11) 

and for γ ≫ 1 we have θ ∼ 1/γ ≪ 1. Thus, light emitted isotropically in the comoving frame 

is observed to be almost entirely collimated along v. This is known as relativistic beaming 

and it can make a source much brighter compared to isotropic emission. 

Consider now light emitted from a source moving at speed v, covering a distance s = vτ 

making an angle θ with the line of sight, in the laboratory frame. The classical Doppler 

effects (for a limited speed of light!) predicts a difference in arrival times 

∆tA = τ − s cos θ = τ(1 − v cos θ). 

If τ = γ/ν0 corresponds to a period of the emitted radiation, including relativistic time 

dilation, the observed frequency will be ν = ∆t−1, that is 

A 

ν = Dν0, D := [γ(1 − v cos θ)] −1 , (1.12) 

where D is called Doppler boosting factor, and we may have D ≫ 1. 


Spacetime intervals 








In 1908 Minkowski put SR in geometrical terms. The second postulate is the same of 

saying that the interval between two spacetime events 

∆s 2 := −∆t 2 + ∆x 2 , (1.13) 

is invariant under Lorentz transformations. Spacelike separations with ∆s 2 > 0 would 

require faster than light signals and are causally disconnected, null separations with 

∆s 2 = 0 characterize light propagation (and it is the equation for the separation surface in 

the figure, known as light cone), timelike separations with ∆s 2 < 0 are either in the causal 

past or future. 

In this latter case it is always possible to define an inertial frame where the two events occur 

at the same place (rest frame) and the quantity 

 

∆τ := −∆s2 = γ −1 ∆t (1.14) 

is called proper time. 





The Minkowski spacetime and the Lorentz group 





In general, any (linear) Lorentz transformation between inertial systems x µ := (t, x) and 

x µ′ 

:= (t ′ , x ′ ), can be written by using the 4 × 4 Jacobian as 

x µ′ 

= Λ µ′ 

µ x µ , (1.15) 

where from now on Greek indices will indicate spacetime 4-vectors 0123, with x0 ≡ t, and 

we shall make use of Einstein’s summation convention on repeated indices. In the case of a 

simple translation of the primed system along x, the Jacobian is 

Λ µ′ 

µ = 

⎛ 

⎜⎝ 

γ −γv 0 0 

−γv γ 0 0 

0 0 1 0 

0 0 0 1 

The interval definition may be rewritten in differential form as 

⎞ 

. (1.16) 

⎟⎠ 

ds 2 := ηµνdx µ dx ν , (1.17) 

where we define the Minkowski metric tensor as the diagonal 4 × 4 array 

ηµν := diag{−1, +1, +1, +1} . (1.18) 

Note that we shall adopt throughout the signature (−1, +1, +1, +1). 









The 4D space generated by this metric is called Minkowski spacetime M4. The Minkowski 

tensor is isotropic, that is invariant in any inertial frame of M4, then det(η) = −1 and 

η −1 ≡ η. Lorentz transformations leave ds ′2 unchanged if and only if 

Λ µ′ 

µ Λ ν′ 

ν ηµ ′ ν ′ = ηµν, (1.19) 

where ηµ ′ ν ′ is the same Minkowski tensor defined above, as can be easily checked 

ds ′2 = ηµ ′ ν ′ dxµ′ dx ν′ 

= ηµ ′ ν ′Λµ′ µ dx µ Λ ν′ 

ν dxν = ηµνdx µ dx ν = ds 2 . 

It is instructive to verify that Lorentz transformations in (1.16) satisfy (1.19). 

Let us now demonstrate that Lorentz transformations form a group in M4 with respect to 

matrix multiplication. In matrix form, the condition (1.19) becomes 

Λ T ηΛ = η . (1.20) 

Since det(Λ) 2 = 1, there is always the inverse transformation Λ −1 , given by 

This is still a Lorentz transformation, as it satisfies 

Λ −1 = ηΛ T η. (1.21) 

(Λ −1 ) T ηΛ −1 = (ηΛη)η(ηΛ T η) = η(ΛηΛ T )η = η. 

Moreover, if Λ1 and Λ2 are Lorentz transformations, then also Λ = Λ2Λ1 

Λ T ηΛ = (Λ2Λ1) T η(Λ2Λ1) = Λ T 

1 (ΛT 

2 ηΛ2)Λ1 = Λ T 

1 ηΛ1 = η, 

hence these two properties ensure that we have a group. 


Basics of tensor calculus 








We now make a mathematical digression, needed to write invariant equations according to 

the Principle of Relativity. Given a vector A(x 1 , x 2 ) defined on a surface, not necessarily a 

plane, and the coordinate basis (e1, e2), this can be decomposed in two ways 

A = A 1 e1 + A 2 e2; A1 = A · e1, A2 = A · e2 (A = A1e 1 + A2e 2 ), 

where upper (lower) indices characterize contravariant (covariant) components. 

It is possible to switch from one representation to another through the metric tensor and its 

inverse, defined as (in any number of dimensions) 

gµν := eµ · eν, g µν gνρ = δ µ ρ, (g µν := e µ · e ν , eµ · e ν = δ ν µ) (1.22) 

where δ µ ν is the Kronecker symbol. Immediate consequences for any vector A = A µ eµ are 

Aµ = gµνA ν ; A µ = g µν Aν; A · B = gµνA µ B ν = g µν AµBν = AµB µ . (1.23) 









Consider now a change of coordinate basis, that we write in the form 

eµ = eµ ′ Mµ′ µ , (1.24) 

where M is the (linear) transformation matrix. Any vector A is an object existing 

independently on the particular basis chosen, hence 

A = A µ eµ = A µ′ eµ ′ = A µ eµ ′ Mµ′ µ ⇒ A µ′ 

= M µ′ 

µ A µ , (1.25) 

which is the rule of transformation for contravariant components. Consider now the scalar 

product of any two vectors A and B, which in turn must be invariant, we have 

A · B = A µ Bµ = A µ′ 

Bµ ′ = Mµ′ µ A µ µ 

Bµ ′ ⇒ Bµ ′ = M 

µ ′ Bµ, (1.26) 

that is covariant vector components transform with (M −1 ) T . 

In the Minkowski spacetime M4 we have M µ′ 

µ = Λ µ′ 

µ and gµν = ηµν. A 4-vector of M4 is any 

object V µ that under a Lorentz transformation changes according to V µ′ = Λ µ′ 

µ V µ , precisely 

as the coordinates in (1.15). Note that the simple form of (1.18) provides the properties 

V µ = (V 0 , V), Vµ = (V0, V) ⇒ V0 = −V 0 . (1.27) 

AµB µ = ηµνA µ B ν = −A 0 B 0 + A · B, (1.28) 

where from now on the vector symbols will indicate spatial 3-vectors alone. 









Let us now provide more general definitions, valid in any differentiable manifold (and thus in 

GR as well). Let x → x ′ be a general coordinate transformation with non-singular Jacobian 

Λ µ′ 

µ := ∂xµ′ 

µ′ 

≡ ∂µx 

µ 

∂x 

, (1.29) 

which only for the linear Lorentz transformation coincides with (1.15), with inverse matrix 

Λ µ 

µ ′ := ∂xµ 

∂x µ′ ≡ ∂µ ′ x µ , Λ µ 

µ ′ Λ µ′ 

ν = δ µ ν . (1.30) 

A contravariant vector has components V µ transforming as the differential displacement 

dx µ′ 

= Λ µ′ 

µ dx µ ⇒ V µ′ 

= Λ µ′ 

µ V µ , (1.31) 

whereas a covariant vector Vµ transforms as the components of a gradient of a function 

µ 

∂µ ′φ = Λ µ ′ µ 

∂µφ ⇒ Vµ ′ = Λ µ ′ Vµ. (1.32) 

For any mixed tensor, say T µν 

λ , we have transformation rules such as 

T µ′ ν ′ 

λ ′ = Λ µ′ 

µ Λ ν′ 

ν Λ λ 

λ 

µν 

′ T λ . 

Analogies with the previous case are apparent if we let eµ ≡ µ 

∂µ, for which eµ ′ = Λ µ ′eµ. 

In particular gµν is a fully covariant tensor, while the Kronecker symbol δ µ ν is a mixed tensor 

(both of rank 2). 









The next step towards writing invariant equations is how to combine different tensors. This 

is accomplished through a few simple algebraic operations, here we just provide one 

example for each and we leave the demonstration of their invariance to the reader. 

1 Linear combinations: 

2 Direct products: 

3 Contractions: 

T µ ν := aA µ ν + bB µ ν. (1.33) 

µ λ 

T ν := A µ νB λ . (1.34) 

T µν µ λν 

:= T λ . (1.35) 

The operation of lowering or raising the indices falls in these categories, since it involves 

contraction of indices with the metric tensor gµν or its inverse g µν . 

An important consequence of these rules is, for example, that the line element 

is effectively an invariant (a scalar): 

ds 2 := gµνdx µ dx ν , (1.36) 

ds ′2 = gµ ′ ν ′ dxµ′ dx ν′ 

= Λ µ 

µ ′Λ ν µ′ 

ν ′ gµνdx dx ν′ 

= gµνdx µ dx ν = ds 2 , 

as already found in Minkowskian spacetime where gµν = ηµν, and equivalently the scalar 

= AµB µ . 

product of any two vectors Aµ ′ Bµ′ 









Not all quantities appearing in relativity are scalars, vectors, or tensors. An important class 

is that of tensor densities. The first example, a scalar density, is the determinant of the 

metric tensor (negative for Lorentzian spacetimes, in particular −1 for gµν = ηµν) 

g := det(gµν), (1.37) 

for which in the transormation rule appears the determinant of the Jacobian 

g ′ = Λ −2 g, Λ := det(Λ µ′ 

µ ) ≡ [det(Λ µ 

µ ′ )] −1 ≡ (Λ ′ ) −1 . 

A tensor density is similar. In particular, a tensor density of weight w transforms as a tensor 

but with an extra factor Λ w . Now, since Λ w = |g| w/2 /|g ′ | w/2 , any tensor density of weight w 

multiplied by |g| w/2 is a tensor. One famous example is the volume element dx 4 , for which 

d 4 x ′ = Λd 4 x ⇒ |g ′ | 1/2 d 4 x ′ = |g| 1/2 d 4 x, (1.38) 

so that |g| 1/2 d 4 x is an invariant volume element. Another case is the Levi-Civita tensor 

density ɛµνρσ, which has weight w = −1. Moreover, we may demostrate that 

ɛµνρσ = gɛ µνρσ , so that we retrieve tensors by defining 

|g| −1/2 ɛµνρσ = −|g| 1/2 ɛ µνρσ = [µνρσ], (1.39) 

where [µνρσ] is the completely antisymmetric symbol, which is +1 for any even 

permutation of [0123], −1 for any odd permutation of the reference sequence, 0 if any 

indices are repeated. 









The derivative of vectors and tensors not necessarily provides another tensor. For example, 

for a contravariant vector V µ′ = Λ µ′ 

µ V µ , we find 

ν′ 

∂µ ′ V = Λ µ 

µ ′ ∂µ(Λ ν′ 

ν V ν ) = Λ µ 

µ ′ Λ ν′ 

ν ∂µV ν + Λ µ 

µ ′ ∂µ∂νx ν′ 

V ν , 

and only for linear transformations such as the Lorentz ones the second term vanishes 

leaving us with a transformation for a rank 2 mixed tensor. Then consider tensors defined 

only along a curve x µ (τ), where τ is any scalar parameter (e.g. the proper time). For 

instance, the usual contravariant vector V µ transforms as 

V µ′ 

(τ) = Λ µ′ 

µ (τ)V µ µ′ dV 

(τ) ⇒ 

dτ 

dV 

= Λµ′ µ 

µ 

µ′ µ dxν 

+ ∂µ∂νx V 

dτ dτ , 

and again only for linear transformations the second term vanishes so that we find a 

tensorial transformation rule. Hence, in Minkowski spacetime, partial derivatives of 

4-vectors and tensors are therefore tensors. 

We are finally ready to make physics. If we manage to write down (differential) equations 

which are valid in a particular inertial frame (for example the rest frame of a particle), and 

these are written in covariant (tensorial) form, the Principle of Relativity insures that these 

will be valid in any inertial frame, since Lorentz transformations leave these equations 

invariant in the Minkowski spacetime M4. 


Particle mechanics 








Consider a particle moving in M4 within its light-cone on a trajectory (world-line) 

x µ := x µ (τ) = (t, x), (1.40) 

where the proper time τ (the time measured in the system comoving with the particle) has 

been here preferred as curve parameter to s. Our goal is to generalize the Newtonian 

definitions v = dx/dt, and a = d 2 a/dt 2 by using 4-vectors of M4. Let us start with 

ds 2 = −dt 2 + dx 2 = −[1 − (dx/dt) 2 ]dt 2 = −(1 − v 2 )dt 2 ⇒ dτ = 

 

−ds 2 = γ −1 dt, 

where t is the time measured in the laboratory system, the appropriate definition for the 

4-velocity of the particle is 

u µ := dxµ 

dτ 

= (γ, γv) , (1.41) 

since it is a covariant expression whose spatial part reduces to the classical definition when 

v ≪ 1 ⇒ γ → 1. The 4-acceleration contains contributions along v and a in its spatial part 

a µ := d2x µ duµ 

≡ 

dτ2 dτ = ( γ4 (v · a), γ 4 (v · a)v + γ 2a) , (1.42) 

and it is easy to demonstrate the two constraints (from the definitions of ds 2 and u µ ): 

uµu µ = −1, uµa µ = 0. (1.43) 









Let m be the rest mass of the particle. The natural SR generalization of Newton’s second 

law of dynamics is 

F µ = ma µ ≡ m duµ 

dτ 

≡ dpµ 

dτ 

, (1.44) 

where F µ is the relativistic 4-force acting on the particle and p µ is the energy-momentum 

4-vector, defined as 

For small velocities we find the limits 

p µ := mu µ ≡ (E, p) = (mγ, mγv) . (1.45) 

E := mγ m + 1 

2 mv2 + . . . , p := mγv mv + . . . , 

and the first relation provides, for v = 0, the famous formula E = mc2 , a result totally 

unpredictable from Newtonian mechanics. If we eliminate v, we find the relation 

 

m2 + p2 , (1.46) 

pµp µ = −m 2 ⇒ E = 

and it is clear that p µ is, as u µ , a time-like vector. Notice that this expression is valid for 

massless particles like photons and neutrinos as well, or in any case where the kinetic 

energy is very large even with respect to the rest-mass energy. In this limit we have 

|p| ≫ m ⇒ E = |p|. 

The use of energy-momentum conservation is of paramount importance for the scattering 

of high-energy particles. In the case of photons (E = |p| = hν), it is straightforward to 

re-derive the relativistic Doppler effect from covariant considerations. 









When there is no external force F µ we recover the Newtonian first law of dynamics, that in 

SR translates into the conservation of u µ (or p µ ). Let us now see how to define the 

relativistic 4-force. If we want to maintain the form of Newton’s law F = dp/dt, the spatial 

part must be written as γF to satisfy (1.44), thus 

F µ := (γ(v · F), γF) , (1.47) 

where F 0 has been derived from the condition F µ uµ = 0. Equation (1.44) leads to 

now containing two components, and to 

F = dp 

dt = mγa + mγ3 (v · a)v, (1.48) 

v · F = dE 

dt = mγ3 (v · a) (1.49) 

which is the generalization of the classical theorem of kinetic energy. Finally, the 4-force 

transforms as a 4-vector, so by applying (1.16) 

F = F ′ , F⊥ = γ −1F ′ ⊥, (1.50) 

where F ′ is the spatial force measured in the particle’s rest system. 





The energy-momentum tensor for particles 





Consider a system of particles, labeled by n, with energy-momentum p µ 

n(t). The density of 

the total p µ and its current are defined as 

 

p µ 

n(t)δ 3 [x − xn(t)], T µi := 

 

p µ 

T µ0 := 

n(t) dxi n(t) 

δ 

dt 

3 [x − xn(t)], 

and they may be united in the symmetric energy-momentum tensor 

T µν 

:= p µ 

n(t) dxν n(t) 

δ 

dt 

3 

[x − xn(t)] = 

 

dτp µ 

n(t) dxν n(t) 

dτ 

δ 4 [x − xn(τ)] , (1.51) 

where the symmetry is apparent from the definition of p µ . The second expression is clearly 

a tensor, and it has been derived multiplying by δ(t ′ − t) and integrating in t ′ = τ. If we 

define the density of 4-force 

G µ µ 

dp 

:= 

n(t) 

δ 

dt 

3 

[x − xn(t)] = 

 

dτ dpµ n(t) 

dτ 

it is possible to derive the following conservation law 

∂iT µi = − 

 

p µ 

n(t) dxi n(t) 

dt 

or, in a fully covariant form 

∂ 

∂x i n 

δ 3 [x − xn(t)] = − 

δ 4 [x − xn(τ)], (1.52) 

 

p µ 

n(t)∂t δ 3 [x − xn(t)] = −∂t T µ0 + G µ , 

∂µT µν = G ν , (1.53) 

and the energy-momentum tensor is conserved when there are no external forces. 





Review of Maxwell equations 





The Maxwell equations for the electromagnetic field are (in Gaussian units with c = 1): 

⎧ 

⎪⎨ 

⎪⎩ ∇ · E = 4πρ 

∇ × B − ∂t E = 4πj 

, 

⎧ 

⎪⎨ 

⎪⎩ ∇ · B = 0 

∇ × E + ∂t B = 0 

, (1.54) 

where the first set contains the equations with sources (charge density and current) and the 

second one the sourceless equations. The sources must obey the conservation of charge 

∂t ρ + ∇ ·j = 0. (1.55) 

Another way to write the equations is to use the scalar and vector potentials, defined by 

for which we have the gauge invariance 

E = −∇φ − ∂t A, B = ∇ × A, (1.56) 

φ → φ − ∂t ψ, A → A + ∇ψ, (1.57) 

that is, the above transformations leave Maxwell’s equations unchanged. If we choose ψ 

such as to satisfy the Lorentz gauge 

∂t φ + ∇ · A = 0, (1.58) 

then the Maxwell equations reduce to the couple of wave equations 

φ = −4πρ, A = −4πj, (1.59) 

where := −∂ 2 t + ∇2 and the general solution may be expressed in terms of retarded 

potentials. 





Covariant form of electrodynamics 





We know that Maxwell’s equations are invariant under Lorentz transformations, however 

this fact must be made apparent. Let us start by noticing that the charge density ρ must 

behave as the time component of a 4-vector, since ρd 3 x must be an invariant charge dq. 

This suggests the definition of a 4-current with the associated conservation condition 

J µ := (ρ,j), ∂µJ µ = 0 , (1.60) 

which are both covariant. It is easy to verify that the 4-current is the source of the covariant 

wave equation for a 4-potential 

A µ := (φ, A), Aµ = −4πJ µ = 0 (∂µA µ = 0) , (1.61) 

where now = η µν ∂µ∂ν = ∂µ∂ µ and within brackets we have the covariant form of the 

Lorentz gauge. 

We now need a tensorial representation for E and B, which must descend from the 

derivation of the 4-potential. Notice that the antisymmetric tensor 

Fµν := ∂µAν − ∂νAµ , (1.62) 

is a good candidate, since it contains 6 independent components, exactly as the 

electromagnetic fields. These definitions lead to the covariant form of Maxwell equations: 

∂µF µν = −4πJ ν , ∂λFµν + ∂µFνλ + ∂νFλµ = 0 . (1.63) 









The (fully covariant) expression of the electromagnetic (or Faraday) tensor is 

⎛ 

0 −Ex −Ey −Ez 

⎞ 

Ex 0 Bz −By 

Fµν = 

⎜⎝ 

Ey −Bz 0 Bx 

Ez By −Bx 0 

, (1.64) 

⎟⎠ 

where µ indicates rows and ν columns. The above form can be verified from (1.62), e.g. 

F0i = ∂t Ai + ∂iφ = −Ei, Fij = ∂iAj − ∂jAi = [ijk]Bk , 

where [ijk] is the 3D completely antisymmetric symbol (+1 for any even number of 

permutations of [123], −1 for an odd number, zero if two indices are repeated). 

From the Lorentz transformation of the Faraday tensor 

F µ′ ν ′ 

= Λ µ′ 

µ Λ ν′ 

ν Fµν = Λ µ′ 

µ F µν (Λ T ) ν′ 

ν ⇒ F ′ = ΛFΛ T 

we find the transformation rules for the electromagnetic fields 

(1.65) 

E ′ = E + γ(E⊥ + v × B), B ′ = B + γ(B⊥ − v × E). (1.66) 

Finally, the relativistic Lorentz 4-force for a charge q in an electromagnetic field is 

since 

dp 

dt = F = q(E + v × B), 

dp µ 

dτ = Fµ = qF µν uν , (1.67) 

dE 

dt = v · F = qv · E. (1.68) 









The energy-momentum tensor for the electromagnetic field 

If the system of particles is made by charges in an external electromagnetic field, then 

dp µ 

dτ = qnF µ dx 

ν 

ν 

dτ ⇒ Gµ = F µ 

ν 

 

dx 

dτqn 

ν 

dτ = Fµ νJ ν 

(1.69) 

where the 4-current has been here re-defined in terms of the contribution of the single 

particles. The energy-momentum tensor is not conserved and we have 

∂µT µν 

m = F νλ Jλ, (1.70) 

where the subscript m stands for matter. In order to retrieve a true conservation law, let us 

define the energy-momentum tensor for the electromagnetic field 

T µν 

em = 1 

4π (FµλF ν 1 

λ − 4 ηµνFλκF λκ ) . (1.71) 

Splitting into temporal and spatial components we find the energy density, Poynting vector, 

and Maxwell stress tensor for the electromagnetic field 

4πT 00 

em = 1 

2 (E2 + B 2 ), 4πT 0i 

em = (E × B)i, 4πT ij 

em = 1 

2 (E2 + B 2 )δij − EiEj − BiBj, 

reshuffling indices in order to use Maxwell equations we find 

4π∂µT µν 

em = F ν λ ∂µF µλ + Fµλ∂ µ F νλ − 1 

2 Fλκ∂ ν F λκ = −4πF ν λ Jλ − 1 

2 Fλκ(∂ ν F λκ + ∂ λ F κν + ∂ κ F νλ ), 

so that only the total energy-momentum tensor is conserved 

∂µT µν 

em = −F νλ Jλ ⇒ ∂µ(T µν 

m + T µν 

em) = 0. (1.72) 





The energy-momentum tensor for hydrodynamics 





In many astrophysical applications, due to extreme rarefaction, the gas or plasma can be 

treated as a perfect fluid, following the laws of hydrodynamics (and magnetohydrodynamics, 

MHD). Due to its importance this subject will be treated separately later, here we just 

provide the definition of the fluid energy-momentum tensor. 

Let us start with the definition for a fluid at rest, in which 

˜T µν 

fl 

= diag(e, p, p, p), 

where e is the proper energy density (here ρ = nm will be the rest mass density for a fluid 

of equal particles with rest mass m and number density n) and p the (isotropic) pressure. 

Under a Lorentz transformation to the laboratory (inertial) frame 

x µ = Λ µ 

λ ˜xλ ⇒ T µν 

= Λµ 

fl λΛν ˜T 

λκ 

κ fl 

we find, for a velocity v of the fluid in the laboratory frame 

T 00 

fl = γ2 (e + pv 2 ), T i0 

fl = γ2 (e + p)vi, T ij 

fl = γ2 (e + p)vivj + pδij. 

If we use the fluid 4-velocity u µ = (γ, γv), the covariant expression is 

T µν 

fl = (e + p)uµ u ν + pη µν , (1.73) 

which reduces correctly to ˜T µν 

fl when uµ = (1, 0). Since the expression is valid in an inertial 

frame and it is written in tensorial form, covariance insures that it is valid in every inertial 

frame. 





The equations of relativistic hydrodynamics 





A similar approach can be used for the matter density. In the rest frame we may define a 

4-current of matter (ρ, 0) that after the usual Lorentz transformation becomes 

ρu µ = (ργ, ργv). (1.74) 

Since in our case m is just a constant, and the number of particles must be conserved, we 

generalize the continuity equation of hydrodynamics as 

to be combined to the conservation of energy and momentum 

where we have supposed that there are no external forces. 

∂µ(ρu µ ) = 0 , (1.75) 

∂µ[(e + p)u µ u ν + pη µν ] = 0 , (1.76) 

The momentum and energy equations are retrieved if we multiply the above equation with 

the projection operator Pµν onto the 3-space orthogonal to u µ 

Pµν := ηµν + uµuν, (1.77) 

and then along u µ . Recalling that uµu µ = −1 ⇒ uµ∂νu µ = 0, we find 

(e + p)u ν ∂νuµ + ∂µp + uµu ν ∂νp = 0, (1.78) 

u µ ∂µe + (e + p)∂µu µ = 0, (1.79) 

and the system is closed once we know an equation of state (EoS) linking p and e. 









Suppose that the EoS is written as p = p(e, s), where s is the specific entropy, that is 

entropy per unit mass (ρs is the entropy per unit volume). It could be demonstrated that the 

system of relativistic fluid equations is hyperbolic, and that the sound speed is 

c : s = (∂p/∂e)s, (1.80) 

where in general e = ρ(1 + ε), with ε the specific internal energy. For a perfect fluid the 

laws of thermodynamics lead to (ρ −1 = V/M, ε = U/M, s = S/M) 

dU = TdS − pdV ⇒ dε = Tds − pd(1/ρ), (1.81) 

which combined to (1.75) and (1.79) gives the additional conservation law 

valid wherever there is no dissipation (e.g. not at shocks). 

u µ ∂µs = 0 ⇒ ∂µ(ρsu µ ) = 0, (1.82) 

As far as the EoS is concerned, often we will adopt the ideal gas γ-law 

p = K(s)ρ γ = (γ − 1)ρε = (γ − 1)(e − ρ) ⇒ c 2 s = (∂ρp)s/(∂ρe)s = γp/(e + p), (1.83) 

with the adiabatic index is γ = 5/3 for a classical gas and γ = 4/3 when radiation 

dominates (ρε = aT 4 , p = aT 4 /3). If radiation is even ultra-relativistic, ε ≫ 1, the EoS is of 

barotropic type p = p(e) = e/3, and cs = 1/ √ 3. The energy equation becomes 

∂µ(e 3/4 u µ ) = 0 ⇒ p = e/3 ∝ ρ 4/3 . (1.84) 

Degenerate matter with T → 0, like in the interiors of a NS, may be modeled by barotropic 

relations too or by polytropic γ-laws with constant K (and γ loses the meaning of the 

specific heats ratio). 









General Relativity: 

the Principle of Equivalence 





General Relativity: introduction 





The two main characters acting in High Energy Astrophysics are certainly neutron stars 

(NSs) and black holes (BHs). Their are the prototypes of compact objects, that is 

astrophysical bodies where the mass M is concentrated within a small radius r, such that 

the density is very high and the compactness parameter 

|φ| GM 

= 

c2 rc2 is not much different (less) than unity. This number is ≈ 10 −6 for the Sun, ≈ 10 −4 for a white 

dwarf, ≈ 0.1 for a NS and 0.5 for a BH (using the Schwarzschild radius rs = 2GM/c 2 ). 

Under these circumstances, the theory of gravitation of Newton must be replaced by the 

General Relativity (GR) theory of Albert Einstein, published in complete form in 1916, for 

which gravity is no longer a force in classical sense but rather a manifestation of the 

curvature of spacetime. A massive object produce a distortion of spacetime around it, and 

in turn this distortion controls the movement of physical objects. 

GR is essentially based on the theory of differential curved manifolds by Riemann, and most 

textbooks first discuss differential geometry and tensor analysis. Here we mainly follow the 

book Gravitation and cosmology by S. Weinberg (Wiley, 1972), more focused on physics 

rather than mathematics. Other books: Misner, Thorn, Wheeler; Shutz; Wald; Straumann. 









The reasons why a new theory was needed after the results of SR were clear: first of all, 

Newton’s law is based on an action at a distance and on an instantaneous propagation of 

information to every point in space. This is something that Einstein had just successfully 

exorcised from other aspects of physics, and clearly Newtonian gravity had to be revised as 

well. Moreover, being the energy and mass equivalent in SR, gravitational energy must be a 

source for gravity itself, so unlike Newton’s laws GR is expected to be a nonlinear theory. 

The three building blocks that led Einstein to GR were: 

1 the Principle of Relativity, 

2 the equivalence of gravitational and inertial masses, 

3 Mach’s principle. 

The first one was to be generalized to generic systems where the laws of nature would have 

to take the same form (general covariance), implying the tensorial language of the theory. 

The second led to the Principle of Equivalence for which gravity can be locally canceled in 

freely falling Minkowskian systems, suggesting the analogy with curved manifolds, where at 

any point an Euclidean geometry can be defined locally, as first suggested by Gauss. 

The third stated that the local inertial properties of physical objects must be determined by 

the total distribution of matter in the universe, that suggested that the geometry of 

spacetime could be determined by the overall distribution of mass (and energy). 









The equivalence of gravitational and inertial masses 

Galileo Galilei (1564-1642) was the first to realize that bodies fall at a rate independent of 

their mass, making experiments with inclined planes and pendulums to reduce friction. 

Isaac Newton (1642-1727) was aware that these conclusions might be only approximately 

true and that the inertial mass appearing in his law of dynamics could be slightly different 

than the gravitational mass in his law of gravitation. If this were the case, we should write 

the two laws as 

F = mia, F = mgg, (2.1) 

where g is a field depending on position and other (gravitational) masses. The acceleration 

at a given point would be 

a = (mg/mi)g, (2.2) 

that could vary for bodies of different mg/mi ratio, for example when changing the 

composition. In particular, pendulums of equal length would have periods T ∼ (mg/mi) −1/2 , 

but differences were never found and Newton concluded that mg ≡ mi ≡ m. 

In 1889 Roland von Eötvös used a torsion pendulum, subject to an appreciable centripetal 

acceleration due to Earth’s rotation, to show that the ratio mg/mi does not change for 

different bodies and substances by more than one part in 10 9 . 





The Principle of Equivalence 





Einstein was much impressed by these experimental evidences, leading in 1907 to his 

Principle of Equivalence of gravitation and inertia, the first step towards the building of his 

relativistic theory of gravity. 

In the famous though experiment of the freely falling elevator, Einstein deduced that in such 

system there is no way to detect the presence of an external static and homogeneous 

gravitational field g. Consider a set of (non-relativistic) particles interacting with a force 

F(xn − xm) (e.g. electrostatic or gravitational). The equations of motion are 

d 

mn 

2xn dt2 = mng + Σmn F(xn − xm). 

However, let us perform the non-Galilean transformation 

x ′ = x − 1 

2 gt2 , t ′ = t, 

then the equation of motion in the freely falling system becomes 

d 

mn 

2x ′ n 

dt ′2 = Σmn F(x ′ n − x ′ m), 

where the field g has been canceled by inertial forces, as a direct consequence of the 

identity between gravitational and inertial masses. The laws of mechanics for the 

accelerated observer using x ′ are exactly the same as those in an inertial system where 

there is no gravity g, as experienced by astronauts orbiting around the Earth. 









However, this is not the end of the story. Had g depended on x or t, we would not have 

been able to cancel gravity out. For example, the Earth is in free fall around the Sun, and for 

the most part we on Earth do not feel the Sun attraction. However, the slight inhomogeneity 

in this field (about 1 part in 6000 from noon to midnight) is enough to raise impressive tides 

in the oceans. 

Therefore Einstein formulated his Principle of Equivalence (EP) in this form: at every 

spacetime point in an arbitrary gravitational field it is possible to choose a locally inertial (or 

freely falling) coordinate system such that, within a sufficiently small region of the point in 

question, the laws of nature take the same form as in unaccelerated Cartesian coordinate 

systems in the absence of gravitation. 

Obviously, for a relativistic new theory of gravitation we need to use SR. The laws of nature 

holding in the local inertial frame must be those of SR, invariant under Lorentz 

transformations, what the EP suggests to us is that we can include the effects of gravitation 

by transforming the laws of nature from a Cartesian inertial system to another (accelerated, 

curvilinear) coordinate system. 

We will later see that EP can be restated in the Principle of General Covariance, which 

extends that of SR: if a physical law is invariant under a general coordinate transformation 

and it is valid in the absence of gravity, then it holds in any coordinate system, with an 

arbitrary gravity field. 


Gravitational forces 








Consider a particle moving freely under the influence of purely gravitational forces. 

According to the Principle of Equivalence, there is a freely falling, locally inertial coordinate 

system ξ α for which the equation of motion is that of SR in the case of no forces applied 

d 2 ξ α 

dτ2 = 0, (2.3) 

where the proper time τ (and the line element) is defined as in SR 

dτ 2 = −ds 2 = −ηαβdξ α dξ β . (2.4) 

For sake of clarity here we use indices α, β, . . . for the local inertial system governed by ηαβ. 

In any another coordinate system x µ (that of the laboratory, or even an accelerated one of 

our choice), (2.3) becomes 

 

d d 

dτ dτ ξα (x µ 

) = d 

 

∂ξα dτ ∂x µ 

If we now multiply by ∂x λ /∂ξ α and recall that 

dx µ 

= 

dτ 

∂ξα 

∂x µ 

d2x µ 

dτ2 + ∂2ξα ∂x µ ∂xν dx µ dx 

dτ 

ν 

= 0. 

dτ 

∂ξα ∂x µ 

∂xλ ∂ξα = δλ µ, (2.5) 

we end up with the equation of motion for a particle subject to purely gravitational forces 

d2x λ 

dτ2 + Γλ dx 

µν 

µ dx 

dτ 

ν 

= 0 , (2.6) 

dτ 





where Γλ µν is the affine connection, defined by 

Γ λ µν := ∂xλ 

∂ξ α 





∂ 2 ξ α 

∂x µ ∂x ν . (2.7) 

The above quantity is clearly symmetric exchanging µ and ν. Notice that the affine 

connection vanishes when ξ α (x µ ) is linear, that is if also x µ is another inertial system 

connected to ξ α by a Lorentz transformation. Thus, the effects of gravity arise when second 

derivatives of the transformation do not vanish. 

The proper time can also be rewritten in the general coordinate system as 

where the general metric tensor is defined as 

dτ 2 ≡ −ds 2 := −gµνdx µ dx ν , (2.8) 

∂ξ 

gµν := ηαβ 

α 

∂x µ 

∂ξβ . (2.9) 

∂xν It may be proved that the values of gµν and Γλ µν at a given point X in the arbitrary system xµ 

provide enough information to determine, up to a Lorentz transformation, the locally inertial 

coordinates ξα in the neighborhood of X. 





Light and massless particles 





For massless particles like photons dτ cannot be used in the equation of motion since the 

right hand side of (2.4) vanishes. However, let us introduce the parameter σ ≡ ξ 0 and recall 

that, in the absence of external forces, SR predicts that E and p = Ev are conserved, that is 

dE 

= 0, 

dσ 

dp dv 

= E 

dσ dσ 

d2ξ 

= E = 0. 

dσ2 Since also d 2 ξ 0 /dσ 2 = 0 clearly holds, the equation of motion for light in the inertial frame 

is written in a covariant way as d 2 ξ α /dσ 2 = 0. The situation is now identical to the previous 

case, so the general equation of motion for light in a gravitational field is 

d2x λ 

dσ2 + Γλ dx 

µν 

µ dx 

dσ 

ν 

dx 

= 0, gµν 

dσ µ dx 

dσ 

ν 

= 0. (2.10) 

dσ 

Basically, both τ and σ just serve as parameters along the trajectory. The second relation 

provides a sort of initial condition for massless particles, and it also gives the time dt taken 

for light to travel a distance dx 

g00dt 2 + 2g0idtdx i + gijdx i dx j = 0 ⇒ dt = {−g0idx i − [(g0ig0j − g00gij)dx i dx j ] 1/2 }/g00, 

whereas for finite distances we just need to integrate along the path. 









Relation between the metric tensor and the affine connections 

We have seen in (2.3) that the motion of particles in a gravitational field is determined by 

the affine connections (2.7), while the proper time and the metric properties are based on 

the metric tensor (2.9). Here we will see how to write the affine connections in terms of 

spatial derivatives of the metric tensor in the system x µ alone, which will then become a 

sort of gravitational potentials. 

Let us start by the definition (2.9) and derive it along x κ : 

gµν = ∂µξ α ∂νξ β ηαβ, ⇒ ∂κgµν = ∂κ∂µξ α ∂νξ β ηαβ + ∂µξ α ∂ν∂κξ β ηαβ, 

where, for sake of simplicity, here we use the compact notation 

∂µ := ∂ 

. 

∂x µ 

Now, the definition of the affine connections can be rewritten by inverting the jacobian and 

recalling the property (2.5) to give 

∂µ∂νξ α = Γ λ µν∂λξ α , 

that plugged in the above relation together with (2.9) yields 

Consider now the combination 

∂κgµν = Γ λ κµ gλν + Γ λ κν gλµ. (2.11) 

∂µgνκ + ∂νgµκ − ∂κgµν = Γ λ µκ gλν + Γ λ µν gλκ + Γ λ νµ gλκ + Γ λ νκ gλµ − Γ λ κµ gλν − Γ λ κν gλµ. 









Due to the symmetry properties of Γ λ µν and gµν, four terms cancel out and the remaining two 

are identical. If we then define g λκ to be the inverse of gλκ, the result is 

Γ λ µν = 1 

2 gλκ (∂µgνκ + ∂νgµκ − ∂κgµν) . (2.12) 

The right hand side is known as Christoffel symbol. The inverse of the metric tensor is 

defined as 

g µν αβ ∂xµ 

:= η 

∂ξα ∂xν , 

∂ξβ (2.13) 

as can be easily verified as follows 

g µν αβ ∂xµ 

gνλ = η 

∂ξα ∂xν ∂ξ 

ηγδ 

∂ξβ γ 

∂xν ∂ξδ ∂xµ ∂ξ 

= ηαβ ηβδ 

∂xλ ∂ξα δ 

∂xµ 

= 

∂xλ ∂ξα ∂ξα = δµ 

∂xλ λ . 

An important consequence that could be easily proved is that 

dx 

gµν 

µ dx 

dτ 

ν 

= −C, (2.14) 

dτ 

where C is a constant of the motion, that is if we derive the above relation along τ we get 

zero. Hence, once the initial condition (2.8) has been chosen, we have C = 1 along the 

whole path. For massless particles C = 0 and another parameter σ must replace τ. 


Geodesics 








An additional consequence is that we may formulate a variational principle for free falling 

particles. Define an arbitrary σ to parametrize the path and calculate the total proper time 

to move from spacetime points A and B as 

B dτ 

TAB = dσ = 

A dσ 

B 

A 

 

−gµν 

dx µ 

dσ 

dxν 1/2 dσ. (2.15) 

dσ 

Now, vary the path from x λ (σ) to x λ (σ) + δx λ (σ), keeping fixed the endpoints, that is 

δx λ (σA ) = δx λ (σB) = 0. The change in TAB is 

A 

B 

δTAB = 

A 

− 1 

 

dx 

−gµν 

2 

µ 

dσ 

dxν −1/2 

λ dxµ dx 

∂λgµν δx 

dσ 

dσ 

ν dδ x 

+ 2gλν 

dσ λ 

dσ 

A 

dxν 

dσ. 

dσ 

The first factor is just dσ/dτ, thus we may eliminate σ in favor of τ everywhere. Then we 

integrate the second term by parts, neglecting the endpoints contribution. This gives 

B 

(− 1 

2 ∂λgµν + ∂µgλν) dxµ dx 

dτ 

ν 

dτ + d2x ν 

dτ gλν 

 

δx λ B 

dτ = Γ κ dx 

µν 

µ dx 

dτ 

ν 

dτ + d2x κ 

gλκδx 

dτ 

λ dτ. 

where (2.12) has been used. Thanks to (2.6) we have 

thus free particles (and photons) follow geodesics in spacetime. 

δTAB = 0, (2.16) 


The Newtonian limit 








Consider now the Newtonian limit for small (and static) gravitational fields and for slowly 

moving particles (dx i /dt ≪ 1 ⇒ dx i /dτ ≪ dt/dτ). The equation of motion (2.3) becomes 

d2x µ 2 dt 

+ Γµ = 0, Γ 

dτ2 00 dτ 

µ 1 = − 00 2 gµν∂νg00. (2.17) 

Suppose a quasi-Minkowskian metric such that 

gµν = ηµν + hµν, |hµν| ≪ 1. (2.18) 

To first order only the term η µν ∂νh00 survives, and its time component vanishes leaving us 

with dt/dτ = const. The final equation for the spatial component is 

d2x i 

1 = 

dτ2 2 ηij 2 dt 

∂jh00 

dτ 

⇒ d2x 1 = 

dt2 2 ∇h00 , (2.19) 

so we have recovered the Newtonian limit if h00 = −2φ + const. Using the Newtonian 

potential outside a spherically symmetric object at a distance r and assuming that also h00 

vanishes at infinity, we may write the distorted time-time component of the metric as 

g00 = −(1 + 2φ) = −(1 − 2GM/r), 

and, since G → G/c 2 = 7.41 × 10 −29 cm/g, φ it is usually negligible (for example on the 

Sun’s surface φ = −2.12 × 10 −6 ). 





Time dilation and gravitational redshift 





Consider a clock in an arbitrary gravitational field. The EP tells us that its spacetime rate 

∆τ is unaffected by gravity if we observe it in the locally flat free falling system. Then, 

combining (2.4) and (2.8) we write 

dτ = (−ηαβdξ α dξ β ) 1/2 = (−gµνdx µ dx ν ) 1/2 . 

The time interval between ticks measured in x µ will be 

dt 

dτ = 

 

dx 

−gµν 

µ dx 

dt 

ν −1/2 (−g00) 

dt 

−1/2 (1 + 2φ) −1/2 1 − φ, 

where the same Newtonian limit as in the last section (low gravity and small velocity) has 

been assumed. Suppose now a photon is emitted by an atomic transition at point x2 and 

then observed at x1, the light frequency ratio will be 

ν2 

ν1 

= dt1 

= 

dt2 

[−g00(x2)] 1/2 

[−g00(x1)] 1/2 1 − φ(x1) + φ(x2). (2.20) 

The observed gravitational redshift for a spherical object of mass M emitting the proton at a 

distance R, supposing |φ(x2)| ≫ |φ(x1)|, is thus 

∆ν/ν = φ = −GM/R , (2.21) 

where ∆ν/ν := ν2/ν1 − 1. Unfortunately, Doppler shifts due to thermal and convective 

motions on the Sun’s surface give a larger contribution. The definitive proof of this GR effect 

was given by Pound and Rebka in a laboratory experiment in 1960. Nowadays gravitational 

redshift is important for GPS and it has been measured in atomic lines from X-ray binaries. 














The Principle of General Covariance 





We have already mentioned that EP establishes a deep analogy between non-Euclidean 

geometry and the theory of gravitation. Here we will make use largely of the language 

common to both, that of tensor analysis. 

Thanks to EP we have learned how to write down physical laws in GR: the law valid in SR is 

extended by performing a coordinate transformation. We could apply this approach to 

mechanics, electrodynamics and even gravitation itself. However, a much more simple, 

elegant yet powerful method is that based on an alternative version of EP: the Principle of 

General Covariance. 

It states that a physical equation holds in a general gravitational field if: 

1 The equation holds in the absence of gravitation; that is, it agrees with SR when 

gµν ≡ ηµν and Γ λ µν = 0. 

2 The equation is generally covariant; that is, it preserves its form under a general 

coordinate transformation x → x ′ . 

This is very similar to the Principle of Relativity, though the geometrical structure of 

spacetime will no longer be that of Minkowski but that of Riemannian curved manifolds. In 

particular, it will be impossible to find, in the presence of gravity, a global transformation 

such that gµν ≡ ηµν everywhere. 





Transformation of affine connections 





The general definitions of contravariant and covariant vectors, tensors and tensor densities 

have been already provided in the SR section and will not be repeated here. Now we are 

going to demonstrate that affine connections do not transform as tensors. Let us start by 

transforming (2.7) expressed in the x ′ system 

Γ λ′ 

µ ′ ∂xλ′ 

ν ′ = 

∂ξα ∂µ ′ ∂ν ′ ξα = Λ λ′ 

λ 

then, making use of the definition of δλ κ , we find 

∂xλ µ 

(Λ 

∂ξα µ ′ Λ ν 

ν ′ ∂µ∂νξ α + ∂κξ α ∂µ ′ ∂ν ′ xκ ) 

Γ λ′ 

µ ′ µ 

ν ′ = Λλ′ 

λ Λ µ ′ Λ ν 

ν ′Γλ µν + Λ λ′ 

λ ∂µ ′ ∂ν ′ xλ = Λ λ′ µ 

λ Λ µ ′ Λ ν 

ν ′Γλ µν − Λ µ 

µ ′ Λ ν λ′ 

ν ′∂µ∂νx , (3.1) 

where only the first term in both relations preserve covariance. The second expression may 

be derived by expanding 0 = ∂µ ′(δλ′ 

ν ′ ) = ∂µ ′ (Λλ′ λ Λ λ 

ν ′ ). 

As a first application of the Principle of General Covariance, let us prove again (2.6). It is 

clearly valid in SR, we need to prove that transforms as a contravariant vector. We have 

d2x λ′ 

= Λλ′ 

dτ2 λ 

d2x λ 

λ′ dxµ 

+∂µ∂νx 

dτ2 dτ 

thus we obtain the desired result 

dxν , Γλ′ 

dτ µ ′ ν ′ 

dx µ′ dx 

dτ 

ν′ 

= Λλ′ 

λ dτ Γλ dx 

µν 

µ 

dτ 

d2x λ′ 

+ Γλ′ 

dτ2 µ ′ ν ′ 

dx µ′ dx 

dτ 

ν′ 

= Λλ′ 

λ dτ 

d 2 x λ 

dτ 2 + Γλ µν 

dx µ 

dτ 

dxν λ′ dxµ dx 

−∂µ∂νx 

dτ dτ 

ν 

dτ . 

dxν 

. (3.2) 

dτ 


Covariant differentiation 








Contrary to what happens in SR (flat metric), in GR (curved manifolds) the standard 

derivative of a vector or tensor is not a tensor. As already anticipated 

ν′ 

∂µ ′ V = Λ µ 

µ ′ ∂µ(Λ ν′ 

ν V ν ) = Λ µ 

µ ′ Λ ν′ 

ν ∂µV ν + Λ µ 

µ ′ ∂µ∂νx ν′ 

V ν , 

and, using the transformation rule for the affine connection (3.1), we can write 

ν′ 

∂µ ′ V + Γ ν′ 

µ ′ λ′ 

λ ′ V = Λ µ′ 

µ Λ ν 

ν ′ (∂µV ν + Γ ν µλV λ ), (3.3) 

which is clearly a covariant transformation rule for mixed tensors. Then we define the 

covariant derivative of a contravariant vector as 

A similar rule may be defined for covariant vectors, that is 

∇µV ν := ∂µV ν + Γ ν µλ V λ . (3.4) 

∇µVν := ∂µVν − Γ λ µν Vλ , (3.5) 

and in general, for any mixed tensor, the covariant derivative is defined as 

∇µT ν ρσ = ∂µT ν ρσ + Γ ν µλT λ ρσ − Γ λ µρT ν λσ − Γλ µσT ν ρλ . (3.6) 

For tensor densities: ∇µD = |g| −w/2 ∇µ(|g| w/2 D), so an extra term w 

2 ∂µln|g| D will appear. 









The combination of covariant differentiation with the algebraic operations previously 

described gives similar results to ordinary differentiation: 

1 Linear combinations: 

2 Direct products (Leibnitz rule): 

∇κ(aA µ ν + bB µ ν) = a∇κA µ ν + b∇κB µ ν. (3.7) 

∇κ(A µ νB λ ) = ∇κA µ ν B λ + A µ ν ∇κB λ . (3.8) 

3 Contractions: 

µ λν µ λν 

∇κT λ = ∂κT λ + Γµ ρ λν 

κρT 

λ + Γν µ λρ 

κρT λ . (3.9) 

We also notice that the covariant derivative of the metric tensor 

∇λgµν = ∂λgµν − Γ κ λµ gκν − Γ κ λν gκµ 

is zero, because of (2.11) and for general covariance, since it vanishes in locally inertial 

coordinates where spatial derivatives of the metric tensor and affine connections are all 

zero and the above relation is a tensor. In general we have 

∇λgµν = ∇λg µν = ∇λδ µ ν = 0, (3.10) 

thus the operations of raising/lowering indices commute with covariant differentiation. 

We see now how it will be easy to extend the laws of physics of SR to the presence of a 

gravitational field (at least on a local scale) by applying the Principle of General Covariance: 

it is enough to replace ηµν with gµν and all standard derivatives with covariant derivatives. 





Example: 2D polar coordinates 





Let us take a short break. To illustrate the meaning of affine connections and covariant 

derivatives, consider the 2D flat space. The position of a point can be determined by either 

Cartesian coordinates ξ α = (x, y) or by polar coordinates x µ = (r, θ), with x = r cos θ and 

y = r sin θ. The line element in the curvilinear system is 

ds 2 = dr 2 + r 2 dθ 2 ⇒ grr = g rr = 1, gθθ = (g θθ ) −1 = r 2 , grθ = g rθ = 0, 

and the non-vanishing Christoffel symbols are 

Γ r 

θθ = −grr ∂r gθθ = −r, Γ θ rθ = gθθ ∂r gθθ = r −1 . 

Consider, for simplicity, a uniform vector field, for example aligned with x, hence 

V x = Vx = A, V y = Vy = 0. In polar coordinates the contravariant components are 

V r = ∂r ∂x 

A = grr 

∂x 

∂r A = A cos θ, V θ = ∂θ 

∂x 

A = gθθ ∂x 

∂θ A = −Ar−1 sin θ, 

which are clearly space dependent with non-vanishing derivatives. How do we recover the 

invariant property that V µ is a uniform field? Let us calculate the covariant derivatives: 

∇r V r = 0, ∇θV r = ∂θV r +Γ r 

θθ V θ = 0, ∇r V θ = ∂r V θ +Γ θ rθ V θ = 0, ∇θV θ = ∂θV θ +Γ θ rθ V r = 0. 

Thus, covariant differentiation takes into account that the system of coordinates itself is 

variable and compensates the changes in the components. We have checked in this simple 

case the tensorial property that when a covariant derivative is zero in the Cartesian system 

ξ α , it must be zero in any curvilinear system x µ . 





Gradient, curl, and divergence 





There are some special cases where the covariant derivative takes a particularly simple 

form. Simplest of all is, of course, the derivative of a scalar, which is just the ordinary 

gradient 

∇µS = ∂µS. (3.11) 

Another simple case is the covariant curl, since for a covariant vector the terms with the 

affine connection vanish and we are left with the ordinary curl 

∇µVν − ∇νVµ = ∂µVν − ∂νVµ. (3.12) 

Consider then the covariant divergence of a contravariant vector 

it is possible to demonstrate that 

so that 

∇µV µ = ∂µV µ + Γ µ 

µλ V λ = ∂µV µ + 1 

2 gµν ∂λgµν V λ , 

Γ µ 1 

µλ = 2 gµν∂λgµν = 1 

2 ∂λln|g| = |g| −1/2∂λ|g| 1/2 , (3.13) 

∇µV µ = |g| −1/2 ∂µ(|g| 1/2 V µ ). (3.14) 

One immediate consequence is a covariant form of Gauss’s theorem: if V µ vanishes at 

infinity then 

∇µV µ |g| 1/2 d 4 x = 0. (3.15) 









For a rank 2 tensor completely contravariant tensor, the divergence is 

while for a mixed rank 2 tensor is 

∇µT µν = |g| −1/2 ∂µ(|g| 1/2 T µν ) + Γ ν µλ T µλ , (3.16) 

∇µT µ ν = |g| −1/2 ∂µ(|g| 1/2 T µ ν ) − Γ µ 

νλ T λ µ . (3.17) 

Let us see some special cases. If T µν = −T νµ is antisymmetric, the first relation gives 

∇µT µν = |g| −1/2 ∂µ(|g| 1/2 T µν ), T µν antisymmetric, (3.18) 

while if T µν = T νµ is symmetric, we may write 

∇µT µ ν = |g| −1/2 ∂µ(|g| 1/2 T µ ν) − 1 

2 T λµ ∂νgλµ, T µν symmetric, (3.19) 

relations that will be useful when discussing hydrodynamics. 

For a completely covariant rank 2 tensor we write the covariant derivative 

so that, cycling the indices 

∇λTµν = ∂λTµν − Γ κ µλ Tκν − Γ κ νλ Tκµ, 

∇λTµν + ∇µTνλ + ∇νTλµ = ∂λTµν + ∂µTνλ + ∂νTλµ, T µν antisymmetric, (3.20) 

which will be used for electrodynamics. 





Vector analysis in orthogonal coordinates 





Let us consider 3D space and a system of orthogonal coordinates, such that 

gij = diag(h 2 

1 

, h2 

2 , h2 

3 ), gij = diag(h −2 

1 , h−2 

2 , h−2 

3 ). 

The usual orthonormal components are Vˆ1 = h1V 1 = v1/h1, and similarly for the other 

components, so that the scalar product between two vectors takes the usual form 

The gradient of a scalar becomes 

V · U = gijV i U j = Vˆ1 

Uˆ1 + Vˆ2 

Uˆ2 + Vˆ3 

Uˆ3 . 

(∇S)i = ∂iS ⇒ ∇S = ( h −1 

1 ∂1S, h −1 

2 ∂2S, h −1 

3 ∂3S, ) 

The contravariant curl may be written as 

(∇ × V) i = ɛ ijk ∇jVk = |g| −1/2 [ijk]∂jVk ⇒ (∇ × V)ˆ1 = (h2h3) −1 [∂2(h3Vˆ3 ) − ∂3(h2Vˆ2 )], 

and similarly for the other components. The divergence of a vector becomes 

∇ · V = (h1h2h3) −1 [∂1(h2h3Vˆ1 ) + ∂2(h1h3Vˆ2 ) + ∂3(h1h2Vˆ3 )]. 

Finally, the Laplacian of a scalar S is the divergence of its gradient, hence 

∇ 2 S = (h1h2h3) −1 [∂1(h2h3/h1∂1 Vˆ1 ) + ∂2(h1h3/h2∂2 Vˆ2 ) + ∂3(h1h2/h3∂3 Vˆ3 )]. 

Other relations of vector analysis could be demonstrated. The reader may check the usual 

formulae of vector analysis for spherical or cylindrical coordinate systems. 





Covariant differentiation along a curve 





So far we have considered tensor fields defined over all spacetime. However, we may have 

tensors defined only along a curve x λ (τ), like the momentum of a particle, and we need to 

define covariant differentiation along this curve. A vector V λ transforms as 

V λ′ 

(τ) = Λ λ′ 

λ (τ)V λ λ′ dV dV 

⇒ = Λλ′ 

λ dτ λ 

λ′ dxµ 

+ ∂µ∂νx 

dτ dτ V ν , 

and thanks to (3.1) we see that if we define the covariant derivative as 

DV λ 

Dτ 

dV λ 

:= 

dτ + Γλ dx 

µν 

µ 

dτ V ν . (3.21) 

The definition for a covariant vector and for mixed tensors is similar: 

DVλ 

Dτ 

dVλ 

:= 

dτ − Γν dx 

λµ 

µ 

dτ Vν , DT µ ν 

Dτ := dT µ ν dx 

+ Γµ 

dτ λκ 

λ 

dτ T κ ν − Γ κ dx 

λν 

λ 

dτ T µ κ. (3.22) 

Notice that, provided the tensor is a field defined everywhere, we may always define 

DV µ 

Dτ 

dxλ 

≡ 

dτ ∇λV µ , DVµ dxλ 

≡ 

Dτ dτ ∇λVµ, DT µ ν dxλ 

≡ 

Dτ dτ ∇λT µ ν. (3.23) 

Finally, when in a locally inertial system the derivative vanishes, this will be true along x λ (τ): 

DV λ dV λ 

= 0 ⇒ 

Dτ dτ = −Γλ dx 

µν 

µ 

dτ V ν . (3.24) 

A vector define as above is said to be defined by parallel transport. 









Particle mechanics in an external gravitational field 

We now return to physics and extend the laws valid in SR by applying the Principle of 

General Covariance, basically replacing standard derivatives by covariant ones. The new 

equations will be valid in an arbitrary gravitational field, at least on a local scale. 

The four-velocity of a particle is constant in SR if not under the influence of any force 

duα dτ = 0, uα := dξα 

. 

dτ 

(3.25) 

If we now define in GR 

u µ := ∂xµ 

∂ξα uα ⇒ u µ = dxµ 

, 

dτ 

(3.26) 

we find again the equation of motion (2.6) obtained with EP as 

Du λ 

Dτ 

= 0 ⇒ duλ 

dτ + Γλ µν uµ u ν = 0, (3.27) 

which is basically the rule for parallel transport of u λ along the curve with tangent vector 

u λ ≡ dx λ /dτ itself. As in SR, we retrieve the normalizing condition 

dτ 2 = −gµνdx µ dx ν = −gµνu µ u ν dτ 2 ⇒ uµu µ = −1. (3.28) 

When a non-gravitational force F λ is applied to a particle of mass m, (3.27) becomes 

m Duλ 

Dτ = Fλ ⇒ m duλ 

dτ = Fλ − m Γ λ µν uµ u ν , (3.29) 

where it is clear that the second term plays the role of a gravitational force. 





Electrodynamics in an external gravitational field 

Maxwell equations in the absence of gravitational fields were 

where the 4-current is 





∂αF αβ = −4πJ β , ∂αFβγ + ∂βFγα + ∂γFαβ = 0, (3.30) 

J α := (ρ,j), ∂αJ α = 0 (3.31) 

If we define, like in the previous case, quantities in a general coordinate system as 

F µν = ∂xµ 

∂ξα ∂xν ∂ξβ Fαβ , J µ = ∂xµ 

∂ξα Jα , (3.32) 

and use gµν to raise and lower the indices (e.g. Fµν = gµρgνσF ρσ ), following the recipe of 

general covariance, the equations in GR are those in SR with the substitutions ∂α → ∇µ: 

∇µF µν = −4πJ ν ⇒ |g| −1/2 ∂µ(|g| 1/2 F µν ) = −4πJ ν , (3.33) 

∇λFµν + ∇µFνλ + ∇νFλµ = 0 ⇒ ∂λFµν + ∂µFνλ + ∂νFλµ = 0 , (3.34) 

and the four-current now satisfies 

∇µJ µ = 0 ⇒ |g| −1/2 ∂µ(|g| 1/2 J µ ) = 0. (3.35) 

Note that we have used the specific relations for the covariant derivative of symmetric and 

antisymmetric tensors. Finally, the electromagnetic (Lorentz) force acting on a charge q is 

F µ = qF µν uν. (3.36) 









Energy-momentum tensor in an external gravitational field 

The density and current of energy and momentum were united in SR into a symmetric 

tensor T αβ , satisfying 

∂αT αβ = G β , (3.37) 

where G β is the density of the external force F α . The generalization to GR is 

∇µT µν = G ν ⇒ |g| −1/2 ∂µ(|g| 1/2 T µν ) = G ν − Γ ν λµ T λµ , (3.38) 

where the second part on the right hand side is the density of gravitational force. Notice 

that, even when G ν = 0, the above equation does not lead in general to conservation of 

quantities like energy, momentum or angular momentum, due to the exchange of energy 

and momentum between matter and gravitation. However, using (3.19) we can write 

|g| −1/2 ∂µ(|g| 1/2 T µ ν) = Gν + 1 

2 T λµ ∂νgλµ, (3.39) 

and we see that under particular symmetries, that is if there is an index ν for which 

∂νgλµ = 0 (and supposing Gν = 0), we retrieve a real conservation law. 

For a system of particles the energy-momentum tensor is, extending the expression in SR 

T µν 

−1/2 

= |g| mn 

n 

µ 

dx n 

dτ dxν nδ 4 (x − xn), (3.40) 

while for the electromagnetic field we have 

T µν 

em = 1 

4π (FµλF ν 1 

λ + 4 gµνFλκF λκ ) , (3.41) 

simply reducing to the corresponding formula in SR if g µν → η µν L. Del Zanna Relativistic Astrophysics. and compact objects 79 / 181




Hydrodynamics in an external gravitational field 





Le laws for hydrodynamics are also easy to obtain. If u µ = dx µ /dτ is the local 4-velocity for 

a comoving fluid element and ρ = nm is the rest mass density (we assume a fluid of equal 

particles with rest mass m and number density n), the covariant continuity equation is 

∇µ(ρu µ ) = 0 ⇒ |g| −1/2 ∂µ(|g| 1/2 ρu µ ) = 0 . (3.42) 

The energy-momentum tensor of the (ideal) fluid is 

T µν 

fl = (e + p)uµ u ν + pg µν , (3.43) 

simply reducing to the corresponding formula in SR if g µν → η µν and obeying the 

conservation law (3.38). The energy density e and the fluid pressure p are those defined in 

the locally inertial comoving frame, and are therefore scalars. The momentum and energy 

equations are retrieved if we multiply (3.38) with the projection operator Pµν onto the 

3-space orthogonal to u µ 

and along u µ . Recalling that uµu µ = −1 ⇒ uµ∇νu µ = 0, we find 

Pµν := gµν + uµuν, (3.44) 

(e + p)u ν ∇νuµ + ∇µp + uµu ν ∇νp = 0, (3.45) 

u µ ∇µe + (e + p)∇µu µ = 0, (3.46) 

and the system is closed once we know an equation of state (EoS) linking p and e. 





Example: barotropic equilibrium 





As a simple application let us solve the equations for a case of a fluid in hydrostatic 

equilibrium in the presence of a static gravitational field. Thus we assume 

u i ≡ 0, ∂0 ≡ 0 ⇒ u µ ∂µ ≡ 0, ∇µu µ ≡ 0, 

so the energy equation is trivially satisfied. From the normalizing condition for the 4-velocity 

we find 

gµνu µ u ν = −1 ⇒ u 0 = (−g00) −1/2 , 

and by plugging into the Euler equation (3.45) written for u µ 

(e + p)Γ µ 

00 u0 u 0 + g µν ∇νp = 0 ⇒ g µν [(e + p) 1 

2 ∂ν(−g00)(−g00) −1 + ∂νp] = 0. 

The spatial part inside square brackets gives the hydrostatic equation 

∇p = −(e + p)∇ ln(−g00) 1/2 , (3.47) 

solvable for a barotropic EoS p = p(e). For example, in the ultrarelativistic limit 

p = e/3 ⇒ e ∝ (−g00) −2 . 

In the Newtonian limit e → ρ ≫ p and g00 → −(1 + 2φ), so that the hydrostatic equation 

(3.47) reduces to Stevino’s law 

∇p = −ρ∇φ. 











The curvature tensor 








We are now going to work out the gravitational field equations by applying the Principle of 

Equivalence to gravitation itself, that is we need a tensorial relation involving the metric 

tensor (and its derivatives) and the distribution of matter and energy. 

Here we treat the problem from a mathematical point of view (as originally done by Gauss 

and Riemann): what is the kind of tensor required? Since first derivatives may be made to 

vanish at any point by changing the system of coordinates, we need second derivatives. 

From the transformation rule for Γλ µν we have 

Γ µ′ 

ν ′ σ ′ = Λ µ′ 

µ Λ ν ν ′Λσσ ′ Γµ νσ − Λ ν ν ′ Λσ µ′ 

σ ′ ∂ν∂σx ⇒ ∂ν∂σx µ′ 

= Λ µ′ 

µ Γ µ νσ − Λ ν′ 

ν Λ σ′ 

σ Γ µ′ 

ν ′ σ ′ . 

If we derive the second expression by ∂ρ, use it again for terms like ∂ρΛ µ′ 

µ ≡ ∂ρ∂µx µ′ , then 

subtract the final result with ρ and σ interchanged, after lengthy calculations we arrive at 

0 = Λ µ′ 

µ (∂ρΓ µ νσ−∂σΓ µ νρ+Γ µ 

λρΓλνσ−Γ µ 

λσΓλνρ)−Λ ν′ 

ν Λ ρ′ 

ρ Λ σ′ 

σ (∂ρ ′ Γµ′ 

ν ′ σ ′ −∂σ ′Γµ′ 

ν ′ ρ ′ +Γ µ′ 

λ ′ ρ ′ Γ λ′ 

ν ′ σ ′−Γµ′ λ ′ σ ′ Γ λ′ 

ν ′ ρ ′ ), 

thus we have obtained a tensorial transformation rule 

where the Riemann curvature tensor is 

R µ′ 

ν ′ ρ ′ σ ′ = Λ µ′ 

µ Λ ν ν ′ Λρ 

ρ ′ Λ σ σ ′ Rµ νρσ 

(4.1) 

R µ νρσ := ∂ρΓ µ νσ − ∂σΓ µ νρ + Γ µ 

λρ Γλ νσ − Γ µ 

λσ Γλ νρ , (4.2) 

and no other tensor can be built from the metric tensor and its first and second derivatives. 





Algebraic properties of the curvature tensor 

Let us start from the fully covariant form of the Riemann tensor 





Rµνρσ ≡ gµκR κ νρσ = gµκ∂ρΓ κ νσ − gµκ∂σΓ κ νρ + gµκ(Γ κ λρ Γλ νσ − Γ κ λσ Γλ νρ). (4.3) 

The derivatives of the affine connection can be transformed using 

∂ρδ η µ = 0 ⇒ gµκ∂ρg κη = −g κη ∂ρgµκ = −g κη (Γ λ ρµ gλκ + Γ λ ρκ gλµ), 

where the last equality comes from (2.11). The first term in (4.3) is then 

gµκ∂ρΓ κ νσ = 1 

2 ∂ρ(∂νgµσ + ∂σgµν − ∂µgνσ) − Γ κ νσ(Γ λ ρµ gλκ + Γ λ ρκ gλµ), 

and the second term is the same if we swap ρ and σ. We finally obtain 

Rµνρσ = 1 

2 (∂ρ∂νgµσ − ∂ρ∂µgνσ + ∂σ∂µgνρ − ∂σ∂νgµρ) + gλκ(Γ λ µσΓ κ νρ − Γ λ µρΓ κ νσ). (4.4) 

From the above relation we easily demonstrate the three algebraic properties below. 

1 Symmetry relation: 

2 Antisymmetry relation: 

3 Cyclic relation: 

Rµνρσ = Rρσµν. (4.5) 

Rµνρσ = −Rνµρσ = −Rµνσρ = +Rνµσρ. (4.6) 

Rµνρσ + Rµρσν + Rµσνρ = 0. (4.7) 





Contractions of the curvature tensor 





The symmetries of the Riemann tensor imply that, in n dimensions, only n 2 (n 2 − 1)/12 out 

of n 4 components are actually independent: for n = 4 we go down from 256 to just 20! 

Let us now see the contractions of the Riemann tensor. We define the Ricci tensor as 

Rµν := R λ µλν , (4.8) 

which is symmetric (1) with 10 independent components and it is the only independent rank 

2 contraction, since (2) implies that the contraction of any other two indices leads to either 

±Rµν or to zero. Then, we define the scalar curvature as 

R := R µ µ , (4.9) 

and again (2) implies that other contractions leads to either ±R or zero. Notice that (3) gives 

|g| −1/2 ɛ µνρσ Rµνρσ = 0, 

so there is no other way to build a scalar from the Riemann tensor. These three tensors will 

be the only building blocks for the theory. 

In 4D, the vanishing of the Riemann tensor is a necessary and sufficient condition to 

exclude gravity, that is one cannot define a global transformation x µ → ξ α to flat space. 

Notice that Rµν = 0 (or R = 0) is not enough to exclude gravity, since the remaining 10 

components of the Riemann tensor could be different from zero. In 3D both the Riemann 

and the Ricci tensors have 6 independent components, so Rµν is enough to describe the 

curvature. In 2D we just need to define the scalar R. 


Example: the 2-sphere 








In order to illustrate the concept of curvature, let us calculate it for the 2-sphere, that is the 

surface of an ordinary sphere. If a is the radius, the (diagonal) metric is 

where x 1 = θ, x 2 = φ, and 

ds 2 = gµνdx µ dx ν = a 2 (dθ 2 + sin 2 θ dφ 2 ), 

gθθ = (g θθ ) −1 = a 2 , gθφ = g θφ = 0, gφφ = (g φφ ) −1 = a 2 sin 2 θ. 

The next step is to calculate the six possible Christoffel symbols, using the procedure 

Γκµν := 1 

2 (∂µgνκ + ∂νgµκ − ∂κgµν), Γ λ µν = g λκ Γκµν. 

The only non vanishing symbols are 

Γθφφ = − 1 

2 ∂θgφφ = −a 2 sin θ cos θ ⇒ Γ θ φφ = gθθ Γθφφ = − sin θ cos θ, 

Γφθφ = 1 

2 ∂θgφφ = a 2 sin θ cos θ ⇒ Γ φ 

θφ = gφφ Γφθφ = cot θ. 

We know that in 2D the scalar curvature is enough, so we need just one non vanishing 

component of the Riemann tensor, for example 

R θ φθφ = ∂θΓ θ φφ − Γθ φφ Γφ 

θφ = sin2 θ − cos 2 θ + sin θ cos θ cot θ = sin 2 θ, 

thus the Ricci tensor and scalar curvature are 

R θ θ 

= Rφ 

φ = Rθφ 

θφ = gφφR θ φθφ = 1/a2 ⇒ R = R θ θ + Rφ 

φ = 2/a2 , 

different from zero, as expected. It is instructive to verify that the curvature vanishes for the 

flat metric ds 2 = dr 2 + r 2 dθ 2 (polar coordinates). 





Round trips by parallel transport 





What are the differences between a flat spacetime, even with curvilinear coordinates and 

non-vanishing affine connections, and a spacetime curved by gravity? 

The first consequence of curvature is that if we parallel transport a vector V µ on a closed 

circuit C, the difference between the initial and final states 

∆V µ 

:= − Γ 

C 

µ νρV ν dx ρ 

is different from zero. Using a sort of Stoke’s theorem it is possible to demonstrate that over 

an infinitesimal circuit the change is 

δV µ = R µ νρσV ν dx ρ dx σ , (4.10) 

thus, by summing all these infinitesimal areal contributions, ∆V µ vanishes if and only if we 

are in flat space. Similarly, parallel transport from one point to another is independent of the 

path if and only if we are in flat space. 


Geodesic deviation 








The presence of curvature can be detected also by observing two nearby freely falling 

particles that travel with trajectories x µ (τ) and x µ (τ) + δx µ (τ). Their equations of motion 

are, respectively 

d2x µ 

+ Γµ 

dτ2 νλ (xµ ) dxν dx 

dτ 

λ 

= 0, 

dτ 

d2 dτ2 (xµ + δx µ ) + Γ µ 

νλ (xµ + δx µ ) d 

dτ (xν + δx ν ) d 

dτ (xλ + δx λ ) = 0, 

and evaluating the difference to first order in δx µ gives 

d2 dτ2 δxµ + ∂ρΓ µ dxν dx 

νλδxρ dτ 

λ dx 

+ 2Γµ 

dτ νλ 

ν dδx 

dτ 

λ 

= 0. 

dτ 

If we now calculate the acceleration of the separation as D 2 δx µ /Dτ 2 , using the above 

relation to eliminate d 2 δx µ /dτ 2 , after some calculation we can obtain 

D2 Dτ2 δxµ = R µ ν dxρ dx 

νρσδx 

dτ 

σ 

dτ 

. (4.11) 

Hence, although a freely falling particle appears to be at rest, curvature (and gravity) 

reveals itself when the separation between the two geodesics is not constant. This is 

basically a tidal effect, which is not in contrast with EP, that arises when the system in 

consideration is not so small with respect to the characteristic length scale of the 

gravitational field. Obviously, this is the physical analogue of violation of the fifth Euclidean 

postulate on a Riemannian curved manifold. 





Commutation of covariant derivatives 





Another important consequence of curvature is that covariant derivatives commute if and 

only if we are in flat space. Let us first calculate 

∇ρ∇σV µ ≡ ∇ρ(∇σV µ ) = ∂ρ(∇σV µ ) − Γ λ ρσ∇λV µ + Γ µ λ 

ρλ∇σV = ∂ρ∂σV µ + ∂ρΓ µ σνV ν + Γ µ σν∂ρV ν − Γ λ ρσ∂λV µ − Γ λ ρσΓ µ 

λν V ν + Γ µ 

ρλ ∂σV λ + Γ µ 

ρλ Γλ σν V ν , 

and then the same relations with ρ and σ exchanged. Only the terms proportional to V ν 

survive, and using (4.2) we find 

∇ρ∇σV µ − ∇σ∇ρV µ = R µ νρσV ν . (4.12) 

Similar relations hold for covariant vectors and for tensors. For instance, given the mixed 

rank 2 tensor T µ ν we have 

∇ρ∇σT µ ν − ∇σ∇ρT µ ν = R µ 

λρσT λ ν − R λ µ 

νρσT λ . (4.13) 

We have thus demonstrated that covariant derivatives of tensors commute if and only if 

R µ νρσ = 0, that is in a flat spacetime. 

Notice the analogy with electrodynamics, for which the fact that 

∂µAν − ∂νAµ 0 

implies the presence of an electromagnetic field Fµν, here when covariant derivatives do not 

commute we are in the presence of a gravitational field. 


The Bianchi identity 








In addition to the algebraic properties, the Riemann tensor satisfies an important differential 

relation, known as the Bianchi identity: 

∇λR µ νρσ + ∇ρR µ 

µ 

νσλ + ∇σR νλρ = 0 . (4.14) 

This expression is fully covariant and it is easy to show that it holds in a locally inertial 

system, where the affine connections vanish (but not their derivatives), and where the 

covariant derivative reduces to the standard one. Hence 

and a cyclic addition leads to the identity. 

∇λR µ νρσ = ∂λ∂ρΓ µ νσ − ∂λ∂σΓ µ νρ, 

A very important consequence of the Bianchi identity is obtained by contracting (4.14) 

twice, first µ and ρ and then ν and σ: 

∇λRνσ + ∇µR µ 

νσλ − ∇σRνλ = 0 ⇒ ∇λR − ∇µR µ 

λ − ∇νR ν µ 1 

λ = 0 ⇒ ∇µ(R λ − 2 δµ 

λ 

from which we obtain the contracted Bianchi identity in the more familiar form 

R) = 0, 

∇µ(R µν − 1 

2 gµν R) = 0 . (4.15) 

Because of the property of having zero divergence, the tensor above will play a crucial role 

in the field equations for GR, as we shall see soon. 





Derivation of the Einstein field equations 





We are finally ready to introduce the long awaited Einstein equations for the gravitational 

field. We expect these equations to be nonlinear, contrary to Maxwell’s equations, since we 

have seen that also gravity carries energy and momentum, thus will be a source for itself. 

We start from the Newtonian Poisson’s law for the potential φ 

∇ 2 φ = 4πGρ, 

where ∇ 2 = δ ij ∂i∂j is the usual Laplacian in 3D space. Since we know that in the Newtonian 

limit g00 −(1 + 2φ) and T00 ρ (p ≪ e ρ), an obvious guess would be 

∇ 2 (−g00) = G00 = 8πGT00 ⇒ Gµν = 8πGTµν. (4.16) 

We then require that Gµν is a rank 2 symmetric tensor containing the second derivatives of 

gµν, thus based on contractions of R µ 

νλρ . The most general choice is 

Gµν = c1Rµν + c2gµνR, 

with c1 and c2 constants. The first aid comes from the contracted Bianchi identity (4.15): 

∇µR µ ν = 1 

2 ∇νR. 

The relation (4.16) tells us that, like Tµν, Gµν must be a (symmetric) divergence-free tensor, 

and the above equation leads to either R = const or to c2 = −c1/2. The first possibility is 

excluded because the trace G µ µ = (c1 + 4c2)R = 8πGT µ µ would be a constant, and this is 

not the case for a general distribution of relativistic matter. Hence we end up with: 

Gµν = c1(Rµν − 1 

2 gµνR). 









The value of c1 is derived from the Newtonian limit, for which we know from (4.16) that 

G00 ∇ 2 (−g00). In this limit and for gµν → ηµν we have |Tij| ≪ |T00| ⇒ |Rij| ≪ |R00|, then 

Moreover 

Rij 1 

2 ηijR, Rii 3 

2 R. 

R η µν Rµν Rii − R00 3 

2 R − R00 ⇒ R 2R00 ⇒ G00 2c1R00. 

We must now calculate R00 from the linearized version of the Riemann tensor, that is from 

(4.3) without the ΓΓ terms. We need 

R00 η µν Rµ0ν0 = η ij Ri0j0 − R0000. 

When the field is static time derivatives vanish, thus R0000 = 0 and 

Ri0j0 = − 1 

2 ∂i∂jg00 ⇒ R00 = − 1 

2 ∇2 g00 ⇒ G00 c1∇ 2 (−g00), 

so that (4.16) can be only satisfied with c1 = 1. 

The final form for Einstein’s field equations (without the cosmological constant) is then 

Rµν − 1 

2 gµνR = 8πGTµν , (4.17) 

where the constant is actually 8πG/c 4 , thus only strong concentrations of mass and energy 

may deform spacetime. An alternative form is sometimes useful. Taking the trace, recalling 

that g µ µ ≡ δ µ µ = 4, and defining T := T µ µ, we can also write 

Rµν = 8πG(Tµν − 1 

2 gµνT). (4.18) 





Gauge conditions on the choice of coordinates 





The knowledge of the Einstein tensor Gµν from (4.17) is unfortunately not enough to 

determine uniquely gµν and thus the coordinate system. Due to the Bianchi identity (4.14) 

Gµν = Rµν − 1 

2 gµνR, ∇µG µ ν = 0, (4.19) 

hence (4.17) does not provide 10 independent equations, but actually only 10 − 4 = 6, 

leaving us with 4 degrees of freedom in the 10 unknowns gµν. Hence, we have a gauge 

invariance on the choice of the coordinate system. 

Notice that this gauge ambiguity in the choice of the coordinates is similar to the situation in 

electrodynamics (in SR), where the Maxwell equations for the vector potential are 

Aµ − ∂µ(∂νA ν ) = −4πJµ, 

clearly invariant under the gauge transformation Aµ → Aµ + ∂µχ, where χ is an arbitrary 

scalar function, and := η µν ∂µ∂ν is the d’Alambertian operator. These are 4 equations, but 

the continuity equation implies a condition of a vanishing divergence, this is why we have 

gauge freedom. The choice usually adopted in SR is the Lorentzian gauge, in which 

∂µA µ = 0 ⇒ Aµ = −4πJµ. 

If initially the vector potential is such that ∂µA µ 0, one needs to solve 

χ = −∂µA µ , 

and adding the gradient of the solution χ we retrieve the Lorentzian gauge. 


Harmonic coordinates 








A particularly convenient choice for the GR metric is that in the harmonic gauge condition 

Γ λ := g µν Γ λ µν = 0. (4.20) 

Notice that for a given a system x µ where Γλ 0, it is always possible to find a system x µ′ 

where Γλ′ = 0, since from (3.1) we find the following set of equations for x µ′ : 

Γ λ′ 

= Γ λ′ 

µ ′ ν ′ gµ′ ν ′ 

= Λ λ′ 

λ Γλ − g µν ∂µ∂νx λ′ 

= 0 ⇒ g µν ∂µ∂νx λ′ 

= Λ λ′ 

λ Γλ . 

Given its nature, there is no way to put the harmonic condition in a covariant form, however 

Γ λ = 1 

2 gµν g λκ (∂µgνκ+∂νgµκ−∂κgµν) = −g µν gνκ∂µg λκ −g λκ |g| −1/2 ∂κ|g| 1/2 = −|g| −1/2 ∂κ(|g| 1/2 g λκ ), 

so that the gauge condition becomes 

Γ λ = −|g| −1/2 ∂κ(|g| 1/2 g λκ ) = 0 . (4.21) 

A scalar function φ is said to be harmonic when (note that in GR has a covariant 

definition) 

φ := g µν ∇µ∇νφ = ∇µ(g µν ∇νφ) = |g| −1/2 ∂µ(|g| 1/2 g µν ∂νφ) = 0, 

so when φ = xλ we have ∂νx λ = δλ ν , and therefore 

x λ = |g| −1/2 ∂µ(|g| 1/2 g µλ ) = −Γ λ . 

We can finally appreciate the meaning of the name. Harmonic coordinates simply satisfy 

x λ = 0 . (4.22) 





Weak fields and linearized field equations 





Since corrections to Newton’s theory are usually very small, it is important to consider the 

weak field limit of GR, that is the linearized theory that arises when the field equations are 

worked out under a first order expansion of a nearly-Minkowskian metric. This approach will 

also lead to the prediction of gravitational radiation. Let us then define 

gµν = ηµν + hµν, |hµν| ≪ 1 , (4.23) 

where hµν is our perturbation. The Minkowski metric ηµν will be used to raise or lower the 

indices of tensors as a background metric (to first order). In this approximation, the 

Christoffel symbols are 

Γ λ µν = 1 

2 ηλκ (∂µhνκ + ∂νhµκ − ∂κhµν), (4.24) 

leading to the linearized Riemann tensor 

Rµνρσ = ηµλ(∂ρΓ λ νσ − ∂σΓ λ νρ) = 1 

2 (∂ρ∂νhµσ + ∂σ∂µhνρ − ∂σ∂νhµρ − ∂ρ∂µhνσ), (4.25) 

with contractions 

and 

Rµν = R λ µλν 

where h := h µ µ, ∂ µ = η µν ∂ν, and = ∂ µ ∂µ. 

= 1 

2 (∂µ∂λh λ ν + ∂ν∂λh λ µ − ∂µ∂νh − hµν), (4.26) 

R = R µ µ = ∂ µ ∂ ν hµν − h, (4.27) 









It is now convenient to impose the condition for harmonic coordinates 

Γ λ ≡ g µν Γ λ µν = 1 

2 ηµν η λκ (∂µhνκ + ∂νhµκ − ∂κhµν) = ∂ µ h λ 

µ − ∂ λ h = 0 ⇒ ∂µh µ ν = 1 

2 ∂νh. 

The Einstein tensor simplifies greatly, leading to the linearized field equations 

Gµν = Rµν − 1 

2 ηµνR = − 1 

2 (hµν − 1 

2 ηµνh) = 8πGTµν. (4.28) 

Finally, it is convenient to introduce the trace reversed field tensor 

¯hµν := hµν − 1 

2 ηµνh (∂µ ¯h µ ν = 0), (4.29) 

for which ¯h = −h, and the linearized field equations are simply 

¯hµν = −16πGTµν . (4.30) 

It is interesting to examine the Newtonian limit. As discussed earlier, the dominant term in 

the energy-momentum tensor is T00 = ρ, so that also ¯h00 will be the leading term, satisfying 

∇ 2¯h00 = −16πGρ. 

Comparing with Newton’s field equation we must have ¯h00 = −4φ ⇒ ¯h = ¯h 0 = 4φ, thus 

0 

inverting (4.29) we find the first order metric for weak fields 

ds 2 = −(1 + 2φ)dt 2 + (1 − 2φ)(dx 2 + dy 2 + dz 2 ), (4.31) 

which is a step forward compared to the result g00 = −(1 + 2φ) derived from the Principle 

of Equivalence. 


Gravitational waves 








The most important consequence of the weak field equation (4.30) is the prediction of 

gravitational radiation. In void we have 

¯hµν ≡ (−∂ 2 t + ∇2 )¯hµν = 0, (4.32) 

which is a set of 10 wave equations for gravitational waves (GWs) propagating at the speed 

of light (c = 1). The simplest solution is that for plane waves 

¯hµν = Aµν exp(ikλx λ ) , (4.33) 

with Aµν the amplitude tensor and k µ := (ω, k) the wave 4-vector. These satisfy 

kµk µ = k · k − ω 2 = 0, Aµνk µ = 0, (4.34) 

the first is the dispersion relation, and since kµ is null it just tells us again that GWs 

propagate at the light speed, the second comes from the harmonic gauge in (4.29), that for 

GWs is also called Lorentz gauge in analogy with electromagnetic waves. 

We have 10 − 4 = 6 independent components, but for weak fields we are free to impose 

extra 4 conditions, e.g. that Aµν is a transverse-traceless (TT) tensor, for which 

Aµνu ν , A µ µ = 0, (4.35) 

where u ν is any arbitrary 4-velocity (i.e. a time-like unit vector). Choosing a Lorentz frame 

where u µ = (1, 0, 0, 0) and propagation along z, the only non-vanishing components are 

A11 = −A22 = A + , A12 = A21 = A × . (4.36) 









What is the effect of GWs and how can they be detected? Let us start by looking at the 

geodesic equation for the metric modified by a plane wave, in the TT gauge where 

¯hµν = hµν. Assuming that the particle is initially at rest, then 

(du µ /dτ)t=0 = −Γ µ 1 = − 00 2 ηµν (2 ˙h0ν − ∂νh00), 

where the dot indicates time derivativation. But in TT h00 = h0µ = 0, hence the trajectory is 

unchanged. This a gauge effect, since in TT we have attached our coordinates to freely 

falling particles, recall that coordinates are just labels. 

More important are the tidal effects, since a GW changes the local curvature of spacetime. 

Consider the relation for geodesic acceleration (4.11) of two nearby particles initially at rest 

with separation vector ξ µ = (0, ξx , ξy , 0): a µ := D2ξ µ /Dτ2 = R µ 

ν00ξν . The required terms in 

the linearized Riemann tensor (4.25) are: 

R x 

x00 = −Rx0x0 = 1 ¨h TT 

2 xx , Rx y00 = −Rx0y0 = 1 ¨h TT 

2 xy , R y 

y00 = −Ry0y0 = 1 ¨h TT 

2 yy , 

thus the equations are 

ax = 1 TT 

2 (¨h xx ξx + ¨h TT 

xy ξy ) = − 1 

2 ω2 (A + ξx + A × ξy ) exp[i(kz − ωt)], 

ay = 1 TT 

2 (¨h xy ξx + ¨h TT 

yy ξy ) = − 1 

2 ω2 (A × ξx − A + ξy ) exp[i(kz − ωt)], 

and it is apparent, for example inspecting the accelerations on an initially circular ring of 

matter, that GWs have two independent transverse (in the x − y plane) polarizations, the 

plus + (A + 0, A × = 0) and the cross × (A × 0, A + = 0). 

(4.37) 









Figure: Effects of + and × polarized GWs on a ring of freely falling particles. 

Here we will not describe how the generation of GWs work. We just provide the so-called 

quadrupole formula for the emitted power of a rotating axisymmetric object, valid for 

ω ≪ L −1 , with L characteristic length-scale of the source: 

P(2Ω) = 32GΩ6I2 e2 5c5 . (4.38) 

Here the emission peaks at twice the rotational frequency Ω, I := I11 + I22 and 

e := (I11 − I22)/I , with the directions 1 and 2 characterizing the two principal axes. For a 

spherical object e = 0 so there is no emission, whereas for the orbit of a mass m at 

distance r we can set e = 1 and I = mr 2 , which starts to be substantial for close binaries. 

Projects like VIRGO, LIGO, GEO600, TAMA struggle with technical difficulties to detect 

GWs directly for the first time. A binary NS-NS or BH-BH merger in the Virgo cluster should 

produce an amplitude of 10 −21 (the amplitude decreases with distance as r −1 ), theoretically 

these instruments should be able to detect the signal through laser interferometry. 









The Schwarzschild solution and classic tests of General Relativity 

There are four classic tests of General Relativity: 

1 The gravitational redshift of spectral lines. 

2 The deflection of light by the Sun. 

3 The precession of the perihelia of the orbits of the inner planets. 

4 The radar echo delay (Shapiro delay). 

The first one only tests the Principle of Equivalence, and it has been already discussed, 

here we will focus on the other three, after introducing the metric for a static and isotropic 

spacetime, derived by Schwarzschild as early as in 1916. 

The most general form of such a metric can only depend on r := (x · x) 1/2 and contain 

rotationally invariant differential terms like x · dx and dx · dx: 

ds 2 = −F(r)dt 2 + 2E(r)dtx · dx + D(r)(x · dx) 2 + C(r)dx 2 . 

It is convenient to introduce spherical coordinates, for which 

and the line element becomes 

x 1 = r sin θ cos φ, x 2 = r sin θ sin φ, x 3 = r cos θ, 

ds 2 = −F(r)dt 2 + 2rE(r)dtdr + r 2 D(r)dr 2 + C(r)(dr 2 + r 2 dΩ 2 ), (4.39) 

where the angular part has been expressed through dΩ 2 := dθ 2 + sin 2 θdφ 2 . By re-defining 

t and r, after some calculations it is possible to arrive to the so-called standard form 

ds 2 = −B(r)dt 2 + A(r)dr 2 + r 2 dΩ 2 , (4.40) 

and we must now derive the r dependence in the unknown functions A and B. 









We now look for a solution of Einstein’s equations in empty space outside a spherical object 

of mass M. The field equations are simply 

where we recall the definitions 

Rµν = 0 , (4.41) 

Rµν = ∂λΓ λ µν − ∂νΓ λ µλ + Γκ µνΓ λ κλ − Γκ µλ Γλ νκ, Γ λ µν = 1 

2 gλκ (∂µgνκ + ∂νgµκ − ∂κgµν). 

The metric is diagonal with 

gtt = 1/g tt = −B, grr = 1/g rr = A, gθθ = 1/g θθ = r 2 , gφφ = 1/g φφ = r 2 sin 2 θ 

(the determinant is g = −ABr 4 sin 2 θ), the non-vanishing Christoffel symbols are 

Γ t tr = B′ /2B, Γ r rr = A ′ /2A, Γ r 

θθ = −r/A, Γr φφ = −r sin2 θ/A, Γ r tt = B′ /2A, 

Γ θ φφ = − sin θ cos θ, Γθ rθ = 1/r, Γφ 

rφ = 1/r, Γφ 

θφ = cot θ, 

and the non-vanishing (diagonal) components of the Ricci tensor are 

Rtt = B′ 

 

B′ A ′ 

B′ 

− + + 

rA 4A A B 

B′′ 

2A , 

A ′ 

B′ A ′ 

B′ 

Rrr = + + − 

rA 4B A B 

B′′ 

2B , 

Rθθ = Rφφ/ sin 2 θ = 1 − 1 

 

r A ′ 

B′ 

+ − . 

A 2A A B 









All these components must be set equal to zero, as in (4.41). Notice that 

 

Rrr Rtt 1 A ′ 

B′ 

+ = + 

A B 2A A B 

 

= 0 ⇒ A(r)B(r) = const, 

and since the asymptotic limit is that of spherical coordinates with A(r) → 1 and B(r) → 1, 

we have A(r) = 1/B(r). Substituting in the Einstein equations 

Rθθ = 1 − B − rB ′ , Rrr = −B ′ /(rB) − B ′′ /(2B) = R ′ θθ /(2rB) 

hence we end up with a differential equation for B(r) by imposing 

Rθθ = 0 ⇒ [rB(r)] ′ = 1 ⇒ B(r) = 1 − 2GM/r, 

where the proportionality constant has been chosen to match the Newtonian limit 

gtt = −B(r) = −(1 + 2φ), with φ = −GM/r. The Schwarzschild metric in standard form is 

ds 2 = −(1 − 2GM/r)dt 2 + (1 − 2GM/r) −1 dr 2 + r 2 dΩ 2 . (4.42) 

Notice that the metric is singular for r → 0 and for the Schwarzschild radius rs := 2GM, 

which however, even for neutron stars, is well hidden inside the object of mass M, and this 

solution is only valid outside the object. It remains present also in the so-called isotropic 

coordinates, a solution alternative to the form (4.40) with A factorizing all spatial metric 

terms 

ds 2 = − 

2 1 − MG/2ρ 

dt 

1 + MG/2ρ 

2 + (1 + GM/2ρ) 4 (dρ 2 + ρ 2 dΩ 2 ); r = ρ(1 + GM/2ρ) 2 , 

but it can be transformed away by mixing t and r terms. 





Geodesics in a Schwarzschild spacetime 

Let us now rewrite the Schwarzschild metric (4.42) for θ = π/2 





ds 2 = −B(r)dt 2 + A(r)dr 2 + r 2 dφ 2 ; B(r) = A(r) −1 = 1 − 2GM/r, (4.43) 

and solve in the equatorial plane the equation of motion 

d2x λ 

dσ2 + Γλ µν 

dx µ dx 

dσ 

ν 

= 0, (4.44) 

dσ 

with σ ≡ τ for a material body or another parameter for massless particles like photons. The 

three differential equations arising from the above formula are 

¨t + B′ 

B ˙r A ′ 

˙t = 0, ¨r + 

2A ˙r2 − r 

A ˙φ 2 + B′ 

2A ˙t 2 = 0, 

¨φ + 2 

r ˙r ˙φ = 0, 

where the dot indicates derivation with respect to σ. The first relation, using ˙B = B ′ ˙r, 

implies B˙t = const, and we normalize σ by choosing ˙t = B −1 . The last two expressions 

define two integrals of motions, namely the specific energy E and angular momentum L 

A ˙r 2 + L 2 /r 2 + 1/B = −E, r 2 ˙φ = L. (4.45) 

The proper time is now derived from the line element, and we find dτ2 = Edσ2 , with E > 0 

for material particles and E = 0 for photons. The parameter σ may be eliminated to get 

2 A dr 

+ 

dt 

L 2 

r 

= −E, 

2 dφ 

= L, (4.46) 

B dt 

B 2 

1 

− 

r2 B 

and by letting A → 1, B → 1 + 2φ we recover the known Newtonian limit. 





Deflection of light by the Sun 





Figure: The deflection of light in Sun’s gravitational field. 

Consider now the deflection of light (E = 0) in the ecliptic plane (θ = π/2) due to the 

gravitational field of the Sun. This is expressed as 

∆φ = 2|φ(r0) − φ∞| − π, (4.47) 

where r0 is the minimum distance, for which dr/dφ = 0. It it is convenient to eliminate both 

σ and t from the motion equations, thus 

A 

r4 2 dr 

+ 

dφ 

1 1 

− 

r2 L 2B = 0 ⇒ φ(r) − φ∞ 

⎡ 

∞ ⎤ 

2 −1/2 

r B(r) 

dr 

= ⎢⎣ 

− 1⎥⎦ 

r r0 B(r0) rB1/2 , (4.48) 

(r) 

where in the second relation we have also used A = B−1 and eliminated L through 

r2 0 = L 2B(r0). Using elliptic integrals, recalling that B(r) = 1 − 2GM/r and leaving only first 

order terms in GM/r, we find ∆φ 4GM/r0. For the Sun the prediction is 

∆φ = (R⊙/r0)θ⊙, θ⊙ := 4GM⊙/c 2 R⊙ 1.75 ′′ , (4.49) 

confirmed at solar eclipses. More spectacular is the gravitational lensing of distant galaxies. 





Precession of perihelia of inner planets 





Figure: The precession of perihelia of inner planets. 

Using similar arguments, but this time for bounded orbits of planets (E > 0), let us calculate 

the precession of perihelia. If r± are the radii where dr/dφ vanishes, we may write 

1 

r2 1 

− 

± L 2 E 

= − 

B(r±) L 2 ⇒ E = r2 + /B(r+) − r2 −/B(r−) 

r2 + − r2 , L 

− 

2 = 1/B(r+) − 1/B(r−) 

1/r2 + − 1/r2 . (4.50) 

− 

The integral to be solved is now 

⎡ 

r r 

φ(r) − φ(r−) = ⎢⎣ 

2 − (B−1 (r) − B−1 (r−)) − r2 + (B−1 (r) − B−1 (r+)) 

r2 + r2 − (B−1 (r+) − B−1 − 

(r−)) 

1 

r2 ⎤−1/2 

dr 

⎥⎦ 

r2B 1/2 (r) , 

r− 

and to first order, the precession of Mercury, the closest planet to the Sun, is 

∆φ = 2|φ(r+) − φ(r−)| − 2π (3πM⊙G/c 2 )(1/r+ + 1/r−) 43.03 ′′ /century. (4.51) 

In 1943, based on observations dating back to 1765, the measured value was as close as 

43.11 ± 0.45 ′′ /century. This was considered the final proof for the validity of Einstein’s GR. 


Shapiro delay 








Figure: Shapiro delay (radar echo from inner planets and delay in pulsar timing). 

The first three tests were originally proposed by Einstein himself, later Shapiro proposed a 

test on timing: the delay in radar echoes reflected by an inner planet. If r0 is the radius 

where the light ray (E = 0) is closest to the Sun (dr/dt = 0), then r 2 

0 = L 2 B(r0) and 

A 

B 2 

dr 

dt 

2 

+ 

 

r0 

2 1 1 

− 

r B(r0) B = 0 ⇒ t(r, r0) = 

r 

r0 

 

r0 

2 −1/2 

B(r) dr 

1 − 

. (4.52) 

r B(r0) B(r) 

The total time is found by summing tE(rE, r0) (Earth-Sun), tp(rp, r0) (Sun-planet), 

multiplying by two and making the usual expansion. The maximum delay (compared to 

straight line propagation) is when the planet is in superior conjunction (the Sun between E 

and p, r0 R⊙): 

(∆t)max (4GM⊙/c 3 )[1 + ln(4rPrE/R 2 ⊙)], (4.53) 

and for Mercury we find 240 µs. The Shapiro delay is also observed in pulsar binaries. 









Part III: applications to compact objects 











Introduction 








We start the part of the course on astrophysical applications with the hydrodynamics of 

spherical explosions. We will briefly review the case of the expansion of the Supernova 

ejected material (the SNR), which can be treated by the non-relativistic equations, and then 

we shall move to the case of GRBs, where the shells of material are expected to propagate 

with Γ ∼ 100, thus Special Ralativity will be used. SNe and GRBs are not directly related to 

compact objects, but are a manifestation of an extremely high-energy event that has 

produced a compact object (NS or BH in a collapse of a massive stellar core). 

In the description of these explosive events we will also need to make use of the physics of 

shock waves. Shocks, especially those of SNRs, are crucial for the acceleration of 

high-energy particles, like the electrons giving rise to the non-thermal emission, for example 

in PWNe, or of the more massive particles observed in cosmic rays (protons, nuclei). 

Shocks and discontinuities are produced naturally in the rarefied astrophysical 

environments, where the fluid is very often made of ionized gas (plasma), highly 

compressible (contrary to liquids). Such plasmas are generally non-collisional, due to the 

extreme low densities (n ∼ 1 cm −3 in the ISM), thus long-range plasma interactions will 

replace direct collisions in non-ideal effects like viscosity, needed at shocks. Typical 

expansion velocities of a SN shock are V 10 4 km s −1 , to be compared to a sound speed 

cs 10 km s −1 in the ISM where T 10 4 K. In the initial phase we thus have a Mach 

number M = V/cs 10 3 ≫ 1, what is usually called a hypersonic motion. 





The equations of classical hydrodynamics 





For non-relativistic velocities, an ideal fluid can be described by the set of equations: 

∂t ρ + ∇ · (ρv) = 0 , (1.1) 

ρ(∂t + v · ∇)v = −∇p − ρ∇φ , (1.2) 

(∂t + v · ∇)s = 0 , (1.3) 

where ρ is the gas (or plasma) mass density, v is the bulk flow velocity, p is the thermal 

pressure, s is the specific entropy (entropy per unit mass), and φ is the external gravitational 

potential (if present). These are, respectively, the continuity equation (conservation of 

mass), the Euler equation (conservation of momentum), and the energy equation, which in 

the absence of dissipation is equivalent to the condition for an adiabatic flow. By combining 

(1.1) and (1.3), we find 

∂t (ρs) + ∇ · (ρsv) = 0, (1.4) 

that is a continuity equation for the entropy per unit volume. Very often in the energy 

equation we should include heating and cooling terms. However, if fluid motions occurs 

over fast enough times, the adiabatic approximation may be still a good one. 

The set of hydrodynamical equations is not closed until an equation of state (EoS) is 

imposed, for example in the form p = p(ρ, s). 


The Bernoulli equation 








Recall the laws of thermodynamics for reversible transformations 

dU = TdS − pdV, dH = TdS + Vdp (H := U + pV), (1.5) 

or, dividing for the mass M of a macroscopic volume V of fluid 

dε = Tds − pd(1/ρ), dh = Tds + dp/ρ (h := ε + p/ρ), (1.6) 

where ρ = M/V, ε = U/M, s = S/M, h = H/M. Then, for adiabatic motions we may 

rewrite the Euler equation as 

s = const ⇒ dh = dp/ρ ⇒ (∂t + v · ∇)v = −∇(h + φ). (1.7) 

The above relation is very useful in case of a stationary (and adiabatic) flow. Recalling the 

vectorial relation 

(v · ∇)v = ∇( 1 

2 v2 ) − v × (∇ × v), (1.8) 

it is straightforward to derive the Bernoulli equation 

v · ∇( 1 

2 v2 + h + φ) = 0 ⇒ 1 

2 v2 + h + φ = const , (1.9) 

where the constant refers to each streamline. The constant will be the same even across 

streamlines if and only if the flow is irrotational (i.e. vanishing vorticity: ω = ∇ × v ≡ 0). 


Sound waves 








Consider small perturbations around a static equilibrium of the fluid 

ρ = ρ0 + δρ, v = δv, p = p0 + δp, 

and linearize the hydrodynamical equations by neglecting all second order terms. It is easy 

to verify that all perturbations satisfy the wave equation 

[∂ 2 t − c2 s ∇ 2 ]f(x, t) = 0 ⇒ f(x, t) = A exp[i(k · x − ωt)], (1.10) 

where A ≪ 1 and the wave vector and pulsation satisfy the dispersion relation 

ω 2 = c 2 s k 2 . (1.11) 

The characteristic velocity is the sound speed, defined as 

c 2 

∂p 

s := = γ 

∂ρ s 

p 

, (1.12) 

ρ 

where the second relation holds for an ideal gas EoS, with γ := cp/cV the ratio of specific 

heats. Usual forms of the ideal gas EoS and related properties are 

p = (kB/m)ρT, p = (γ − 1)ρɛ, s = cV ln(p/ρ γ ), cp − cV = (kB/m), (1.13) 

where m = M/N = ρ/n is the particle mass. 









Conservative form of the hydrodynamical equations 

The continuity equation (1.1) expresses, in differential form, the conservation of mass. For a 

generic quantity u the conservation law is 

∂t u + ∇ · f = 0 ⇒ d 

 

u dV = − f · ndS, 

dt V 

∂V 

where f = f(u) is the flux of u and we have applied the Gauss theorem. Basically, the 

integral of u over a control volume V increases in time because of the incoming flux across 

the boundaries S = ∂V. Since we expect that also momentum and energy are to be 

conserved, we can make it apparent. 

Neglecting external forces for simplicity, let us now add (1.2) to (1.1) multiplied by v. Using 

∇ · (A B) = (A · ∇)B + (∇ · A)B, 

we find the equation for the momentum conservation 

∂t (ρv) + ∇ · (ρvv + pI) = 0 , (1.14) 

where I is the identity tensor with components δij. The flux of the momentum component i 

along the direction j is thus given by the Reynolds stress tensor 

which is clearly symmetric. 

Rij := ρvivj + pδij, (1.15) 









The derivation of the conservation of energy is more elaborate. The contribution of the 

kinetic energy is obtained by multiplying (1.2) by v. The left hand side member is 

ρv · (∂t + v · ∇)v = ∂t ( 1 

2 ρv2 ) − 1 

2 v2 ∂t ρ + ρv · [∇( 1 

2 v2 ) − v × (∇ × v)], 

where the last term vanishes. Using the continuity equation we end up with 

For an adiabatic flow (Tds = 0) we have 

hence 

dε = −p d(1/ρ) ⇒ ρ dε 

dt 

∂t (ρ 1 

2 v2 ) + ∇ · (ρv 1 

2 v2 ) = −(v · ∇)p. 

p dρ d 

1 dρ 

= ⇒ (ρε) = (ρε + p) = −(ρε + p)(∇ · v), 

ρ dt dt ρ dt 

∂t (ρ ε) + ∇ · (ρv ε) = −(∇ · v)p, 

The energy equation in conservative form is then obtained summing the two contributions 

∂t [ρ( 1 

2 v2 + ε)] + ∇ · [ρv ( 1 

2 v2 + h)] = 0 , (1.16) 

in which the thermal contribution to the flux is ερv + pv = hρv, with h replacing ε due to the 

work of pressure forces. 





Discontinuity surfaces and shocks 





Hydrodynamics has the peculiarity to admit discontinuous solutions. Obviously, only the 

integral form of the equations can be used across discontinuity surfaces. Consider a 

stationary and planar discontinuity surface along x. For a generic conservation law 

xs +η 

dJ 

dJ 

= 0 ⇒ [J] := 

dx xs −η dx dx ≡ J2 − J1 = 0, 

where J is the conserved flux and J1 = J(xs − η), J2 = J(xs + η), the quantities just before 

and after the discontinuity at x = xs. If the surface moves, we consider a reference frame 

centered on xs. 

Applying the above rule to the hydrodynamics equations we have the following relations: 

mass flux conservation 

momentum flux conservation 

energy flux conservation 

[ρvx] = 0, (1.17) 

[ρv 2 x + p] = 0, [ρvxvy] = 0, [ρvxvz] = 0, (1.18) 

which must be solved to find the jumps in ρ, p, and v. 

[ρvx( 1 

2 v2 + h)] = 0, (1.19) 









Two classes of solutions exist: contact discontinuities and shocks. In the first case there is 

no flow across the surface (v1x = v2x = 0), then all relations are automatically satisfied by 

any jumps [ρ] 0, [vy] 0, [vz] 0, the only condition is the continuity of pressure [p] = 0. 

In the case of shocks, v1x 0, v2x 0, then we are left with 

[ρvx] = 0, [ρv 2 x + p] = 0, [ 1 

2 v2 + h] = 0, (1.20) 

and [vy] = [vz] = 0. Given the continuity in the transverse velocity, we can choose a 

reference frame in order to have vy = vz = 0. If we label with 1 the unshocked (upstream) 

medium, then v1 = Vs if the shock moves with speed Vs in the laboratory frame. After 

some calculations, the Rankine-Hugoniot (RH) jump conditions are 

ρ2 

ρ1 

p2 

p1 

T2 

T1 

M 2 

2 

= v1 

= 

v2 

(γ + 1)M2 

= 2γM2 

1 

γ + 1 

= [2γM2 

1 

1 

(γ − 1)M 2 

1 

= 2 + (γ − 1)M2 

1 

2γM 2 

1 

+ 2 , (1.21) 

γ − 1 

− , (1.22) 

γ + 1 

− (γ − 1)][(γ − 1)M2 

1 

(γ + 1) 2 M 2 

1 

+ 2] 

, (1.23) 

− (γ − 1) , (1.24) 

where we have supposed an ideal gas law, for which h = c 2 s /(γ − 1), and M = v/cs is the 

Mach number. Downstream quantities are functions of the shock Mach number M1 alone. 









Shocks naturally arise in fluid mechanics due to the non-linearity of the equations. Any finite 

disturbance, even a linear sound wave, after some time will steepen into a shock. The RH 

relations for shocks have two solutions, one for M1 < 1 and one for M1 > 1 (for M1 = 1 

there is no transformation). It may be demonstrated that the specific entropy is also 

discontinuous (hence viscosity or other non-ideal effects intervene) and we have 

s2 > s1 if and only if M1 > 1, (1.25) 

that is entropy (disorder) is produced post-shock if and only if the shock is supersonic. The 

second law of thermodynamics ensures that the solution of the RH relations with M1 > 1 is 

the only physically valid one, because otherwise T2 < T1 and v2 > v1, that is work would be 

produced out of heat, without any other consequence. On the contrary, shocks basically 

convert ordered kinetic energy (v2 < v1) into disordered motions (heating), as T2/T1 can be 

quite large as M1 increases. 

In the case of Supernova explosions the shock is said to be hypersonic, that is M1 ≫ 1. In 

this limit the RH relations become 

ρ2 = 

γ + 1 

γ − 1 ρ1, 

2 2ρ1v1 p2 = 

γ + 1 , T2 = 

2(γ − 1)(m/kB)v 2 

(γ + 1) 2 

1 

, M 2 

2 

γ − 1 

= , (1.26) 

2γ 

where m is the average mass of particles. For γ = 5/3 we have a maximum density jump 

ρ2 = 4ρ1. Notice that the downstream pressure and temperature are unbounded. 





Self-similar flows: the Sedov-Taylor solution 





Consider a sudden explosion releasing an energy E within a small volume, that we consider 

negligible, at r = 0 and t = 0. If the external medium is homogeneous and static (density 

ρ0 and temperature T0 → 0, we suppose a cold gas condition), the motion of the shock 

front, with position Rs(t) and velocity Vs(t), will be radially symmetric and can be found by 

self-similarity arguments using the only data at our disposal E and ρ0: 

 

Rs(t) = β Et2 

1 

5 

∝ t 

ρ0 

2/5 , Vs(t) ≡ dRs 2Rs 

= 

dt 5t ∝ t−3/5 , (1.27) 

where β is a non-dimensional quantity that will be derived later on. The form of Rs has been 

found by purely dimensional arguments, without solving any equation. 

Right after the passage of the (strong) shock, the fluid quantities at r = R− s are provided by 

the RH relations (1.26) with ρ1 = ρ0 and v1 = −Vs, since we must now reintroduce the 

shock motion as the jump conditions had been derived in the shock frame 

ρ(R − s ) = ρ2 = 

γ + 1 

γ − 1 ρ0, v(R − s ) = Vs + v2 = 2Vs 

γ + 1 , p(R− s ) = p2 = 2ρ0V 2 s 

, (1.28) 

γ + 1 

so basically Vs + v2 is the relative velocity of the shocked fluid compared to the (static) 

background. 









We are now ready to solve for the post-shock flow. Let us introduce the non-dimensional 

distance to the front 

ξ = r 

 

ρ0 

= r 

Rs(t) β Et2 1 

5 

(1.29) 

and normalize all fluid quantities against their values right past the shock at R − s 

ρ(r, t) = 

γ + 1 

γ − 1 ρ0R(ξ), v(r, t) = 2Vs 

γ + 1 V(ξ), p(r, t) = 2ρ0V 2 s 

P(ξ). (1.30) 

γ + 1 

We can now integrate the fluid equations (in spherical symmetry) backwards from ξ = 1 to 

ξ → 0 with initial conditions R(1) = V(1) = P(1) = 1. The equations are 

−ξ ˙R + 2 

˙RV + R ˙V 

γ + 1 

+ 4 RV 

= 0, 

γ + 1 ξ 

−2ξ ˙V − 3V + 4V ˙V γ − 1 2 ˙P 

= − 

γ + 1 γ + 1 R , 

−3 − ˙Pξ 

P 

2 V ˙P ˙R 2γ ˙R 

+ + γξ − V = 0, (1.31) 

γ + 1 P R γ + 1 R 

where the dot indicates derivation in ξ. This set of first order ODEs can be easily integrated 

with a simple Runge-Kutta method, available in any numerical package. 









Figure: The self-similar Sedov-Taylor solution for γ = 5/3. 

Notice that once we know the post-shock quantities and we rescale the radius with the 

moving Rs(t), the self-similarity insures that the spatial profiles of the normalized quantities 

will not depend on time explicitly. As we can see, most of the mass is concentrated very 

near the front, while the interior is filled with hot and light gas, providing the pressure forces 

needed to make the shock propagate. 

Finally, the value of β can now be calculated by integrating the energy density ρ( 1 

2 v2 + ε) 

over the whole volume occupied by the shock gas 

E = 

Rs 

0 

1 

2 ρv2 + p 

γ − 1 

 

4πr 2 dr ⇒ 1 = β 

γ2 16π 

− 1 25 

1 

that for γ = 5/3, appropriate for a monoatomic gas, provides β 2.02. 

0 

[R(ξ)V 2 (ξ) + P(ξ)]ξ 2 dξ, 





The phases of SNR expansion 





We recall that type II (but most probably also Ib and Ic) SN events are caused by the 

collapse of the iron core of a massive star. When a proto NS is formed, the external 

collapsing matter bounces back and a shock wave is suddenly launched. Initially, the 

motion of the shock front will be ballistic, where the velocity of the ejecta is constant and 

provided by 

E = 1 

2 MejV 2 s , 

and for E = 10 52 erg and Mej = 10 M⊙ we have Vs = 10 4 km s −1 . This free expansion 

phase will end when the collected material (ISM at density ρ0 10 −24 g cm −3 ) has a mass 

4π 

3 ρ0R 3 s = Mej, 

that is for Rs 5 pc and an age of τ 500 yr. The Crab Nebula is still in free expansion, in 

spite of its age of τ 1000 yr, due to the slow expansion velocity Vs < 1000 km s −1 . 

The second phase is characterized by a slow down, due to the increasing mass, and by 

adiabaticity: for temperatures above 10 6 K the cooling time is very long, thus total energy is 

conserved and we may apply the Sedov solution for a self-similar adiabatic flow. For this 

model to be applicable we must also wait until the reverse shock propagating backwards 

due to the slowing down has reached the origin and bounced several times, until all the 

ejected gas has been heated according to the Sedov profiles. 









Figure: The four phases of a SNR expansion: Rs ∝ t, Rs ∝ t 2/5 , Rs ∝ t 1/4 , Rs ∼ const. 

The post-shock temperature decreases rather rapidly as the shock slows down 

T(R − s ) ∝ V 2 s ∝ t −6/5 , 

when the threshold of T 10 6 K ions start to recombine and energy is radiated away 

efficiently (radiative cooling in forbidden lines). Energy cannot be conserved any longer, 

while we may still impose the conservation of momentum, which gives 

4π 

3 ρ0R 3 s Vs = const ⇒ Rs ∝ t 1/4 , Vs ∝ t −3/4 . 

This phase is said of the snowplow, because material is collected post-shock in a very 

narrow layer. Finally, when the local ISM sound speed is reached, we have no longer a 

shock front and there is mixing with the ambient medium at Rs ∼ const. 


Relativistic shocks 








The treatment of the expansion of GRB shells and jets moving at Γ ∼ 100 (here we use the 

capital letter to avoid confusion with the adiabatic index) requires the use of the equations 

for (special) relativistic hydrodynamics 

∂µ(ρu µ ) = 0, ∂µ[(e + p)u µ u ν + pη µν ] = 0, (1.32) 

where ∂µ = (∂t , ∇) and u µ = (Γ, Γv). These may be also written in a more familiar way as 

∂t (ρΓ) + ∇ · (ρΓv) = 0, (1.33) 

∂t [(e + p)Γ 2 v] + ∇ · [(e + p)Γ 2 vv + pI] = 0, (1.34) 

∂t [(e + p)Γ 2 − p] + ∇ · [(e + p)Γ 2 v] = 0. (1.35) 

Notice that the conservative equations for classical hydrodynamics are easily retrieved in 

the limit v ≪ 1 ⇒ Γ → 1 and p ≪ e = ρ(1 + ε) ρ. The only non-trivial relation is the 

energy equation, since we must subtract the continuity equation and approximate 

(e + p)Γ 2 − ρΓ ρΓ[(1 + ε + p/ρ)(1 + 1 

2 v2 ) − 1] 1 

2 ρv2 + ρε + p. 

We remind that as for classical hydrodynamics, also in SR we have a conservation law for 

the entropy in the absence of dissipative terms 

u µ ∂µs = 0, ∂t (ρsΓ) + ∇ · (ρsΓv) = 0, (1.36) 

entropy will increase at shocks, where ordered bulk flow motion is converted into heat and 

microscopic disordered motion. 









Let x be the direction normal to a planar shock front and suppose for simplicity that the flow 

occurs along x. For a stationary situation, the Taub shock conditions are found by imposing 

the conservation of fluxes at the two sides of the shock written in the comoving system of 

the front, namely 

[ρΓvx] = 0, [(e + p)Γ 2 v 2 x + p] = 0, [(e + p)Γ 2 vx] = 0. (1.37) 

We are interested in strong shock conditions, for which Γ1 ≫ 1, the ISM is a cold medium 

with p1 = 0 and e1 = ρ1, and the shocked gas becomes relativistically hot with p2 = e2/3 

and cs 2 = 1/ √ 3 (cs ≡ dp/de). In this case the Taub conditions are easily solved 

v2 = 1/3, Γ 2 

2 = 9/8, Γ2s = Γ 2 

1 /2, ρ2 = √ 8ρ1Γ1, p2 = (2/3)ρ1Γ 2 

1 , (1.38) 

where Γs is the shock Lorentz factor, corresponding to the shock speed obtained by the 

relativistic composition rule vs = (v1 − v2)/(1 − v1v2). Notice that the downstream Mach 

number is M2 = v2/cs 2 = 1/ √ 3 < 1, as expected. 









Relativistic self-similar flows: the Blandford-McKee solution 

Consider a sudden release of a confined relativistically hot material (a fireball) with 

η = E/M = e/ρ ≫ 1, 

producing the propagation of a spherical relativistic blast wave front with position Rs(t) and 

Lorentz factor Γs(t) from the origin r = 0 at t = 0, and restrict to the ultra-relativistic 

conditions (1.84) 

p = e/3 ≫ ρ ⇒ p ∝ ρ 4/3 . 

The total energy density in the fixed laboratory frame is T 00 = (e + p)Γ 2 − p, then the 

energy contained within a spherical shell at Rs is 

E 

Rs 

0 

16π pΓ 2 r 2 dr ∝ Γ 2 s R3 s . (1.39) 

If we suppose that the flow is initially adiabatic, as for the Sedov solution, then E is 

conserved, Γ 2 s R3 s Γ 2 s t3 ∼ const and 

Γs(t) ∼ t −3/2 ⇒ Rs(t) t [1 − 1/(8Γ 2 s)], (1.40) 

where the second relation has been obtained integrating Vs 1 − 1/(2Γ 2 s) in time. We have 

thus derived how a fireball decelerates adiabatically while expanding in the ISM without 

solving any equation, as for the Sedov solution. The exact expression will be only available 

after the full set of relativistic equations for the shocked fluid has been solved. As for the 

non-relativistic case, a self-similar flow will be considered. 









We must now look for a self-similarity variable appropriate for ultra-relativistic shells. By 

inspecting the Taub conditions, we see that ρ2 ∼ Γs. However, due to the contraction of 

lengths, in the laboratory frame the density would rather go as ρ ′ 2 = Γ2ρ2 ∼ Γ2 s , so the 

swept-up material is collected within a narrow shell of width ∼ Rs/Γ2 s . An appropriate new 

coordinate to describe the post-shock flow is thus (Blandford & McKee, 1976) 

or even more conveniently 

ξ := (Rs − r)/(Rs/Γ 2 s) = (1 − r/Rs)Γ 2 s, (1.41) 

χ := 1 + 8ξ = 1 + 8(1 − r/Rs)Γ 2 s, (1.42) 

which is 1 at the shock position and increases towards the origin (note that we are zooming 

into the narrow shell of shocked material). If we switch from the variables r and t to χ and 

Γ2 s in the (adiabatic) relativistic fluid equations in spherical symmetry in the Γ ≫ 1 we find 

Γ 2 = 1 

2 Γ2 s χ −1 , ρ = √ 8Γsρ1 χ −5/4 , p = 2 

3 Γ2 s ρ1 χ −17/12 , 

where the Taub relations (1.38) have been employed to normalize the functions at χ = 1. 

Substitution in the expression for E allows us to recover the constant of proportionality: 

 

17 E 

Γs(t) = 

8π ρ1c 5 . (1.43) 

t3 Notice that thanks to the hypothesis Γ ≫ 1 we have managed to obtain algebraic equations 

for solutions in simple power-law form. These solutions can be further generalized to a 

continuous energy injection from the central engine and to propagation in stratified media. 





The phases of the expansion of a GRB 





Let us now try to model all the phases of the expansion of a GRB shell. Conservation of 

mass, energy and momentum or a relativistically hot gas with p = ρε/3 ∝ ρ 4/3 says that 

R 2 ρΓ ∼ const, R 2 p 3/4 Γ ∼ const, R 2 (ρ + 4p)Γ 2 ∼ const, 

and two initial phases after the fireball release can be distinguished: 

free acceleration. As long as p ≫ ρ (η = e/ρ ≫ 1) the internal pressure forces 

accelerate the shell according to 

Γ ∝ R, ρ ∝ R −3 , p ∝ R −4 . (1.44) 

coasting. After the radius RL = ηR(t = 0) we start to have p ≪ ρ, so the solution is 

with a constant shell speed 

Γ = const, ρ ∝ R −2 , p ∝ R −8/3 . (1.45) 

While propagating into the ISM, the GRB shell is collecting material and deceleration is 

supposed to start. This phase corresponds to the X-ray afterglow. The condition is that the 

incoming swept-up momentum ΓM is comparable to the mass of the ejecta, locally at rest 

Mej. For large Γ, M ∼ Mej/Γ much earlier than the classical condition M ∼ Mej for SNRs. 









Figure: The four phases of a GRB expansion: Γs ∝ R, Γs ∼ const, Γs ∝ R −3/2 , and the final Sedov stage. 

The last two phases of a GRB shell (adiabatic) expansion are then 

relativistic self-similar deceleration. This is the afterglow phase described analytically 

by the Blandford-McKee model, for which 

Γ ∝ R −3/2 , ρ ∝ R −3/2 , p ∝ R −3 . (1.46) 

Sedov-Taylor deceleration. The flow is sub-relativistic and decelerates as V ∝ t −3/4 , 

until mixing with ISM when the velocity becomes sub-sonic. 











Introduction 








For relativistic stars we mean stars where Newtonian dynamics is not sufficient to account 

for their global structure and properties. The prototypes are certainly Neutron Stars (NSs), 

but they can be generalized to various types of degenerate matter EoSs, up to the limit of 

quark stars (or strange stars), where hadrons are deconfined in a quark-gluon plasma 

(QGP) state under the effect of the strong gravity, rather than of high temperature. 

Observations of double NS systems, pulsars, and X-ray binaries lead to NS masses 

M ∼ 1.3 − 1.5 M⊙ and radii R ∼ 10 − 12 km, with matter above the nuclear density 

n ∼ 2.8 10 14 g cm −3 . The compactness parameter is thus about 

ξ := |φ|/c 2 = GM/(c 2 R) ∼ 20%, 

and GR effects should be taken into account for realistic modeling. More complex is the the 

problem of which EoS to employ: astrophysical observations are used to constrain nuclear 

physics models! Relativistic stars are considered to be cold, in the sense that their thermal 

internal energy is much smaller than the Fermi energy due to a degenerate neutron gas: 

E ∼ kT < 1 MeV ∼ 10 10 K ≪ EF ∼ 30 MeV. 

For such degenerate relativistic stars we often consider a barotropic EoS of the kind 

p = p(n), e = e(n) ⇒ p = p(e), (2.1) 

and for simple applications even a polytropic EoS (here ρ = nm) 

may be accurate enough if K and γ are carefully chosen. 

p = Kρ γ , (2.2) 





Structure equations for a static, spherical star 





Let us now assume the simplest case of a static (non-rotating) and spherically symmetric 

relativistic star with a given cold gas EoS. The appropriate choice for the GR metric is again 

that assumed for Schwarzschild solution, and we are free to adopt its standard form (4.40) 

ds 2 = −B(r)dt 2 + A(r)dr 2 + r 2 dΩ 2 , (2.3) 

where as usual dΩ 2 := dθ 2 + sin 2 θdφ 2 for an isotropic space. We must now derive the r 

dependence in the unknown functions A and B, precisely as we did for Schwarzschild 

solution, with the difference that we are no longer in void. The Einstein field equations to 

solve are 


2 gµνT) , (2.4) 

where as usual T := T µ µ is the trace of the energy-momentum tensor. 

For a static fluid we have u i = 0, and since the metric is diagonal we may also write 

ur = uθ = uφ = 0. 

The normalizing condition for the 4-velocity says that 

so we are left with 

g µν uµuν = −1 ⇒ ut = −(−g tt ) −1/2 = −(−gtt ) 1/2 = −B 1/2 , 

Ttt = eB, Trr = pA, Tθθ = p r 2 , Tφφ = p r 2 sin 2 θ, 

and the trace is T = Ttt /gtt + Trr /grr + Tθθ/gθθ + Tφφ/gφφ = 3p − e. 









The non-vanishing (diagonal) components of the Ricci tensor have been already calculated 

for the Schwarzschild solution, thus the system of differential equations is the following: 

Rtt = B′ 

 

B′ A ′ 

B′ 

− + + 

rA 4A A B 

B′′ 

= 4πG(e + 3p)B, 

2A 

A ′ 

B′ A ′ 

B′ 

Rrr = + + − 

rA 4B A B 

B′′ 

= 4πG(e − p)A, 

2B 

Rθθ = 1 − 1 

 

r A ′ 

B′ 

+ − = 4πG(e − p)r 

A 2A A B 

2 , 

where the equation along φ is useless since Rφφ = Rθθ sin 2 θ and also Tφφ = Tθθ sin 2 θ. An 

equation for A(r) alone is easily obtained combining the equations above 

Rrr 

2A 

Rθθ Rtt 

+ + 

r2 2B 

= A ′ 

rA 

1 1 

+ − 

2 r2 r2A 

r ′ 

= 8πGe ⇒ = 1 − 8πGer 

A 

2 , 

and the solution with finite A(0) is very similar to the corresponding one in void 

 

A(r) = 1 − 2Gm(r) 

−1 , m(r) := 

r 

r 

e(r 

0 

′ )4πr ′2 dr ′ . (2.5) 

Notice that we recover the Newtonian result, where also a function m(r) of the total mass 

contained within a radius r was defined, though here the mass function takes into account 

any form of internal energy. 









Let us now use the equation for hydrostatic equilibrium (3.47), which yields 

∇p = −(e + p)∇ ln(−gtt ) 1/2 ⇒ B ′ /B = −2p ′ /(e + p). (2.6) 

Together with (2.5), the above relation between B and p can be put in the Einstein equation 

for Rθθ to provide an expression for p ′ . It is convenient to use B = −gtt = exp(2φ), hence 

the final set of first order ODEs for GR stellar structure is 

dp Gme 

= − 

dr r2 

1 + p 

 

1 + 

e 

4πr3 

p 

1 − 

m 

2Gm 

−1 , (2.7) 

r 

dm 

dr = 4πr2e, (2.8) 

 

dφ 

1 + 

dr 4πr3 

p 

1 − 

m 

2Gm 

−1 , (2.9) 

r 

= Gm 

r 2 

known as the TOV equations, derived in 1939 by Tolman, Oppenheimer, and Volkoff. Given 

a barotropic EoS p = p(e), the equations are integrated from r = 0, where m(0) = 0 and 

p(0) = p(ec), up to the stellar surface radius r = R, where p(R) = 0. Then we have that 

ec alone is enough to provide the global solution for stellar structure, in analogy with 

polytropic stars like white dwarfs where p = p(ρ) and ρc determine the solution. 

Compared to the Newtonian case (p ≪ e ρ, 2Gm/r ≪ 1), TOV also accounts for: 

1 the pressure is an additional source for gravity, 

2 since gravity also acts on p, the inertia ρ is replaced by e + p, 

3 gravity increases faster than r −2 for a decreasing r, by a factor (1 − 2Gm/r) −1 . 





Gravitational mass and exterior solution 





Notice that one of the Einstein equations have not been used, this is because of the Bianchi 

identity implying that G µν is divergenceless, which leads to the equation of momentum 

conservation, that is the equation for hydrostatic equilibrium used above. 

At this stage, we also must specify the concept of mass further. We define gravitational 

mass of our relativistic star the quantity M used above 

R 

M := m(R) ≡ e(r)4πr 

0 

2 dr. (2.10) 

This mass is the same encountered for the Schwarzschild solution, the one producing the 

gravitational field in void. This is true also for the field of relativistic stars beyond r = R, 

where e → 0 and p → 0, since 

dφ 

dr 

= GM 

r 2 

 

1 − 2GM 

−1 r 

(r > R), 

and imposing asymptotic flatness for r → ∞ we recover the Schwarzschild solution 

φ(r) = 1 

2 ln 

 

1 − 2GM 

r 

 

⇒ B(r) = A(r) −1 = 1 − 2GM 

r 

(r > R), (2.11) 

for a point source of mass M located at r = 0. This is a consequence of a more general 

Birkhoff’s theorem of GR: the unique (stable) solution in void ouside a spherically symmetric 

distribution of matter is Schwarzschild’s spacetime. This is true even for a non-stationary 

configuration, for example a pulsating star. 





Baryonic mass and binding energy 





The definition of the gravitational mass M as the total mass-energy contained inside the star 

for r ≤ R is not as straightforward as it may look like. Notice that the proper volume of a 

spherical shell of width dr is actually not simply 4πr 2 dr, but rather 

dV := (AB) 1/2 4πr 2 dr, 

hence it is clear that the expression for the gravitational mass also contains the contribution 

of the (negative) gravitational potential energy, which will decrease the actual value of M. 

Even more informative is to compare the gravitational mass with the baryonic mass, that 

arising from the conservation law 

∇µ(ρu µ ) ≡ mn∇µ(nu µ ) = 0, (2.12) 

where ρ = nmn and mn = 1.66 10 −24 g is the rest-mass of a nucleon. For a static fluid 

gtt (u t ) 2 = −1 ⇒ u t = B −1/2 , thus the baryonic mass is defined as 

M0 := mnN, N := 

R 

nu 

0 

t R 

dV = 

0 

n(r) 

 

1 − Gm(r) 

r 

−1/2 

4πr 2 dr, (2.13) 

where N is the total number of nucleons in the star, thus the baryonic mass is simply the 

total contribution by non-interacting baryons at infinity. The difference 

E := M − M0 = M − mnN, (2.14) 

is the internal binding energy of the star, and we must have E < 0 for gravity to win against 

thermal pressure and insure stability. For neutron stars typically we find |E| 0.2M⊙c 2 . 





If we now recall that the thermal energy density is 

the binding energy can be written as 

where 

R 

T := 

0 

 

ρɛ 1 − 2Gm 

−1/2 4πr 

r 

2 dr, V := 





ρɛ = e − ρ = e − mnn, (2.15) 

E = T + V, (2.16) 

R 

e 

0 

⎡ 

⎢⎣ 1 − 1 − 2Gm 

⎤⎥⎦ 

−1/2 

r 

4πr2dr, (2.17) 

that are the thermal and gravitational energies, recognized by their Newtonian limits 

R 

T ρɛ 4πr 

0 

2 dr, 

R 

V − 

0 

Gmρ 

 

4πr 

r 

2 dr. (2.18) 

It may be demonstrated that generalizations of the virial theorem can be defined in GR as 

well, even in the rotating case. 

We have learned that the knowledge of ec is enough to build a stellar model. It may be 

proved that the turning points discriminating stable and unstable equilibria to radial 

perturbations are derived from 

dE dN 

= 0, = 0, (2.19) 

dec dec 

and that the stability branches for NS equilibria are those for which 

dM 

> 0, 

dec 

dM 

< 0. (2.20) 

dR 





Relativistic stars with constant energy density 





A simple analytical solution can be found even in GR for spherical non-rotating stars in the 

incompressible limit, that is stars with uniform energy density 

e = const ⇒ m = (4π/3)r 3 e = M(r/R) 3 , (2.21) 

corresponding to ρ = const in the Newtonian limit. The TOV equation for the pressure 

becomes 

p ′ [(e + p)(e/3 + p)] −1 = 4πGr[1 − 8πGr 2 e/3] −1 , 

which can be integrated from r = R, where p = 0, inwards. This gives 

e + p 

e + 3p = 

−1/2 

2 1 − 8πGr e/3 

, 

1 − 8πGR 2 e/3 

and finally the pressure is, replacing e for M using (2.21) 

p(r) = 3M 

4πR 3 

(1 − 2GMr 2 /R 3 ) 1/2 − (1 − 2GM/R) 1/2 

3(1 − 2GM/R) 1/2 − (1 − 2GMr2 /R3 . (2.22) 

) 1/2 

The metric functions A(r) and B(r) can be calculated similarly 

 

A(r) = 1 − 2GMr2 

R3 −1 

, B(r) = 1 

⎡ 

⎢⎣ 

4 

3 

 

1 − 2GM 

1/2 

− 1 − 

R 

2GMr2 

R3 1/2⎤⎥⎦ 2 

, 

where these solutions are valid inside the star and match Schwarzschild’s solution for 

r = R, valid outside the star in void according to Birkhoff’s theorem. 









Such stars are certainly non-physical, and the main interest is that it is possible to find an 

analytical solution. However, we can derive some general results. Notice that the pressure 

may become infinite for 

r 2 ∞ := 9R 2 − 4R 3 /(GM), (2.23) 

thus, in order to avoid this unphysical situation, the star must have 

ξ := GM/R < 4/9 ⇒ rs := 2GM < (8/9)R. (2.24) 

It is quite interesting that this result is actually quite general: it is possible to demonstrate 

that any spherically symmetric non-rotating star with e(r) ≤ 0 from 0 to R has the above 

condition as an upper limit for its compactness. Similarly, we could demonstrate that the 

stable stars with the smallest central pressure are those with uniform density, so p(0) from 

(2.22) is an upper limit (again valid for GM/R < 4/9). 

Notice that this condition readily translates into an upper limit for the gravitational redshift at 

the star’s surface. If a photon is emitted at re = R and observed at infinity ro = ∞ where 

gravitation is negligible, then 

νe 

νo 

= [−g00(re)] 1/2 

[−g00(ro)] 1/2 = B(R)1/2 = 

so that the highest redshift observable for ξ < 4/9 is 

 

1 − 2GM 

1/2 , (2.25) 

R 

z := ∆λ/λ = λe/λo − 1 = νo/νe − 1 = (1 − 2GM/R) −1/2 − 1 < 2. (2.26) 

Redshift from atomic transitions of accreting materials onto a NS in a binary system has 

been really measured, with values corresponding to z ∼ ξ ∼ 20%, well below the limit. 


Neutron stars 








Figure: The complex inner structure of a neutron star. 

The detailed microphysics in the interior of a NS is one of the most challenging questions in 

Astrophysics. The first TOV model in 1939 supposed a star made up by degenerate cold 

non-interacting neutrons, produced by reactions like 

p + e → n + νe, 

where neutrinos are free to escape (actually enough protons and electrons must remain so 

that the Pauli principle prevents neutron beta decay). 









We recall that for a Fermi gas in the T → 0 limit, the number density is 

n = 8π 

h3 kF 

0 

k 2 dk = 8π 3 

k 

3h3 F ⇒ kF = 

1/3 3 3h n 

, (2.27) 

8π 

where kF is the Fermi momentum. Energy density and degenerate pressure are provided 

by the expressions 

e = 8π 

h3 kF 

(k 

0 

2 + m 2 n) 1/2 k 2 dk, p = 8π 

3h3 kF 

k 

0 

2 (k 2 + m 2 n) −1/2 k 2 dk, (2.28) 

then it is clear that the final EoS will be of the barotropic type 

e = e(n), p = p(n) ⇒ p = p(e), (2.29) 

as anticipated and used in the TOV system. In particular, in the non-relativistic case 

kF ≪ mn we find a polytropic p ∼ n5/3 , while in the ultra-relativistic case p ∼ n4/3 . 

Moreover, when kF ≫ mn we find 

p = e 8π 

= 

3 3h3 kF 

0 

k 3 dk = 2π 4 

k 

3h3 F , (2.30) 

and we retrieve the EoS expected for ultra-relativistic particles. Amazingly, when plugged 

into the TOV equations we can find an exact solution (approximately valid in the core region) 

which is diverging for r → 0 but still integrable with m(r) → 0. 

p(r) = (56πGr 2 ) −1 , (2.31) 









Figure: Stability branches for white dwarfs and neutron stars. 

Without assuming any particular limit, the TOV equations may be still integrated provided a 

central value ec. The mass M of a degenerate gas neutron star reaches a maximum for 

Mmax = 0.7M⊙, Rmax = 9.6 km, 

corresponding to a surface redshift and compactness parameter z ∼ ξ ∼ 0.13. 

If electrons and protons are included, the results are not that different. What really makes a 

difference is to include nuclear forces, leading to very different EoS and to a range of limiting 

masses from 0.37M⊙ to 2.4M⊙. Moreover, crystalline crust, superfluid interior, powerful 

magnetic field, and rapid rotation should be included for any realistic model of NS structure. 





Neutron stars and numerical relativity 





Figure: Simulation of a merger of magnetized NSs setting the stage for a short GRB (Rezzolla 2011). 

Complex EoS, magnetic field, rotation, and especially a fully multi-D case and time 

dependence (e.g. pulsations of a newly formed proto-NS) are certainly impossible to treat 

analytically and nowadays numerical simulations have reached very high standards. 

Numerical relativity relies on the solution of the combined Einstein equations and 

conservation laws for hydrodynamics (or MHD) in a curved spacetime. In order to do so, the 

so-called 3 + 1 formalism is employed, where time is singled out to follow evolution at the 

price of losing the elegance of the covariant form. Simulations are also helpful in providing 

templates of GW emission for a wide variety of physical situations. 











Introduction 








We have seen that after the core collapse of a massive star, a neutron star is expected to 

form, with an equilibrium between gravity and degenerate (neutron) gas pressure. This 

equilibrium configurations are stable up to a certain mass, depending on the details of the 

EoS (unknown). The TOV limit is 0.7 M⊙, more realistically Mmax ∼ 2 − 3 M⊙. Above this 

limit, unless some new exotic equilibrium is reached (strange or quark stars), gravity wins 

and collapse is expected to proceed until a singularity with ρ → ∞ is reached, or at least up 

to the quantum limit of the Planck density 

ρP := c5 

G 3 = 5.1 × 1093 g cm −3 . 

We are going to see that before this stage is reached, General Relativity predicts that an 

event horizon will form, hiding the singularity (and the related unknown physics) from the 

exterior world, where GR continues to hold. 

Amazingly, the most general solution for a stationary black hole is known analytically (the 

Kerr-Newman geometry). It depends on only three (observable!) parameters: mass M, 

angular momentum J, and charge Q (often neglected), thus all other information about the 

initial state and about the detailed physics involved is radiated away during collapse in the 

form of EM or GR waves. This situation is often summarized by Wheeler’s aphorism a black 

hole has no hair. 


Birkhoff’s theorem 








Let us now investigate the physics of gravitational collapse, following the pioneering 

approach by Oppenheimer and Snyder (1939), who considered the evolution of a 

homogeneous sphere of (pressureless) gas in GR. 

Adopting the assumption of a spherically symmetric spacetime, as already done for 

Schwarzschild’s metric, the line element can be written in the most general form as 

ds 2 = −F(r, t)dt 2 + 2rE(r, t)dtdr + r 2 D(r, t)dr 2 + C(r, t)(dr 2 + r 2 dΩ 2 ), (3.1) 

where the angular part has been expressed through dΩ 2 := dθ 2 + sin 2 θdφ 2 , as usual. By 

re-defining t and r, the analogous of the metric element in standard form is then 

ds 2 = −B(r, t)dt 2 + A(r, t)dr 2 + r 2 dΩ 2 , (3.2) 

where the unknown functions A and B now depend on time as well. 

After the Ricci tensor terms are calculated, it is possible to see that only additional terms 

proportional to ˙A and Ä appear. In particular, Rtr ∝ ˙A, so that in void A = A(r) and the 

Einstein equations reduce to those in the static case. 

This proves Birkhoff’s theorem, for which a spherically symmetric gravitational field always 

produces the static Schwarzschild solution outside the compact object, even in the time 

dependent case. As a consequence, no gravitational radiation will be emitted during a 

spherically symmetric collapse. 





Evolution equations in comoving coordinates 





Instead of using the metric in standard form, gravitational collapse is best followed by 

introducing comoving coordinates, attached to an imaginary set of freely-falling particles. If 

clocks are in free-fall, then the proper time interval between two points (x, t) and (x, t + dt) 

of a particle’s trajectory is 

dτ 2 = −gµνdx µ dx ν = −gtt dt 2 , 

and dt = dτ ⇒ gtt = −1, thus F(r, t) = 1. Moreover, the particle’s trajectory x = const and 

t = τ satisfies 

0 = d2x i 

dτ2 + Γi dx 

µν 

µ dx 

dτ 

ν 

dτ = Γi tt = gij∂t gtj, 

then ∂t gtj = 0, and for our isotropic metric gtr = 2rE(r), gtθ = gtφ = 0. This term can be 

set to zero if we redefine the coordinates as t ′ = t − 2 rE(r)dr, r ′ = r, so that 

g ′ ∂t 

tr = gtr 

∂t ′ 

∂r 

+ gtt 

∂r ′ 

∂t 

∂t ′ 

∂t 

∂r ′ = gtr − ∂t 

= 0. 

∂r ′ 

Dropping the primes and replacing C, D with U, V, the metric in comoving coordinates can 

be finally expressed in the so-called Gaussian normal form as 

ds 2 = −dt 2 + U(r, t)dr 2 + V(r, t)dΩ 2 , (3.3) 

with the new unknowns U, V are to be determined by solving Einstein’s equations. 





The non-vanishing (diagonal) metric coefficients are 





grr = 1/g rr = U, gθθ = 1/g θθ = V, gφφ = 1/g φφ = V sin 2 θ, gtt = 1/g tt = −1, 

with determinant g = −UV 2 sin 2 θ. The Einstein equations for a perfect fluid are 


2 gµνT); Tµν = (e + p)uµuν + pgµν , (3.4) 

where, in comoving coordinates, the fluid 4-velocity is 

u r = ur = u θ = uθ = u φ = uφ = 0, u t = 1, ut = −1. 

The stress tensor terms are T r r = T θ φ t 

θ = T φ = p, Tt = −e, and the trace is then 

T = T λ λ = 3p − e. After some lengthy algebra the relevant equations are (Rφφ = Rθθ sin 2 θ): 

V ′′ ′2 V 

Rrr = − + 

V 2V 2 + U′ V ′ 

2UV 

Ü ˙U 2 

+ − 

2 4U + ˙U ˙V 

= 4πG(e − p)U, 

2V 

V ′′ 

Rθθ = 1 − 

2U + V ′ U ′ ¨V 

+ 

4U2 2 + ˙V ˙U 

= 4πG(e − p)V, 

4U 

Rtt = − Ü ¨V ˙U 2 ˙V 2 

− + + = 4πG(e + 3p), 

2U V 4U2 2V 2 

˙V ′ 

Rtr = − 

V + V ′ ˙V ˙UV ′ 

+ = 0, 

2V 2 2UV 

where the dot indicates a time derivative and the prime a radial derivative. 









Suppose now our perfect fluid is homogeneous, thus energy, density, and pressure 

depends on time alone. Then it is convenient to seek a separable solution for our unknown 

functions U and V. A convenient choice satisfying Rtr = 0 is 

U(r, t) := R 2 (t)f(r), V(r, t) := R 2 (t)r 2 . (3.5) 

The first Einstein equation, divided by U, becomes 

[r 2 f(r)] −1 f ′ (r) + R(t)¨R(t) + 2 ˙R 2 (t) = 4πGR 2 (t)[e(t) − p(t)], 

and only if the radial term is a constant the equation will be satisfied. The solution is 

f(r) = (1 − kr 2 ) −1 , (3.6) 

and the second Einstein equation is also satisfied. The final form of the metric is then 

ds 2 = −dt 2 + R 2 2 dr 

(t) 

1 − kr2 + r2dΩ 2 

 

, (3.7) 

which is the famous Robertson-Walker metric for an isotropic and homogeneous universe, 

derived in 1935-36. The remaining Einstein equations are the Friedmann equations 

¨R = − 4πG 

3 (e + 3p)R, ˙R 2 + k = 8πG 

3 eR2 , (3.8) 

providing a first set of equations for the coupled evolution of metric and matter. 









Homogeneous and isotropic collapse of a dust sphere 

We are now ready to specify the evolution to the homogeneous and isotropic collapse of a 

dust sphere, as in the original work of 1939. The last equation is the energy conservation 

for a perfect fluid, that is 

u µ ∇µe + (e + p)∇µu µ = 0 ⇒ ˙e + (e + p)3 ˙R/R = 0, (3.9) 

which in the case of a pressurless fluid (or dust), when p ≪ ρ and e → ρ coincides with 

mass conservation. If we normalize with R(0) = 1, the density increases in time like 

ρ(t) = ρ(0)R(t) −3 . (3.10) 

Moreover, if at t = 0 we have ˙R(0) = 0, the Friedmann equations provide 

˙R 2 (t) = k[R −1 (t) − 1], k = (8πG/3)ρ(0), (3.11) 

for which a parametrized solution is that of a cycloid: 

R = (1 + cos ψ)/2, t = (2 √ k) −1 (ψ + sin ψ). (3.12) 

Note that R(t) vanishes when ψ = π, and hence when t = T, where 

T = π 

2 √ 1/2 π 3 

= = 

k 2 8πGρ(0) 

π 

⎛ 

r 

⎜⎝ 

2 

3 ⎞1/2 

0 

⎟⎠ , 

2GM 

M := 4π 

3 ρ(0)r3 0 . (3.13) 

Thus, a uniform dust sphere of initial radius r0 and mass M will collapse from rest to a state 

of infinite rest mass and energy density in the (proper) finite time T. 





Formation of a Schwarzschild black hole 





We have learned that a singularity is expected to form in a finite time T as the result of 

gravitational collapse. However, we know from Birkhoff’s theorem that outside the 

collapsing body the metric must match the Schwarzschild solution (4.42) (here G = c = 1) 

ds 2 = −(1 − 2M/r)dt 2 + (1 − 2M/r) −1 dr 2 + r 2 dΩ 2 , (3.14) 

where rR(t) → r and gtt −1. Before the singularity is reached, the star will entirely fall 

within the event horizon characterized by the Schwarzschild radius rS := 2M where the 

coordinate system is singular. Notice that in the natural units adopted, time, space, and 

mass have all the same unit of measure, for example mass. 

In particular, note that light cones (ds2 = 0) of events occurring for constant θ and φ have 

an increasingly steeper slope 

dt/dr = ±(1 − rS/r) −1 

(3.15) 

as r approaches rS, so that an event occurring near the event horizon will take an 

asymptotically infinite time to reach a distant observer. 

Therefore, also the finite proper time T for gravitational collapse will appear infinite from 

outside, and in general no signals, not even light, can escape from the interior of the event 

horizon: a black hole has formed. By inspecting the curvature tensor we could see that only 

r = 0 is a true singularity, the one at r = rS can be removed by changing coordinates (e.g. 

Kruskal), thus any infalling body will not notice anything special when crossing the horizon, 

but the above statements concerning a distant observer remain true. 





Orbits around a Schwarzschild black hole 





Consider now the problem of finding the orbits of a test particle around a Schwarzschild 

black hole (in the equatorial plane θ = π/2). One way to do that is to derive the Christoffel 

symbols and work out the equations of motions (see the section on classical tests of GR), 

or to look for geodesics. Using the Lagrangian L, here we seek the minima of 

B B √ 

TAB := dτ = −2L dσ; 2L := gµν ˙x 

A 

A 

µ ˙x ν , (3.16) 

where the dot indicates a σ derivative. This implies the solution of the Lagrange equations 

 

d ∂L 

dσ ∂ ˙x µ 

 

− ∂L 

= 0, (3.17) 

∂x µ 

for x µ = (t, r, θ, φ). Invariance in t and φ leads to the definition of two constant of motions 

E := −∂L/∂˙t = −gtt ˙t = (1 − 2M/r)˙t, L := ∂L/∂ ˙φ = gφφ ˙φ = r 2 ˙φ. (3.18) 

For a material particle, σ ≡ τ, ˙x µ ≡ u µ , E = −ut is the specific energy of the particle at 

infinity and L = uφ is its specific angular momentum. Putting these constants into the 

definition 2L ≡ uµu µ = −1 we find the equation 

(1 − 2M/r) −1 E 2 − (1 − 2M/r) −1 ˙r 2 − L 2 /r 2 = 1, (3.19) 

or, rearranging the terms and introducing the effective potential 

˙r 2 + V 2 

eff (r) = E2 , V 2 

eff (r) := (1 − 2M/r)(1 + L 2 /r 2 ) . (3.20) 









Figure: Plots of Veff(r), in the case L > 2 √ 3M (left panel) and for various values of L (right panel). 

Let us now study the possible orbits. The extrema of Veff are 

dVeff 

dr 

2 

L 

= 0 ⇒ r = 1 ± 1 − 12(M/L) 

2M 

2 

 

, (3.21) 

which provides two real solutions (both with r > rS) only if L > 2 √ 3M. In this case bound 

orbits are possible, though not closed (periastron precession). Turning points are those 

where Veff = E, the minimum in the potential gives the stable circular orbits. When 

L = 2 √ 3M we have the innermost stable circular orbit, with r = 6M = 3rS. At this radius, a 

particle has a fractional binding energy of 1 − E = 1 − √ 8/9 5.72%, that is the efficiency 

for accretion onto a Schwarzschild BH (to be compared with 0.7% for nuclear reactions 

inside stars). Finally, a particle with L < 2 √ 3M will inevitably cross the event horizon and 

fall into the BH. 









Notice that the Newtonian limit (E − 1 ≪ 1, M/r ≪ 1, L/r ≪ 1) is recovered by writing 

v 2 /2 + V N eff (r) = (E2 − 1)/2, V N 2 

eff (r) := (Veff − 1)/2 −M/r + L 2 /2r 2 , 

and basically the only difference is the missing term −ML 2 /r 3 , which is thus the one 

responsible for deviations from Keplerian orbits. 

Consider now radial infall from rest at r0 for a particle with L = 0. The equation of motion is 

˙r 2 + 1 − 2M/r = E 2 = 1 − 2M/r0 ⇒ ˙r 2 = rS/r − rS/r0, 

exactly the same as (3.11) once we set R = r/r0 and k = rS/r 3. 

Thus, the proper time for 

0 

infall is the same for the collapse of a homogeneous dust sphere with M = (4π/3)ρ(0)r3 

T = dτ = 

0 

r0 

dr 

π 

= 

[rS/r − rS/r0] 1/2 

2 √ π 

= 

k 2 r0 

r0 

rS 

0 : 

1/2 

. (3.22) 

In both cases, the Schwarzschild time for a distant observer will instead diverge for r = rS, 

since dt/dτ = E(1 − rS/r) −1 , and the event horizon is only reached for t → ∞. 

Figure: Time to fall in a BH. For a distant observer not even rS is ever reached. 









Orbits around a Schwarzschild black hole: photons 

For massless particles like photons dτ = 0 and we must use the affine parameter σ. The 

equation of motion becomes 

(1 − 2M/r) −1 E 2 − (1 − 2M/r) −1 ˙r 2 − L 2 /r 2 = 0, (3.23) 

and dividing by L 2 = r 4 ˙φ 2 and rearranging some terms, it can be written as 

1 

r 4 

dr 

dφ 

2 

+ B −2 (r) = b −2 ; B −2 (r) := 1 

r 2 

 

1 − 2M 

r 

 

. (3.24) 

Here the effective potential B −2 (r) has only an extremum (maximum) for r = 3M (a circular 

unstable orbit for light), with value 1/27M 2 regardless of the value of b := L/E (the impact 

parameter). A photon with b ≤ 3 √ 3M will be thus captured by the BH. 

Figure: Plot of B−2 eff (r), where the dashed lines indicate possible values of b (left panel), and orbits of 

photons with the same initial position but different b (right panel). 


Kerr black holes 








We have seen that Schwarzschild black holes are born from the collapse of an isotropic 

matter distribution. However, in Astrophysics this is a very rare condition, since usually an 

angular momentum J is initially present (while the total charge Q is typically screened in a 

plasma). We then expect to find in nature rotating black holes, though no direct proof has 

been found yet (iron lines in X-ray binary systems are the only available clue). 

Any stationary and axially symmetric metric can be written in the general form 

ds 2 = −Adt 2 + B(dφ − ωdt) 2 + Cdr 2 + Ddθ 2 , (3.25) 

and solving Einstein equations for the unknown functions of r and θ, an analytical solution 

for a BH with mass M and specific angular momentum a = J/M was found in 1963 by Kerr 

ds 2 

= − 1 − 2Mr 

ρ2 

dt 2 2aMr sin θ2 

− 2 

ρ2 dtdφ + ρ2 

∆ dr2 + ρ 2 dθ 2 + Σ2 

ρ2 sinθ2 dφ 2 , (3.26) 

here expressed in the so-called Boyer and Lundquist form, where 

ρ 2 := r 2 + a 2 cos 2 θ, ∆ := r 2 + a 2 − 2Mr, Σ 2 := (r 2 + a 2 ) 2 − ∆a 2 sin 2 θ 2 . (3.27) 

Notice the specific angular momentum has also dimension of a length when c = G = 1, 

and we must have 0 ≤ a ≤ M. For a = 0 we recover the static Schwarzschild metric, when 

a = M we have a maximally rotating black hole. The novel feature is the off-diagonal term 

gtφ = −2aMr sin θ 2 /ρ 2 

(ω := −gtφ/gφφ = 2aMr/Σ 2 ), (3.28) 

leading to a frame dragging with angular velocity ω in the positive φ direction. 









This form of the metric has an event horizon when ∆ = 0, leading to a spherical surface 

with radius 

r+ := M + (M 2 − a 2 ) 1/2 , (3.29) 

and limiting values 2M = rS when a = 0 and M when a = M. In analogy with the 

non-rotating case we could prove that this is a removable singularity (while the inner one at 

r− is not), though here we will simply assume that anything falling within the horizon 

becomes casually disconnected from the rest of the universe. Notice that differently from 

the static case, there is no counterpart of Birkhoff’s theorem for Kerr BHs. 

Consider now a stationary observer orbiting with fixed r, θ, and constant angular velocity 

Since this observer must follow a time-like worldline, then 

Ω := dφ/dt = ˙φ/˙t = u φ /u t . (3.30) 

gµνu µ u ν = (u t ) 2 [gtt + 2Ωgtφ + Ω 2 gφφ] = −1 ⇒ gtt + 2Ωgtφ + Ω 2 gφφ < 0, (3.31) 

then the angular velocity of the observer must be in the range Ωmin < Ω < Ωmax, with 

Ωmin/max := [−gtφ ± (g 2 tφ − gtt gφφ) 1/2 ]/gφφ = ω ± (ω 2 − gtt /gφφ) 1/2 . (3.32) 

Notice we have a condition Ω > Ωmin = 0 at the static limit gtt = 0, for which 

r0(θ) := M + (M 2 − a 2 cos 2 θ) 1/2 

(≥ r+), (3.33) 

where no static observers (Ω = 0) can exist. The region r+ < r < r0, where gtt > 0, is 

named ergosphere, and orbits are dragged by the angular velocity ω of inertial frames. 









Figure: The ergosphere of a Kerr black hole, limited by the event horizon r+ and the static limit r0. 

We could also think in terms of photons emitted in the azimuthal direction ±φ with constant 

r and θ. Their initial angular velocity Ω = dφ/dt for a distant observer is derived from 

ds 2 = gtt dt 2 + 2gtφdt dφ + gφφdφ 2 = 0, (3.34) 

that is Ω = Ωmin/max. At the static limit gtt = 0 the solutions are 0 and 2ω, then photons 

emitted in the −φ direction will appear static. When gtt > 0 (inside the ergosphere) both 

solutions have the same sign as ω, thus photons are inevitably dragged by the BH rotation. 

In other words, light cones are deformed in such a way to always point in the ω direction. 

Thanks to the property gtt > 0 it is possible in principle to extract energy from a Kerr BH at 

expense of its angular momentum (Penrose, 1969), due to the existence of particle 

trajectories falling in the BH with negative energy. If an external magnetic field is present, a 

powerful mechanism of electromagnetic energy extraction has also been suggested 

(Blandford-Znajek, 1977), likely to provide enough power for AGNs and GRB sources. 





Orbits around a Kerr black hole 





In analogy to static black holes, let us study briefly the orbits (geodesics) around Kerr black 

holes, for simplicity in the equatorial plane θ = π/2 normal to the BH rotational axis. The 

Lagrangian (per unit mass) is 

2L = −(1 − 2M/r)˙t 2 − 4aM/r˙t ˙φ + r 2 /∆˙r 2 + (r 2 + a 2 + 2aM 2 /r) ˙φ 2 , (3.35) 

where ˙t = dt/dσ and so on. Correspondingly to the two ignorable coordinates t and φ we 

obtain the usual first integrals of motion 

E := −∂L/∂˙t = −gtt ˙t, L := ∂L/∂ ˙φ = gφφ ˙φ, (3.36) 

where the constants are the particle specific energy and angular momentum. 

Let us now specialize to circular orbits with ˙r = 0. The radial Lagrange equation for 

constant r (∂L/∂r = 0) can be turned into an equation for Ω = ˙φ/˙t as 

(−M/r 2 )˙t 2 + (2aM/r)˙t ˙φ + (r − a 2 M/r 2 ) ˙φ 2 = 0 ⇒ (r 3 − a 2 M)Ω 2 + 2aMΩ − M = 0, (3.37) 

whence Kepler’s third law in Kerr metric is 

Ω = ± 

M 1/2 

r3/2 , (3.38) 

± aM1/2 where the upper (lower) sign is for co-rotating (counter-rotating) orbits. Note that in the 

static limit we recover precisely the Newtonian result r 3 Ω 2 = const. 









If we now impose 2L = −1 for particles with mass, we can still write the motion equation as 

˙r 2 + V 2 

eff (r) = E2 , (3.39) 

where the potential retains the same functional form as in the static case 

V 2 

2M 

eff (r) := 1 − 

r + L 2 − a2 (E2 − 1) 

r2 − 

2M(L − aE)2 

r3 , (3.40) 

which is recovered for a = 0. Circular orbits occur where ˙r = 0 always (perpetual turning 

points), that is when the two conditions 

V 2 

eff = 0, 

dV 2 

eff 

= 0 (3.41) 

dr 

both hold. After some algebra one could express E and L in terms of the orbit radius r. 

Of particular interest is the marginally stable circular orbit, that is the circular orbit with 

r = rms corresponding to the minimum of the potential. The expression for the radius is 

complicated (see Bardeen et al. 1972), here we just mention that in the maximally rotating 

case a = M we have rms = M for co-rotating orbits. The corresponding fractional binding 

energy is 1 − E = 1 − 1/ √ 3 42.3%, and this enormous efficiency for gravitational energy 

conversion is the main reason why the existence of super-massive rotating black holes in 

the centers of AGNs has been suggested already quite early. 





Hydrodynamic accretion: the Bondi solution 





Accretion onto compact objects like black holes is powerful source of energy. Consider 

matter ∆M = ˙M∆t falling radially from infinity and converting its kinetic energy into heat 

and radiation when it stops at a radius R. The available power is, in Newtonian dynamics 

L = 1 

˙Mv 

2 

2 ff = GM ˙M 

R = ξ ˙Mc 2 , 

 

ξ := GM 

c2 

, 

R 

where ˙M is the mass accretion rate, vff is the free fall velocity and ξ is the efficiency for 

accretion. We have see that detailed GR calculations actually show that we can reach 

almost ξ ∼ 42% for a maximally rotating black hole, though in that case we have considered 

accretion from the last stable circular orbit of an accretion disk. By simply replacing R = rS 

in the Newtonian definition one would find ξ = 50%. 

So far we have considered test particles (or dust) orbiting and eventually accreting onto 

black holes. However, we know that the interstellar medium is mainly composed by ionized 

gas (plasma), at typical conditions (assuming an ideal adiabatic gas (p ∼ ρ γ ) 

ρ∞ 10 −24 g cm −3 , T∞ 10 4 K, a∞ = (γkB T∞/m) 1/2 10 km s −1 

where thermal effects and compressions are not negligible, in general. In the following we 

are going to study the (classical) hydrodynamic stationary radial accretion onto a compact 

object, leading to the Bondi transonic solution. Later the same problem will be generalized 

to consider GR effects, adopting the Schwarzschild metric for a non-rotating BH. 









The hydrodynamic equations for a stationary, adiabatic, and radial flow are the continuity 

equation 

1 

r2 d 

dr (r2ρv) = 0, (3.42) 

and the Euler equation with the gravity term due the accreting central object of mass M 

v dv 

dr 

1 dp 

= − 

ρ dr 

GM 

− . (3.43) 

r2 It is convenient to replace the pressure gradient using the sound speed 

a 2 

dp 

:= = γ 

dρ s 

p 

ρ = γ kB T 

m 

1 dp 1 dρ 

⇒ = a2 , 

ρ dr ρ dr 

(3.44) 

where the density gradient can be obtained by deriving the continuity equation as 

2 

r 

1 dρ 

+ 

ρ dr 

1 dv 

+ = 0. (3.45) 

v dr 

The combination of all these ingredients allows us to rewrite Euler equation in the form 

(v 2 − a 2 ) 1 dv 2a2 GM 

= − 

v dr r r2 , (3.46) 

which provides the flow velocity in terms of r once a 2 is known. For an isothermal gas this 

is straightforward, since a = a∞ everywhere, and the equation can be readily solved. 









Figure: The transonic Bondi inflow (here u = |v|) and other solutions allowed by the parameter space. 

For the case of accretion (v < 0) from r → ∞, we must look for a flow with −v small at large 

distances (and with vanishing gradient) and large near the accreting body, possibly a 

smooth monotonic function. We thus require a positive gradient everywhere, but apparently 

it must vanish for 2a 2 /r = GM/r 2 and become negative for smaller radii. 

There is actually another option: if for this critical radius rc we have 

v 2 c = a 2 c = GM 

, (3.47) 

2rc 

with vc and ac respectively the velocity and sound speed at the critical, then the gradient 

can remain positive while the inflow becomes supersonic, with |v| > a for r < rc. This is the 

transonic accretion solution found by Bondi in 1952. The transonic outflow is the 

corresponding Parker-type solution for a wind (v > 0 and positive gradient everywhere), 

with asymptotically large supersonic velocities for r → ∞. 









In the general adiabatic case, the actual solution can only be found by solving (3.46) 

together with the continuity equation and the assumption p ∼ ρ γ . Useful relations come 

from the integral form of hydrodynamic equations, namely the Bernoulli equation 

1 

2 v2 + a2 GM 

− 

γ − 1 r = const = a2 ∞ 

, (3.48) 

γ − 1 

and the mass conservation law for a given accretion rate ˙M > 0 

At the critical point we have the condition 

1 

2 a2 c + a2 c GM 

− = 

γ − 1 rc 

5 − 3γ 

while from the adiabatic condition 

The accretion rate is then 

4πr 2 ρv = const = − ˙M. (3.49) 

2(γ − 1) a2 c ⇒ ac 

= 

a∞ 

a 2 ∼ ρ γ−1 ⇒ ρ ∼ a 2/(γ−1) ⇒ ρc 

ρ∞ 

 

= 

 

2 

5 − 3γ 

2 

5 − 3γ 

˙M = 4πr 2 c ρcac = π(GM) 2 ρca −3 

c = π(GM) 2 f(γ)ρ∞a −3 

 

∞ , f(γ) := 

1/2 

, (3.50) 

1/(γ−1) 

. (3.51) 

2 

5 − 3γ 

5−3γ 

γ−1 

, (3.52) 

written in terms of the known parameters M, ρ∞, and a∞. The function f is of the order of 

one for the usual values 1 ≤ γ ≤ 5/3. 





In numerical terms, the accretion rate is 

˙M = 8.77 × 10 −16 

2 

M 

M⊙ 

ρ∞ 

10 −24 g cm −3 





 

ρ∞ 

10 km s−1 −3 M⊙ yr −1 , 

assuming a pure hydrogen gas with m = mp/2 and γ = 5/3. It could be proved that the 

Bondi solution provides the largest steady accretion rate for a gas with a given γ, with 5/3 

being an upper value to have a transonic solution. 

Notice that for other values of the accretion rate, a transonic inflow is not found and the flow 

may be subsonic everywhere, though time-dependent numerical simulations show that the 

Bondi regime is reached from unstable steady solutions. 

The physics of the transonic solution is that of de Laval nozzle, where compression due to 

the varying section provides the possibility of supersonic acceleration. The inflow is initially 

partially slowed down by thermal pressure, at the sonic point all terms in the Bernoulli 

integral are of the same order, for r ≪ rc the gravity term dominates and the speed 

approaches the free fall value v → − √ 2GM/r. Correspondingly, the density piles up as 

r −3/2 . 

Finally, if near the central object, within the critical radius, one imposes some extra 

boundary condition other than those at infinity, like a given temperature or pressure, the 

supersonic inflow must stop abruptly with the creation of a steady shock, followed by a 

subsonic inflow on a branch of the other family of solutions. 









Hydrodynamic steady accretion onto a Schwarzschild black hole 

Let us briefly investigate the GR extension of Bondi solution, that is the relativistic 

hydrodynamic accretion onto a Schwarzschild black hole, as proposed by Michel (1972). 

We shall use the same approximations as in the Newtonian case by Bondi, namely a 

steady, radial, and polytropic flow. The metric is characterized by 

gtt = g −1 

rr = 1 − 2GM/r, |g| 1/2 = r 2 , 

and the relevant hydrodynamic equations are the conservation of mass and energy, simpler 

to use with respect to the radial momentum equation and equivalent to it for a polytropic 

flow. Recall that here u θ = u φ = 0 and ∂t = ∂θ = ∂φ = 0. These are 

where u ≡ u r , and 

from which 

∇µ(ρu µ ) = 1 

r2 d 

dr (r2ρu) = 0, (3.53) 

∇µ[(e + p)u µ ut + g µ 1 

t p] = 

r2 d 

dr [r2 (e + p)uut ] = 0, (3.54) 

r 2 ρu = const, h 2 u 2 t = const, (3.55) 

where h := (e + p)/ρ is the specific enthalpy. The ut term is derived from the normalizing 

condition 

uµu µ = u 2 t /gtt + grr u 2 = −1 ⇒ u 2 t = 1 − 2GM/r + u2 . (3.56) 









For a polytropic gas EoS of index n we introduce the temperature T ∼ p/ρ and we set 

Moreover, the specific enthalpy and sound speed are 

p ∼ ρ γ = ρ 1+1/n ⇒ ρ ∼ T n . (3.57) 

h = 1 + (1 + n)T, a 2 := γp 

ρh = 

(1 + n)T 

. (3.58) 

n[1 + (1 + n)T] 

The two constants of motion are conveniently written in the form 

r 2 T n u = C1, [1 + (1 + n)T] 2 (1 − 2GM/r + u 2 ) = C2, (3.59) 

and the flow is determined once C1 and C2 have been provided. After differentiation, by 

eliminating the T derivatives, we get the analogue of the Bondi equation 

 

u 2 − 

 

1 − 2GM 

 

+ u2 a 

r 2 

 

du 

dr = 

 

2a 2 

 

1 − 2GM 

 

+ u2 − 

r GM 

 

u 

r r 

, (3.60) 

where again we are looking for a monotonic inflow solution. As in the Newtonian case, there 

is a critical radius rc where the gradient vanishes unless 

u 2 c = GM 

2rc 

= 

a2 c 

1 + 3a2 , (3.61) 

c 

which are the conditions for transonic relativistic accretion. The non-relativistic case is 

recovered for GM/r ≪ 1, u 2 ≪ 1, and a 2 ≪ 1. 









The presence of the critical point beyond the BH event horizon is ensured for any EoS 

obeying the causality constraint a 2 < 1. This is because for r → ∞ the flow is subsonic and 

the square brackets on the left hand side contain a negative term. For r = rS = 2GM the 

same term is instead u 2 (1 − a 2 ) > 0, then a critical point where the term vanishes for r > rS 

must exist. 

Figure: The transonic Michel inflow solution: radial four velocity and density profiles. 

By computing the inflow four-velocity u we actually see that close to the event horizon the 

function ceases to be monotonic. This is due to the coordinates singularity. If one uses the 

locally measured vˆr = √ grr vr , where (1 − v2 ˆr )−1/2v r = u, than the profiles returns 

monotonic and |vˆr | = 1 at r = rS, since 

vˆr = 

u 

(1 − 2GM/r + u2 . (3.62) 

) 1/2 














A unified scheme for high energy sources? 

We have seen in the introduction that pulsars, 

magnetars, SGRs, X-ray binaries, short and long 

GRB engines, and even the supermassive cores of 

AGNs are all characterized by the presence of a 

compact object causing high energy emission. 

Most likely the primary source of energy is gravity 

(accretion, especially for black holes), though we 

do not know exactly how conversion to other forms 

of energy occurs. 

We know for certain that fast rotation and strong 

magnetic fields are key ingredients for pulsars, 

where EM winds are able to make objects like the 

Crab Nebula shine at all wavelegths. On the other 

hand, rotating black holes are the best candidates 

for efficient engines for X-ray binaries, GRBs, and 

AGNs (high accretion luminosity), and even if they 

do not posses a proper magnetosphere, the 

magnetized plasma accreting from disks may be 

the answer. Finally, the magnetic field is essential 

to collimate the spectacular ultrarelativistic jets 

propagating along the poles. 





Figure: The Crab Nebula (optical 

filaments and X-ray synchrotron core), 

and Cen A (a radio galaxy) with two 

collimated polar jets. 





Pulsars: the Pacini-Gold model 

Let us now describe the Pacini-Gold model of the tilted 

rotating magnetic dipole in vacuum. The field is Let , then 

and the polar surface field is 

B(r) = [3(m · ˆr)ˆr − m]/r 3 , (4.1) 

Bp = 2m/R 3 , (4.2) 

where m is the moment, ˆr = r/r, and R is the stellar 

radius. When this moment is tilted by an angle α and 

rotates with pulsation Ω around the z axis, the moment is 

m = (BpR 3 /2)(sin α cos Ωt, sin α sin Ωt, cos α), (4.3) 

and the radiated electromagnetic energy in time is 

˙E = −(2/3c 3 )| ¨ m| 2 = −B 2 p R6 Ω 4 sin 2 α/6c 3 . (4.4) 

This energy output must be compensated somehow, 

namely by the spindown due to the magnetic braking 

˙Erot = d 1 ( 

dt 2 I Ω2 ) = IΩ ˙Ω. (4.5) 





Figure: Cartoon of a pulsar. 

The pulsed radio emission is 

produced at the poles of the 

magnetosphere. 









If we then impose ˙E = ˙Erot, the precise law Ω = Ω(t) can be obtained by integrating 

˙Ω = −KΩ 3 , (4.6) 

with K directly proportional to B2 p and inversely proportional to the moment of inertia I. 

Observable quantities are the period P = 2π/Ω and its spindown rate ˙P, which can be used 

to infer the values of ˙E, Bp sin α, and τ: 

˙E ∼ ˙P/P 3 

, Bp sin α ∼ P ˙P, τ ∼ P/ ˙P. 

For the Crab Nebula we measure P = 33 ms and ˙P = 4.3 10 13 s s −1 , and for a typical NS 

radius (R = 10 km) 

˙E ∼ 10 38 erg s −1 , Bp sin α ∼ 10 12 G, τ ∼ 1000 yr, 

which fit nicely with the expected values. In particular, this estimate for the energy emission 

by a rotating dipole was made by Franco Pacini in 1967 just before the discovery of pulsars, 

for a model to explain the synchrotron luminosity (L ≤ | ˙E|) of the Crab Nebula. 

Notice that this is true for the dipole mechanism, while more generally one could suppose 

˙Ω = −KΩ n . Actually the majority of pulsars have a measured braking index n between 2 

and 3, so either the field is not exactly a dipole n = 3, or this emission mechanism can only 

provide simple estimates. What we are going to learn next is: 

1 pulsars are not surrounded by void but by plasma-filled magnetospheres, 

2 extraction of electromagnetic energy is possible even for aligned rotators. 





Pulsars: the Goldreich-Julian model 





Consider a stationary unipolar inductor, that is a perfect conductor inside a dipolar magnetic 

field with m Ω. Then net EM force on a charge must vanish in the NS interior 

E + 1 

c v × B = 0, v = Ω × r. (4.7) 

Charges will accumulate on the NS surface, causing a jump in the radial electric field, while 

Eθ must be continuous. This component is, along the surface 

Eθ = − 1 

c BpΩR cos θ sin θ. (4.8) 

If outside we had void, then E = −∇φ and the potential φ will satisfy ∇ 2 φ = 0. In order to 

match the condition at the NS surface, the appropriate solution of the Laplace equation is 

φ = − 

5 BpΩR 

r3 P2(cos θ) (4.9) 

and the force on an electron along the magnetic field at the surface is as large as 

E · B = −(B 2 p ΩR/c) cos 3 θ 0, eE/(GMme/R 2 ) 10 12 . (4.10) 

Clearly, the assumption of a void atmosphere must be wrong and we will have a dense 

magnetosphere. Charges and currents will distribute in order to screen the EM fields 

reaching a stationary force-free equilibrium 

ρE + 1 

c j × B = 0. (4.11) 









Figure: Sketch of the Goldreich-Julian magnetospheric model. 









Goldreich and Julian (1969) produced the qualitative description of the previous slide: the 

external magnetic field will have a near zone with a dense plasma trapped in a corotating 

closed dipolar field inside the light cylinder, where 

R := r sin θ = RL : v := vφ = ΩR → c ⇒ RL := c/Ω. (4.12) 

For the force-free condition, like in the NS interior we must have 

E + 1 

c v × B = 0 ⇒ E = − 1 

c ΩReφ × B, (4.13) 

thus the electric field is purely poloidal. It is interesting to calculate the charge density 

ρ = 1 

4π ∇ · E = c 

2π 

ΩBz 

, (4.14) 

1 − (R/RL ) 2 

and since Bz ∝ 3 cos 2 θ − 1, there will be a critical angle cos 2 θc = 1/3 at which the net 

charge density will change sign, that is the model assumes a regime of charge separation: 

electrons will be extracted by the polar regions, while the equatorial closed field will be 

mainly filled by positrons. 

In the far region, beyond the light cylinder, corotation at v > c is no longer possible, poloidal 

fieldlines open up and a toroidal Bφ ∼ 1/r component forms (reversing sign at the equator 

→ current sheet → reconnection?). In this region, inertial effects by relativistically 

accelerates particles start to dominate over electromagnetic forces, and this is why where 

force-free no longer apply fieldlines can be deformed and dragged by particles. A relativistic 

pulsar wind has thus formed, carrying away inertia and EM fields (even for α = 0). 


The pulsar equation 








The magnetospheric model of Goldreich and Julian is rather qualitative, but we can make 

things more quantitative than that. Let us still assume an aligned rotator, a surface dipolar 

field and look for stationary and axisymmetric solutions (∂t = ∂φ = 0) in the force-free 

regime. The equations are 

∇ × E = 0, ∇ · B = 0, (∇ · E)E + (∇ × B) × B = 0, (4.15) 

with 4πρ = (∇ · E) and 4πj = (∇ × B). Let us start with E. Because of axisymmetry, Eφ 

must be constant on circles centered on the polar axis and perpendicular to it. However, 

Stokes’ theorem tells us that its line integral must vanish (since ∂t B = 0), thus E is purely 

poloidal. Imposing E · B = 0, a convenient way to write it is 

E = − 1 

c ωReφ × B, (4.16) 

with ω an unknown function of R and z. Moreover, eliminating the electric field 

(j − 1 

c ωReφ) × B = 0 ⇒j = ρωReφ + κB, (4.17) 

and the current has a rotating component plus a field aligned component, with κ another 

unknown function. As far as the magnetic field is concerned, like in all magnetostatic or 

MHD stationary and symmetric equilibria, it is convenient to introduce the Euler potential f 

∇ · B = 1 ∂(RBR) ∂Bz 

+ 

R ∂R ∂z = 0 ⇒ Bp = ∇f 

R × eφ, (4.18) 

and basically f(R, z) = Aφ. Notice that surfaces of constant f, the magnetic surfaces, 

contain the poloidal fieldlines. 









We now go back to E. Thanks to the expression of Bp we may write 

E = − 1 

c ω∇f ⇒ ∇ × E = − 1 

c ∇ω × ∇f = 0, (4.19) 

and clearly ω = ω(f), then each magnetic surface may rotate at a characteristic angular 

velocity. However, fieldlines are anchored to the NS surface, rotating at the constant Ω 

speed, thus we set 

E = − 1 

c ΩReφ × B = − 1 

c Ω∇f, (4.20) 

so that f(R, z) is also proportional to the electrostatic potential. Moreover 

2 ∂ f 

∇ · E = 4πρ = − 1 

c Ω∇2 f = − 1 

RL 

∂R2 + ∂2f 1 

+ 

∂z2 R 

∂f 

∂R 

 

, (4.21) 

and similarly, from the definition of Bp, we have 

(∇ × B) · eφ = 4π 

c jφ = − 1 

2 ∂ f 

R ∂R2 + ∂2 

f 1 ∂f 

− . (4.22) 

∂z2 R ∂R 

At this stage we just need to rewrite jφ in terms of ρ and κBφ. The integral form of Ampere’s 

law involves the total current across a circle of radius R. We get 

2πRBφ = 4π 

c I ⇒ Bφ = 2I 

, 

cR 

(4.23) 

and we could demonstrate that 

κ = κ(f), I = I(f), κ = 1 dI 

. 

2π df 

(4.24) 





Putting all together, the pulsar equation for the potential f is 

∂2f ∂R2 + ∂2f 1 

− 

∂z2 R 

R2 + R2 

L 

R 2 

L 

− R2 

∂f 

∂R 

= − 

4R 2 

L 

c 2 (R 2 

L − R2 ) 





I dI 

df 

(4.25) 

Once I = I(f) is known, the above equation is an 

elliptic-type second order PDE. Unfortunately, f is not 

known until the equation is solved, so this is an implicit 

problem which can only be faced numerically. 

Notice that this equation has a singularity for R = RL , 

unless the regularity condition 

R = RL : 

∂f 

∂R 

2RL dI 

= I 

c2 df 

. 

(4.26) 

holds at the light cylinder. Then, the total current I(f) is not 

exactly a free function, since it must be such as to allow a 

sort of transonic solution, as in the Bondi (or better, Parker) 

problem. Here, since EM fields dominate over matter, the 

sound speed is replaced by the speed of light c. 

Figure: A numerical solution of 

the pulsar magnetosphere 

obtained by Spitkovsky. 









Modern numerical solutions are obtained by solving the time dependent Maxwell equations, 

subject to the force-free condition in (4.15), waiting for the system to relax. When this 

approach is followed, there are no problems of regularity at the light cylinder. 

Energy losses may be finally calculated on top of numerical simulations. These are 

proportional to the angular momentum losses RTRφ, where T µν is dominated by the EM 

terms, to be integrated over a sphere of radius r: 

dE 

dt 

dLz 

= Ω , 

dt 

dLz 

dt = 

 

RTRφr 2 sin θdθdφ. (4.27) 

The integration for the numerical solution of the axisymmetric steady problem gives 

dE 

dt = − m2Ω4 c3 , (4.28) 

where m is the magnetic dipole moment, to be compared with the result of the Pacini model 

dE 2 

= − 

dt 3 sin2 α m2Ω4 c3 . (4.29) 

Numerical 3D simulations of a tilted rotating dipole leads to the result 

dE 

dt = −(1 + sin2 α) m2Ω4 c3 , (4.30) 

which highlights the possibility of energy losses for α = 0, impossible in the vacuum model. 

This Poynting-dominated wind is possible due to the slowdown of the neutron star, as in the 

other case. At larger distances the inertia of particles will be important, and (relativistic) 

MHD must replace the force-free regime. 





Kerr Black Holes: the Blandford-Znajek model 





The extreme case of a compact source of EM energy and angular momentum losses at the 

expense of rotation is that of rotating black holes. If the net charge is zero, the BH cannot 

posses a magnetic field of its own, while it must be drawn in from outside, typically by the 

magnetized plasma falling from the accretion disk. 

Once we have a poloidal field, for example an initial vertical uniform field, the frame 

dragging induced velocity in the ergosphere is able to produce an electric field 

and then an outward Poynting flux 

E = − 1 

c v × B, (4.31) 

S = 4π 

c E × B (4.32) 

carrying EM energy (and angular momentum) losses to infinity becomes possible. As for 

pulsars, we need to suppose the presence of a dense magnetosphere created by the 

charges induced by the unscreened electric field. 

The same calculation done for a pulsar can be repeated within Kerr metric in 

Boyer-Lundquist coordinates. We could look for steady and axisymmetric equilibria and find 

a second-order PDE for the Euler potential f describing the poloidal magnetic field surfaces, 

now with two unknown functions: the usual current I(f) and the angular velocity Ω(f) 

measured at infinity. This model is named after Blandford and Znajek. 









Figure: Cartoon of the currents and fields induced by a uniform magnetic field around a Kerr black hole. 

Here we need two regularity conditions, one at the event horizon (∆ = 0) and the other at 

the generalization of the light cylinder, where constraints on the free functions must be 

imposed to avoid singularities. A crude estimate for the EM losses is 

dE 

dt = 1045 erg s −1 

 

a 

M 

M 

10 9 M⊙ 

Bp 

10 4 G 

which provides enough energy to power AGNs. Time-dependent numerical simulations in 

the force-free regime fully confirm this scenario, by relaxing to the steady-state 

Blandford-Znajek solution. 

2 

, 





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