Testing the Equivalence Principle

Testing the Equivalence Principle 

Claus Lämmerzahl 

Centre for Applied Space Technology and Microgravity (ZARM), 

University of Bremen, 28359 Bremen, Germany 

DFG Research Training Group “Models of Gravity” 

Course “Relativistic Astrophysics and Experimental Gravitation” 

Tübingen, June 19, 2012 

C. Lämmerzahl (ZARM, Bremen) Testing the Equivalence Principle Tübingen, June 19, 2012 1 / 99

The Bremen drop tower 


The Bremen drop tower 

Space Science 

Fundamental Physics 

Key Technologies 

Control systems 

Space technology 

Micro satellites 

Fluid mechanics 

Fluid dynamics 

Energy and propulsion 

Computational fluid 

dynamics 

Experimental fluid 

mechanics 


Outline 

1 Introduction 


Outline 


2 A more detailed analysis 

Implications of the UFF 

Operational foundation of Riemannian structures 


Outline 





3 More degrees of freedom 


Outline 






4 The Equivalence Principle in the quantum domain 

Interference experiments 

Accelerated frames 

Apparent violations of the Universality of Free Fall 


Outline 










5 Order of equations of motion 


Outline 











6 * Existence of inertial systems 


Outline 












7 Creation of the gravitational field 

Newton potential 

The Universality of the Gravitational Field 


Outline 















8 Further issues 

Relation to Newton’s axioms 

* Operational foundation of Riemannian structures 1 


Outline 


















9 Summary 


Outline 


Introduction 

















9 Summary 


Geometrization 

Introduction 

There are 2 aspects of geometry 

1 configuration space and/or internal space 

2 gravity = geometrization of interaction 



Introduction 

There are 2 aspects of geometry 

1 configuration space and/or internal space 

2 gravity = geometrization of interaction 

most important formula for physics (and mathematics) 

m i ẍ = F 

model for force 

F = −m g ∇U U = Newton potential , ∆U =4πρ 

acceleration 

ẍ = − m g 

m i 

∇U 


Introduction 

Observation: Universality of Free Fall 



Introduction 

acceleration 

observation 

a = ẍ = − m g 

m i 

∇U 

η = a 2 − a 1 

1 

1 

2 

2 (a 1 + a 1 ) = ( 

mg 

( 

mg 

m i 

− 

)2 

m i 

(( ( 

)1 

mg mg 

m i 

+ 

m 

)2 i 

)1 

idealization: Equivalence Principle m g = m i 

⇒ path does not depend on particle ↔ geometry 

) ≤ 2 · 10 −13 

Equivalence Principle = geometrization of gravitational interaction 

More advanced question: Why can relativistic gravity be described by a 

Riemannian geometry? 

Answer: universality principles, Equivalence Principles, ... 

Remark: Will point to usually hidden assumltions ... 


Universality principles 

Introduction 

Gravity acts always and everywhere 

Gravity can be transformed away 

Gravity acts on all kinds of matter 

Gravity acts on all kinds of matter in the same way 

Gravity acts on all kinds of clocks 

Gravity acts on all kinds of clocks in the same way 

Gravity is created from all kinds of matter 

Gravity is created from all kinds of matter in the same way 



Introduction 











Introduction 











Introduction 











Introduction 











Introduction 











Introduction 











Introduction 











Introduction 











Introduction 









gravity ⇔ universality principles ⇔ gravity = geometry 


Introduction 

Structure of standard physics 

Einstein Equivalence Principle 

Universality 

of Free Fall 

Universality of 

Gravitat. Redshift 

Lorentz 

Invariance 

Equations of motion for matter 

Maxwell–equation 

Dirac–equation 

Matter determines 

gravity 

Gravity determines 

dynamics 

General Relativity 

Gravity = Metric 

Einstein–equations 


The present situation 

Introduction 

All aspects of Lorentz invariance are experimentally well tested and confirmed 

Foundations 

Postulates 

c = const 

Principle of Relativity 



Introduction 

All aspects of Lorentz invariance are experimentally well tested and confirmed 

Foundations 

Postulates 

c = const 

Principle of Relativity 

Tests 

Independence of c from 

velocity of the source 

Universality of c 

Isotropy of c 

Independence of c from 

velocity of the laboratory 

Time dilation 

Isotropy of physics 

(Hughes–Drever 

experiments) 

Independence of physics from 

the velocity of the laboratory 



Introduction 

Many aspects of the Universality of Free Fall are experimentally well tested and 

confirmed 

Postulate 

In a gravitational field all 

structureless test particles fall in 

the same way 



Introduction 

Many aspects of the Universality of Free Fall are experimentally well tested and 

confirmed 

Postulate 

In a gravitational field all 

structureless test particles fall in 

the same way 

Tests 

UFF for 

Neutral bulk matter 

Charged particles 

Particles with spin 

No test so far for 

Anti particles 



Introduction 

Many aspects of the Universality of the Gravitational Redshift are experimentally 

well tested and confirmed 

Postulate 

In a gravitational field all clocks 

behave in the same way 

g 



Introduction 

Many aspects of the Universality of the Gravitational Redshift are experimentally 

well tested and confirmed 

Postulate 

In a gravitational field all clocks 

behave in the same way 

g 

Tests 

UGR for 

Atomic clocks: electronic 

Atomic clocks: hyperfine 

Molecular clocks: vibrational 

Molecular clocks: rotational 

Resonators 

Nuclear transitions 

No test so far for 

Anti clocks 



Introduction 

All predictions of General Relativity are experimentally well tested and confirmed 

Foundations 

The Einstein Equivalence 

Principle 

Universality of Free Fall 

Universality of Gravitational 

Redshift 

Local Lorentz Invariance 



Introduction 


Foundations 





Redshift 


Implication 

⇓ 

Gravity is a metrical theory 



Introduction 


Foundations 





Redshift 


Implication 

⇓ 


⇒ 

Predictions for metrical theories 

Solar system effects 

Perihelion shift 

Gravitational redshift 

Deflection of light 

Gravitational time delay 

Lense–Thirring effect 

Schiff effect 

Strong gravitational fields 

Binary systems 

Black holes 

Gravitational waves 



Introduction 


Foundations 





Redshift 


Implication 

⇓ 


⇒ 

Predictions for metrical theories 

Solar system effects 

Perihelion shift 

Gravitational redshift 

Deflection of light 

Gravitational time delay 

Lense–Thirring effect 

Schiff effect 

Strong gravitational fields 


Black holes 

Gravitational waves 

⇓ 

General Relativity 


Outline 


A more detailed analysis 

















9 Summary 


Outline 




















9 Summary 



Consequences of the UFF 


Trajectories 

Trajectory of a particle x = x(p; t) 

p = particle parameter (e.g. mass, charge, etc) 

UFF ⇒ trajectory does not depend on particle parameters x = x(t) 

This is already the geometrization of the gravitational interaction. 

But: which geometry? 

The set of all trajectories is a path structure, 

no differential equation fixed, may depend on its history 





Principle of scientific determinism 

future of particle determined by its past (Hille & Phillips 1957) 

through integral equation or differential equation of infinite order ... 

∫ t 

x(t) =x 0 + f(x(t ′ ),t ′ )dt ′ 

t 0 

⇒ 0= 

One can in principle calculate geodesic deviation, and all that ... 

∞∑ 

i=0 

g(x(t),t) di 

dt i x(t) 





One needs more to arrive at our well known equations of motion ... 

Order of equations of notion / Cauchy problem 

Newton’s setup: trajectory determined through (also interacting particles) 

initial position x 0 = x(t 0) and 

initial velocity v 0 =ẋ(t 0). 

⇒ ordinary differential equations of second order: 

ẍ µ = H µ (p; x, ẋ) 

Here we do not consider consequences of fundamental equations of motion 

Radiation back reaction: differential equation of higher order 

Extended bodies: differential equation of higher order 

We are dealing with the fundamental equations of motion themself 

Question: Why the fundamental equations of motion are of second order? 

Equivalent to questioning Newton’s second axiom 





UFF + second order 

equation of motion 

ẍ µ = H µ (x, ẋ) 

equation of motion does not depend on particle parameter p 

equation of motion is of second order 

this defines a curve structure 

Gravity cannot be transformed away: 

Acceleration towards the center of Earth 

depends on horizontal velocity 

Condition to be able to transform away 

gravity is stronger then pure UFF. 

Implies several effects: G(T ), violationof 

UGR, ... 

a 1 

a 2 

v 



The free fall: The notions 



∃ coordinate system ∀ particles : ẍ =0 

Then in an arbitrary coordinate system 

ẍ µ = −Γ µ ρσ(x)ẋ ρ ẋ σ 

autoparallel equation (without back reaction!) 

Need still relation between the connection Γ µ ρσ(x) and the metric g µν 

g µν related to LLI: LLI ⇒ exists coordinate system so that light is 

characterized by η µν l µ l ν =0. 

Defines conformal class of Riemannian metrics: [g µν ]={g ′ µν|g ′ µν = e φ g µν }. 

Connection established through requirement of causality. 

definition: reading of clock ds 2 = ∫ g µν dx µ dx ν along path of clock 

φ = const from UGR. 


Outline 




















9 Summary 




Description of tests of the universality principles 

Purpose: parametrization of deviations, comparison of different experiments 

Haugan formalism (Haugan, AP 1979) 

Ansatz: effective atomic Hamiltonian (from modified Dirac and modified Maxwell) 

( ) 

H = mc 2 + 1 

( 

δ ij + δmij i 

p i p j + m δ ij + δm ) 

gij 

U ij (x)+... 

2m m 

m 

with 

δm iij = anomalous inertial mass tensor (depends on atom and state) 

δm gij = anomalous gravitational mass tensor ( ” ) 

∫ (x − x 

U ij ′ ) i (x − x ′ ) j ρ(x ′ ) 

= G 

|x − x ′ | 3 d 3 x ′ , δ ij U ij = U 

for vanishing anomalous terms 

H = mc 2 + p2 

2m + mU(x)+... 




Part of a larger scheme: Hierarchy of theories 

Full 

theory 

Quantum Gravity 

Experiment 

Observation 

Clock readout 

Interference fringes 

acceleration 

counting events 

... 





Full 



derived 

Effective 


Dilaton scenarios 

Axion fields 

Torsion 

... 

Experiment 

Observation 

Clock readout 


acceleration 


... 





Full 



derived 

Effective 


Phenomenology 

Test theory 

Experiment 

Observation 


Axion fields 

Torsion 

... 

Standard Model Extension 

PPN formalism 

c 2 –formalism 

Robertson–Mansouri–Sexl 

THɛµ–formalism 

χ − g–formalism 

... 

Clock readout 


acceleration 


... 





Full 



derived 

Effective 


⊂ 

Phenomenology 


Experiment 

Observation 


Axion fields 

Torsion 

... 


PPN formalism 





... 

Clock readout 


acceleration 


... 





Full 



derived 

Test theories 

mediate between 

experiment and full theory 

Effective 


⊂ 

Phenomenology 


Experiment 

Observation 


Axion fields 

Torsion 

... 


PPN formalism 





... 

Clock readout 


acceleration 


... 






Canonical equations 

a i = δ ij ∂ j U(x)+ δmij i 

m ∂ jU(x)+δ ij δm gkl 

m 

∂ jU kl (x) 

For diagonal mass tensors δm iij = δm i δ ij , δm gij = δm g δ ij : 

a i = δ ij m g 

m i 

∂ j U 

Comparison of acceleration of two different particles: Eötvös coefficient 

η = a 2 − a 1 

1 

2 (a 2 + a 1 ) = (m g/m i ) 2 

− (m g /m i ) 

( 1 

) 

1 (mg /m i ) 2 

+(m g /m i ) 1 

2 


Tests of UFF 



1 Tests with bulk matter 

Method Grav field Accuracy Experiment 

Torsion pendulum Sun η ≤ 2 · 10 −13 Adelberger 2006 

2 Tests with quantum matter 


Atom interferometry Earth η ≤ 10 −9 Chu, Peters 1999 

3 Gravitational self energy 


Torsion pendulum and LLR Sun η ≤ 1.3 · 10 −3 Baessler et al 1999 

4Chargedparticles 


Free fall of electron Earth η ≤ 10 −1 Witteborn & Fairbank 1967 


Tests of UFF 



3 Gravitational self energy 


Torsion pendulum and LLR Sun η ≤ 1.3 · 10 −3 Baessler et al 1999 

4Chargedparticles 


Free fall of electron Earth η ≤ 10 −1 Witteborn & Fairbank 1967 

5 Particles with spin 


Weighting polarized bodies Earth η ≤ 10 −8 Hsie et al 1989 

6Anti–particles 


Free fall of anti–Hydrogen Earth η ≤ 10 −3 − 10 −5 (estimate) 



Universality of Free Fall in space 




MICROSCOPE: The Mission 


French space mission with participation of 

CNES, ESA, ZARM and PTB 

Mission goal: Test of Equivalence 

Principle with an accurary of η =10 −15 

Mission overview: 

Micro-satellite of CNES Myriade series 

Drag–free satellite 

Sun–synchronous orbit 

Altitude about 800 km 

Mission lifetime of 1 year 

Payload: 

Two high–precision capacitive differential 

accelerometers 

Science sensor: Ti and Pt test mass 

Reference sensor: two Pt test masses 

Test of accelerometers at ZARM drop 

tower 





Universality of kinetic and Gravitational Redshift 

frequency comparison 

( ( 

ν 2 

= ν0 (2) 

2 ∆δm 

iij 

ν 1 ν1 

0 1+ 

m (2) 

) 

− ∆δm(1) iij 

m (1) 

} {{ } 

frequency ratio depends on velocity 

v i v j + 

( ) 

(2) ∆δm gij 

− ∆δm(1) gij 

U ij (x) 

m (2) m (1) 

} {{ } 

frequency ratio depends on position 

1st term: singles out a certain frame of reference ⇒ violation of Local 

Lorentz invariance 

2nd term: singles out a certain position ⇒ violation of Local Position 

invariance ⇔ violation of Universality of Gravitational Redshift 

for diagonal anomalous mass terms 

ν 

( 

2 

= ν0 2 

ν 1 ν1 

0 

1+(α clock 2 − α clock 1 )v 2 

} {{ } 

violation of LLI, MS 

) 

+(β clock 2 − β clock 1 ) U(x) 

} {{ } 

violation of UGR = LPI 

) 


Redshift tests 



Tests of the Universality of Kinematical Redshift 

no real comparison experiment exists: all experiment just test the kinematical 

redshift and look for deviations of |α − 1| 

Method Accuracy Experiment 

Doppler shift 1 · 10 −2 Ives & Stilwell 1938 

2–photon spectroscopy 1.4 · 10 −6 Riies et al, PRL 1988 

saturation spectroscopy 8 · 10 −8 Saathoff et al, PRL 2003 

Tests of the Universality of Gravitational Redshift 

Comparison Accuracy Experiment 

Cs – Resonator 2 · 10 −2 Turneaure & Stein 1987 

Mg – Cs (fine structure) 7 · 10 −4 Godone et al 1995 

Resonator – I 2 (electronic) 4 · 10 −2 Braxmaier et al, PRL 2002 

Cs – H-Maser (hf) 2.5 · 10 −5 Bauch et al, PRD 2002 

Cs – Hg 5 · 10 −6 Fortier et al, PRL 2007 


Clocks 



Atomic clocks 

Based on principal transitions 

Based on fine structure 

Based on hyperfine transitions 

Molecular clocks 

Based on rotational 

transitions 

Based in vibrational 

transitions 

Light clocks 

Gravitational clocks 

Planetary motion 


Rotation 

Earth 

Pulsars 

Decay of particles 

All based on different physical principles, laws. 

Clock tests are tests of the coupling of interaction fields (Maxwell, weak, 

strong) to gravity. 

Clocks of different nature exhibit a different dependence on fundamental 

constants. 

UGR ↔ time– and space–dependence of fundamental constants. 


More tests: LLI 



Universality of velocity of light 

is a postulate of SR 

The meaning of the universality: c does not depend on 

the velocity of the source ⇒ uniqueness of c 

its frequency 

its polarization 

the direction of propagation 

Compatibility with other particles 

For all particles the velocity of light is the limiting velocity 

c = c + = c − = c ν = v max 

p 

= v max 

e 

= v grav 

Implication: c is universal and can, thus, be interpreted as geometry 




Independence of c from the velocity of the source 

normal 

v 

orbital plane 

ψ 

R 

r 

Earth 

Description and result (Brecher, PRL 1977) 

Model c ′ = c + κv 

Time of flight consideration: May happen 

Reversal of chronological order (one light ray may overtake the other) 

Multiple images 

... 

Result |κ| ≤10 −11 





γ 

π 0 

v =0 

γ 





γ 

π 0 

v =0 

γ 

proton source 

p 

detector detector 

L 

π 0 γ 

v 

Be–target 

clock 





γ 

π 0 

v =0 

γ 

proton source 

p 

detector detector 

L 

π 0 γ 

v 

Be–target 

clock 

Description and result (Alväger et al 1964) 

Model c ′ = c + κv 

Result |κ| ≤10 −6 at v =0.99975 c 



Tests of the universality of c 


t 

δt 

c − δc 

c 

Polarization / dispersion of light 

Method: Comparison of arrival times of 

light with different polarization / 

different frequency from distant galaxies 

l 

x 

Result 

Carrol, Field and Jackiw 1992, Kauffmann and Haugan 1995, Kostelecky and 

Mathews 2002 

∣ ∣ ∣∣∣ c + − c − ∣∣∣ 

≤ 10 −32 

c + 

Schafer 1999, Biller 1999 ∣ ∣ ∣∣∣ c ν − c ν ′ ∣∣∣ 

≤ 6 · 10 −21 

c ν 





Velocity of neutrinos 

Method: Comparison of arrival times of photons and neutrinos from supernova 

SN1987a 

Result 

Longo 1987, Stodolsky 1988 

Limiting velocity of protons 

c − v ν 

∣ c 

∣ ≤ 10−8 

Result 

Coleman and Glashow 1998 

∣ 

v max 

p 

− c 

c ∣ ≤ 10−21 





Limiting velocity of electrons 

deflector 

e source 

e − 

accelerator 

γ 

e − 

target 

L 

e − γ 

p detector 

e detector 

Result 

Giurogossian et al, PRD 1965 

∣ 

v max 

e 

− c 

c ∣ ≤ 10−6 

Other tests by Brown et al, PRD 1963, Alspector et al, PRL 1976, Kalbfleisch et 

al, PRL 1976 with result of same order of accuracy 


Outline 


More degrees of freedom 

















9 Summary 


Spin 


What is spin? 

Avectorattachedtotheparticle,additionalparameter 

In standard theory from ordinary coupling (Mathiesson–Papapetrou–Dixon 

formalism): a µ = λ C R µ νρσv ν S ρσ ⇒ violation of UFF at the order 

10 −20 m/s 2 ,beyondexperiment 

general ansatz for dynamics of spin D u S =ΩS 

Spin equation of motion valid for all spins: Ω does not depend on spin → 

geometry ⇒ torsion T (Audretsch & C.L. QCG 1987) 

Coupling of orbital angular momentum: Consistency with ordinary minimal 

coupling to metric Ni, PRL 2010 

as WKB from field equations for higher spin particles: D u S = f(T )S: 

effective equations do not fulfill universality (Spinosa 1982, ...) 

underlying equations of motion do fulfill universality 

underlying action principle fulfills universality 



The gravitomagnetic field 


The Schiff–effect 



UFF and spin 


Anomalous coupling 

Speculations: violation P , C, andT symmetry in gravitational fields (Leitner 

&Okubo1964, Moody & Wilczek, PRD 1974) suggest 

V (r) =U(r)[1+A 1 (σ 1 ± σ 2 ) · ̂r + A 2 (σ 1 × σ 2 ) · ̂r] , 

One body (e.g., the Earth) is unpolarized → 

V (r) =U(r)(1+Aσ · ̂r) . 

Hyperfine splittings of H ground state: A p ≤ 10 −11 , A e ≤ 10 −7 

Hari Dass 1976, 1977, includesvelocityoftheparticles 

[ 

V (r) =U 0 (r) 1+A 1 σ · ̂r + A 2 σ · v ( 

c + A 3̂r · σ × v )] 

c 

still CPT–invariant 


UFF and charge 


Standard theory 

In standard theory from ordinary coupling (deWitt & Brehme, AP 1968) 

a µ = αλ C R µ νv ν ∼ 10 −35 m/s 2 

Anomalous coupling 

Anomalous coupling (Dittus, C.L., Selig, GRG 2004) 

H = p2 

p2 

+ mU(x)+κeU(x) = 

(1+κ 

2m 2m + m e ) 

U(x) . 

m 

Charge dependent anomalous gravitational mass 

Can be generalized to charge dependent anomalous inertial mass (e.g. 

Rohrlich 2000) 

⇒ Charge dependent Eötvös factor 

It is possible to choose κ’s such that for neutral composite matter UFF is 

fulfilled while for isolated charges UFF is violated 


Outline 


The Equivalence Principle in the quantum domain 

















9 Summary 


Outline 




















9 Summary 



The general phase shift 


mirror 1 

analyzer 

∇U 

I 

II 

h 

source 

beam 

splitter 

l 

mirror 2 

wave functions at analyzer 

( 

ϕ I (x) = exp − 1 2 

( 

ϕ II (x) = exp − 1 2 

∫ x 

) 

θds 

x 

∫ 

0 

x 

) 

θds 

x 0 

( 

exp −i 

( 

exp −i 

∫ x 

p µ dx µ ) 

a 0 

x 

∫ 

0 

x 

p µ dx µ ) 

a 0 

x 0 



The general phase shift 


The intensity 

Intensity of the two interfering wave functions at one port of the analyzer 

I = |ϕ I + ϕ II | 2 

= |ϕ I | 2 + |ϕ II | 2 +2Re (ϕ ∗ I ϕ II ) 

= 2(1+cos∆φ) |a 0 | 2 , 

with phase shift 

∮ 

∆φ = 

p 

integration along classical trajectory 

attenuation factor ∫ x θds does not play any role 



Interference experiment 


Schrödinger equation 

quasiclassical limit 

i ∂ψ 

∂t = − 2 

∆ψ + mUψ 

2m 

const. = E = p2 

2m + mU 

then one obtains phase shift for U = gh + O(h 2 ) (Colella & Overhauser, PRL 

1974) 

δφ = mglh 

v 

depends on mass −→ Equivalence Principle? 



On the Equivalence Principle 


Model 

Schrödinger equation in gravitational 

field 

i ∂ψ 

∂t = − 2 

∆φ + mUψ 

2m 

Phase shift 

For pure gravitational acceleration 

atom interferom. (Bordé 1989) 

δφ = k · g T 2 

neutron interf. (CL, GRG 1996) 

δφ = C · g T 2 

Discussion 

Exact quantum result 

UFF exactly fulfilled 

Does not depend on 

comes in by introducing 

classical notions 

height = h = v zT = k m T 

length = l = v 0T 

then 

δφ = k z gT 2 = mghl 

v 0 

classical notions are operationally 

not realized 

δφ = k z gT 2 contains experimentally 

given quantities only 





Model 

Schrödinger equation in gravitational 

field 

i ∂ψ 

∂t = − 2 

2m i 

∆φ + m g Uψ 

Phase shift 

For pure gravitational acceleration 

atom interferom. (Bordé 1989) 

δφ = m g 

m i 

k · g T 2 

neutron interf. (CL, GRG 1996) 

δφ = C · g T 2 

Discussion 

Exact quantum result 

UFF exactly fulfilled 

Does not depend on 

comes in by introducing 

classical notions 

height = h = v zT = k m T 

length = l = v 0T 

then 

δφ = k z gT 2 = mghl 

v 0 

classical notions are operationally 

not realized 

δφ = k z gT 2 contains experimentally 

given quantities only 





can be used for an operational definition of the equivalence principle in the 

quantum domain, even in curved space–time (C.L., GRG 1996) 

it can be shown that this operational definition is equivalent to theminimal 

coupling procedure (C.L., APP 1997) 

minimal coupling ←→ geometry 


Outline 




















9 Summary 




Quantum mechanics in accelerated frame 

Schrödinger equation in accelerated frame 

i ∂ψ 

∂t = − 2 

∆ψ − mg · xψ 

2m 

Transformation 

x → ξ = x+ 1 2 gt2 and ψ(z,t) → ϕ(ξ,t)=ψ(x+ 1 2 gt2 ,t)exp 

(ig · xt + i ) 

3 g2 t 3 

free Schrödinger 

i ∂ϕ 

∂t = − 2 

2m ∆ϕ 






i ∂ψ 

∂t = − 2 

2m ∆ψ − mg · xψ + g|ψ|2 ψ 



(ig · xt + i ) 

3 g2 t 3 


i ∂ϕ 

∂t = − 2 

2m ∆ϕ + g|ϕ|2 ϕ 

Also holds for nonlinear Schrödinger equation with arbitrary nonlinearity 






i ∂ψ 

∂t = − 2 

2m ∆ψ − mg · xψ + g|ψ|2 ψ 



(ig · xt + i ) 

3 g2 t 3 


i ∂ϕ 

∂t = − 2 

2m ∆ϕ + g|ϕ|2 ϕ 

Also holds for nonlinear Schrödinger equation with arbitrary nonlinearity 

Gravity can be transformed away ⇒ 

equivalence principle for the GPE, solitons in the sense of the Einstein elevator 

(Chen & Liu, PRL 1976) 


Outline 




















9 Summary 



The basic equations 


The model 

Klein–Gordon equation 

g µν D µ D ν ϕ + m 2 ϕ =0, D = ∂ + { ··· } 

Fluctuating metric 

g µν = η µν + h µν , |h µν |≪1 

noise 

〈h µν (x)〉 st = γ µν , δ ρσ 〈h µρ (x)h νσ (x)〉 st = σ 2 µν 

small amplitude of fluctuations 

frequency might be large, wavelength might be small 

〈·〉 st = averaging over a space–time volume 

we do not require the h µν to obey a wave equation 



The basic equations 


Approximations 

Weak field up to second order ˜h µν = h µρ h ρ 

ν 

Relativistic approximation of metric and quantum field (á la Kiefer & Singh, 

PRD 1994) 

Hψ = −( (3) g) 1 2 ( ) 

4 

2m ∆ cov ( (3) g) − 1 4 ψ 

′ 

+ m 2 

− 1 { 

} 

i∂ i ,h i0 i0 

(1) − ˜h (1) ψ 

2 

manifest hermitean w.r.t. flat scalar product 

only second order terms do not vanish by averaging 

Dirac equation 

(˜h00 

) 

(0) − h00 (0) ψ 



Short wavelength 


Spatial average 

spatial average 

〈Aψ〉 s (x) := 1 

V x 

∫ 

short wavelength of fluctuations: V small 

spatial average of Schrödinger equation 

with α(x) =〈˜h ij − h ij 〉 s (x) 

V x 

A(y)ψ(y)d 3 y 

H = 1 ( 

δ ij + α ij (x) ) p i p j + α 0 

2m 

α ij (x): small variation w.r.t. x, fluctuations w.r.t. t. 

decompose α ij (x) =˜α ij (x)+γ ij (x) with 〈γ ij 〉 t =0 

˜α ij (x) acts like an anomalous inertial mass tensor 



Space–time fluctuations 


Fluctuation model 

α ij ↔ spectral noise density of fluctuations 

particular model: 

˜α ij (x) = 1 ∫ 

V ˜hij (x,t)d 3 x= 1 ∫ 

x V x 

V x 

model: power law spectral noise density 

1/V x 

(S 2 (k,t)) ij d 3 k 

(S 2 (k,t)) ij =(S 2 0n) ij |k| n integration 

=⇒ α ij (x) =(S 2 0n) ij λ −(6+n) 

p 

with dim(S 2 0n) ij =length 3+ n 2 

V x ∼ λ 3 p 

λ p = invariant length scale of quantum object = λ Compton 

λ p = de Broglie wave length 

λ p = geometric extension l p of quantum object (Bohr radius of atom) 



Space–time fluctuations 


Fluctuation model 

assumption: S 0n ∼ l 3+ n 2 

Planck ,then 

α ij (x) ∼ 

effective Hamiltonian 

( 

H = 1 δ ij + 

2m 

( 

lPlanck 

l p 

) β 

a ij (x) , β =6+n, a ij (x) =O(1) 

( 

lPlanck 

l p 

) β 

a ij (x) 

) 

p i p j = 1 ( 

) 

δ ij + δmij (x) 

p i p j 

2m m 

δm ij = anomalous inertial mass tensor, depends on particle 

δm ij leads to violation of Universality of Free Fall 

β = 1 2 

↔ random walk 

β = 2 3 

↔ holographic noise 


Result 



Result 

metric fluctuations ⇒ anomalous inertial mass → apparent violation of UFF 

alternative route for violation of UFF and LLI 

need of quantum tests 

Example 

for Cesium and Hydrogen and geometric extension of atoms 

η β=1 =10 −20 , η β=2/3 =10 −15 , η β=1/2 =10 −12 

accuracy 10 −15 is planned for the next years 

(Göklü & C.L. CQG 2008) 


Outline 


Order of equations of motion 

















9 Summary 



Order of equation of motion? 

Usual framework 

L = L(t, x, ẋ) 

⇒ 

Most important equation in physics! 

d 

p = F (t, x, ẋ) with p = mẋ 

dt 

More general equations? 

p = mẋ is a constitutive law. Can be more general (as is many cases) 

p = f(ẋ, ẍ, ... x,...) 

Then equations of motion of higher order 

Influence of external fluctuations (e.g. space–time fluctuations, gravitational 

wave background, Göklü, C.L., Camacho & Macias, CQG 2009): generalized 

Langevin equation with extra force term 

∫ t 

0 

C(t − t ′ )ẋ(t ′ )dt ′ 



Order of equation of motion? 

Generalized framework 

Our specific model 

L = L(t, x, ẋ, ẍ) 

⇒ 

d 2 

... 

(ɛẍ) =F (t, x, ẋ, ẍ, x) 

dt2 Gauge procedure in order to invent structure of interactions 

L(t, x, ẋ, ẍ) =L 0 (t, x, ẋ, ẍ) −q 0 A a ẋ a 

} {{ } 

+ q 1 A ab ẋ a ẋ b 

} {{ } 

1st order gauge fields 2nd order gauge fields 

with (Pais–Uhlenbeck oscillator) 

L 0 (t, x, ẋ, ẍ) =− ɛ 2ẍ2 + m 2 ẋ2 

dimɛ =kgs 2 

ɛ QG ∼ m Planck t 2 Planck ∼ 10−95 kg s 2 ɛ Ce ∼ m Ce t 2 Ce ∼ 10−71 kg s 2 



Equation of motion 

simplest case: constant electric field 

ɛ .... 

x +mẍ = qE 0 

solution in 1D with initial conditions x(0) = 0, ẋ(0) = 0, ẍ(0) = 0, and ... x (0) = 0 

x(t) = q ( 1 

m E 0 

2 t2 + ɛ ) 

m (cos (ωt) − 1) small deviation 

ẋ(t) = q ( √ ) ɛ 

m E 0 t − 

m sin (ωt) small deviation 

ẍ(t) = q m E 0 (1 − cos (ωt)) 

O(1) deviation 

... 

x (t) = q √ m 

m E 0 

ɛ 

sin (ωt) ω 

= √ m 

ɛ 

large deviation 

zitterbewegung 

Limit ɛ → 0 exists for x and ẋ, notfor ... x 


Search for ɛ 


Accelerated flight 

Flight through accelerator 

〈ẋ(L)〉−ẋ 0 

= ɛ 

ẋ 0 4m 

Ion interferometric measurement of acceleration 

phase shift 

− + 

ẋ 2 0 

∆φ = A(ω)k · ẍ(ω)T 2 

L 

with transfer function 

A(ω) = sin2 (ωt) 

ω 2 


Search for ɛ 


Electronic devices 

Zitterbewegung of a charged particle induces voltage noise 

1 

2 C〈U 2 〉 t = m〈ẋ 2 〉 = 1 ( q 

) 2 

2 ɛ m E 0 

R 

C 

I 

General estimate: ɛ ≤ 10 −50 kg s 2 . 

Applies to mirrors in gw interferometers? 

Adding a small higher derivative term is a mathematical method toanalyze 

differential equations. 

C.L. & Rademaker 2009 


Outline 


* Existence of inertial systems 

















9 Summary 



Finsler geometry 

Motivation 

generic generalization of GR 

leads to deformed light cones and mass shells 

has been discussed within Quantum Gravity (Jacobson, Liberati & Mattingly) 

Very Special Relativity (Cohen & Glashow) 

Metric geometries 

Euclidean distance 

Riemannian distance 

dl 2 = δ ij dx i dx j 

dl 2 = g ij (x)dx i dx j 

by coordinate transformation it can be brought locally into Euclidean form 




Finsler space 

Finsler length function 

dl 2 = F (x, dx) , 

F(x, λdx) =λ 2 F (x, dx) 

Finsler metric tensor f µν (x, dx) is defined as 

dl 2 = g µν (x, dx)dx µ dx ν , where g µν (x, y) = 1 ∂ 2 F (x k ,y m ) 

2 ∂y µ ∂y ν 

Light cones 

Light cone defined by 

ds 2 = dt 2 − dl 2 




Euclidean light cone Riemannian light cone Finslerian light cone 












Geodesics 

∫ 

δ ds =0 ⇒ 0= d2 x µ 

ds 2 + { ρσ µ } (x, ẋ) dxρ 

ds 

with 

dx σ 

ds 

{ µ 

ρσ } (x, ẋ) =g µν (x, ẋ)(∂ ρ g σν (x, ẋ)+∂ σ g ρν (x, ẋ) − ∂ ν g ρσ (x, ẋ)) 

Main characteristics of geodesic motion 

Geodesic equation fulfills Universality of Free Fall 

{ ρσ µ } (x, ẋ) cannot be transformed to zero ∀ẋ 

⇒ gravity cannot be transformed away locally 

⇔ Einstein’s elevator does not hold: if for one particle we have force–free 

motion, other freely falling particles experience a fictitious force 

⇒ fictitious forces cannot be transformed away 

⇔ there is no inertial system 




Possible effects 

Radial acceleration different from horizontal acceleration 

¨r = G 1(r)M r 3 

r 2 T 2 = G 2(r)M 

4π 2 

Condition to be able to transform away 

gravity is stronger than pure UFF. 

Acceleration toward the Earth depends on 

horizontal velocity. 

Formalism may host the Pioneer anomaly 

(escape orbits behave different as bound 

orbits) 

Speculation: violation of UGR, G = G(T ) 

C.L. & Perlick 2012 

a 1 

a 2 

v 



Parametrizing deviations from Riemann/Minkowski 

Special case: “power law” metrics 

dl 2 =(g µ1µ 2...µ 2n 

(x)dx µ1 dx µ2 ···dx µ2n ) 1 r 

Deviation from Riemann/Minkowksi 

dl 2r = ( ) 

g µ1µ 2 ···g µ2r−1µ 2r 

+ φ µ1...µ 2r dx 

µ1 

···dx µ2r 

Gives anisotropic light propagation (C.L., Lorek & Dittus, GRG 2009) 



Quantum mechanics in Finsler space 

Finslerian Hamilton operator 

H = H(p) with H(λp) =λ 2 H(p) 

“Power–law” ansatz (non–local operator) 

Simplest case: quartic metric 

Deviation from standard case 

H = 1 ( 

g 

i 1...i 2r 

) 1 

r 

∂ i1 ···∂ i2r 

2m 

H = 1 ( 

g ijkl ) 1 

2 

∂ i ∂ j ∂ k ∂ l 

2m 

H = − 1 ( 

∆ 2 + φ ijkl ) 1 

2 

∂ i ∂ j ∂ k ∂ l 

2m 

= − 1 

√ 

2m ∆ 1+ φijkl ∂ i ∂ j ∂ k ∂ l 

∆ 2 



Quantum mechanics in Finsler space 

H = − 1 (1+ 

2m ∆ 1 φ ijkl ) 

∂ i ∂ j ∂ k ∂ l 

2 ∆ 2 

Hughes–Drever: H tot = H + σ · B 

Atomic interferometry, atom–photon interaction 

δφ ∼ H(p + k) − H(p) = k2 

2m + 1 ( 

) 

δ il + φijkl p j p k 

m p 2 p i k l 

modified Doppler term: gives different Doppler term while rotating the whole 

apparatus (even in Finsler light still propagates on straight lines, anisotropy – 

deformed mass shell) 


Outline 


Creation of the gravitational field 

















9 Summary 


Outline 




















9 Summary 



Test of Newton potential I 


(1+αe −r/λ) 

U = GM r 

long range limits 

short range limits 



Test of Newton potential II 


Kostelecky framework (Kostelecky, PRD 2005): anisotropy of Newtonian potential 

Experiments 

Result 

U = MG 

r 

(1+ ri c ij r j 

r 2 ) 

Atomic interferometry (Müller et al, PRL 2007) 

LLR (Battat, Chandler & Stubbs, PRL 2007) 

|c ij |≤10 −5 ...10 −9 


Outline 




















9 Summary 


actio = reactio ? 





Active and passive mass 


Gravitationally bound two–body 

system (Bondi, RMP 1957) 

m 1i ẍ 1 = m 1p m 2a 

x 2 − x 1 

|x 2 − x 1 | 3 

m 2i ẍ 2 = m 2p m 1a 

x 1 − x 2 

|x 1 − x 2 | 3 

center–of–mass and relative 

coordinate 

X := m 1i 

x 1 + m 2i 

x 2 

M i M i 

x := x 2 − x 1 

M i = m 1i + m 2i = total inertial 

mass. Then 

m 1i 

m 1a 

m 1p 

F 

x 1 

x 2 

O 

F 

m 2i 

m 2a 

m 2p 



Active and passive mass 


Decoupled dynamics of relative coordinate 

Ẍ = m 1pm 2p 

M i 

C 21 

x 

|x| 3 with C 21 = m 2a 

m 2p 

− m 1a 

m 1p 

ẍ = − m ( 

) 

1pm 2p m 1a m 2a 

m 1i + m 2i 

m 1i m 2i m 1p m 2p 

x 

|x| 3 

C 21 =0: ratio of the active and passive masses are equal for both particles 

C 21 ≠0: ⇒ self–acceleration of center of mass 

Interpretation 

Ẍ ≠0 ⇔ C 12 ≠0 ⇔ 

Violation of law of reciprocal action or of actio = reactio for gravity 

The gravitational field created by masses of same weight depends on its 

composition. Has the same status as the Weak Equivalence Principle. 

Requires experimental tests ... 




Experiment testing m ga = m gp 

Measurement of relative acceleration 

Step 1: Take two masses with m pg1 = m pg2 (equal weight) 

Step 2: Test active equality of these two masses with torsion balance 

Experimental setup: Torsion balance with equal passive masses reacting on m ag1 

and m ag2 

No effect has been seen: C 12 ≤ 5 · 10 −5 (Kreuzer, PR 1868) 





Measurement of relative acceleration 

Step 1: Take two masses with m pg1 = m pg2 (equal weight) 

Step 2: Test active equality of these two masses with torsion balance 

Experimental setup: Torsion balance with equal passive masses reacting on m ag1 

and m ag2 

No effect has been seen: C 12 ≤ 5 · 10 −5 (Kreuzer, PR 1868) 





geometric 

center 

Measurement of center–of–mass 

acceleration 

Earth 






acceleration 

geometric 

center 

s 

center of 

mass 

14 ◦ 

ŝ 

Earth 






acceleration 

geometric 

center 

F self M M rEM 

2 s ρ 

= C Al−Fe 

s 

F EM M ⊕ r 

center of 

M 

2 r M ∆ρŝ 

mass 

Effect of tangential part: increase of 

orbital angular velocity 

14 ◦ 

ŝ 

∆ω 

ω 

=6π F self 

F EM 

sin 14 ◦ per month 

From LLR ∆ω 

ω 

≤ 10−12 per month 

⇒ 

C Al−Fe ≤ 7 · 10 −13 

Earth 

Bartlett & van Buren, PRL 1986 

significant improvement with new 

LLR data and moon orbiter data 

possible 




Active and passive charges: Dynamics 

C.L., Macias, Müller, PRA 2007 

Dynamics of two electrically bound particles (E = external electric field) 

Similar phenomena 

m 1i ẍ 1 = q 1p q 2a 

x 2 − x 1 

|x 2 − x 1 | 3 + q 1pE(x 1 ) 

m 2i ẍ 2 = q 2p q 1a 

x 1 − x 2 

|x 1 − x 2 | 3 + q 2pE(x 2 ) 

New feature: Active and passive neutrality (actively neutral system can react 

on external em field – passivley neutral systems can create an em field) 

Very good neutrality measurements ⇒ C 12 ≤ 10 −21 

Other approach through fine structure constant for H and He + 

Also: active and passive magnetic moment 

Theory: no Hamiltonian for total system, only for relative motion 

One should do dedicated experiments! 


Outline 


Further issues 

















9 Summary 


Outline 




















9 Summary 


Newton’s axioms 



1 Existence of absolute time 

2 Existence of inertial systems 

violated by Finsler space–time or, more generally, by nonlinear connections 

3 Force free motion in inertial systems: uniform and straight 

cannot hold for all particles in Finsler space–time or with nonlinear 

connections 

4 Motion with force 

d 

(mv) =F (x, v) 

dt 

Higher order? Defines the notion of force. 

5 Law of reciprocal actions 

(actio = reactio) 

6 Linear superposition of forces 

is not really fundamental 


Outline 


Further issues * Operational foundation of Riemannian structures 1 

















9 Summary 


Light 


Axiom L1 

1 For a particle trajectory P 

through p and a nearby event q 

there are two light rays 

connecting P and q. 

2 For a parametrization t of P , 

the function g p : q ↦→ 

1 

2 (t(p 2) − t(p))(t(p 1 ) − t(p)) in 

U p is C 2 . 

Axiom L2 

1 The set of all light rays divides 

T p (M ) into two connected 

components. 

2 The set of all light rays consists 

of two connected components. 

Ehlers, Pirani and Schild 1972 

t 

p 2 

L 

p 

L 

p 1 

P 

q 

U p V p 


Light 


Consequences 

Define g(p, q) = 1 2 (t(p 2) − t(p))(t(p 1 ) − t(p)) 

Exists 2nd rank tensor 

g µν symmetric 

g µν =lim 

q→p 

det g µν ≠0non singular tensor 

g(l, l) =0for l = light ray l µ = g µν k ν 

∃S : k ∼ dS 

l fulfills the geodesic equation D l l =0 

∂ 

∂q µ 

∂ 

g(p, q) 

∂qν Result 

g µν is a space–time metric. 


Particles 


Freely falling particles 

1 All particles fall along the same path 

2 The path is uniquely given by its initial position and initial velocity 

3 ∀p ∈ M ∃ coordinate system so that ∀ freely falling particles 

Consequence 

Equation of motion for particles 

d 2 x µ (t) 

dt 2 

d 2 x(τ) 

dτ 2 ∣ ∣∣∣p 

=0 

+Π µ ρσ(x) dxρ 

dt 

Projective structure = equivalence class 

dx σ 

dt 

= α dxµ (t) 

dt 

[ 

Π 

µ 

ρσ 

] 

:= 

{ 

¯Πµ ρσ | ¯Π µ ρσ =Π µ ρσ + B (ρ δ µ σ)} 

. 


Particles 


Freely falling particles 

1 All particles fall along the same path 

2 The path is uniquely given by its initial position and initial velocity 

3 ∀p ∈ M ∃ coordinate system so that ∀ freely falling particles 

Consequence 

Equation of motion for particles 

d 2 x µ (t) 

dt 2 

d 2 x(τ) 

dτ 2 ∣ ∣∣∣p 

=0 

+Π µ ρσ(x) dxρ 

dt 

Projective structure = equivalence class 

dx σ 

dt 

= α dxµ (t) 

dt 

[ 

Π 

µ 

ρσ 

] 

:= 

{ 

¯Πµ ρσ | ¯Π µ ρσ =Π µ ρσ + B (ρ δ µ σ)} 

. 


Compatibility 


Axiom C: Compatibility 

No particle is faster than light 

The velocity of particles can be arbitrarily close to the velocity of light 

Consequence 

Particles move in a Weylian space–time 

0= d2 x µ 

ds 2 

Property of D: 

dx ρ dx σ 

+Γµ ρσ 

ds ds 

with Γ µ ρσ = { ρσ µ } + 1 ( 

δ 

µ 

2 ρ a σ + δ σa µ ρ − g ρσ g µν ) 

a ν 

D µ g ρσ = a µ g ρσ 

It is possible to build light clocks: transport of time unit 


Riemannian structure 


Axiom: 2nd clock effect 

There is no second clock effect (implied by UGR). 

Consequence 

Particles move in a Riemannian space–time 

0= d2 x µ 

ds 2 

dx ρ dx σ 

+Γµ ρσ 

ds ds 

with Γ µ ρσ = { µ 

ρσ } , 

ds 2 = g µν dx µ dx ν 

Properties of particles and light rays define the space–time. 

More precise: space–time structures (modulo reparametrizations and 

projective transformations) 

What can be questioned 

Condition g p is C 2 .Relaxingthisleadstodirection–dependenceofmetric. 

Cauchy problem → higher order equations of motion 

Gravity can be transformed away → non–linear connections 






Consequence 


0= d2 x µ 

ds 2 

dx ρ dx σ 

+Γµ ρσ 

ds ds 


ρσ } , 






Condition g p is C 2 .Relaxingthisleadstodirection–dependence of metric. 








Consequence 


0= d2 x µ 

ds 2 

dx ρ dx σ 

+Γµ ρσ 

ds ds 


ρσ } , 














Consequence 


0= d2 x µ 

ds 2 

dx ρ dx σ 

+Γµ ρσ 

ds ds 


ρσ } , 










Outline 


Summary 

















9 Summary 


Main theme 

Summary 

Test particles 

Gravity and its structure can only be explored through the motion 

of test particles 

Orbits and clocks 

Massive particles and light 

quantum fields 

What is gravity depends on the structure of the equation of motion 

Existence of inertial systems 

Order of differential equation 

Dependence on particle parameters 


Summary 

Summary 

What determines gravity? 

GR = UFF + CP + LLI + UGR + Newton potential + UGF + ... 

scheme not complete as far as Einstein’s equations are concerned 

part of it can be interpreted as test of Newton’s axioms: IS + CP + UGF 

fundamental violation of principle vs. apparent violation of principle 


Summary 

Summary 

All fundamental postulates have to be tested as good as possible 

UFF, UGR, and LLI are questioned from quantum gravity 

Further motivation for tests: Observational and theoretical hints from 

quantum gravity and from gravity at large distances 

Small accelerations: Hints from unexplained observations (dark matter) 

Finsler: from quantum gravity (Jacobson, Liberati & Mattingly, PRD 2004) 

Higher order equations of motion from, e.g., loop quantum gravity (Alfaro, 

Morales–Tecotl & Urrutia, PRL 2000, PRD 2000, Horavagravity),or 

correlated influence from space–time fluctuations (Göklü, C.L., Camacho & 

Macias, CQG 2009) 

No model until now (cannot be derived from, e.g., string theory) for active ≠ 

passive mass / charge. 

This is a symmetry of physics which unfortunately is not yet well analyzed. 

But may be of interest for discussing quantization procedure. 


Summary 

Summary 

Thank you! 

Thanks to 

H. Dittus 

E. Göklü 

D. Lorek 

A. Macias 

Z. Marijewic 

H. Müller 

V. Perlick 

P. Rademaker 

DLR 

DFG 

Center of Excellence QUEST

Testing the Equivalence Principle

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