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1-10 - Fysikum - Stockholms universitet

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GENERAL RELATIVITY<br />

1 Introduction<br />

Lecture notes<br />

by<br />

Kjell Rosquist<br />

<strong>Fysikum</strong>, <strong>Stockholms</strong> <strong>universitet</strong><br />

VT 2011<br />

Galileo demonstrated in his famous experiment in Pisa that particles move on identical spacetime<br />

trajectories in the gravitational field independently of their mass. Newton went further and showed<br />

that the gravitational acceleration of a body does not depend on its composition. This follows from<br />

the formula m1a = Gm1m2r −2 which gives the acceleration (a) of a mass m1 in the gravitational field<br />

of another mass m2 at a distance r (G is Newton’s gravitational constant). By dividing both sides of<br />

the equation by m1 we see that the acceleration of m1 does not depend on the mass m1 itself. The<br />

equality between inertial and gravitational mass is noted more or less as a curious fact in Newtonian<br />

gravity. Einstein, however, regarded this observation as a special case of a more general postulate,<br />

the principle of equivalence, which he took as the basis of general relativity (GR). The principle of<br />

equivalence states that all physical laws if written in tensor form have the same form in any freely<br />

falling reference system. Such a reference frame is by definition a system which is unaffected by all<br />

forces except gravity. Mathematically this means that the theory should be invariant under arbitrary<br />

transformations of the spacetime coordinates. The fundamental observation that the gravitational<br />

interaction does not depend on the mass or composition of a body makes it natural to interpret<br />

gravity as an aspect of the geometry of the spacetime rather than as an ordinary force. In Newtonian<br />

or special relativistic mechanics free particles move along straight lines. Two free particles do not<br />

influence each other (unless they collide). Therefore, in order to describe for example the motion of<br />

the earth in the gravitational field of the sun in geometrical terms we clearly need a more general<br />

concept of geometry than that of ordinary flat Euclidean space. It is here that Riemann’s theory of<br />

1


Kjell Rosquist 8<br />

Figure 4: Differentials in curvilinear coordinates. The vector with components v α at x α<br />

has components v α + dv α at x α + dx α . To take the differential we first parallel transport<br />

the vector from x α to x α + dx α . The transported vector has components v α + δv α .<br />

A scalar is unchanged by parallel transport, δφ = 0, i.e. Dφ = dφ or φ;α = φ,α. We can use this<br />

to derive the change in the components of a covector under parallel transport. Using δ(u α vα) =0we<br />

obtain<br />

Then since the u α are arbitrary it follows that<br />

u α δvα = −vαδu α<br />

= vαu γ Γ α γβdx β<br />

= vγu α Γ γ αβdx β .<br />

(31)<br />

δuα = uγΓ γ αβdx β . (32)<br />

It follows from this that the covariant derivative of a covector is given by<br />

uα;β = uα,β − uγΓ γ αβ . (33)<br />

Next we show that the connection coefficients are symmetric in their lower indices, Γ α βγ =Γ α γβ.<br />

To that end let φ be a scalar and define a covector with components uα = φ,α. Thenuα,β = uβ,α from<br />

which it follows<br />

uα;β − uβ;α = −(Γ γ αβ − Γ γ βα)uγ . (34)<br />

The left hand side of this expression is a tensor. Therefore the right hand side must also be a tensor.<br />

But then the right hand side being zero in Cartesian coordinates must be zero in any coordinate<br />

system. Thus the connection coefficients are symmetric as claimed.<br />

Let us now take a look at the relation between the connection and the metric. The covariant<br />

derivative of the metric is a tensor with components given by<br />

gαβ;γ = gαβ,γ − gδβΓ δ αγ − gαδΓ δ βγ . (35)<br />

In Cartesian coordinates the right hand side is zero. Thus since it is a tensor gαβ;γ = 0 in any<br />

coordinate system. In other words the metric is covariantly constant. We shall express the connection<br />

in terms of the metric by solving (35) as a system of linear equations in Γ α βγ. Definingfirst<br />

we write (35) in three equivalent ways by permuting indices as<br />

Γαβγ = gαδΓ δ βγ , (36)<br />

gαβ,γ =Γαβγ +Γβαγ ,<br />

gγα,β =Γγαβ +Γαγβ ,<br />

−gαβ,γ = −Γβγα − Γγβα ,<br />

(37)

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