MODERN COSMOLOGY

Aspects of general relativity 11

2.1.1 The equivalence principle

The problem of how to generalize the laboratory laws of special relativity is solved
by using the equivalence principle, in which the physics in the vicinity of freely
falling observers is assumed to be equivalent to special relativity. We can in fact
obtain the full equations of general relativity in this way, in an approach pioneered
by Weinberg (1972). In what follows, Greek indices run from 0 to 3 (spacetime),
Roman from 1 to 3 (spatial). The summation convention on repeated indices of
either type is assumed.
Consider freely falling observers, who erect a special-relativity coordinate
frameξμin their neighbourhood. The equation of motion for nearby particles is
simple:
d^2 ξμ
dτ^2

= 0 ; ξμ=(ct,x,y,z),

i.e. they have zero acceleration, and we have Minkowski spacetime

c^2 dτ^2 =ηαβdξαdξβ,

whereηαβis just a diagonal matrixηαβ =diag( 1 ,− 1 ,− 1 ,− 1 ). Now suppose
the observers make a transformation to some other set of coordinatesxμ.What
results is the perfectly general relation

dξμ=

∂ξμ

∂xν

dxν,

which, on substitution, leads to the two principal equations of dynamics in general
relativity:

d^2 xμ

dτ^2

+μαβ

dxα

dτ

dxβ

dτ

= 0

c^2 dτ^2 =gαβdxαdxβ.

At this stage, the new quantities appearing in these equations are defined only in
terms of our transformation coefficients:

μαβ=

∂xμ

∂ξν

∂^2 ξν

∂xα∂xβ

gμν=

∂ξα

∂xμ

∂ξβ

∂xν

ηαβ.

This tremendously neat argument effectively uses the equivalence principle
to prove what is often merely assumed as a starting point in discussions of
relativity: that spacetime is governed by Riemannian geometry. There is a metric
tensor, and the gravitational force is to be interpreted as arising from non-zero
derivatives of this tensor.