Introduction to Cosmology

32 Special Relativity

includes the coordinate d푥^0 ≡푐d푡so that the invariant line element in Equation (2.7)
can be written

d푠^2 =푐^2 d휏^2 =푔휇휈d푥휇d푥휈. (2.14)

The 16 components ofgin flat Minkowski space-time are given by the diagonal matrix
휂휇휈, a generalization of the Kronecker delta function to four-space-time,

휂 00 = 1 ,휂jj=− 1 ,푗= 1 , 2 , 3 , (2.15)

all nondiagonal components vanishing. The choice of signs in the definition of휂휇휈is
not standardized in the literature, but we shall use Equation (2.15).
The path of a body, or its world line, is then described by the four coordinate func-
tions푥(휏),푦(휏),푧(휏),푡(휏), where the proper time휏is a new absolute parameter, an
invariant under Lorentz transformations. A geodesic in the Minkowski space-time is
also a straight line, given by the equations

d^2 푡

d휏^2

= 0 , d

(^2) R
d휏^2

= 0. (2.16)

In the spherical coordinates [Equation (2.11)] the Minkowski metric [Equa-
tion (2.7)] takes the form

d푠^2 =푐^2 d푡^2 −d푙^2 =푐^2 d푡^2 −d푅^2 −푅^2 d휃^2 −푅^2 sin^2 휃d휙^2. (2.17)

An example of a curved space is the two-dimensional surface of a sphere with radius
푅obeying the equation

푥^2 +푦^2 +푧^2 =푅^2. (2.18)

This surface is called a two-sphere.
Combining Equations (2.6) and (2.18) we see that one coordinate is really super-
fluous, for instance푧, so that the spatial metric [Equation (2.6)] can be written

d푙^2 =d푥^2 +d푦^2 +

(푥d푥+푦d푦)^2

푅^2 −푥^2 −푦^2

. (2.19)

This metric describes spatial distances on a two-dimensional surface embedded in
three-space, but the third dimension is not needed to measure a distance on the sur-
face. Note that푅is not a third coordinate, but a constant everywhere on the surface.
Thus measurements of distances depend on the geometric properties of space, as
has been known to navigators ever since Earth was understood to be spherical. The
geodesics on a sphere are great circles, and the metric is

d푙^2 =푅^2 d휃^2 +푅^2 sin^2 휃d휙^2. (2.20)

Near the poles where휃= 0 ∘or휃= 180 ∘the local distance would depend very little
on changes in longitude휙. No point on this surface is preferred, so it can corre-
spond to a Copernican homogeneous and isotropic two-dimensional universe which
is unbounded, yet finite.
Let us write Equation (2.20) in the matrix form

d푙^2 =

(

d휃 d휙

)

g

(

d휃

d휙

)

, (2.21)