Introduction to Cosmology

Metrics of Curved Space-time 31

C

θ

φ

P

z

R

x

y

S

r

Figure 2.2A two-sphere on which points are specified by coordinates(휃,휙).

have used the convention to leave out the summation sign; it is then implied that
summation is carried out over repeated indices. One commonly uses Roman letters in
the indices when only the spatial components푥푖,푖= 1 , 2 ,3, are implied, and Greek let-
ters when all the four space-time coordinates푥휇,휇= 0 , 1 , 2 ,3, are implied. Orthogonal
coordinate systems have diagonal metric tensors and this is all that we will encounter.
The components ofgin flat Euclidean three-space are

푔ij=훿ij,

where훿ijis the usual Kronecker delta.
The same flat space could equally well be mapped by, for example, spherical or
cylindrical coordinates. The components푔ijof the metric tensor would be different,
but Equation (2.10) would hold unchanged. For instance, choosing spherical coordi-
nates푅,휃,휙as in Figure 2.2,

푥=푅sin휃sin휙, 푦=푅sin휃cos휙, 푧=푅cos휃, (2.11)

d푙^2 takes the explicit form

d푙^2 =d푅^2 +푅^2 d휃^2 +푅^2 sin^2 휃d휙^2. (2.12)

Geodesics in this space obey Newton’s first law of motion, which may be written as

R̈= 0. (2.13)

Dots indicate time derivatives.

Minkowski Space-time. In special relativity, symmetry between spatial coordinates
and time is achieved, as is evident from the Minkowski metric (2.7) describing a flat
space-time in four Cartesian coordinates. In tensor notation the Minkowski metric