13.2 The Cosmological FLRW Metric
211
ds
2
= c
2 dt
2
− a(t)
2
dr
2
1 − kr 2 + r
2
(dθ
2
+ sin
2
θ dϕ
2
)
,
(13.12b)
and the function a(t) is dimensionless. The function a(t) is called the scale factor
and is at the heart of cosmological theory. The choice (13.12b) has a nice advantage:
since only the product of the scale factor times the bracket in (13.12b) has physical
meaning we can consistently take a(t) to be unity at the present cosmic time t 0 .
Then the square bracket in (13.12b) is the square of the current physical distance to
a nearby galaxy. We can think of the curvature parameter k as the inverse square of
the maximum coordinate distance allowed for an observed galaxy.
The cosmological metric in either the form (13.12a) or (13.12b) is known as the
Friedmann-Lemaitre-Robertson-Walker or FLRW metric, after its various discoverers (Schutz 2009). It is the basis of relativistic cosmological theory, and a primary
task of theoretical cosmology is to determine the nature of the scale factor and the
curvature parameter to compare with observations.
13.3 Consequences of the Metric
Before we go on to apply the field equations of general relativity to the cosmological
problem we will show several cosmological consequences that follow from the FLRW
metric alone, independent of the dynamics imposed by the field equations. The first
consequence provides a remarkably simple picture of the motion of particles in the
metric, of the galaxies in the cosmic fluid: we will show that they remain at fixed
locations with respect to the coordinate system. This is of course an approximation
that ignores local or peculiar motion of galaxies.
A galaxy is assumed to follow in general relativity a geodesic
d
2 x
α
ds 2 +
α
βγ
dx
β
ds
dx
γ
ds
= 0.
(13.13)
For its spatial motion we take α = i and calculate the acceleration,
d
2 x
j
ds 2 = −
j
βγ
dx
β
ds
dx
γ
ds
.
(13.14)
For a galaxy that is initially at rest in the coordinate system let us see what this
acceleration is. For such a galaxy the spatial components of the 4-velocity are zero,
so
d
2 x
j
ds 2 = −
j
00
dx
0
ds
dx
0
ds
.
(13.15)
211
ds
2
= c
2 dt
2
− a(t)
2
dr
2
1 − kr 2 + r
2
(dθ
2
+ sin
2
θ dϕ
2
)
,
(13.12b)
and the function a(t) is dimensionless. The function a(t) is called the scale factor
and is at the heart of cosmological theory. The choice (13.12b) has a nice advantage:
since only the product of the scale factor times the bracket in (13.12b) has physical
meaning we can consistently take a(t) to be unity at the present cosmic time t 0 .
Then the square bracket in (13.12b) is the square of the current physical distance to
a nearby galaxy. We can think of the curvature parameter k as the inverse square of
the maximum coordinate distance allowed for an observed galaxy.
The cosmological metric in either the form (13.12a) or (13.12b) is known as the
Friedmann-Lemaitre-Robertson-Walker or FLRW metric, after its various discoverers (Schutz 2009). It is the basis of relativistic cosmological theory, and a primary
task of theoretical cosmology is to determine the nature of the scale factor and the
curvature parameter to compare with observations.
13.3 Consequences of the Metric
Before we go on to apply the field equations of general relativity to the cosmological
problem we will show several cosmological consequences that follow from the FLRW
metric alone, independent of the dynamics imposed by the field equations. The first
consequence provides a remarkably simple picture of the motion of particles in the
metric, of the galaxies in the cosmic fluid: we will show that they remain at fixed
locations with respect to the coordinate system. This is of course an approximation
that ignores local or peculiar motion of galaxies.
A galaxy is assumed to follow in general relativity a geodesic
d
2 x
α
ds 2 +
α
βγ
dx
β
ds
dx
γ
ds
= 0.
(13.13)
For its spatial motion we take α = i and calculate the acceleration,
d
2 x
j
ds 2 = −
j
βγ
dx
β
ds
dx
γ
ds
.
(13.14)
For a galaxy that is initially at rest in the coordinate system let us see what this
acceleration is. For such a galaxy the spatial components of the 4-velocity are zero,
so
d
2 x
j
ds 2 = −
j
00
dx
0
ds
dx
0
ds
.
(13.15)