Chapter 12
The Einstein Field Equations
for Cosmology
Abstract Cosmology seeks to answer the rather grandiose question “What is the
universe?” After about half a century of being mainly a theoretical and mathematical
subject cosmology entered the mainstream of physics only in the last half of the twentieth century thanks to a viable theoretical structure provided by general relativity and
diverse observations of distant galaxies and measurements of the cosmic microwave
background (CMB) radiation. In this chapter we begin our study of cosmology by
applying general relativity to the entire universe, in which the source of gravity is
taken to be a cosmic fluid. The gravitational field equations naturally allow for one
component of the fluid to be the “dark energy” that has been found by observation
to be the dominant component.
12.1 The Field Equations and Energy-Momentum
Conservation
In Chap. 8 we obtained the Einstein equations for the gravitational field. We also
mentioned the simplest energy-momentum tensor of interest, the so-called dust tensor
that describes a fluid characterized by only a mass density and velocity. In particular
the dust fluid has no pressure. In this chapter we want to include fluid pressure to
describe the large-scale universe of cosmology (Adler 1975; Schutz 2009). That is we
want the appropriate cosmological energy-momentum tensor. We must also include
the cosmological constant that describes the so-called dark energy that appears to
pervade the universe and which has a large effect on its dynamics.
Recall from Chap. 8 that the field equations relate the Einstein tensor to the
energy-momentum tensor of the source by
G
μν
= R
μν
−
1
2
g
μν R = CT
μν
, C = −
8π G
c 2 .
(12.1a)
Since the Einstein tensor has a zero divergence the energy-momentum tensor must
also have a zero divergence, so we include a subsidiary condition,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_12
193
The Einstein Field Equations
for Cosmology
Abstract Cosmology seeks to answer the rather grandiose question “What is the
universe?” After about half a century of being mainly a theoretical and mathematical
subject cosmology entered the mainstream of physics only in the last half of the twentieth century thanks to a viable theoretical structure provided by general relativity and
diverse observations of distant galaxies and measurements of the cosmic microwave
background (CMB) radiation. In this chapter we begin our study of cosmology by
applying general relativity to the entire universe, in which the source of gravity is
taken to be a cosmic fluid. The gravitational field equations naturally allow for one
component of the fluid to be the “dark energy” that has been found by observation
to be the dominant component.
12.1 The Field Equations and Energy-Momentum
Conservation
In Chap. 8 we obtained the Einstein equations for the gravitational field. We also
mentioned the simplest energy-momentum tensor of interest, the so-called dust tensor
that describes a fluid characterized by only a mass density and velocity. In particular
the dust fluid has no pressure. In this chapter we want to include fluid pressure to
describe the large-scale universe of cosmology (Adler 1975; Schutz 2009). That is we
want the appropriate cosmological energy-momentum tensor. We must also include
the cosmological constant that describes the so-called dark energy that appears to
pervade the universe and which has a large effect on its dynamics.
Recall from Chap. 8 that the field equations relate the Einstein tensor to the
energy-momentum tensor of the source by
G
μν
= R
μν
−
1
2
g
μν R = CT
μν
, C = −
8π G
c 2 .
(12.1a)
Since the Einstein tensor has a zero divergence the energy-momentum tensor must
also have a zero divergence, so we include a subsidiary condition,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_12
193