12.3 The Cosmological Constant as Vacuum or Dark Energy
199
significant, and must be determined by observation. One might guess, on the basis
of dimensional analysis, that its value might be comparable to the inverse square of
the size of the universe.
Some theorists, notably Einstein who invented it, have objected to the cosmological term on esthetic grounds: the field equations are simpler without it. The dominant
viewpoint at present is that its nonzero observed value makes it quite important; the
present standard model of cosmology includes it as a major ingredient of the universe.
We will discuss this further in following chapters.
The introduction of the cosmological constant in the above was by purely formal
mathematical means: it is allowed by the mathematical structure of the equations.
There is however an alternative physical interpretation of the cosmological term that
is interesting. If we simply move the cosmological term to the right side of the field
equations,
G μν = C
T μν −
C
g μν
,
(12.19)
then we may interpret it as a contribution to the total energy-momentum tensor. In the
absence of any ponderable material it may be thought of as the energy-momentum
tensor of empty space, that is of the vacuum. However the cosmological term corresponds to a peculiar energy-momentum tensor. Comparison with the perfect fluid
energy-momentum tensor in (12.14) shows that it may be viewed as a perfect fluid if
−
C
g
αβ
= ρu
α u
β
+
p
c 2
u
α u
β
− g
αβ
.
(12.20)
This is only consistent if we take the mass density and pressure of the vacuum to be
p
c 2 = −ρ, ρ = −
C
=
c
2
8π G
, mass density.
(12.21)
If instead we use the energy density ρc
2 these look a bit simpler
p = −ρ V , ρ V =
c
4
8π G
, energy density.
(12.22)
That is, the pressure is the negative of the energy density, much in contrast to the
situation for an ideal gas in which the pressure is positive and smaller than the
energy density. The vacuum fluid is thus quite peculiar and is now an important part
of present cosmological theory. Because it does not interact directly with light it is
widely called dark energy. This name also allows a more general view of its nature;
the dark energy is presently the subject of intense observational and theoretical study
(Amendola 2010).
199
significant, and must be determined by observation. One might guess, on the basis
of dimensional analysis, that its value might be comparable to the inverse square of
the size of the universe.
Some theorists, notably Einstein who invented it, have objected to the cosmological term on esthetic grounds: the field equations are simpler without it. The dominant
viewpoint at present is that its nonzero observed value makes it quite important; the
present standard model of cosmology includes it as a major ingredient of the universe.
We will discuss this further in following chapters.
The introduction of the cosmological constant in the above was by purely formal
mathematical means: it is allowed by the mathematical structure of the equations.
There is however an alternative physical interpretation of the cosmological term that
is interesting. If we simply move the cosmological term to the right side of the field
equations,
G μν = C
T μν −
C
g μν
,
(12.19)
then we may interpret it as a contribution to the total energy-momentum tensor. In the
absence of any ponderable material it may be thought of as the energy-momentum
tensor of empty space, that is of the vacuum. However the cosmological term corresponds to a peculiar energy-momentum tensor. Comparison with the perfect fluid
energy-momentum tensor in (12.14) shows that it may be viewed as a perfect fluid if
−
C
g
αβ
= ρu
α u
β
+
p
c 2
u
α u
β
− g
αβ
.
(12.20)
This is only consistent if we take the mass density and pressure of the vacuum to be
p
c 2 = −ρ, ρ = −
C
=
c
2
8π G
, mass density.
(12.21)
If instead we use the energy density ρc
2 these look a bit simpler
p = −ρ V , ρ V =
c
4
8π G
, energy density.
(12.22)
That is, the pressure is the negative of the energy density, much in contrast to the
situation for an ideal gas in which the pressure is positive and smaller than the
energy density. The vacuum fluid is thus quite peculiar and is now an important part
of present cosmological theory. Because it does not interact directly with light it is
widely called dark energy. This name also allows a more general view of its nature;
the dark energy is presently the subject of intense observational and theoretical study
(Amendola 2010).