Environmental Engineering Reference
In-Depth Information
where we have used the Einstein mass-energy relation to write the energy density
ˁ
=
ˁ
M
c
2
and redefined the time
t
ₒ
ct
. In GR all forms of energy gravitate,
therefore the use of the energy density instead of the matter density allow us to
consider other forms of energy besides matter as sources of gravity. One interesting
point is that Eq. (
2
) is the same that the obtained in GR. The root of this apparent
coincidence is the equivalence principle: InGREq. (
2
) is obtained in a specific chosen
coordinates, in these coordinates the free fall observers have fixed space coordinates
and as a consequence, about these observers there is a neighborhood where the laws
of Special Relativity hold.
Note that we can solve Eq. (
2
)for
a
.
Then, we need at least one more equation to close the system. Consider an adiabatic
expansion of the same configuration, the thermodynamical Gibbs equation is then
dE
(
t
)
once we know
ˁ(
t
)
, or alternatively
ˁ(
a
)
3
3
the volume enclosed by the sphere. Giving the dependence on time
t
we can write
E
=−
PdV
, where
P
is the pressure of the considered fluid and
V
=
4
ˀ
GR
(
t
)
/
+
ˁ
V
P V
,or
=
ˁ
V
=−
ˁ
+
3
H
(ˁ
+
P
)
=
0
,
(3)
which is the continuity equation. To finally solve the system of equations (
2
) and (
3
)
we need an Equation of State (EoS) that relates the energy density with the pressure.
In general this can be written as
P
, where
S
is the entropy of the fluid.
But since in this scenario we are restricted to adiabatic processes, the EoS can take
the barotropic form
P
=
P
(ˁ,
S
)
=
(ˁ)ˁ
(ˁ)
is called the EoS parameter function.
Consider for the moment the case of constant
w
, in such a situation the continuity
equation can be integrated to give
w
, where
w
)
=
ˁ
0
a
−
3
(
1
+
w
)
ˁ(
a
(4)
where
ˁ
0
≡
ˁ(
a
0
)
and we have normalized
a
0
≡
a
(
t
0
)
=
1; in this work, as usually,
t
0
denotes the present time. For example, the case
w
0 corresponds to a very
dilute fluid (
dust
) for which the energy density decays as the inverse of the volume,
ˁ
m
=
=
ˁ
m
0
a
−
3
; the case
P
3 corresponds to radiation for which the energy
density decays as the fourth power of the scale factor,
=
ˁ/
ˁ
r
=
ˁ
r
0
a
−
4
—three powers
for the dilution of the photons and one more for their redshift.
In GR the constant
K
is related to the curvature of spacetime, and due to the
assumption of homogeneity and anisotropy of space, there are only three possibilities
that correspond to flat space which is the case of
K
=
0, spherical space (
K
=
1)
and hyperbolic space for (
K
1). Consider the case in which
K
equals zero, we
can insert the solution given by Eq. (
4
) into Eq. (
2
) to obtain
=−
8
3
ˁ
0
1
/
2
ˀ
G
a
−
3
(
1
+
w
)/
2
+
1
˙
=
,
a
(5)
Search WWH ::
Custom Search