Eur. Phys. J. C (2019) 79 :106 Page 3 of 13 106
For the spherically symmetric metric (1)wehave
H =
1
3
2
˙
Y
Y
+
˙
B
B
, ˙u
μ
=
˜
∇
μ
(
ln N
)
= Aδ
r
μ
, A ≡
N
,r
N
,
(5)
and the space-like symmetric trace-free tensors σ
μ
ν
and E
μ
ν
can be written as
σ
μ
ν
= Σ e
μ
ν
, E
μ
ν
= We
μ
ν
. (6)
Here W and Σ are scalar functions,
Σ =
1
3
˙
Y
Y
−
˙
B
B
, W =−Ψ
2
, (7)
with Ψ
2
the conformal invariant of Petrov type D space-
times, and e
μ
ν
= h
μ
ν
− 3n
μ
n
ν
= Diag
[
0, −2, 1, 1
]
is the
tensor basis that serves as eigenframe for spacelike sym-
metric trace free tensors in Petrov type D spacetimes, with
h
μν
= g
μν
+u
μ
u
ν
and n
μ
=
√
g
rr
δ
r
μ
the projection tensor
and a spacelike normal vector tangent to the orbits of SO(3)
(note that ˙e
μ
ν
= 0).
The 4-velocity of the other non-comoving components
are related to u
μ
via the relative velocity measured by the
fundamental observers v
μ
(i)
, defined such that v
μ
(i)
u
μ
= 0.
Then, the 4-velocity of the i-th fluid reads,
u
μ
(i)
= γ
(i)
u
μ
+ v
μ
(i)
, with γ
(i)
=
1 − v
2
(i)
−
1
2
, (8)
where “i” labels the components and v
2
(i)
= g
μν
v
μ
(i)
v
ν
(i)
.
The total energy-momentum tensor is made up of all the
contributions from the different species, and in general it will
no longer be the energy-momentum tensor of a perfect fluid,
but
T
μν
=
i
T
μν
(i)
= ρ u
μ
u
ν
+ ph
μν
+ 2q
(μ
u
ν)
+ π
μν
, (9)
where ρ, p, π
μν
and q
μ
are the energy density, isotropic
and anisotropic pressures,
3
and the energy flow measured
by the fundamental observers along u
μ
. These components
are determined by projecting the energy-momentum tensor
parallel and orthogonal to u
μ
[78,79]:
ρ = T
μν
u
μ
u
ν
, q
μ
=−T
μν
u
ν
− ρu
μ
, p =
1
3
T
μν
h
μν
,
(10a)
π
μν
= T
μν
=
h
η
(μ
h
υ
ν)
−
1
3
h
μν
h
ηυ
T
ηυ
. (10b)
3
In this setup, the cosmological constant is implicitly considered by
the substitution ρ → ρ + Λ and p → p − Λ.
Although the total energy-momentum tensor is always con-
served, the energy-momentum tensors of the individual
components are not necessarily conserved. If there are
non-gravitational interactions between them, they satisfy
∇
ν
T
μν
(i)
= J
μ
(i)
, where J
(i)
is the rate of energy and momen-
tum densities transfer between species i as measured in
the u
μ
-frame. In absence of non-gravitational interaction
these energy-momentum tensors are separately conserved:
J
μ
(i)
= 0 for all i.
2.1 A mixture of non-interacting perfect fluids
We now focus on the case of a mixture of non-interacting
fluids, each one a perfect fluid with a suitable equation of
state in its intrinsic frame (denoted with
∗
):
p
∗
(i)
= w
(i)
ρ
∗
(i)
, in general w
(i)
= w
(i)
(t, r ). (11)
In this way, the total energy-momentum tensor follows from
adding up the corresponding tensors of the dynamically sig-
nificant species as seen from the u
μ
frame (Eqs. (9) and (10)).
Explicitly, if we choose the fundamental observers those
along u
μ
(0)
, then the contribution to the total energy-
momentum tensor of the “0” component reads
T
μν
0
= ρ
∗
0
u
μ
u
ν
+ p
∗
0
h
μν
. (12)
On the other hand, the energy-momentum tensor of the i-th
component comoving with velocity u
μ
(i)
(see Eq. (8)) takes
the form
T
μν
(i)
= ρ
(i)
u
μ
u
ν
+ p
(i)
h
μν
+ 2q
(μ
(i)
u
ν)
+ π
μν
(i)
, (13)
with the dynamical quantities given by [78,79]:
ρ =
i
ρ
(i)
,ρ
(i)
= γ
2
(i)
(1 + w
(i)
v
2
(i)
)ρ
∗
(i)
, (14a)
p =
i
p
(i)
, p
(i)
=
w
(i)
+
1
3
γ
2
(i)
v
2
(i)
(1 + w
(i)
)
ρ
∗
(i)
,
(14b)
q
μ
=
i
q
μ
(i)
, q
μ
(i)
= γ
2
(i)
(1 + w
(i)
)ρ
∗
(i)
v
μ
(i)
, (14c)
π
μν
=
i
π
μν
(i)
,π
μν
(i)
= γ
2
(i)
(1 + w
(i)
)ρ
∗
(i)
v
μ
(i)
v
ν
(i)
, (14d)
where for spherical symmetry spacetimes the anisotropy ten-
sor can be written as π
μ
(i)ν
= Π
(i)
e
μ
ν
, with Π
(i)
to be deter-
mined from Eq. (14d).
The dynamics of this fluid mixture can be determined from
the first order “1 +3” fluid flow representation of Einstein’s
field equations given in “Appendix A”, by direct substitution
of ρ, p,π
μν
, q
μ
by (14a)–(14d), with
q
(i)μ
= Q
(i)
δ
r
μ
,v
(i)μ
= V
(i)
δ
r
μ
, (15)
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