150 Page 4 of 20 Eur. Phys. J. C (2020) 80 :150
˜ρ =−
2
f
R
δT
0
0
−
1
3
δT
+
2α
f
R
δθ
0
0
−
δθ
3
+
δ Q
6
−
δ R
3γ c
2
.
(25)
On the other hand by using Eqs. (14) and (15)wehave
δT
0
0
−ρc
2
,δT −ρc
2
,δQ = δθ = δθ
0
0
ρ
2
c
4
,
(26)
where we have applied the condition p ρc
2
in the weak
field limit. In GR we have f
Q
= 0. Moreover in this case
we have δ R =−γδT . Consequently it is easy to show that
˜ρ = ρ, andEq. (24) recovers the standard Poisson’s equation.
For another special case, the EMSG model studied in [41]is
given by f (R, Q) = R − ηT
2
= R − ηQ. For this model
we have f
RR
= 0, f
R
= 1, and f
Q
=−η = αγ . Moreover
from Eq. (16), one may simply verify that δ R =−γ(δT +
α(2δ Q − δθ)). Therefore Eq. (25)gives
˜ρ = ρ(1 + 2αρ c
2
), (27)
in this special case, the effects of EMSG can be included in
the effective density and pressure defined as, see [43]
ρ
eff
= ρ +
αc
2
2
8ρ
p
c
2
+ ρ
2
+ 3
p
2
c
4
, (28)
p
eff
= p +
αc
4
2
ρ
2
+ 3
p
2
c
4
, (29)
More specifically, it has been shown in [43] that the governing
equations of EMSG, are completely similar to GR and the
only difference is that ρ and p are replaced with ρ
eff
and p
eff
.
In this case, in the weak field limit we can rewrite Eq. (27)as
˜ρ = ρ
eff
+ 3p
eff
/c
2
. In other words the Poisson’s equation,
as one may expect, takes the following form
∇
2
Φ =
γ c
4
2
ρ
eff
+ 3
p
eff
c
2
. (30)
This is similar to the corresponding equation in GR, where
we take into account pressure as a source for gravity (see [58]
for more details).
Now before moving on to discuss the Euler equation, let us
summarize the weak field limit and write the modified Pois-
son’s equation for two different categories, namely EMSG
models with f
RR
= 0 and f
RR
= 0. For the first case, using
Eqs. (16) and (24)–(26), we arrive at
∇
2
Φ =
γ c
4
2 f
R
ρ + 2αρ
2
c
2
, (31)
and similarly for the second case, using Eqs. (16) and (23)–
(26), we find a more complicated Poisson’s equation
∇
2
Φ =
γ c
4
6 f
R
4ρ + 5αρ
2
c
2
−
M
2
4π
e
−M|r−r
|
|r − r
|
ρ(r
) − αρ
2
(r
)c
2
d
3
r
.
(32)
4 Hydrodynamics equations in weak-field limit
To find the Newtonian limit of the hydrodynamics equations,
one can take the covariant derivative of the field Eq. (2)as
below
∇
μ
f
R
R
μν
+ f
R
∇
μ
R
μν
−
1
2
g
μν
∇
μ
f
= γ ∇
μ
T
μν
+
(
∇
ν
−∇
ν
)
f
R
−∇
μ
f
Q
θ
μν
. (33)
To simplify the third term in the above equation, we recall
that f (R, Q) = f
1
(R) + f
2
(Q). Therefore one can easily
verify that
g
μν
∇
μ
f = g
μν
f
R
∇
μ
R + f
Q
∇
μ
Q
. (34)
Also, the fifth term can be simplified as below [59]
(
∇
ν
−∇
ν
)
f
R
= R
μν
∇
μ
f
R
. (35)
Using the Bianchi identity, and after some manipulations,
one can find the perturbed form of Eq. (33)as
∇
μ
δT
μν
= α
∇
μ
δθ
μν
−
1
2
η
μν
∇
μ
(
δ Q
)
, (36)
where δ Q = δT
μν
δT
μν
. Note that, the background quantities
are shown without the “0” index here. To achieve the hydro-
dynamics equations in the Newtonian limit, one can ignore
the terms containing the pressure compared with the simi-
lar terms containing the density. In fact, the pressure plays
role in the relativistic situations, which are not, of course, of
interest in this study.
Let us look at the order of magnitudes. What we have
assumed is: firstly, as mentioned before, the pressure can
be ignored comparing with the density in our background
system. Secondly, the gravitational field assumed to be weak.
And finally, the velocities inside the background are slow.
Using a small parameter , these assumptions can read
p
ρc
2
v
2
c
2
Φ
c
2
∝
2
(37)
On the other hand, considering the Newtonian form of the
Euler’s equation, one can see that ∂v/∂t (v ·∇)v ∇Φ
and, therefore
123
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