Post-Newtonian Spherically Symmetrical Accretion

Gilberto M. Kremer1, ∗ and Leandro C. Mehret1, †

1Departamento de Fısica, Universidade Federal do Parana, Curitiba 81531-980, Brazil

The objetive of this work is to investigate the influence of the corrections to the spherical symmet-rical accretion of an infinity gas cloud characterized by a polytropic equation into a massive objectdue to the post-Newtonian approximation. Starting with the steady state post-Newtonian hydrody-namic equations for the mass, mass-energy and momentum densities, the post-Newtonian Bernoulliequation is derived. The post-Newtonian corrections to the critical values of the flow velocity, soundvelocity and radial distance are obtained from the system of hydrodynamics equations in sphericalcoordinates. It was considered that the ratio of the sound velocity far the massive body and thespeed of light was of order a∞/c = 10−2. The analysis of the solution led to following results: theMach number for the Newtonian and post-Newtonian accretion have practically the same values forradial distances of order of the critical radial distance; by decreasing the radial distance the Machnumber for the Newtonian accretion is bigger than the one for the post-Newtonian accretion; thedifference between the Newtonian and post-Newtonian Mach numbers when the ratio a∞/c� 10−2

is insignificant; the effect of the correction terms in post-Newtonian Bernoulli equation is moreperceptive for the lowest values of the radial distance; the solutions for a∞/c > 10−2 does not leadto a continuous inflow and outflow velocity at the critical point; the comparison of the solutionswith those that follow from a relativistic Bernoulli equation shows that the dependence of the Machnumber with the radial distance of the former is bigger than the Newtonian and post-NewtonianMach numbers.

I. INTRODUCTION

An important area of research in astrophysics is related to the steady state problem of spherically symmetricalaccretion of a perfect gas into a massive object. The pioneers works in this subject were published by Hoyle andLyttleton [1, 2], Bondi [3, 4] and Michel [5]. Nowadays this problem is still a subject of several investigations wherea relativistic fluid accretes into a massive body described by other metrics: Schwarzschild – de Sitter, Reissner-Nordstrom, Reissner-Nordstrom – de Sitter (see e.g. [6–20] and the references therein).

The aim of this paper is to investigate the influence of the corrections to the Newtonian accretion due to the post-Newtonian approximation. We start from the post-Newtonian hydrodynamic equations for the mass, mass-energyand momentum densities and derive the post-Newtonian Bernoulli equation. The post-Newtonian corrections to thecritical values of the flow velocity, sound velocity and radial distance are obtained from the system of hydrodynamicsequations. It is shown that due to the post-Newtonian corrections the critical point does not correspond to thetransonic point as in the Newtonian accretion. The solution for the Mach number as function of a dimensionlessradial distance depends on the ratio of the sound velocity far the massive body and the speed of light a∞/c. Thisratio was fixed to be a∞/c = 10−2 which is of relativistic order. For values of this ratio greater than 10−2 thereis no continuity in the inflow and outflow velocity at the critical point, while for values smaller than 10−2 there isno difference between the Newtonian and post-Newtonian solutions. For a∞/c = 10−2 the Mach number for theNewtonian and post-Newtonian accretion have practically the same values for radial distances of order of the criticalradial distance, but by decreasing the radial distance the Mach number for the Newtonian accretion is bigger than theone for the post-Newtonian accretion. The weak field limit of the relativistic Bernoulli equation [5] is also developedand a comparison of the solutions are investigated.

The paper is organized as follows: in Section II the post-Newtonian hydrodynamic equations are introduced, whilein Section III the post-Newtonian mass density accretion rate, the Bernoulli equation and the critical values for theflow velocity, sound speed and radial distance are obtained. In section III B the post-Newtonian equation for theMach number as function of a dimensionless radial distance is deduced. In Section IV the relativistic sphericallysymmetrical accretion is developed. The analysis of the Newtonian and post-Newtonian solutions is developed inSection V. We close the paper with a summary of the results in Section VI.

∗ kremer@fisica.ufpr.br† lcmehret@gmail.com

arX

iv:2

010.

0404

0v2

[gr

-qc]

24

Jun

2021

2

II. POST-NEWTONIAN HYDRODYNAMIC EQUATIONS

The post-Newtonian approximation is a method of successive approximations in 1/c2 powers of the light speed forthe solution of Einstein’s field equations. It was proposed in 1938 by Einstein, Infeld and Hoffmann [21] and the corre-sponding hydrodynamic equations in the first post-Newtonian approximation (1PN) were obtained by Chandrasekhar[22, 23].

In the post-Newtonian approximation Einstein’s field equations are solved for an Eulerian fluid characterized bythe energy-momentum tensor

Tµν = (ε+ p)UµUν

c2− pgµν . (1)

Here Uµ denotes the four-velocity (with UµUµ = c2), gµν the metric tensor while ε and p the energy density andpressure of the fluid, respectively. The energy density has two parts ε = ρc2(1 + ε/c2) one associated with the massdensity ρ = mn and another to the internal energy density ε. The internal energy density for a non-relativistic perfectfluid is given by ε = p/(γ − 1)ρ, where γ = cp/cv is the ratio of the specific heats at constant pressure and constantvolume. For a fluid of monatomic molecules cv = 3k/2m with k denoting Boltzmann constant and m the rest massof a fluid molecule.

The solution of Einstein’s field equations leads to the following components of the metric tensor

g00 = 1 +2φ

c2+

2

c4(φ2 + ψ

)+O(c−6), (2)

g0i = − 1

c3ζi +O(c−5), (3)

gij = −(

1− 2φ

c2

)δij +O(c−4), (4)

while the corresponding components of the four-velocity in 1PN are

U0 = c

[1 +

1

c2

(V 2

2− φ

)+

1

c4

(3V 4

8− 5φV 2

2+φ2

2− ψ + ζiVi

)], (5)

with U i = U0Vi/c. Above φ is the Newtonian gravitational potential which satisfies the Poisson equation∇2φ = 4πGρ.The corresponding Poisson equations for the scalar ψ and the vector ζi gravitational potentials, read

∇2ζi = 16πGρVi, (6)

∇2ψ = 8πGρ

(V 2 − φ+

ε

2+

3p

2ρ

)+∂2φ

∂t2, (7)

The gravitational potentials φ, ζi and ψ are those introduced by Weinberg [24] and their connection with the gravi-tational potentials U , Ui and Φ of Chandrasekhar [22] are

φ = −U, ζi = −4Ui +1

2

∂2χ

∂t∂xi, ψ = −2Φ, (8)

where χ is a super-potential which obeys the equation ∇2χ = −2U .The hydrodynamic equation for the mass density is obtained from the particle four-flow balance law Nµ

;µ = 0together with the representation Nµ = nUµ where n denotes the particle number density, yielding

∂ρ∗∂t

+∂ρ∗Vi∂xi

= 0, where ρ∗ = ρ

[1 +

1

c2

(V 2

2− 3φ

)]. (9)

This equation is the 1PN approximation of the continuity equation and corresponds to eq. (117) of Chandrasekhar[22].

The mass-energy density hydrodynamic equation in the 1PN approximation follows from the time component ofthe energy-momentum tensor balance law T 0ν

;ν = 0, resulting

∂σ

∂t+∂σVi∂xi

− 1

c2

(ρ∂φ

∂t+∂p

∂t

)= 0, where σ = ρ

[1 +

1

c2

(V 2 − 2φ+ ε+

p

ρ

)]. (10)

The above equation corresponds to eq. (9.8.14) of Weinberg [24] and eq. (64) of Chandrasekhar [22]. Note that wehave to identify ε with ρ in Weinberg’s book [24] and take c = 1.

3

From the spatial components of the energy-momentum tensor balance law T iν ;ν = 0 follows the 1PN approximationfor the momentum density hydrodynamic equation, namely

∂σVi∂t

+∂σViVj∂xj

+∂

∂xi

[p

(1− 2φ

c2

)]+ ρ

∂φ

∂xi

[1 +

2

c2

(V 2 − φ+

ε

2+

3

2

p

ρ

)]−4ρ

c2

(∂φVi∂t

+ Vj∂φVi∂xj

)+ρ

c2∂ψ

∂xi+ρ

c2

(∂ζi∂t

+ Vj∂ζi∂xj− Vj

∂ζj∂xi

)= 0, (11)

which matches eq. (9.8.15) of Weinberg [24] and eq. (68) of Chandrasekhar [22].In the Newtonian limiting case the hydrodynamic equations for the mass density (9) and the mass-energy density

(10) coincide and become the continuity equation

∂ρ

∂t+∂ρVi∂xi

= 0, (12)

while eq. (11) reduces to the Eulerian momentum density hydrodynamic equation

∂ρVi∂t

+∂ρViVj∂xj

+∂p

∂xi+ ρ

∂φ

∂xi= 0. (13)

III. POST-NEWTONIAN ACCRETION

A. Post-Newtonian Bernoulli Equation

In the analysis of the spherically symmetrical accretion a massive object of mass M is surrounded by an infinitegas cloud and is moving with a velocity V relative to it. The gas cloud at large distances from the star is at rest withuniform density and pressure denoted by ρ∞ and p∞, respectively. The gas motion is steady spherically symmetricaland it is not taken into account the increase in the massive object. The gas is characterized by a polytropic equationof state and by a sound velocity a given by

p = κργ , a =

√dp

dρ=

√γp

ρ, (14)

where κ is a constant and γ is related to the polytropic index n by γ = (n+ 1)/n.For steady states the hydrodynamic equations for mass density (9), mass-energy density (10) and momentum density

(11) become

∂ρ∗Vi∂xi

= 0, (15)

∂σVi∂xi

= 0, (16)

σVj∂Vi∂xj

+ ρ∂φ

∂xi

[1 +

2

c2

(V 2 − φ+

3γ − 2

2(γ − 1)

p

ρ

)]+

∂

∂xi

[p

(1− 2φ

c2

)]−4ρ

c2Vj∂φVi∂xj

+ρ

c2∂ψ

∂xi+ρ

c2Vj

(∂ζi∂xj− ∂ζj∂xi

)= 0. (17)

In the steady state momentum density hydrodynamic equation (17) we have used the corresponding mass-energydensity hydrodynamic equation (16) and the relationship ε = p/(γ − 1)ρ.

In spherical coordinates the fields depend only on the radial coordinate r and due to the fact that we are dealingwith a spherically symmetrical flow, the components of the hydrodynamic velocity are Vi = (V (r), 0, 0). Hence,equations and (15) – (17) become

d{r2ρ

[1 + 1

c2

(V 2

2 − 3φ)]V}

dr= 0, (18)

d{r2ρ

[1 + 1

c2

(V 2 − 2φ+ γ

γ−1pρ

)]V}

dr= 0, (19)

ρ

[1 +

1

c2

(V 2 − 6φ+

γ

γ − 1

p

ρ

)]VdV

dr+

d

dr

[p

(1− 2φ

c2

)]+ρ

dφ

dr

[1− 2

c2

(V 2 + φ− 3γ − 2

2(γ − 1)

p

ρ

)]+ρ

c2dψ

dr= 0. (20)

4

For the analysis of the flow velocity it is more convenient to introduce the proper velocity of the flow vr which ismeasured by a local stationary observer (see e.g [25, 26]). The proper velocity is defined by

vr =Ur

U0/c=

Ur

(U0/c) (1 + 2φ/c2). (21)

Since Vi = U i/(U0/c), we have that

vr =V

(1 + 2φ/c2), =⇒ V = vr

(1 + 2φ/c2

). (22)

By taking into account the relationship (22) the system of differential equations (18) – (20) can be rewritten interms of the proper velocity vr as

d{r2ρ

[1 + 1

c2

(v2r2 − φ

)]vr

}dr

= 0, (23)

d{r2ρ

[1 + 1

c2

(v2r + a2

γ−1

)]vr

}dr

= 0, (24)[1 +

1

c2

(v2r − 2φ+

a2

γ − 1

)]vrdvrdr

+dρ

dr

a2

ρ

(1− 2φ

c2

)+dφ

dr

[1− 1

c2

(2φ− a2

γ − 1

)]+

1

c2dψ

dr= 0. (25)

Above we have used to the expression for the sound speed (14).The integration of (23) and (24) imply the mass density and mass-energy accretion rates

Mρ∗ = 4πρr2[1 +

1

c2

(v2r2− φ

)]vr, (26)

Mσ = 4πρr2[1 +

1

c2

(v2r +

a2

γ − 1

)]vr. (27)

From these equations we obtain a relationship between both accretion rates

Mσ = Mρ∗

[1 +

1

c2

(v2r2

+ φ+a2

γ − 1

)]. (28)

Here we can use the Newtonian Bernoulli equation

v2r2

+a2

γ − 1+ φ =

a2∞γ − 1

, (29)

for the underlined term, since it is of order 1/c2 in (28, resulting

Mσ = Mρ∗

[1 +

1

γ − 1

a2∞c2

]. (30)

Hence, the mass density and mass-energy accretion rates differ from each other by a 1/c2 term, i.e., in the Newtonian

limiting case both coincide, i.e., Mσ = Mρ∗ = 4πρr2V .The multiplication of the momentum density (25) by

1

ρ

[1 +

1

c2

(2φ− a2

γ − 1

)]leads to the following differential equation

vrdvrdr

[1 +

v2rc2

]+dρ

dr

a2

ρ

[1− a2

c2(γ − 1)

]+dφ

dr+

1

c2dψ

dr= 0. (31)

The post-Newtonian Bernoulli equation follows from the integration of (31), yielding

v2r2

[1 +

v2r2c2

]+

a2

γ − 1

[1− a2

2c2(γ − 1)

]+ φ+

ψ

c2=

a2∞γ − 1

[1− a2∞

2c2(γ − 1)

]. (32)

5

Here we have assumed that the proper velocity vr and the gravitational potentials φ, ψ vanish far from the massiveobject. Note that (32) reduces to the Newtonian Bernoulli equation (29) by neglecting the terms in 1/c2.

We can rewrite the hydrodynamic equations in spherical coordinates (23), (24) and (31) as

2

r+ρ′

ρ+v′rvr

+1

c2(vrv

′r − φ′) = 0, (33)

2

r+ρ′

ρ+v′rvr

+1

c2

(2vrv

′r + a2

ρ′

ρ

)= 0, (34)(

1 +v2rc2

)vrv′r +

[1− a2

c2(γ − 1)

]a2ρ′

ρ+ φ′ +

ψ′

c2= 0, (35)

where the prime denotes the differentiation with respect to r.The system of differential equations (33) – (35) can be solved as a system of algebraic equations for V ′, ρ′ and ψ′,

yielding

v′rvr

=2

r

a2(

1− rφ′

2c2

)− rφ′

2

v2r(1− a2

c2

)− a2

,ρ′

ρ= −2

r

v2r

(1− rφ′

c2

)− rφ′

2

v2r(1− a2

c2

)− a2

, (36)

ψ′ =a4[rφ′ − 2v2r

(1− rφ′

c2

)]− 2a2(γ − 1)v4r

(1− rφ′

2c2

)+ rφ′v4r(γ − 1)

r(γ − 1)[v2r(1− a2

c2

)− a2

] . (37)

We infer from (36) that the solution must pass through a critical point defined by a critical radius rc, a criticalproper velocity Vc and a critical sound velocity ac when the nominator and denominator of these equations vanish.The existence of a critical point prevent singularities in the flow solution and guarantees a smooth monotonic increaseof the flow velocity when r decreases. At the critical point we have

a2c =rcφ′c

2(

1− rcφ′c2c2

) ≈ rcφ′c

2

(1 +

rcφ′c

2c2

), (38)

V 2c =

a2c(1− a2c

c2

) ≈ a2c (1 +a2cc2

), (39)

V 2c =

rcφ′c

2(

1− rcφ′cc2

) ≈ rcφ′c

2

(1 +

rcφ′c

c2

). (40)

The above approximations are valid since we are working with a first post-Newtonian theory.The value of ψ′c at the critical point is obtained from the substitution of (40) into (37) yielding

ψ′cc2

=φ2c

2c2rc=G2M2

2c2r3c, (41)

by taking into account the expression for the Newtonian gravitational potential

φ = −GMr

= −rφ′. (42)

Another way to determine ψ′ is to observe that this potential is of order 1/c2 and we can approximate (37) by

ψ′

c2=−a4

[φ+ 2v2r

]− 2a2(γ − 1)v4r − φv4r(γ − 1)

c2r(γ − 1) [v2r − a2], (43)

by neglecting the terms proportional to 1/c2 and considering the relationship for the Newtonian potential φ′r = −φ.Now by taking into account the virial theorem 2K + W = 0, where K and W represent the kinetic and potentialenergies, we can assume that 2v2r + φ = 0 and (43) reduces to

ψ′

c2=

φ2

2c2r. (44)

The gravitational potential ψ is obtained from the integration of (44) by using the Newtonian gravitational potential(42) resulting

ψ

c2= −G

2M2

4c2r2= − φ2

4c2, so that

ψcc2

= − φ2c4c2

. (45)

6

Above it was considered that ψ vanishes at r →∞. Note that (45)2 is the integral of (41) with respect to rc.For the determination of the critical values we make use of Bernoulli equation (32) and the expressions of the sound

speed (38), proper velocity (40) and gravitational potential (45)2. The resulting equation is an algebraic equation forthe determination of φc at the critical point. Its value up to order 1/c2 is

φc = − 4a2∞(5− 3γ)

[1 +

15− 11γ

8(γ − 1)(5− 3γ)

a2∞c2

], (46)

which implies the expression for the critical radius

rc =(5− 3γ)

4

GM

a2∞

[1− 15− 11γ

8(γ − 1)(5− 3γ)

a2∞c2

], (47)

thanks to the relationship φc = −GM/rc.The critical values of the sound speed ac and proper velocity Vc are obtained from (38) and (40) by using (46) for

the elimination of φ′c = −φc/rc, resulting

a2c =2a2∞

(5− 3γ)

[1− 1− 5γ

8(γ − 1)(5− 3γ)

a2∞c2

], (48)

V 2c =

2a2∞(5− 3γ)

[1− 17− 21γ

8(γ − 1)(5− 3γ)

a2∞c2

]. (49)

Furthermore, by using the expression for the sound speed a2 = γp/ρ, the polytropic equation of state p = κργ andthe critical value for the sound speed (48) it follows the critical value of the mass density

ρcρ∞

=

(aca∞

) 2γ−1

=

(2

5− 3γ

) 1γ−1

[1− 1− 5γ

8(γ − 1)2(5− 3γ)

a2∞c2

]. (50)

The mass-density accretion rate (26) can be written as

Mρ∗ = 4πρcr2c

[1 +

1

c2

(V 2c

2− φc

)]Vc = 4πλc

(GM

a2∞

)2

ρ∞a∞, (51)

where the so-called critical accretion eigenvalue [25] is given by

λc = (5− 3γ)3γ−52γ−2 2

9−7γ2γ−2

[1 +

121− 216γ + 103γ2

16(5− 3γ)(γ − 1)2a2∞c2

]. (52)

In the Newtonian limiting case (46) – (52) reduce to the well-know values found in the literature (see e.g. [25]),namely

φc = − 4a2∞(5− 3γ)

, rc =(5− 3γ)

4

GM

a2∞, (53)

a2c = V 2c =

2a2∞(5− 3γ)

,ρcρ∞

=

(2

5− 3γ

) 1γ−1

, (54)

λc = (5− 3γ)3γ−52γ−2 2

9−7γ2γ−2 . (55)

B. Mach number as Function of the Radial Distance

For the determination of the dependence of the flow velocity as function of the radial distance Bondi [4] introducedthe following dimensionless quantities

r∗ =ra2∞GM

, s∗ =V

a∞, t∗ =

ρ

ρ∞, (56)

which are related to the radial distance, flow velocity and mass density, respectively. Another dimensionless quantitywhich is useful in this analysis is the ratio of the flow velocity and the speed of sound u∗ = V/a, which is the Machnumber.

7

From the Newtonian mass density accretion rate we have in these dimensionless quantities

Mρ = 4πρr2V = 4πλ

(GM

a2∞

)2

ρ∞a∞, (57)

where λ = r2∗s∗t∗ is a constant.For the post-Newtonian approximation the Bondi dimensionless quantities (56) are written as

r∗ =ra2∞GM

(1 +

GM

2rc2

), s∗ =

vra∞

(1 +

v2r2c2

), t∗ =

ρ

ρ∞. (58)

Solving (58) for r and vr and considering terms up to 1/c2 we obtain

r =GM

a2∞r∗

(1− β2

2r∗

), vr = a∞s∗

(1− β2

2s2∗

), (59)

where β = a∞/c denotes a relativistic parameter which is the ratio of the value of the sound speed far from themassive object and the light speed.

With respect to the new variables (58), the mass density accretion rate (26) becomes

Mρ∗ = 4πλ

(GM

a2∞

)2

ρ∞a∞, (60)

by taking into account the Newtonian potential φ = −GM/r. Here we have also that λ = r2∗s∗t∗.The dependence of the proper velocity as a function of the radial velocity is obtained from the Bernoulli equation

(32) written in terms of the dimensionless quantities (r∗, u∗). We begin by writing the dimensionless parameters s∗and t∗ as functions of (r∗, u∗)

s∗ = u2γ+1∗

(λ

r2∗

) γ−1γ+1

1 +β2

γ + 1u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

, (61)

t∗ =

(λ

u∗r2∗

) 2γ+1

1− β2

γ + 1u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

, (62)

where we have taken into account (58)2, u∗ = vr/a, λ = r2∗s∗t∗ and t∗ = ρ/ρ∞ = (a/a∞)2

γ−1 .

Next we rewrite vr and a in terms of (r∗, u∗) from (59)2 and a/a∞ = tγ−12∗ , yielding

vr = a∞u2γ+1∗

(λ

r2∗

) γ−1γ+1

1− β2(γ − 1)

2(γ + 1)u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

, (63)

a = a∞

(λ

u∗r2∗

) γ−1γ+1

1− β2(γ − 1)

2(γ + 1)u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

. (64)

The last step is to rewrite the gravitational potential φ and ψ as functions of r∗

φ = −GMr

= −a2∞r∗

(1 +

β2

2r∗

),

ψ

c2= − φ2

4c2= −a

2∞

4r2∗β2. (65)

Note that in (61) – (65) we have considered only terms up to the order 1/c2.The final equation which gives the dependence of the Mach number u∗ with the dimensionless radial distance r∗ is

obtained from Bernoulli equation (32) together with (63) – (65) resulting

u4γ+1∗

2

(λ

r2∗

) 2(γ−1)γ+1

[1 +

β2

2(γ + 1)u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

]− 1

r∗

(1 +

3β2

4r∗

)

+1

γ − 1

(λ

u∗r2∗

) 2(γ−1)γ+1

1−(

1 +(γ + 1)

2(γ − 1)2u2∗

)β2(γ − 1)

γ + 1u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

=1

γ − 1

(1− β2

2(γ − 1)

). (66)

8

In the Newtonian limiting case we get – by neglecting the β2 terms – eq. (14) of Bondi [4], namely

u4γ+1∗

2

(λ

r2∗

) 2(γ−1)γ+1

+1

γ − 1

(λ

u∗r2∗

) 2(γ−1)γ+1

=1

r∗+

1

γ − 1. (67)

In the next section we shall analyze the relativistic spherically symmetrical accretion. This analysis will be basedon the work of Michel [5] and on the book by Shapiro and Teukolsky [25] .

IV. RELATIVISTIC ACCRETION

A. Relativistic Bernoulli Equation

We begin by writing the line element in spherical coordinates in the Schwarzschild metric (r, θ, ϕ)

ds2 =

(1− 2GM

rc2

)(dx0)2 − 1(

1− 2GMrc2

) (dr)2 − r2

[(dθ)

2+ sin2 θ (dϕ)

2], (68)

where rS = 2GM/rc2 is the Schwarzschild radius which defines the event horizon of a Schwarzschild black hole.The perfect fluid is characterized by particle four-flow Nµ = nUµ and energy-momentum tensor (1) and the balance

equations for the particle four-flow and energy-momentum tensor are given by

Nµ;µ =

1√−g

∂√−gNµ

∂xµ= 0, (69)

Tµν;ν =

1√−g

∂√−g Tµν

∂xν− 1

2T νσ

∂gνσ∂xµ

= 0. (70)

In the analysis of the spherically symmetrical accretion the non-vanishing components of the four-velocity are

(Uµ) =

(U0 =

dx0

dτ, Ur =

dr

dτ, 0, 0

). (71)

From the constraint gµνUµUν = c2 the component U0 is connected with Ur by

U0

c=

√1− 2GM

rc2 +(Ur

c

)21− 2GM

rc2

. (72)

The integration of the balance equation for the particle four-flow (69) and the time component of the energy-momentum tensor (70) lead to

√−gnUr = constant,

√−g (p+ ε)

Ur

c

U0

c= constant. (73)

Combining the above equations the following relationship holds(p+ ε

ρ

)2[

1− 2GM

rc2+

(Ur

c

)2]

= constant. (74)

Let us introduce the sound speed a, which for a relativistic fluid is defined by

a2

c2=

ρ

p+ ε

(∂p

∂ρ

). (75)

We recall that the polytropic equation of state and the energy density equation are given by

p = κργ , ε = ρc2 +κργ

γ − 1, (76)

so that we can write from the above equations that

p+ ε

ρ=κγργ−1

γ − 1+ c2 =

κγργ−1

a2/c2, (77)

9

which implies the following relationships

κγργ−1 =(γ − 1)a2

γ − 1− a2/c2,

p+ ε

ρ=

c2

1− a2/(γ − 1)c2. (78)

The relativistic Bernoulli equation follows from (74) and (78), yielding(1− a2

c2(γ − 1)

)2

=

(1− a2∞

c2(γ − 1)

)2[

1− 2GM

rc2+

(Ur

c

)2]. (79)

Here it was supposed that far from the massive body 2GM/rc2 and Ur vanish while the sound speed becomes a∞.The determination of the critical points are obtained from the differentiation of (73) and elimination of dρ, yielding

dUr

Ur

[a2

c2− (Ur/c)

2

1− 2GM/rc2 + (Ur/c)2

]

+dr

r

[2a2

c2− GM/rc2

1− 2GM/rc2 + (Ur/c)2

]= 0. (80)

The expressions for the critical gas flow velocity and sound speed are determined when both expressions in theparenthesis in (80) vanish resulting

(Urc )2 =GM

2rc, a2c =

(Urc )2

1− 3(Urc /c)2, (Urc )2 =

a2c1 + 3(ac/c)2

. (81)

The above equations correspond to the equations (8) – (14) of the work of Michel [5].From now one we shall restrict the analysis to the weak field limit of the relativistic case, since we are interested

in comparing it with the post-Newtonian approximation developed in the previous section. We begin by writing theBernoulli equation (79) at the critical point thanks to (81) as(

1 + 3a2cc2

)(1− a2c

c2(γ − 1)

)2

=

(1− a2∞

c2(γ − 1)

)2

, (82)

which is a third order algebraic equation for the determination of the critical sound speed a2c . This equation wassolved in [27] but here we are interested in its weak field approximation which reads

a2c =2a2∞

5− 3γ

[1− 3(3γ + 1)

2(5− 3γ)(γ − 1)

a2∞c2

]. (83)

From the knowledge of the critical sound speed the critical values for the flow velocity, mass density and radialdistance read

(Urc )2 =2a2∞

5− 3γ

[1 +

3(7− 11γ)

4(5− 3γ)(γ − 1)

a2∞c2

], (84)

ρcρ∞

=

(2

5− 3γ

) 1γ−1

[1− 3(3γ + 1)

2(5− 3γ)(γ − 1)2a2∞c2

], (85)

rc =(5− 3γ)

4

GM

a2∞

[1− 3(7− 11γ)

4(γ − 1)(5− 3γ)

a2∞c2

]. (86)

Furthermore, from the mass accretion rate M = 4πr2cρcVc it follows the critical accretion eigenvalue

λc = (5− 3γ)3γ−52γ−2 2

9−7γ2γ−2

[1 +

3(17− 66γ + 33γ2)

8(γ − 1)2(5− 3γ)

a2∞c2

]. (87)

Note that the above expressions differ from those obtained in the post-Newtonian approximation.From the relativistic Bernoulli equation one may obtain its weak field limit by considering terms up to the 1/c2

order in (79), yielding

(Ur)2

2

[1−

(Ur

c

)2

− 4φ

c2

]+

a2

(γ − 1)

[1− a2

2c2(γ − 1)− 2φ

c2−(Ur

c

)2]

+φ

(1− 2φ

c2

)=

a2∞(γ − 1)

(1− a2∞

2c2(γ − 1)

), (88)

10

where we have introduced the Newtonian potential φ = −GM/r. Without the 1/c2 – terms (88) reduces to thenon-relativistic Bernoulli equation, however this expression differs from the post-Newtonian Bernoulli equation (32).

Let express the weak filed approximation of the Bernoulli equation in terms of the proper velocity of the flow vrdefined by (21). The relationship between the components Ur and U0 follows from UµU

µ = c2, yielding

(U0

c

)2

=1 +

(1− 2φ

c2

) (Ur

c

)21 + 2φ

c2

, (89)

and the proper velocity (21) becomes

vr =Ur√

1 + 2 φc2

√1 +

(1− 2φ

c2

) (Ur

c

)2 . (90)

By retaining terms up to 1/c2 the expression of the radial four-velocity component in terms of the proper velocityreads

Ur = vr

[1 +

φ

c2+

v2r2c2

]. (91)

If we insert (91) into (88) and consider terms up to 1/c2 order we find the weak field Bernoulli equation given interms of the proper velocity, namely

v2r2

[1− 2φ

c2

]+

a2

(γ − 1)

[1− a2

2c2(γ − 1)−2φ

c2− v2rc2

]+ φ

(1− 2φ

c2

)− a2∞

(γ − 1)

(1− a2∞

2c2(γ − 1)

)=v2r2

[1− 2φ

c2

]+

a2

(γ − 1)

[1 +

3a2 − 4a2∞2c2(γ − 1)

]+ φ

(1− 2φ

c2

)− a2∞

(γ − 1)

(1− a2∞

2c2(γ − 1)

)= 0. (92)

For the underlined term above we have used the Newtonian Bernoulli equation (29), since it is of 1/c2 order.

B. Mach Number as Function of the Radial Distance

The mass density accretion rate for the weak field is obtained from (73) which in terms of the proper velocity reads

M = 4πρr2Ur = 4πρr2vr

[1− GM

rc2+

v2r2c2

]. (93)

Following the same methodology of the previous section we introduce the dimensionless quantities

r∗ =ra2∞GM

(1− GM

2rc2

), s∗ =

vra∞

(1 +

v2r2c2

), t∗ =

ρ

ρ∞, (94)

so that the mass density accretion rate becomes

M = 4πλ

(GM

a2∞

)2

ρ∞a∞, where λ = r2∗s∗t∗. (95)

From (94) we can write

r =GMr∗a2∞

(1 +

β2

2r∗

), vr = a∞s∗

(1− β2s∗

2

). (96)

Due to the fact that the expression for s∗ above is the same as the one in the post-Newtonian approximation (58)we can use the (63) and (64) for the proper velocity and sound speed as a function of the Mach number u∗ anddimensionless radial distance r∗, respectively. For the gravitational potential we have

φ = −a2∞r∗

(1− β2

2r∗

). (97)

11

The expressions of the dependence of the Mach number u∗ as function of the dimensionless radial distance r∗ isobtained from the weak field Bernoulli equation (92) together with (63) , (64) and (97), resulting

u4γ+1∗

2

(λ

r2∗

) 2(γ−1)γ+1

[1− β2(γ − 1)

γ + 1u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

+2β2

r∗

]− 1

r∗

(1 +

3β2

2r∗

)+

1

γ − 1

(λ

u∗ r2∗

) 2(γ−1)γ+1

[1− 2β2

γ − 1

−(

1− 3(γ + 1)

2(γ − 1)2u2∗

)β2(γ − 1)

(γ + 1)u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

]=

1

γ − 1

(1− β2

2(γ − 1)

). (98)

For the relativistic Bernoulli equation (79) we can use the Bondi dimensionless quantities (56) and write it as1− β2

γ − 1

(λ

U∗r2∗

) 2(γ−1)γ+1

2

=

(1− β2

γ − 1

)21− 2β2

r∗+ β2U2

∗

(λ

U∗r2∗

) 2(γ−1)γ+1

. (99)

Here U∗ = Ur/a is the Mach number with respect to the radial component of the four-velocity. The Mach numberwritten in terms of the proper velocity follows from (90) and reads

u∗ =vra

=U∗√

1− 2β2

r

√1 +

(1 + 2β

2

r

)β2U2

∗

(λ

U∗r2∗

) 2(γ−1)γ+1

. (100)

V. ANALYSIS OF THE SOLUTIONS

In this section we shall compare the solutions for the Mach number u∗ = vr/a as function of the dimensionlessradial distance r∗ which follow from the different approximations of the Bernoulli equation.

In the tables and figures below the Newtonian solution of (67) is denoted by (N), the post-Newtonian solution of(66) by (PN) and the weak field approximation solution of (98) by (WF). For the relativistic accretion – denoted by(R) – the Bernoulli equation (99) was solved for the Mach number with respect to the radial four-velocity and from(100) the Mach number for the proper velocity was obtained.

In the determination of the Mach number u∗ as a function of the dimensionless radial distance r∗ it was consideredthat the ratio of the sound velocity far from the massive body and the light speed is equal to β = a∞/c = 10−2, whichis of relativistic order.

In Table I the values for the Mach number u∗ as function of the dimensionless radial distance r∗ are given in therange 5× 10−4 ≤ r∗ ≤ 2.5× 10−2 for a ultra-relativistic Fermi gas where γ = 4/3. In the Newtonian approximationthe critical radius is r∗ = 0.25 where the critical Mach number assumes the value u∗ = 1. We infer from this table thatby decreasing the dimensionless radial distances r∗ from the massive body the Mach number increases. Furthermore,the values of the Mach number for the relativistic case are bigger than the Newtonian ones. The Mach number valuesfor the post-Newtonian and weak field approximations are practically the same and are smaller than those for theNewtonian case. The difference between the Newtonian, post-Newtonian and weak field solutions becomes very smallby increasing the dimensionless radial distance and the solutions practically coincide at r∗ = 10−3. In Figure 1 it isplotted the contour plot for the Bernoulli equations: Newtonian (67), post-Newtonian (66) and weak field (98). TheNewtonian solution is represented by a dashed line and the post-Newtonian and weak field approximations by the samestraight line, since they practically coincide. It is shown that the difference between the Newtonian, post-Newtonianand weak field are very small and coincide by increasing the dimensionless radial distance.

r∗ u∗ (N) u∗ (PN) u∗ (WF) u∗ (R)5 × 10−4 10.29 9.66 9.66 20.26

10−3 8.53 8.24 8.26 11.265 × 10−3 5.40 5.37 5.37 6.07

10−2 4.35 4.34 4.34 4.975 × 10−2 2.40 2.41 2.41 3.31

2.5 × 10−2 1.00 1.07 1.02 2.09

TABLE I. Mach number u∗ = vr/a as function of the dimensionless radial distance r∗ for a ultra-relativistic Fermi gas γ = 4/3.

The Mach number u∗ as function of the dimensionless radial distance r∗ for a diatomic gas where γ = 7/5 isdisplayed in Table II in the range 5×10−4 ≤ r∗ ≤ 2×10−2. The same conclusions as in the former case can be drawn,

12

γ = 43

0.001 0.002 0.003 0.004 0.005 0.006

5.0

5.5

6.0

6.5

7.0

7.5

8.0

r*

u*

N

PN=WF

FIG. 1. Contour plot showing the Mach number u∗ as function of the dimensionless radial distance r∗ for a ultra-relativisticFermi gas γ = 4/3. The Newtonian solution is represented by a dashed line while the post-Newtonian and weak field solutionsby a straight line.

i.e., in comparison with the Newtonian solutions the dependence of Mach number with respect to the dimensionlessradial distance for the relativistic case is bigger, the post-Newtonian and the weak field solutions are smaller and bothhave practically the same values. For the Newtonian case the critical radius is r∗ = 0.2 where the Mach number attainsthe value u∗ = 1. In Figure 2 the contour plots of the Newtonian (dotted line), the post-Newtonian (straight line) andthe weak field (dotted line) solutions are displayed showing that the values of the Mach number for the Newtoniansolution is bigger that those for the post-Newtonian and weak field solutions and that the difference between thembecomes very small by increasing the dimensionless radial distance.

r∗ u∗ (N) u∗ (PN) u∗ (WF) u∗ (R)5 × 10−4 7.21 6.83 6.81 14.22

10−3 6.16 5.99 5.98 8.225 × 10−3 4.16 4.14 4.13 4.77

10−2 3.43 3.43 3.43 4.025 × 10−2 2.00 2.00 2.00 2.862 × 10−2 1.00 1.02 1.06 2.21

TABLE II. Mach number u∗ = V/a as function of the dimensionless radial distance r∗ for a diatomic gas γ = 7/5.

For a non-relativistic Fermi gas or a monatomic gas γ = 5/3 and in this case the contour plots of the Newtonian andweak field solutions are shown in Figure 3. The critical dimensionless radial distance for the Newtonian case is r∗ = 0where the the Mach number becomes equal to u∗ = 1. We note that r∗ = 0 is a turning point for the Newtoniansolution where a transition occurs from an accretion flow to a wind flow. The weak field solution is smaller than theNewtonian ones and the turning point is about r∗ ≈ 4× 10−3.

Here it is important to comment the behaviors of the post-Newtonian and weak field solutions found in the aboveanalysis when compared with the Newtonian and relativistic solutions. It was expected that the post-Newtonianand weak field solutions should be more close to the relativistic one and not smaller than the Newtonian solution.By inspecting the Newtonian (67), the post-Newtonian (66) and the weak field (98) equations we infer that the twolatter equations have corrections from the Newtonian one and their solutions should furnish different results for thedependence of the Mach number as function of the dimensionless radial distance. But why the values of the Machnumber for the post-Newtonian and weak field are smaller than in the Newtonian case? The only clue is to look atthe expression for the proper velocity (63) for the post-Newtonian and weak field which can be written as

vr = a∞u2γ+1∗

(λ

r2∗

) γ−1γ+1

1− β2(γ − 1)

2(γ + 1)u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

= vNr

1− β2(γ − 1)

2(γ + 1)u

4γ+1∗

(λ

r2∗

) 2(γ−1)γ+1

, (101)

13

γ = 75

0.001 0.002 0.003 0.004 0.005

4.0

4.5

5.0

5.5

6.0

r*

u* N

PN

WF

FIG. 2. Contour plot showing the Mach number u∗ as function of the dimensionless radial distance r∗ for a diatomic gasγ = 7/5. The Newtonian solution is represented by a dashed line, the post-Newtonian by straight line and weak field by adotted line.

γ = 5

3

0.000 0.005 0.010 0.015 0.020

0.6

0.8

1.0

1.2

1.4

r*

u*

N

WF

FIG. 3. Contour plot showing the Mach number u∗ as function of the dimensionless radial distance r∗ for a non-relativisticFermi gas γ = 5/3. The Newtonian solution is represented by a dashed line while the weak field solution by a straight line.

where vNr is the Newtonian expression for the proper velocity. One infers from the above equation that the propervelocities for the post-Newtonian and weak field should be smaller than the one for the Newtonian case, which couldexplain the difference in the behavior of the solutions.

VI. SUMMARY

In this work we have analyzed the influence of the first post-Newtonian approximation in the spherical symmetricalaccretion of an infinity gas cloud characterized by a polytropic equation of state into a massive object. The starting

14

point was the steady state post-Newtonian hydrodynamics equations for mass, mass-energy and momentum densities.The integration of the system of equations in spherical coordinates – where the fields depend only on the radialcoordinate – lead to the determination of the mass accretion rate and the Bernoulli equation in the post-Newtonianapproximation. From the system of differential equations for mass density, flow velocity and post-Newtonian potentialsthe critical point was identified. The critical point prevent singularities in the flow solution and guarantees a smoothmonotonic increase of the flow velocity along the trajectory of the particle so that through the critical point acontinuous inflow and outflow velocity happen. The critical point in the accretion Newtonian theory corresponds tothe transonic point where the flow velocity matches the sound speed. In the post-Newtonian approximation the criticalflow velocity is connected with the sound speed but their expression are not the same. From the post-NewtonianBernoulli equation an equation for the Mach number was obtained as a function of a dimensionless radial coordinate.Similar expressions were derived for the relativistic Bernoulli equation and its weak field approximation based on thework by Michel [5]. For the solution of the post-Newtonian equation it was considered that the ratio of the soundvelocity far the massive body and the speed of light was of order a∞/c = 10−2 which is of relativistic order. The resultsobtained were: (i) the Mach number for the Newtonian, post-Newtonian and weak field accretions have practicallythe same values for radial distances of order of the critical radial distance; (ii) by decreasing the radial distance theMach number for the Newtonian accretion is bigger than the one for the post-Newtonian and weak field accretions;(iii) the effect of the correction terms in post-Newtonian and weak field Bernoulli equations are more perceptive forthe lowest values of the radial distance; (iv) practically there is no difference between the Newtonian, post-Newtonianand weak field Mach numbers when the ratio a∞/c � 10−2; (v) the solutions for a∞/c > 10−2 does not lead to acontinuous inflow and outflow velocities at the critical point; (vi) from the comparison of the solutions with thosethat follow from the relativistic Bernoulli equation shows that the Mach number of the former is bigger than theNewtonian, post-Newtonian and weak field Mach numbers.

ACKNOWLEDGMENTS

G. M. K. has been supported by CNPq (Conselho Nacional de Desenvolvimento Cientıfico e Tecnologico), Braziland L. C. M. by CAPES (Coordenacao de Aperfeicoamento de Pessoal de Nıvel Superior), Brazil. We thank thereferee for suggestions and comments.

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