- Email: [email protected]
Exact formulae for a general relativistic fluid sphere
PHYSICS
Volume 25A, number 6
EXACT
FORMULAE
A GENERAL
FOR
25 September 1967
LETTERS
RELATIVISTIC
FLUID
SPHERE
Br. EUCHOWICZ Department
of Radiochemistry,
University
of Warsaw,
Warsaw,
Poland
Received 3 August 1967
internal solution for a general relativistic fluid sphere has been generalized - still for a static and spherically symmetric matter distribution. Formulae for the metric tensor, density and pressure are derived.
The well known Schwarzechild
The internal solution given by Schwarzschild [l] already half a century ago has been generalized later by Volkoff [2]. For Volkoff’s sphere there was e-h -t UJfor Y + 0. Another generalization is possible, with a finite value of e-h in the centre of the fluid sphere. This generalization was obtained following a procedure outlined in the present author’s report [3]. The standard form of the line element squared in canonical Schwarzschild coordinates, and relativistic units with G = c = 1 will be used. The following expressions for e-x and ev are obtained: e-k = 1 - a/K - (1 - l/K) ax2
(1)
and
I
ev =
.=a
[C, (l+_))Q
1 1
2
+‘C2 (l-Jl,p;Tz)c]2
l+J 1+(1-K@ 1-Jl+(l-K)x2 K ln l+-]
x
2 (1
+ E2 cos [K ln 1’mj/2 x
for
OCa/KCd
for
K = 2a
for
i < a/K C 1
(2)
The matter density p is P
(3)
To save space we give below the pressure only for a/K < i: 8Tfi$ _ 2-2ff+(a/K)(2+3) x2
+ (3-201) (i
- 1)a - 2a (l-l/K)
a Jl,p;;z
C1(l+Jlh8x2)o-l cl(l+~,o
+ c2(l-&$)e-l + C2(1-G,”
(4)
In the formulae above the dimensionless radial Variable x = y/^/b is used, where yb denotes the geometrical radius of the sphere. The parameter “an denotes the ratio of the Schwarzschild radius ‘Ye of the sphere to its geometrical radius yb; it cannot exceed unity. K is an arbitrary parameter 2 $. The formulae above are valid only for K f 1. The formulae for K = 1 (with e -A being constant throughout the sphere) reduce to Tolman’s solution VI [4] and to its generalization (with C = 0) presented in ref. 5. The parameters a, p and K are expressed in terms of a/K: p = (l/K-l)a (l-a/K)
, KzkF
l-a K
(5)
The integration constants Cl.. . E2 are to be determined from boundary conditions (continuity of e* and its derivative across the sphere surface). We obtain e.g. for a/K C i:
419
PHYSICS
Volume 25A, number 6
Cl = 4a(l;K_l)
[I + &#-‘([2
- 5a + k/K
LETTERS
25 September 1967
- 2ct(l-a)] dl-a+
[3a - 2 + 2u (l-u)]
m}
(6)
The class of solutions presented here is characterized by the parameter K; in the limit K -m one arrives at the already known internal Schwarzschild solution. The latter result is already evident if we look at the formulae (1) for e-k and (3) for p. The expressions (2) for e” cannot be used in this case, since already from the beginning one has a degenerate type of differential equation for e”; it is proved that its solution is just the well known Schwarzschild expression. Only for K = 3 the density at the sphere boundary drops to zero; the solution in this case may be regarded as one of the very few solutions in closed form for a gaseous sphere, and it leads to an equation of state which may be presented in the form of an analytical expression for pressure p (in terms of density p). An analysis of this equation, together with a derivation of the present results and those for other types of fluid spheres, will be published elsewhere since it takes much place. The aim of the present author’s work (of which this publication constitutes only a stage) is to find some new solutions of the gravitational field equations which might give the internal structure of fluid spheres (as models of dense, contracted stars) under their own gravitation. This is important since there are at our disposal only few exact solutions describing the gravitational field inside a spherical mass distribution, and in canonical coordinates we have no easily surveyable model for a gaseous static sphere *. The solution presented in this paper constitutes the simplest regular generalization of the Schwarzschild internal solution. esx remains finite in the centre (though no longer Galilean here) even with the density going here to infinity. The behaviour of the solution is just opposite to the behaviour of the Volkoff sphere, another generalization of the Schwarzschild internal solution, where density remained constant while e-x was going to infinity. It should be added, finally, that this solution exhausts the possibilities of arriving at closed exact formulae for e-& and e”, when starting with two leading terms only in the Laurent series for the function U(T) = 4 (l-re-x) in the vicinity of the centre of the sphere. etc.
References 1. K.Schwarzschild, Sitz. Preuss. Akad. Wiss. 18 (1916) 424. 2. G.M.Volkoff. Phys. Rev. 55 (1939) 55. 3. Br.Kuchowicz. Some New Solutions of the Gravitational Field Equations, Radiochem. (November 1966). 4. R.C.Tolman, Phys. Rev. 75 (1939) 364. 5. Br.Kuchowicz, Phys. Letters 24A (1967) 221. 6. H.A.Buchdahl, Astrophys. J. 140 (1964) 1512; 147 (1967) 310.
* Two simple nates.
420
models
derived
by Buchdahl [6] in isotropic
coordinates
Report,
Univ. of Warsaw.
appear complicated
Dept. of
in the canonical
coordi-
Volume 25A, number 6
EXACT
FORMULAE
A GENERAL
FOR
25 September 1967
LETTERS
RELATIVISTIC
FLUID
SPHERE
Br. EUCHOWICZ Department
of Radiochemistry,
University
of Warsaw,
Warsaw,
Poland
Received 3 August 1967
internal solution for a general relativistic fluid sphere has been generalized - still for a static and spherically symmetric matter distribution. Formulae for the metric tensor, density and pressure are derived.
The well known Schwarzechild
The internal solution given by Schwarzschild [l] already half a century ago has been generalized later by Volkoff [2]. For Volkoff’s sphere there was e-h -t UJfor Y + 0. Another generalization is possible, with a finite value of e-h in the centre of the fluid sphere. This generalization was obtained following a procedure outlined in the present author’s report [3]. The standard form of the line element squared in canonical Schwarzschild coordinates, and relativistic units with G = c = 1 will be used. The following expressions for e-x and ev are obtained: e-k = 1 - a/K - (1 - l/K) ax2
(1)
and
I
ev =
.=a
[C, (l+_))Q
1 1
2
+‘C2 (l-Jl,p;Tz)c]2
l+J 1+(1-K@ 1-Jl+(l-K)x2 K ln l+-]
x
2 (1
+ E2 cos [K ln 1’mj/2 x
for
OCa/KCd
for
K = 2a
for
i < a/K C 1
(2)
The matter density p is P
(3)
To save space we give below the pressure only for a/K < i: 8Tfi$ _ 2-2ff+(a/K)(2+3) x2
+ (3-201) (i
- 1)a - 2a (l-l/K)
a Jl,p;;z
C1(l+Jlh8x2)o-l cl(l+~,o
+ c2(l-&$)e-l + C2(1-G,”
(4)
In the formulae above the dimensionless radial Variable x = y/^/b is used, where yb denotes the geometrical radius of the sphere. The parameter “an denotes the ratio of the Schwarzschild radius ‘Ye of the sphere to its geometrical radius yb; it cannot exceed unity. K is an arbitrary parameter 2 $. The formulae above are valid only for K f 1. The formulae for K = 1 (with e -A being constant throughout the sphere) reduce to Tolman’s solution VI [4] and to its generalization (with C = 0) presented in ref. 5. The parameters a, p and K are expressed in terms of a/K: p = (l/K-l)a (l-a/K)
, KzkF
l-a K
(5)
The integration constants Cl.. . E2 are to be determined from boundary conditions (continuity of e* and its derivative across the sphere surface). We obtain e.g. for a/K C i:
419
PHYSICS
Volume 25A, number 6
Cl = 4a(l;K_l)
[I + &#-‘([2
- 5a + k/K
LETTERS
25 September 1967
- 2ct(l-a)] dl-a+
[3a - 2 + 2u (l-u)]
m}
(6)
The class of solutions presented here is characterized by the parameter K; in the limit K -m one arrives at the already known internal Schwarzschild solution. The latter result is already evident if we look at the formulae (1) for e-k and (3) for p. The expressions (2) for e” cannot be used in this case, since already from the beginning one has a degenerate type of differential equation for e”; it is proved that its solution is just the well known Schwarzschild expression. Only for K = 3 the density at the sphere boundary drops to zero; the solution in this case may be regarded as one of the very few solutions in closed form for a gaseous sphere, and it leads to an equation of state which may be presented in the form of an analytical expression for pressure p (in terms of density p). An analysis of this equation, together with a derivation of the present results and those for other types of fluid spheres, will be published elsewhere since it takes much place. The aim of the present author’s work (of which this publication constitutes only a stage) is to find some new solutions of the gravitational field equations which might give the internal structure of fluid spheres (as models of dense, contracted stars) under their own gravitation. This is important since there are at our disposal only few exact solutions describing the gravitational field inside a spherical mass distribution, and in canonical coordinates we have no easily surveyable model for a gaseous static sphere *. The solution presented in this paper constitutes the simplest regular generalization of the Schwarzschild internal solution. esx remains finite in the centre (though no longer Galilean here) even with the density going here to infinity. The behaviour of the solution is just opposite to the behaviour of the Volkoff sphere, another generalization of the Schwarzschild internal solution, where density remained constant while e-x was going to infinity. It should be added, finally, that this solution exhausts the possibilities of arriving at closed exact formulae for e-& and e”, when starting with two leading terms only in the Laurent series for the function U(T) = 4 (l-re-x) in the vicinity of the centre of the sphere. etc.
References 1. K.Schwarzschild, Sitz. Preuss. Akad. Wiss. 18 (1916) 424. 2. G.M.Volkoff. Phys. Rev. 55 (1939) 55. 3. Br.Kuchowicz. Some New Solutions of the Gravitational Field Equations, Radiochem. (November 1966). 4. R.C.Tolman, Phys. Rev. 75 (1939) 364. 5. Br.Kuchowicz, Phys. Letters 24A (1967) 221. 6. H.A.Buchdahl, Astrophys. J. 140 (1964) 1512; 147 (1967) 310.
* Two simple nates.
420
models
derived
by Buchdahl [6] in isotropic
coordinates
Report,
Univ. of Warsaw.
appear complicated
Dept. of
in the canonical
coordi-