On Schwarzschild’s Interior Solution and Perfect Fluid Star Model

: We solve the boundary value problem for Einstein’s gravitational ﬁeld equations in the presence of matter in the form of an incompressible perfect ﬂuid of density ρ and pressure ﬁeld p ( r ) located in a ball r ≤ r 0 . We ﬁnd a 1-parameter family of time-independent and radially symmetric solutions (cid:8)(cid:0) g a , ρ a , p a (cid:1) : − 2 m < a < a 1 (cid:9) satisfying the boundary conditions g = g S and p = 0 on r = r 0 , where g S is the exterior Schwarzschild solution (solving the gravitational ﬁeld equations for a point mass M concentrated at r = 0) and containing (for a = 0) the interior Schwarzschild solution, i.e., the classical perfect ﬂuid star model. We show that Schwarzschild’s requirement r 0 > 9 κ M (cid:14) ( 4 c 2 ) identiﬁes the “physical” (i.e., such that p a ( r ) ≥ 0 and p a ( r ) is bounded in 0 ≤ r ≤ r 0 ) solutions { p a : a ∈ U 0 } for some neighbourhood U 0 ⊂ ( − 2 m , + ∞ ) of a = 0. For every star model { g a : a 0 < a < a 1 } , we compute the volume V ( a ) of the region r ≤ r 0 in terms of abelian integrals of the ﬁrst, second, and third kind in Legendre form.


A Boundary Value Problem
The interior Schwarzschild solution (cf. K. Schwarzschild, [1]) r ≤ r 0 ,R 2 = 3c 2 8πκρ , provides a simple model of a star, shaped as a homogeneous sphere of incompressible fluid with radius r 0 . The Lorentzian metric (1) is a time-independent and radially symmetric solution to the boundary value problem for gravitational field equations g αγ = g S αγ and p = 0 on r = r 0 , for nonempty space, whose matter and energy content is described by the energy-momentum tensor where ρ and p are the density and pressure fields, respectively, and g S is the exterior Schwarzschild solution, i.e., m = κM c 2 , r ≥ r 0 . Lorentzian metric g S is only defined in the region of r > 2m. In particular, it must be r 0 > 2m. (6) Solutions to (2) and (3) are looked for in the form g = e ν(r) c 2 dt 2 − e λ(r) dr 2 + r 2 dθ 2 + sin 2 θ dϕ 2 (7) where the unknown functions ν(r) and λ(r) are determined from the field equations under the additional assumption that the fluid matter is at rest, i.e., the components of the velocity four vector u α are u 0 , 0, 0, 0 with g 00 u 0 2 = 1. If we set u α = g αβ u β then u 0 = g 0β u β = g 00 u 0 = √ g 00 , u i = 0, hence, T αβ becomes (the energy-momentum tensor for the perfect fluid at rest). The nonzero components of the Ricci tensor of the metric (7) are Then, scalar curvature R = g αβ R αβ is Substitution from (8) - (12) into (2) turns (2) and (3) into the boundary value problem where C = −8πκ/c 2 . There are but three independent field Equations (13)-(15), for Equations (2) with (α, γ) ∈ {(2, 2), (3, 3)} coincide. So, (13)-(15) is a system of ODEs with four unknowns (ν, λ, ρ , p) and a physically reasonable solution should satisfy ρ(r) > 0 and p(r) ≥ 0. The addition of (13) and (14) − yields ν + λ > 0 [for any physically reasonable solution (ν, λ, ρ , p)]. Let us eliminate p between (14) and (15), respectively differentiate (14), so that to obtain Elimination of ν among (18) and (19) then leads to Lastly, let us eliminate λ between (17) and (20) so that to obtain which readily determines ν, provided that ρ and p are known. The main result in [1] is that ODEs system (13)-(15) may be integrated under the additional assumption that the fluid is incompressible, i.e., ρ(r) = ρ = constant. The assumption that ρ is constant is justified by its advantages in the mathematical handling of Equations (13)-(15). Indeed, if ρ ∈ (0, +∞), then (13) readily furnishes where a ∈ R is a constant of integration. If a ∈ R \ {0} then λ(0) = −∞ hence [by (7)] det g αβ = 0 at r = 0 i.e., the space-time described by g is singular on the 2-surface r = 0, at least from the nineteenth-century perspective adopted in [2] (pp. 1-3). To derive a solution without singularities of the sort, K. Schwarzschild chose (cf. [1]) a = 0 (and then λ is readily determined from (22)). It is our purpose in the present paper to determine the remaining (a fortiori unbounded) solutions (ν, λ, ρ , p) to the boundary value problem (13)-(16) corresponding to choices a ∈ R \ {0}. Other interior Schwarzschild-type solutions were determined by P.S. Florides (cf. [3]), A.L. Mehra (cf. [4]), N.K. Kofinti (cf. [5]), and O. Gron (cf. [6], further generalizing the work in [3]). Let us set Next, let us integrate (21) (under the assumption that ρ is constant) to obtain where D ∈ R is a constant of integration. Moreover, we eliminate p/c 2 + ρ between (24) and (17) which may be written Let us replace e −λ from (23) to obtain the following ODE with the unknown ν By a change of dependent variable γ(r) = e ν(r)/2 , Equation (26) may be written To solve the family of ODEs (27), one should integrate the associated homogenous equation For every λ ∈ C and n ∈ N, we adopt customary notation n √ λ = |λ| 1/n exp i arg(λ)/n , where arg : C → [0, 2π). Let I 3 = {ζ ∈ C : ζ 3 = 1} = {ω 0 , ω 1 , ω 2 }. To compute the integral in (29), let us consider the polynomial P a (r) = r 3 −R 2 r − aR 2 ∈ R[r] and set We need to distinguish among the following cases: In Case (I), one has a = 0 and {λ 1 , where ω k = exp 2kπi 3 ∈ I 3 . Consequently, one has the decomposition where B ∈ R is a constant of integration (the general solution to (28)). For a = 0, the solution obtained in [1] i.e., the general solution to homogenous equation [coinciding with u 0 as given by (30)]. In Case (II), one has λ 1 , λ 2 ∈ C \ R, λ 2 = λ 1 , and set X a is so that once again U a (r) dr = log P a (r) r + α leading to (30). In Case (III), and P a decomposes as P a = r ±R The associated inhomogeneous equation (i.e., (27) with a = 0) admits the obvious (constant) solution γ 0 = 1 2 DR 2 ; hence, the general solution to (32) is The determination of a particular solution to (27) is a considerably more difficult problem, involving hyperelliptic integrals. Equation (27) reads is a particular solution to inhomogeneous Equation (27). Hence, the general solution to (27) is Let g a be the metric tensor defined by (7) with g 00 = e ν and g 11 = −e λ , respectively given by (23) and (35), i.e., (0 < r ≤ r 0 ). Let us set A = 1 2 DR 2 and J a (r) = r 5/2 dr P a (r) 3/2 , so that γ a (r) = −A sign P a (r) P a (r) r J a (r), and then As is customary, the constants of integration A, B ∈ R are determined by the boundary conditions that the coefficients of the Lorentzian metrics (36) and (5) coincide on r = r 0 , and p(r 0 ) = 0 (p vanishes at the boundary, so that it matches continuously with the zero pressure at the exterior of the fluid star). The identification of g 11 coefficients (in (36) and (5) at r = r 0 ) This is equivalent toR 2 = r 3 0 (a + 2m) and yields a > −2m.
By taking into accountR 2 = 3c 2 8πκρ and m = κM c 2 , we may rewrite (38) as and density ρ is determined [under the requirement of regularity (i.e., looking for a bounded solution λ) a = 0 (as in [1]), and the boundary condition (40) leads to We are left with the determination of constants A and B in (36). To this end, let us return to (24) and substitute e ν/2 from (35) so that to obtain which evaluated at r = r 0 gives (by the boundary condition p(r 0 ) = 0) Moreover, by identifying the g 00 coefficients (of Lorentzian metrics (36) and (5))at r = r 0 where a = sign . Let us substitute from (43)-(74) into (36)-(37) so that to obtain For a = 0, Let us eliminate ρ between (40) (with a = 0) andR 2 = 3c 2 8πκρ so that to obtain The calculation of the dimensionless quantity (46) for α-Centauri and Sun (at the present age t p = 4.5 × 10 9 years, respectively, at age t p + ∆t where ∆t = 0.5 Gyr) is reported in Table 1 (gravitational constant κ = 6.67 × 10 −11 kg −1 ·m 3 ·s −2 and speed of light in vacuum c = 299, 792, 458 m·s −1 ). Let M (t) and R (t) be, respectively, Sun's mass and radius at age t, so that M (t p ) = M = 1.9884 × 10 30 kg and R (t p ) = R = 6.957 × 10 5 km. Table 1. Calculation of r 0 /R for Sun and α-Centauri.
Data used for α-Centauri are R α = 0.145 R and M α = 0.123 M (cf. e.g., [7]); R (t p + ∆t) R = 1.03712 after ∆t Gyr from present age. Dimensionless quantity r 0 /R decreases at later ages both as a function of r 0 (cf. Figure 1) and as a function of M (as Sun expels 2.5 × 10 −14 M per year from emission of electromagnetic energy and ejection of matter with solar wind, cf. [8]).
One has (with a satisfying (38)) In particular, g a is only defined in (a region contained in) P a (r) < 0 (and P a (r 0 ) = −R 2 r 0 − 2m < 0 because of g 00 > 0 for any r ≥ r 0 for the exterior Schwarzschild solution), so that The hyperelliptic integral in (49) may be derived from the integral in (60), thought of as a parametric integral ∂ ∂m while the calculation of the antiderivative in (60) is addressed in Appendix A.

Pressure Field
To fully solve the boundary value problem (13)-(16), one is left with the determination of pressure field p(r). To this end, we recall (24), implying (by p(r) ≥ 0 and ρ > 0) that yielding a = −1. Note that (by (6) and (39)) P a (r 0 ) = 2m − r 0 a + 2m As a consequence of (40), ρ = ρ a where Let us substitute from (52), (50) (with a = −1) and (59) into (24). We obtain p = p a where where we set For a = 0, Formula (53) becomes (expressing the pressure field in the case considered by K. Schwarzschild, cf. op. cit.) (see Figure 2). We prove the following theorem: the following statements are equivalent: (i) r 0 > 9m/4 and (ii) there is an open neighbourhood of the origin U 0 ⊂ (−2m, +∞), such that, for any a ∈ U 0 , pressure field p a (r) is a bounded function in the interval 0 ≤ r ≤ r 0 . With notations (54) and (55), pressure p a is bounded in 0, r 0 if and only if I a (r) = v(a) for any 0 ≤ r ≤ r 0 . We first attack the implication (ii) =⇒ (i). In particular (under Hypothesis (ii)), p 0 is bounded in [0, r 0 ], yielding where v(0) = 2 3 which is (i). This is precisely the requirement (on structural parameters r 0 and M) discovered by K. Schwarzschild (cf. op. cit., or formula (9.163) in [10] (p. 292)) that ought to hold in order that pressure p 0 (r) may never become infinite inside the fluid. To prove the implication (i) =⇒ (ii) is to show that the previous result persists under a small 1-parameter variation of p 0 about a = 0. Let us assume that requirement (57) is satisfied. Then, (56) holds, so that (as I 0 (r) is strictly decreasing in 0 ≤ r ≤ r 0 )one has I 0 (0) < v(0). Let us consider auxiliary function ψ(a) = I a (0) − v(a) defined in a > −2m. By the continuity of ψ, property ψ(0) < 0 persists in some open neighbourhood 0 ∈ U 0 ⊂ (−2m, +∞) i.e., ψ(a) < 0 or I a (0) < v(a) for any a ∈ U 0 . Lastly, as I a is strictly decreasing in 0, r 0 , we may conclude that Q.e.d. As a corollary, we may show that the maximum domain Ω ⊂ R 4 of the "physical" solution g a , ρ a , p a (obeying to the requirements ρ a (r) ≥ 0, p a (r) ≥ 0, and p a (r) is bounded in 0 ≤ r ≤ r 0 )is Ω = R × B r 0 (0), for any a ∈ U 0 . Here B r (x) denotes the closed ball of radius r > 0 and center x ∈ R 3 . Indeed, as g 00 > 0 and g 11 < 0 in Ω, the domain is determined by the requirements r = 0, P a (r) < 0 and I a (r) = v(a). To compute Ω, note first that P a (r) = 3r 2 − r 3 0 a + 2m and Consequently, if i) a ≥ 0 then P a (r) < 0 for any r ∈ 0, r 0 , while if ii) −2m < a < 0, there is a unique r(a) ∈ 0, r 0 such that P a r(a) = 0, P a (r) > 0 for any r ∈ 0, r(a) , and P a (r) < 0 for any r ∈ r(a), r 0 .

Volume Calculations
The "true" physical density is not constant as an effect of curvature i.e., the metric g a (with a = r 3 0 R 2 − 2m) varies in the fluid ambient, and the very notion of density depends on g a .

Conclusions and Open Problems
In the present paper, we examined a fundamental issue, present in most textbooks nowadays on general relativity and gravitation theory, i.e., the derivation of the interior Schwarzschild solution g 0 (cf. e.g., [10] (pp. 280-295)) and the resulting simple model of a star. To determine g 0 (and actually (g 0 , ρ 0 , p 0 ) where ρ 0 is the (constant) density and p 0 the (radial) pressure field), one looks for a line element ds of the form ds 2 = e ν(r) c 2 dt 2 − e λ(r) dr 2 + r 2 dθ 2 + sin 2 θ dϕ 2 and needs to solve the boundary value problem (13)-(16) for (λ, µ, ρ, p). The general solution of the ODE system (13)-(15) contains three constants of integration A, B, a ∈ R, and the given boundary conditions determine the first two, while the third is but subject to the constraint a > −2m. This leads to a 1-parameter family of solutions λ a , µ a , ρ a , p a a>−2m to (13)-(16) with the property that λ a (r) is bounded at r = 0 if and only if a = 0. K. Schwarzschild discarded the solutions with a = 0 on the ground that the determinant of each g a with a = 0 vanishes at r = 0; this is, of course, also the case for g 0 , and the question arises whether any of the additional solutions g a is physically acceptable, thus leading to alternative geometric descriptions of the interior of a star of incompressible fluid. We explicitly determine the solutions g a , ρ a , p a a>−2m and show that the requirement (57) i.e., r 0 > 9m/4, which in Schwarzschild's work is equivalent to p 0 (r) staying finite in r ≤ r 0 , is also equivalent to the boundedness of p a (r) in r ≤ r 0 , though only for a lying in some open neighbourhood U 0 ⊂ (−2m, +∞) of a = 0. It should be observed that estimate (6) i.e., r 0 > 2m is slightly improved by estimate (57); When the geometry is described by g 0 , the parameter r 0 /R is << 1, as may be argued by invoking experimental evidence, and development (77) gives a fair approximation to volume V(0). We slightly improved Schwarzschild's result by developing V(0) to arbitrary fixed order (cf. our (70) in § 3) and providing a mathematically elegant estimate on the Taylor rest (cf. (71) in § 3) On the basis of our discussion of the parameter r 0 /R in § 1 for the Sun, one has r 0 /R << 1 beyond Sun's present age, and Formulas (70) turned out to be more involved, as depending upon the evaluation of an abelian integral that may not be expressed in terms of algebraic functions, and which we can only reduce to the canonical Legendre form (cf. (60) and (69) in § 3) confined to case a > a 0 for some constant a 0 > 0 (whose precise description is given in § 3). Going back to (38), one has r 0 R = a + 2m r 0 hence lim a→+∞ r 0 R = +∞.
Of course, one should take into account the upper bound on a imposed by (51) implying However, the right-hand side of (78) is in general > 1 (hence truncated Taylor approximation of V(a) as a function of r 0 /R is meaningless). As a byproduct of (77), one obtains (cf. [1], or (9.178) in [10] (p. 294)) the truncated Taylor approximation of the mass defect and uses (79) (together with the classical mechanics calculation of the surface potential on a sphere) to attribute V(0) ρ 0 − M to the loss of energy in packing the matter under its own gravitational energy. A similar formula for the mass defect V(a) ρ a − M is not known. That spherically symmetric solutions such as {g a : −2m < a < a 1 } where a 1 = 9r 2 0 4 (r 0 − 2m) − 2m, cf. (51) are appropriate for the calculation of mass defect of strange stars appears to be accounted for by [11].
Author Contributions: The authors have equally contributed to the elaboration and writing of the paper.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. An Abelian Integral
Our purpose in this section is bringing to the canonical form the abelian integral where R(x, y) is the rational function R(x, y) = x 3 y. Starting from one sets as customary and distinguishes three cases as (I) is abelian in Cases (I)-(II) (while in Case (III) it reduces to a rational integral). To apply the classical scheme of reduction to the canonical (Legendre) form, one needs a decomposition of (A2) into second-degree factors P(x) = x 2 + px + q x 2 + p x + q . The required algebraic information is summarised in Table A1.