Generalized transformations and coordinates for static spherically symmetric general relativity

We examine a static, spherically symmetric solution of the empty space field equations of general relativity with a non-orthogonal line element which gives rise to an opportunity that does not occur in the standard derivations of the Schwarzschild solution. In these derivations, convenient coordinate transformations and dynamical assumptions inevitably lead to the Schwarzschild solution. By relaxing these conditions, a new solution possibility arises and the resulting formalism embraces the Schwarzschild solution as a special case. The new solution avoids the coordinate singularity associated with the Schwarzschild solution and is achieved by obtaining a more suitable coordinate chart. The solution embodies two arbitrary constants, one of which can be identified as the Newtonian gravitational potential using the weak field limit. The additional arbitrary constant gives rise to a situation that allows for generalizations of the Eddington–Finkelstein transformation and the Kruskal–Szekeres coordinates.


Introduction
The first exact solution to the empty space field equations of general relativity is due to Karl Schwarzschild [1]. The derivation is now commonplace and can be readily found in the literature (e.g. [2][3][4][5][6][7]). It describes the space-time outside a spherically symmetric, static and asymptotically flat body of mass M.
The line element in Schwarzschild geometry in spherical coordinates x μ = (ct, r, θ, ϕ) is given by where dΩ 2 = dθ 2 + sin 2 θ dϕ 2 , r 0 = −2GM/c 2 , and the speed of light in vacuum and Newton's gravitational constant are given by c and G, respectively, and here we adopt a time scale in which c = 1.
The standard approach in deriving the Schwarzschild solution is to consider one of the following spherically symmetric line elements of the form: ds 2 = a(r) dt 2 − c(r) dr 2 − r 2 dΩ 2 , (1.2) ds 2 = a(r, t) dt 2 − c(r, t) dr 2 − r 2 dΩ 2 (1.3) and ds 2 = a(r, t) dt 2 + 2b(r, t) dr dt − c(r, t) dr 2 − r 2 dΩ 2 , (1.4) where a, b and c denote unknown functions of either space or both space and time which are to be determined. Throughout the literature, convenient coordinate transformations [8,9] involving the introduction of a new time coordinate, allow for the removal of the non-orthogonal component in equation (1.4) and physical arguments such as a static space-time [7] inevitably lead to the Schwarzschild solution.
The aforementioned assumptions then guarantee that the only solution to the empty space field equations (see §2) is given by equation (1.1) giving rise to the so-called Birkhoff's theorem [10][11][12]. The key point of this paper is that if these simplifying coordinate transformations are not made, another solution opportunity presents itself which suggests an improved coordinate chart to that of Schwarzschild which contains only one global singularity at the origin r = 0 and not a coordinate singularity which is often studied in Schwarzschild geometry. Furthermore, the alternative solution allows for the generalization of the Eddington-Finkelstein and Kruskal-Szekeres coordinate transformations. By making the usual simplifying assumptions and coordinate transformations at the outset, the solution presented below is excluded.
In §2, a description of the governing equations which leads to both solutions is presented. In §3, we present a novel derivation of the Schwarzschild solution using the governing equations presented in the prior section. In §4, we derive a new solution to the empty space field equations of general relativity and corresponding generalizations of the Eddington-Finkelstein transformation and the Kruskal-Szekeres coordinates. Finally, the non-zero, independent Christoffel symbols and mixed and covariant Einstein tensor components associated with equation (1.4) are presented in appendices A-D.

Governing equations
Throughout the remainder of the present paper, attention is restricted to the space-time metric given by equation (1.4). The metric tensor g ab and the inverse metric tensor (g ab ) −1 are given by where a, b and c are functions of both space and time and ω(r, t) ≡ ac + b 2 . We determine the unknown functions a, b and c when applied to the empty space field equations of general relativity [7] G ab = 0 and G ab ≡ R ab − 1 2 g ab R, (2.2) where R ab and R are the Ricci tensor and Ricci scalar, respectively (cf. [3,7]). The first two equations under consideration are equations (B 2) and (B 3) which are given by and G 01 = a(ωc t + abc r + a r bc + 2b 2 b r ) respectively. After some simplification and introducing u(r, t) = ra/ω, we may rewrite equations (B 2) and (B 3) as a system of linear partial differential equations where we have introduced the subscript notation to indicate partial derivatives with respect to the respective coordinate. The above system of equations can be immediately solved to give u r = 1 and u t = 0. We obtain an expression for a(r, t) by integrating u r = 1, thus where r 0 is introduced as a constant of integration. Next, consider equation (A 4) which, upon simplification yields ωc t + bω r = 0.
which, after some rearranging can be expressed as by introducing expressions v = rb/ω and w = rc/ω and making use of u r = 1, the above now reads Finally, upon integration, an expression for the unknown function b(r, t) is obtained, given by where f (r) is obtained after integrating equation (2.6) with respect to time. By substituting equations (2.4) and (2.7) into ω(r, t) = ac + b 2 , we determine the unknown function c(r, t) in terms of f (r), which is given by It is clear from this equation that c(r, t) is at most a function of r only, so that c(r, t) = c(r). From equation (2.5), assuming that b(r, t) is non-zero and using the fact that c t = 0, we can deduce that ω(r, t) is at most a function of time, so that ω(r, t) = ω(t). By substituting equation Finally, by multiplying equation (B 5) by −4ω(t) 2 /r, the remaining equation of interest is given by where, after some simplification and using ω = ω(t) and c t = 0 the above equation becomes Dividing the above equation by 2ω(t) 3/2 , we deduce To simplify the above expressions further, consider the expression for b(r, t) which is given by equation ( denominator in both the first and third terms. Also, the second term in the above expression is a second derivative of a linear function, hence, the only non-zero term in the above equation becomes (2.10) Evidently, this can be satisfied in three different ways: In the first two cases, Schwarzschild-like solutions are obtained which, under certain conditions, give precisely the Schwarzschild solution. More importantly, case (iii) gives rise to a new solution opportunity which leads to generalizations of the Eddington-Finkelstein transformation and the Kruskal-Szekeres coordinates.

Schwarzschild solution
The three situations in which equation ( The precise Schwarzschild solution is realized by imposing the conditions α = 1 and ψ 0 = 0.

Derivation of new solution and generalized transformations and coordinates
The third and final case to be considered is of particular interest as it produces an alternative solution to the empty space field equations of general relativity. Case The new derived expressions for the functions a, b and c constitute an alternative exact solution of the field equations of general relativity where the line element is For β = 0, the line element becomes (ds) 2 = 1 + r 0 r dτ 2 + 2 dr dτ − r 2 dΩ 2 . (4.5) Using the computer algebra system (CAS) Maxima [13], the three line elements given by equations (4.3)-(4.5) can all be independently verified as bona fide solutions of the field equations, where α and β denote arbitrary constants and ω(t) denotes an arbitrary function of time. We observe that equation (4.5) corresponds to the outgoing Eddington-Finkelstein coordinates and since we have arbitrarily assigned the positive root for b in equation (4.2), we might similarly retrieve the ingoing Eddington-Finkelstein coordinates by adopting the negative sign. We further observe that both the Schwarzschild solution and equation (3.4) with α = 1 have the common structure that can be written as where ψ has the constant value ψ 0 for the Schwarzschild solution given by equation ( where dr and sin ψ are given by dr = dr 1 + r 0 /r , sinψ = 1 − β(r + r 0 ) r 2 1/2 (4.8) and ψ = π/2 in the case of equation (4.5). By performing the transformation τ = τ + ρ(r ), it is clear that equation (4.7) becomes (ds) 2 = 1 + r 0 r ((dτ ) 2 + 2(sin ψ + ρ (r )) dτ dr + [(sin ψ + ρ (r )) 2 − 1](dr ) 2 ) − r 2 dΩ 2 , (4.9) where the prime denotes differentiation with respect to r . From the structure of equation (4.9), it is apparent that ρ(r ) can be chosen to produce any desired equation of the form of equation (4.7). Thus, as an example we may obtain the Schwarzschild line element by making dρ dr + sin ψ = sin ψ 0 , (4.10) from which we may deduce that the function ρ(r ) is determined by performing the integration dρ = (r sin ψ 0 − [r 2 − β(r + r 0 )] 1/2 ) dr r + r 0 , (4.11) which clearly admits a range of analytical expressions depending upon the value of β, noting the greatly simplified form arising from the special case β = −4r 0 . As an illustration of the above, we derive the unknown function ρ(r ) which allows for the coordinate transformation from Schwarzschild to Eddington-Finkelstein coordinates. In Schwarzschild geometry ψ 0 = β = 0 and hence, equation (4.11) becomes ρ(r) = ± r r + r 0 dr = ±(r − r 0 ln(r + r 0 ) + C), (4.12) where C is a constant of integration. Substituting equation (4.12) into τ = τ + ρ(r) and applying to the Schwarzschild line element gives precisely the outgoing and ingoing Eddington-Finkelstein coordinates depending on the choice of sign. We note that τ = τ + ρ(r) together with equation ( To extend the Kruskal-Szekeres coordinate transformation, we define the variables ξ and η through the differential relations dξ = dr + dτ + sin ψ dr and dη = dr − dτ − sin ψ dr , (4.13) where r , τ and sin ψ are all as given above. Explicitly, ξ and η are given by (derivation given in appendix D) where ξ 0 and η 0 are introduced as arbitrary constants of integration, noting that ξ and η are essentially r + τ and r − τ , respectively, where τ is precisely as defined in the previous section. In the above integral evaluations, it is assumed that the arguments of all logarithms are positive; in other cases, slightly modified formulae may apply. On evaluating the product dξ dη, we see that equation (4.7) becomes (ds) 2 where we can identify the product dξ dη as the generalized ingoing/outgoing Eddington-Finkelstein coordinates. Let us propose generalized coordinates R and T such that and on adopting the values ξ 0 = η 0 = r 0 ln(r 0 ), we may deduce the relations for the standard Kruskal-Szekeres coordinates and, therefore, assuming we adopt the same values for the arbitrary constants ξ 0 and η 0 .

Conclusion
We have derived a spherically symmetric, static solution of the empty space field equations of general relativity, where the line element given by equation (4.4) involves two arbitrary constants. By consulting the weak field limit [3], we can immediately identify r 0 = −2GM/c 2 as the Schwarzschild radius. The second arbitrary constant allows for the generalizations of the well-known Eddington-Finkelstein transformation and the Kruskal-Szekeres coordinates. Furthermore, a variety of analytical forms for the extended Eddington-Finkelstein transformation and the Kruskal-Szekeres coordinates are derivable depending on the value of β, and one derivation is provided in appendix D. − r 0 ln [r 2 − β(r + r 0 )] 1/2 − r − β(r + r 0 ) 2r 0 + ξ 0 (D 6) and the expression for η may be similarly obtained.