General Spherically Symmetric Solutions of Einstein Vacuum Field Equations With Lamba
| Scientific Paper | |
|---|---|
| Title | General Spherically Symmetric Solutions of Einstein Vacuum Field Equations With Lamba |
| Read in full | Link to paper |
| Author(s) | Amir M Abbassi |
| Keywords | General Relativity, Exact Solutions, Point Mass, osmological Constant. |
| Published | 2002 |
| Journal | Apeiron |
| Volume | 9 |
| Number | 3 |
| No. of pages | 19 |
Read the full paper here
Abstract
A one-parameter family of inequivalent non-static spacetimes for a point mass in the presence of cosmological constant are investigated.
Overview
This Apeiron paper of July 2002 is by A. H. Abbassi and S. Gharanfoli of Tarbiat Modarres University together with A. M. Abbassi of Tehran University. It is a technical exercise in exact solutions of the Einstein vacuum field equations, and its target is Birkhoff's theorem — or rather the confidence usually placed in it. The Schwarzschild metric is normally held to be the unique vacuum spherically symmetric field of a point mass, and the Schwarzschild–de Sitter metric its unique counterpart when the Cosmological Constant Λ is non-zero. The authors argue that this uniqueness is an artefact of a conventional coordinate choice, and construct instead a one-parameter family of inequivalent spacetimes labelled by a dimensionless constant a.
The line element they propose replaces the Schwarzschild radial function by r + aM, so that the area of a surface of constant r is no longer 4πr2. For a = 0 the Schwarzschild solution is recovered; for a > 2 the coordinate singularity at the horizon is removed, and for any a ≠ 0 the curvature invariant stays finite at the origin, so the intrinsic singularity vanishes as well. The authors then set Λ ≠ 0 and derive a non-static metric with both features, everywhere analytic, asymptotically approaching the non-static de Sitter metric appropriate to a Λ-dominated universe. The dissident payload is stated in their closing remarks: a is "a dimensionless fundamental constant of theory and nature", not a coordinate convention, and if it is non-zero then "the documents in the litrature on the observation of black holes should be revisited."
The argument
The one-parameter family
The starting point is the line element, in Schwarzschild coordinates,
- ds2 = (1 − 2M/(r + aM))dt2 − (1 − 2M/(r + aM))−1dr2 − (r + aM)2dΩ2
with the source at r = 0 and a a dimensionless parameter. The obvious objection is that r′ = r + aM turns this straight back into Schwarzschild, making the family a set of subspaces of the Schwarzschild manifold. The authors answer with a Riemannian counter-example rather than an assertion. Take the plane R2 with ds2 = dr2 + r2dθ2 and the space R′2 with ds2 = dr2 + (r + a)2dθ2 over the same coordinate range. For any finite bound b the disc areas are πb2 and π(b2 + 2ab), so the restricted region of R2 is contained in that of R′2; taking b → ∞ gives R2 ⊂ R′2, and since R2 is complete and non-extendable, R2 = R′2. The inclusion runs the opposite way to the naive argument. They add a second point in the Remarks: the map r → r + aM translates the centre of symmetry and so breaks spherical symmetry, and does not belong to SO(3), the group whose action defines spherical symmetry in the first place. They also reject the usual definition of r by the area of constant-r surfaces, on the ground that "before fixing the metric there is no possibility of speaking the distance, so in the same way, there is no possibility of speaking the area", taking instead the Cartesian r = √(x2 + y2 + z2).
Bounding a by the classical tests
Because a is a physical parameter on this reading, it must be constrained by measurement, and the authors turn to the two classical tests. Following Weinberg's photon-trajectory integral, the light deflection to second order for a < 1 is
- Δφ = 4x + (15π/16 − 2)(1 + a)x2 + …, x = M/r0
or in the weak-field form Δφ = Δφfo[1 + (M/r0)(15π/16 − 1)(1 + a)], with Δφfo = 4M/r0. Against the VLBI measurement of radio-wave deflection by the Sun, Δφ ≈ (0.9998 ± 0.0008)Δφfo, and with M/r0 ≈ 10−6 for the Sun, this gives a < 103. The perihelion precession, Δφ = Δφfo[1 + (M/L)(19/6 + e2/2 − ... )(1 + a)] compared with the radar-ranging and long-baseline result Δφ ≈ (1.003 ± 0.005)Δφfo, gives the looser bound a < 105. The tighter of the two stands: the weak-field tests permit a anywhere up to about 103. A separate expression, valid for a > 3, is derived for the strong-field regime, where the closest approach tends to zero and the deflection can become very large — offered as a target for strong-field Gravitational Lensing observation, a field the authors trace to Fritz Zwicky's 1937 proposal and the 1979 double-QSO discovery.
Removing the singularities
For the Λ = 0 family the Riemann scalar invariant is
- RabcdRabcd = 48M2/(r + aM)6
which is finite over the whole range of r whenever a ≠ 0 — the intrinsic singularity at the origin is gone, leaving at most a coordinate singularity at r = (2 − a)M, which disappears for a > 2. Geodesic completeness for a > 2 is argued from the fact that gtt never changes sign, so t remains timelike everywhere and the constant-t hypersurfaces carry a positive-definite distance function under which every Cauchy sequence converges.
The Λ ≠ 0, a = 0 case is examined next. The non-static metric proposed in Abbassi's earlier work has invariant 48M2/(R(t)ρ)6 + (8/3)Λ2, still intrinsically singular at the origin. The authors argue for discarding the Schwarzschild–de Sitter metric on four counts: it has two coordinate singularities rather than one; a non-zero Λ of any size already suppresses the r ≈ 2M coordinate singularity in their form; setting M = 0 fails to recover the homogeneous isotropic FRW background; and it "shows a redshift-magnitude relation that contradicts the observational data."
The general solution
Section IV solves the general case. Writing ds2 = B(r,t)dt2 − R2(t)[A(r,t)dr2 + D(r)dΩ2], the condition Rtr = 0 gives AB = 1 after imposing the FRW boundary condition at large distance; the combination Rtt/B + Rrr/AR2 = 0 gives (D1/2)″ = 0, hence
- D1/2 = r + aM
reproducing the same parameter; and Rθθ + Λgθθ = 0 integrates to A−1/2(1 − Λρ2/3) + ... = 2M, the constant fixed by the post-Newtonian limit. The resulting metric has no apparent singularity anywhere and invariant 48M2/(D3R6(t)) + (8/3)Λ2, reducing to the earlier results when a or Λ is switched off.
Geodesics
Integrating the geodesic equations on the equatorial plane, with J the angular-momentum constant and the free-fall boundary conditions dr/ds = 0, dt/ds = 1 at infinity, yields d2ρ/ds2 = −M/ρ2 + Λρ/3, i.e. the gradient of the potential
- Φ = −M/ρ − Λρ2/6
At the location of the source, ρ = aMR(t), the potential is "very large, but it is finite." The authors suggest that the collapse of massive objects may supply a physical mechanism fixing a, and that ultra-high-energy phenomena in active galactic nuclei and cosmic rays could give a lower bound on it.
Assessment
The paper's strengths are mathematical and, within their scope, real. The counter-example against the "it's just a shifted radial coordinate" objection is neat and to the point: it demonstrates concretely that reasoning about inclusion of manifolds from coordinate ranges is unreliable, and the SO(3) argument reinforces it independently. The authors do the honest work of confronting their parameter with data rather than leaving it free, deriving second-order expressions for both light bending and perihelion precession and reading bounds off VLBI and radar-ranging measurements — and they report the weaker bound alongside the stronger. The Λ ≠ 0, a ≠ 0 solution is derived, not guessed, from the field equations with stated boundary conditions, and it does what is claimed: it is analytic everywhere and asymptotes to the non-static de Sitter metric. The referee's point, which the authors incorporate as Remark 5, sharpens the whole paper: either a is a coordinate artefact, in which case nothing follows, or it is a measurable constant of nature, in which case the standard picture of gravitational collapse and the observational case for black holes both need re-examination. The paper takes the second horn deliberately.
The difficulties are correspondingly serious, and to their credit the authors flag several themselves. The most damaging is one they raise but leave open at the end of Section II: within the interval 0 < r < (2 − a)M the coordinate r functions as a time coordinate, so the "location of the point mass at r = 0" is not a place at all, and it is unclear how one may then evaluate D there. They write that "this ambiguity particularly needs to have a satisfactory explanation" and do not supply one. The choice D1/2(0) = aM is justified only by the observation that M is the sole fundamental length available — a dimensional argument, not a derivation, and it is what puts the parameter into the theory in the first place. Geodesic completeness is argued for a > 2 only, using an equivalence between metric and geodesic completeness that holds for positive-definite metrics; extending that reasoning to a Lorentzian manifold requires more than is given, and the authors themselves note earlier that "the proof of completenss for a pseudo-Riemannian sapce is not an easy task."
The confrontation with the singularity theorems is likewise gestured at rather than settled. Remark 7 concedes that the "most serious objection" comes from Hawking and Penrose, whose proofs assume the existence of closed trapped surfaces, and replies only that in this metric "it can be checked this does not necessarily occur" — with no calculation of the expansion of outgoing null congruences shown. Since the whole claim to have eliminated the intrinsic singularity rests on evading those theorems, this is where a demonstration was most needed. There is also an unaddressed empirical difficulty in the bound itself: the analysis constrains a through the solar field, where M/r0 ≈ 10−6, so a bound of a < 103 permits a shift of order 103M — comparable to the Schwarzschild radius times a thousand, which for compact objects is not a small perturbation. Yet the paper offers no confrontation with the measurements that probe exactly that regime: the orbital-decay rate of binary pulsars, the near-horizon orbits of stars around the Galactic Centre, or the strong-field lensing it recommends as a future test. Finally, the assertion that the Schwarzschild–de Sitter metric gives a redshift–magnitude relation contradicting observation is carried entirely by a citation to the authors' own earlier work and is not reproduced here, though it does substantial work in motivating the replacement.
Judged as what it is — a careful exact-solutions paper making a well-posed claim about the scope of Birkhoff uniqueness — it is competent and its central geometric argument deserves an answer. Judged by its wider ambition, to unseat black holes as observed objects, it has not yet met the singularity theorems on their own ground.