Defending the Schwarzschild singularity

Since the paper by Oppenheimer and Snyder1 was published, the myth that the Schwarzschild singularity is merely a coordinate singularity and so is of little consequence has become ubiquitous2345. Pejorative language is a part of this, by describing the Schwarzschild metric as a pathological  function and the singularity as merely an apparent singularity whilst the singularity at the centre is described as being a real singularity that cannot be eliminated by any coordinate transformation. Yet such a real singularity is impossible with all known physics and a belief in its existence leads inevitably to further paradoxes. The fact is, that once you accept the impossible, almost anything becomes possible. Time travel, wormholes, other universes, and many other ideas straight from the pages of science fiction become worthy of serious academic discussion.

The first thing to remember is that for any given geodesic, it will still be comprised of the same set of points in spacetime no matter which coordinate system is used. There are never two different geodesics for the same particle and for the particular case of a particle falling into a Schwarzschild black hole, how can it be that in one set of coordinates, the particle is hovering for a time approaching infinity whilst in another, it has already reached its total demise? The answer to this conundrum is that in Schwarzschild coordinates, the central singularity can only be reached after an infinitely long time has passed. This does not lead to a denial of a central singularity - just that we will never, from our viewpoint, observe it's an effect on our, or our descendent's lifetimes.

But first let us pick apart the assertion in Oppenheimer and Snyder's paper, that the Schwarzschild singularity can be eliminated by a coordinate transformation.

Consider the line element for the metric of a non-rotating spherical object in Schwarzschild coordinates:

\[D^2=\left(1-\frac{r_s}{r}\right)c^2dt^2-\left(1-\frac{r_s}{r}\right)^{-1}dr^2\]

\[-r^2(d\theta^2+sin^2\theta d\varphi^2) \]

which, at the event horizon, approaches infinity.

Now \(D\) in the above equation is an invariant quantity and so is unchanged in any coordinate system. If you were unaware of this fact, we can prove this quite simply. In a different (primed) coordinate system, we have by definition

  \[ D'^2=g_{\mu'\nu'}dx^{\mu'} dx^{\nu'}\]

\[= \frac{ \partial x^\mu}{ \partial x^{\mu '}} \frac{ \partial x^\nu}{\partial x^{\nu '}} g_{\mu\nu}\frac{\partial x^{\mu '}}{\partial x^\mu}dx^\mu \frac{\partial x^{\nu '}}{\partial x^\nu} dx^\nu \]

\[=g_{\mu\nu}dx^\mu dx^\nu =D^2\]

This result seems to contradict long-established assertions by showing that solutions in Kruskal-Szekeres coordinates, or other comoving coordinates, have an infinite discontinuity at the event horizon, or see one of the points in counterarguments for a reason for discounting this view entirely. It is natural to wonder how this could have been missed by other investigators. The reason is that comoving coordinates systems effectively smooth out the curvature of spacetime by dividing out the curvature. At the event horizon this becomes \(\infty/ \infty\) which is undefined. With the spacetime being shown to be flat on either side of the event horizon, it seemed at the time, a very reasonable extension to assume that it is also flat across the event horizon. Now, with clear evidence that this is not the case, we have to recognise that there is a discontinuity that is not admissible in a Riemannian manifold. In comoving coordinates, the interior and exterior solutions are thus valid in separate coordinate patches, separated by the event horizon. There is no reason to assume that the internal solution is physically correct. By comparison, Schwarzschild coordinates are valid throughout the spacetime from infinity up to and including the event horizon, but not inside.

This work was undertaken in the hope of reaffirming the validity of the internal structure of a non-rotating black hole proposed in an earlier paper. This model proposed that the field inside the event horizon of a black hole is infinite everywhere. The question to be resolved is - "Can a region that is infinite everywhere be a Riemannian manifold?". A necessary requirement for this is that all derivatives of the field remain finite. This may not be true for a field of infinite values, but physically, these infinities are a result of a limiting process. It will take infinite time for them to be fully realised. In the current epoch, they will still be extremely large values but not yet infinite. Let \(r=r_s - \delta r\) then the metric is given by 

\[D^2=\left(1-\frac{r+\delta r}{r}\right)c^2 dt^2 -\left(1-\frac{r+\delta r}{r}\right)^{-1}dr^2\]

\[-r^2(d\theta^2+sin^2\theta d\varphi^2) \]

\[=\frac{\delta r}{r}c^2 dt^2+\frac{ r}{\delta r}dr^2-r^2(d\theta^2+sin^2\theta d\varphi^2) \]

This is differentiable, and in the limit as \(\delta r \rightarrow 0\) we can assume that it remains differentiable, making the interior a Riemannian manifold. Taking into account, the warning we have had, of the doubtful validity of assuming a limit in a physical situation, we can still assert that this becomes almost true in the future.  By joining this with the Schwarzschild metric, we can thus cover the whole of space.

To explain this again graphically, 

KruskalKoords.gif

Courtesy of Wikipedia

which shows infalling Kruskal–Szekeres coordinates, shows a blue hyperbola as the surface where the Schwarzschild radial coordinate is constant (and with a smaller value in each successive frame, until it ends at the singularities). To understand this, note that each hyperbola is a line of constant \(r\) in Schwarzschild coordinates with the event horizon being the 45° lines through the origin. The point to note is that as a particle 'sails' through the event horizon, it occupies every point along these axes. This is just a visual representation of the value of \(\infty/ \infty\), an indeterminate quantity.

Rigidity->

See also the excellent work of Weller in debunking the myth of traversing the event horizon.

1 On Continued Gravitational Contraction, Oppenheimer, J. R. and Snyder, H., Phys. Rev., 56, issue 5, pages 455--459, Sep 1939, {http://link.aps.org/doi/10.1103/PhysRev.56.455}

2 Gravitation, W Misner, K. S. Thorne and J. A. Wheeler, - 1973 - Macmillan

3 General Relativity. R. M. Wald - 1984 - University of Chicago Press

4 Black Holes: a student text, D. Raine and E. Thomas, 2015 - Imperial College 
Press

5 A Most Incomprehensible Thing: Notes towards a very gentle introduction to the mathematics of relativity, P Collier, - 2013 - Incomprehensible Books

 

Agree or disagree, or have any questions or observations about this, and I would love to hear from you, so please This email address is being protected from spambots. You need JavaScript enabled to view it., or leave a comment. Your views are always most welcome.

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