Friday, October 4, 2013

Black Hole Information Paradox 3. Look for the information where you lost it

After I reviewed briefly the so-called black hole wars, and expressed my doubts about black hole complementarity, there are still many things to be said. However, I would like to skip over various solutions proposed in the last decades, and discuss the one that I consider most natural.

All the discussions taking place within the last year around black hole complementarity and firewall are concentrated near the event horizon. But why looking for the information at the event horizon, when it was supposed to be lost at the singularity?

Remember the old joke with the policeman helping a drunk man searching his lost keys under a streetlight, only to find later that the drunk man actually lost them in the park? When asked why did he search the keys under the streetlight, the drunk man replied that in the park was too dark. In science, this behavior is called the streetlight effect.

By analogy, the dark place is the singularity, because it is not well understood. The lightened place is the event horizon. This is Schwarzschild's equation describing the metric of the black hole:
$${d} s^2 = -(1-\frac{2m}{r}){d} t^2 +(1-\frac{2m}{r})^{-1}{d} r^2 + r^2{d}\sigma^2,$$
where ${d}\sigma^2 = {d}\theta^2 + \sin^2\theta {d} \phi^2$ is the metric of the unit sphere $S^2$, $m$ the mass of the body, and the units were chosen so that $c=1$ and $G=1$.

Schwarzschild's metric has two singularities, one at the event horizon, and the other one at the "center".

But in coordinates like those proposed by Eddington-Finkelstein, or by Kruskal-Szekeres, the metric becomes regular at the event horizon, showing that this singularity is due to the coordinates used by Schwarzschild. Fig. 1. represents the Penrose-Carter diagram of the Schwarzschild black hole. The yellow lines represent the event horizon, and we see that the metric is regular there.

Figure 1. Penrose-Carter diagram of the Schwarzshild black hole.


While at the event horizon the darkness was dispersed by finding appropriate coordinates, it persisted at the central singularity, represented with red. This is a spacelike singularity, and it is not actually at the center of the black hole, but in the future. This kind of singularity could not be removed completely, because it was not due exclusively to the coordinates.


However, in my paper Schwarzschild Singularity is Semi-Regularizable, I showed that we can eliminate the part of the singularity due to coordinates, by the transformation $r = \tau^2$, $t = \xi\tau^4$. The Schwarzshild metric in the new coordinates becomes

$${d} s^2 = -\frac{4\tau^4}{2m-\tau^2}{d} \tau^2 + (2m-\tau^2)\tau^4(4\xi{d}\tau + \tau{d}\xi)^2 + \tau^4{d}\sigma^2.$$

The metric is still singular, because it is degenerate, but the coordinate singularity was removed. The metric extends analytically through the singularity $r=0$, and the Penrose-Carter diagram becomes as in Fig. 2.
Figure 2. Penrose-Carter diagram of the extended Schwarzshild black hole.

In the new coordinates, the singularity behaves well. Although the metric is degenerate at the singularities, in arXiv:1105.0201 I showed that this kind of metric allows the construction of invariant geometric objects in a natural way. These objects can be used to write evolution equations beyond the singularity.

The Schwarzschild metric is eternal, but in the case relevant to the problem of information loss, the black hole is created and then evaporates. The analytic extension through the singularity presented earlier also works for this case, and the Penrose-Carter diagram is shown in Fig. 3.B.
Figure 3. A. Penrose diagram for the evaporating black hole, standard scenario.
B. Penrose diagram for the evaporating black hole, when the solution is analytically extended through the singularity (as in arXiv:1111.4837). In the new solution, the geometry can be described in term of finite quantities, without changing Einstein's equation. Fields can go through the singularity, beyond it.

Information is no longer blocked at the singularity. The physical fields can evolve beyond the singularity, carrying the information, which is therefore recovered if the black hole evaporates.

This is not a modification of General Relativity, it is just a change of variables. The proposed objects remain finite at singularity, and the standard equations can be rewritten in terms of these new, finite objects. These objects are natural, and don't require a modification of Einstein's General Relativity. The proposed fix is not made by changing the theory, but by changing our understanding of the mathematics expressing the theory.

One principal inspiration for me when finding this solution is the work of David Finkelstein, especially the brilliant solution to the problem of the apparent singularity on the event horizon. Imagine how happy I was when I received by email, at the end of December 2012, the following encouragements from him:

Dear Cristi Stoica,

I write concerning your paper "Schwarzschild Singularity is Semi-Regularizable" (arXiv 1111.4837v2).
I write first to thank you for the deep pleasure that this paper afforded me.
Your regularization of the central true singularity of the Schwarzschild metric is a remarkable and beautiful example of thinking outside the box. It is a natural, generally covariant, and deep result on a problem that has drawn wide attention, that of gravitational singularities. You found your solution easily once you conceived the idea, and yet it has been overlooked for these many decades by the truly great minds in the field.

[...]

With good wishes for your future explorations,
David Finkelstein
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