Redefining the black holes

Recently, both Nature (Stephen Hawking: 'There are no black holes') and New Scientist (Stephen Hawking's new theory offers black hole escape)  covered Hawking's recent paper, Information Preservation and Weather Forecasting for Black Holes, discussed earlier on this blog. Since then, I've seen several times articles re-blogging the idea that Hawking said there are no black holes. Some hurried to say that Hawking considers this his "greatest blunder" (making reference to Einstein's regret that he conjectured the existence of dark energy for the wrong reasons, this preventing him to realize the expansion of the universe).

What Hawking said in fact was that

The absence of event horizons mean that there are no black holes - in the sense of regimes from which light can't escape to infinity.

But he continues that black holes exist, but they are not as he originally defined them:

There are however apparent horizons which persist for a period of time. This suggests that black holes should be redefined as metastable bound states of the gravitational field.

After a regular person makes a claim about something, he hardly changes his mind. Especially since that claim is part of what made him famous. We find difficult to withdraw our positions, because we are afraid to look weak. One reason I admire Hawking is that he had in several occasions the courage to change his mind, and even to admit he was wrong. He made several bets with his fellows Kip Thorne and John Preskill, concerning the existence of black holes, of naked singularities, and regarding the information loss. He eventually conceded all these bets, even though no clear cut evidence was discovered for either of the sides.

Hawking's first great discovery was the big bang singularity theorem, according to which the universe started from a singularity. It is difficult to later reject the very thing that made you famous in the first place, but Hawking, together with James Hartle, replaced the initial singularity with the famous no-boundary proposal, which doesn't have this singularity (although, technically, the positive defined metric they put at the beginning of the universe is separated by the Lorentzian one by a space slice which is in fact singular).

At various points of his career, Hawking expresses his doubts about string theory. For instance, in his debate with Penrose, he said

I think string theory has been over sold.

and

it seems we don’t need string theory even for the beginning of the universe.

 and

If this is true it raises the question of whether string theory is a genuine scientific theory. Is mathematical beauty and completeness enough in the absence of distinctive observationally tested predictions. Not that string theory in its present form is either beautiful or complete.

But in few years, he became a major supporter of string theory, as follows from this paper and this book.

Arguably, most of the fame of Hawking comes from his results concerning the black holes. But I don't think it is true as it is claimed now that, after a lifetime dedicated to the study of black holes, he arrived at the conclusion that they don't exist. He only rejects the existence of black holes defined as objects surrounded by event horizons, defined in their turn in a particular way. And in fact, he rejects that notion of event horizon. The notion of event horizon exists for long time, but at some point, Hawking redefined it, as the surface separating the points in spacetime which can't be seen from the future null infinity. Before that, the event horizon was known from stationary black holes, like the Schwarzschild, the Reissner-Nordström, and the Kerr-Newmann ones, and was generalized to trapped null surfaces. Hawking opposed to this general definition, because it would depend on the observer. Such apparent horizons are therefore not invariant, and Hawking proposed a global definition. The problem with the global definition is that it depends on the entire future, to establish whether a given point will eventually be visible from the null infinity or not. But if the black holes evaporate in a way compatible with the AdS-CFT conjecture, they have to respect the CPT symmetry. Since a global notion of event horizon violates this symmetry, Hawking proposes to reject it.

Hawking did not change his mind about the existence of the black holes, but only about his own definition of black holes, as those regions in spacetime which can't be seen from the future null infinity. He proposes instead to consider again the black holes to be regions surrounded by apparent horizons.

Hawking breaks the firewall

Hawking finally uploaded the paper containing his Skype talk at the Fuzz or fire workshop, named Information Preservation and Weather Forecasting for Black Holes. The paper, whose body has two pages, is an almost verbatim transcription of the 9' talk, with a tiny paragraph inserted before the final one. The talk was very dense, with great qualitative arguments, but almost no quantitative ones, and I kind of hoped that the paper will be more detailed in this respect.

The first argument Hawking brought against firewalls is that 

if the firewall were located at the event horizon, the position of the event horizon is not locally determined but is a function of the future of the spacetime.

Hawking defined long time ago the event horizon as being the surface separating the events that will eventually be seen from the future infinity, from those that will never be. Thus, we can know the event horizon only if we know the entire future history of the universe.This rules out any special structure which one may try to attach to the horizon, being it firewalls, stretched horizons, bits containing the information from the black hole etc. This argument is technically correct, but this doesn't rule out alternative local definitions of the horizon, and on which the firewall may live. I think this argument comes from the usage of different definitions.

 One thing I find particularly intriguing is that Hawking doesn't discuss the singularities. Singularities are predicted by Penrose's black hole singularity theorem, which inspired Hawking in coming up with  his own big bang singularity theorem. Also singularities are a necessary part of Hawking's original argument for the information loss. So, it is a bit strange that he doesn't say much about them. Well, he referred to the paper in which he proposed the resolution of the information paradox, and said that "the correlation functions from the Schwarzschild anti deSitter metric decay exponentially with real time". So, he considers that the contribution from the Schwarzschild singularities is negligible.


I find more interesting Hawking's argument that the ADS-CFT correspondence requires the black holes to be symmetric in time:

the evaporation of a black hole is the time reverse of its formation (modulo CP), though the conventional descriptions are very different. Thus if one assume quantum gravity is CPT invariant, one rules out remnants, event horizons, and firewalls.

Of course, again, one can imagine a way by which the firewalls are time symmetric, and use a different definition of the event horizon. But the reason I find interesting this argument of Hawking is that it doesn't preclude singularities, only the singularities that are not time symmetric. For instance, fig. A. depicts the Penrose diagram of the evaporating black hole that is not time symmetric, while fig. B. depicts a time symmetric one, obtained by analytic extension beyond the singularity. I give more details about this in Black Hole Information Paradox 3. Look for the information where you lost it.

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.

So, I think Hawking's argument based on the ADS-CFT correspondence is compatible with the approach to the black hole singularities which I proposed, and excludes the standard solution, which is not time symmetric.

Defending my PhD Thesis

Update:

My Ph.D. Thesis Singular General Relativity was published at Minkowski Institute Press and can be ordered at Amazon.

 ___________________________________________

Last week, on December 6, 2013, I defended my PhD Thesis.
The Thesis is named Singular General Relativity, and can be found at arXiv:1301.2231.

Thesis Abstract:

This work presents the foundations of Singular Semi-Riemannian Geometry and Singular General Relativity, based on the author's research. An extension of differential geometry and of Einstein's equation to singularities is reported. Singularities of the form studied here allow a smooth extension of the Einstein field equations, including matter. This applies to the Big-Bang singularity of the FLRW solution. It applies to stationary black holes, in appropriate coordinates (since the standard coordinates are singular at singularity, hiding the smoothness of the metric). In these coordinates, charged black holes have the electromagnetic potential regular everywhere. Implications on Penrose's Weyl curvature hypothesis are presented. In addition, these singularities exhibit a (geo)metric dimensional reduction, which might act as a regulator for the quantum fields, including for quantum gravity, in the UV regime. This opens the perspective of perturbative renormalizability of quantum gravity without modifying General Relativity.

The Thesis is based on a series of papers, from which the following are published or accepted:

[1] C. Stoica On Singular Semi-Riemannian Manifolds, Int. J. Geom. Methods Mod. Phys., 0(0):1450041, March 2014, arXiv:1105.0201.

[2] C. Stoica Schwarzschild Singularity is Semi-Regularizable, Eur. Phys. J. Plus, 127(83):1–
8, 2012, arXiv:1111.4837.
[3] C. Stoica Analytic Reissner-Nordstrom Singularity, Phys. Scr., 85(5):055004, 2012, arXiv:1111.4332.
[4] C. Stoica Kerr-Newman Solutions with Analytic Singularity and no Closed Timelike
Curves, To appear in U.P.B. Sci. Bull., Series A, arXiv:1111.7082.
[5] C. Stoica Spacetimes with Singularities, An. St. Univ. Ovidius Constanta, 20(2):213–238, July 2012, arXiv:1108.5099.

[6] C. Stoica Einstein Equation at Singularities, Cent. Eur. J. Phys., 12 (2014), 123-131, arXiv:1203.2140.

[7] C. Stoica Beyond the Friedmann-Lemaitre-Robertson-Walker Big Bang singularity,
Commun. Theor. Phys., 58(4):613–616, March 2012, arXiv:1203.1819.
[8] C. Stoica On the Weyl Curvature Hypothesis, Annals of Physics, 338:186–194, November 2013, arXiv:1203.3382.

[9] C. Stoica The Geometry of Black Hole Singularities, Advances in High Energy Physics, Volume 2014 (2014), Article ID 907518  arXiv:1401.6283.
[10] C. Stoica Metric dimensional reduction at singularities with implications to Quantum Gravity, Annals of Physics 347C (2014), pp. 74-91, arXiv:1205.2586.


Others are not yet published:

[11] C. Stoica Warped Products of Singular Semi-Riemannian Manifolds, arXiv:1105.3404.
[12] C. Stoica Cartan's Structural Equations for Degenerate Metric, Balkan J. Geom. Appl., Vol. 19, No. 2, (2014), p. 118-126,  arXiv:1111.0646.
[13] C. Stoica Big Bang singularity in the Friedmann-Lemaitre-Robertson-Walker spacetime, arXiv:1112.4508.

FQXi contest 2013 "It From Bit or Bit from It", results announced

The results of this year's FQXi contest are announced.

Here is how the top looked at the end of the community voting:
http://fqxi.org/community/forum/category/31419?sort=community

Now, the members of the jury made their choices too, and here are the results:

http://fqxi.org/community/essay/winners/2013.1

In addition to the winning essays, there are many interesting entries, including some of those that were not among the finalists.

My essay, The Tao of It and Bit (arXiv:1311.0765), got a fourth prize.

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