Tuesday, July 29, 2014

A confused sleeping beauty

The Sleeping Beauty problem

 A recent post by Sean Carroll reignited a debate about the "Sleeping beauty problem".
This is a simple problem of probabilities, involving tossing a coin. But for some reason, it seems to be no agreement about its solution.

Consider the following experiment:
- On Sunday, put Sleeping Beauty to sleep.
- Toss a fair coin.
- We are interested to ask Sleeping Beauty the question 
Q. What is your belief now for the proposition that the coin landed heads?
- If the coin comes up heads, wake up Sleeping Beauty on Monday and ask her the question. Then drug her to forget that awakening.
- If the coin comes up tails, wake up Sleeping Beauty both on Monday and Tuesday and ask her the question. Each time drug her to forget that awakening.
- In both cases, don't forget to wake her on Wednesday and end the experiment.

If you have trouble convincing a Beauty to let you put her to sleep and drug her, you can try your luck with people who already have very short memory, like Lucy Whitmore from "50 First Dates", Leonard from "Memento", or Dory from "Finding Nemo". Or you can make the experiment with Dory from "Finding Nemo". 

You can also perform the experiment with Dory from "Finding Nemo".

Those thinking they know the answer are mainly in one of two camps: halfers, who think she should answer 1/2, and thirdirs, who think she should answer 1/3. Thirdirs say that when Beauty is waken and interviewed, she thinks she can be in one of three situations. Since only in one of the cases the coin turned up heads, the answer must be 1/3. Halfers say that this answer is wrong, being probably caused by drug abuse, and since the coin is fair, the answer should be 1/2. There is nothing that can provide new information to Sleeping Beauty, so this answer should remain 1/2.

I will not detail here the debates still ongoing on the net, and the articles which are written about this. I just want to explain why I think that this debate is based on different understandings of the question.

Another experiment

Consider the following experiment.
- Prepare a large box, in which you can put apples and oranges, without seeing its content.
- Toss a fair coin.
- If the coin comes up heads, put one orange in the box.
- If the coin comes up tails, put two apples in the box.
- Repeat this many times.
- At the end, randomly extract a fruit from the box. Unless the experiment took too long, the fruits are not yet rotten, so you can extract a fruit.
- Then answer the following questions:
1. What is the probability that the fruit you will extract was introduced after the coin landed heads?
2. What fraction of the total times the coin was tossed, it landed heads?

The answer to question 1 is of course 1/3, because 1/3 of the fruits are oranges, and oranges were placed in the box when the coin landed heads.

The answer to question 2 is of course 1/2, because the coin is supposed to be fair.

My claim is that thirdirs were actually answering question 1, and halfers were answering question 2.

The question Sleeping Beauty was asked can be seen as equivalent to both question 1 and question 2.

To see how it can be seen as equivalent to question 1, consider a combination of the two experiments. Say that Sleeping Beauty is not only asked the question, but also it is given a fruit to put in the box. If the coin landed heads, she will receive an orange, and if it landed tails, she will receive an apple. She will put them in the box, then she will be put to sleep and forget about the awakening. Say the experiment is ran a large number of times. At the end, she can just count the fruits, and she will find that 1/3 of them are indeed oranges, so she will know that indeed the answer to the question is 1/3. Asking her about her belief that the coin landed heads that time is the same as asking her about her belief that she will receive an orange.

It is true that for every time the coin landed tails, she gets two apples, while every time it landed heads, she gets only one orange. This is why some tend to understand the question as being actually question 2.

Removing the confusion

So the dispute between thirdirs and halfers is due to the fact that they interpreted the question differently, and consequently answered different questions.

Instead of asking Sleeping Beauty the question as originally stated, we could just ask her two questions:
Q1. What is your belief that this awakening occurred following an event in which the coin landed heads?
Q2. What is your belief that when the coin was tossed, it landed heads?

Tuesday, March 25, 2014

Impossibility theorems, a counterexample (the seven bridges problem)

In mathematics and physics there are some results called no-go theorems, or impossibility theorems. To name just a few: Euler's solution to the problem of the seven bridges of Königsberg, Gödel's incompleteness theorems, Bell's theorem, Kochen-Specker theorem, Penrose and Hawking's singularity theorems.

Research is an adventurous activity - you can spend years on researching a dead end, or you can stumble by luck upon something worthy without even knowing (for example, the discovery by Penzias and Wilson of the cosmic microwave background radiation). To avoid spending years looking in the wrong places, researchers use various guiding lines. Impossibility results are some of them, which are by far the most reliable. Other guidelines are following the trends of the moment (also dictated by the need to publish and receive citations), following the opinions of authorities in the field, reading only what they read etc. I personally consider misguided the idea of interpreting the results and filtering what you read and research by using the eyes of the authorities, no matter who they are. But it is understandable that they may seem the best we have, and that anyway the "mainstream" follows them, so if you want to fit in, you have to do the same.

What about the impossibility theorems, aren't they more objective than just fashion trends dictated by authority figures? Of course they are. However, they apply to specific situations, contained in the hypothesis of the theorems. Moreover, they rely on a mathematical model of reality, and not on reality itself. While I think that the physical world is isomorphic to a mathematical model, this doesn't mean that it is isomorphic to the models we use.

I will give just a simple example. Remember the problem of the Seven Bridges of Königsberg. It was solved negatively by Euler in 1735, and led to graph theory and anticipated the idea of topology. The problem is to walk through the city by crossing each bridge once and only once. Here is a map, which is of course an idealization:

Euler reduced the problem to an even more idealized one. He denoted the shores and the islands by vertices, and the bridges by edges, and obtained probably the first graph in the history:

Euler was then able to show immediately that there is no way to walk and cross each of the bridges once and only once (without jumping like Mario or swimming in the river, or being teleported!). The reason is that an even number of edges have to meet at the vertices which are not those where you start or end the trip. But there are no such vertices in the above graph, so all four have to be starting or ending vertices. But at most two vertices can be used to start and end, so the problem has negative answer.

This illustrates the main point of this article. The problem has a negative answer, but this doesn't mean that in reality the answer is negative too. The mathematical model is an idealization, which forgets one thing: that the Pregel river has a spring, a source of origin. If we add the spring to the map, we obtain a different problem:

This problem has a simple solution, which is obtained by "going back to the origin":

This is "thinking outside the box", literally, because you have to go outside the original picture box. I came up with this solution years ago, when I was in school and read about Euler's solution. Of course, it doesn't contradict Euler's theorem, because the resulting graph is different than the one he considered, as we can see below:

Hence, Euler's theorem itself tells us how to solve the problem associated with this graph. The problem is solved by the very theorem which one considers to forbid the existence of a solution.

The main point of this simple example is that even in simple cases we don't actually know the true settings in which we apply the no-go theorems, or we ignore them to idealize the problem. We are applying the no-go theorems in the dark, so perhaps, rather than being guidelines, they are blocking our access to the real solutions of the real problems. While most researchers try to avoid being in contradiction with impossibility theorems, maybe it is good to reopen closed cases from time to time.

Sunday, January 26, 2014

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

Thursday, January 23, 2014

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.

Sunday, December 15, 2013

Defending my PhD Thesis

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
, 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 Quantum Gravity from Metric Dimensional Reduction at Singularities, to appear in Annals of Physics, 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 MetricarXiv:1111.0646.
[13] C. Stoica Big Bang singularity in the Friedmann-Lemaitre-Robertson-Walker spacetime, arXiv:1112.4508.