This paper can be downloaded for the following 30 days from the Physica Scripta website.
Link to the arXiv version.
Abstract:
An analytic extension of the Reissner–Nordström solution at and beyond the singularity is presented. The extension is obtained by using new coordinates in which the metric becomes degenerate at r = 0. The metric is still singular in the new coordinates, but its components become finite and smooth. Using this extension it is shown that the charged and non-rotating black hole singularities are compatible with the global hyperbolicity and with the conservation of the initial value data. Geometric models for electrically charged particles are obtained.
Friday, April 20, 2012
Sunday, January 22, 2012
Is semi-classical gravity wrong?
Semi-classical gravity is not considered fundamental, yet it escaped to experimental falsification. Of course, maybe we don't have yet the technology, or at least ideas of experiments we can do, to falsify it. It would be nice to be able to differentiate it experimentally from various quantum gravity approaches. But theoretically, it stands pretty well: being the most straightforward union between general relativity and quantum mechanics, it inherits their successes.
Are we sure that the theoretical reasons to reject it are so good? The regularization works promising for the semi-classical Einstein equation. The main problem seems to be that of singularities, but is there any evidence that this will not be solved?
One possibility is to rewrite Einstein's equation in a different way, which is equivalent to the original, but works in singularities. A simple type of metric singularity is when the metric becomes degenerate. The metric can be smooth (hence its components in a chart remain finite), yet the Kretschmann scalar can diverge, sure sign of a singularity. In arXiv:1105.0201, arXiv:1105.3404, arXiv:1111.0646 is developed the mathematics of such metrics, and it is consistent and without infinities, if the proper variables are used (for example, we have to use $g_{ab}$ and $R_{abcd}$, but not $g^{ab}$ and $R^a{}_{bcd}$).
Once we have this extension of the semi-Riemannian geometry developed, we need to show that we can apply it to the singularities of the Schwarzschild, Reissner-Nordstrom and Kerr-Newman singularities. In the standard expressions of these solutions, some components of the metric diverge. But there are coordinates which make the metric smooth - similar to how the Eddington-Finkelstein coordinates removed the apparent singularity on the event horizon, only that in our cases the metric becomes degenerate at the singularities. So, we can now write an equation equivalent to Einstein's, valid even at the singularities of these black holes, or at more general black holes which change in time, for example by Hawking evaporation (arXiv:1111.4837, arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099). We can write field equations on such spacetimes, and the information can now pass through these singularities. Similarly, we can write this extend version of Einstein's equation through a FLRW singularity, without having problems with the infinities (arXiv:1112.4508).
About the problem of the wavefunction collapse. If it is discontinuous, it will lead to violations of the energy conservation. It will also imply (never observed) violation of the conservation of other quantities like spin or electric charge. So, maybe the wavefunction remains all the way unitary. How can this be possible, when the projection postulate seems to tell that it is discontinuous? A possibility is described here, here, and at arxiv:1309.2309.
I don't want by this to claim that General Relativity, and semi-classical gravity, have no problems, or that they are all solved by the solutions presented here. What I want to say is that, to justify other more radical theories, people frequently make claims about how General Relativity fails in one place or another. GR will most likely be replaced by a more complete theory, but it is important to know how much we can keep from it, and how much we should change.
Are we sure that the theoretical reasons to reject it are so good? The regularization works promising for the semi-classical Einstein equation. The main problem seems to be that of singularities, but is there any evidence that this will not be solved?
One possibility is to rewrite Einstein's equation in a different way, which is equivalent to the original, but works in singularities. A simple type of metric singularity is when the metric becomes degenerate. The metric can be smooth (hence its components in a chart remain finite), yet the Kretschmann scalar can diverge, sure sign of a singularity. In arXiv:1105.0201, arXiv:1105.3404, arXiv:1111.0646 is developed the mathematics of such metrics, and it is consistent and without infinities, if the proper variables are used (for example, we have to use $g_{ab}$ and $R_{abcd}$, but not $g^{ab}$ and $R^a{}_{bcd}$).
Once we have this extension of the semi-Riemannian geometry developed, we need to show that we can apply it to the singularities of the Schwarzschild, Reissner-Nordstrom and Kerr-Newman singularities. In the standard expressions of these solutions, some components of the metric diverge. But there are coordinates which make the metric smooth - similar to how the Eddington-Finkelstein coordinates removed the apparent singularity on the event horizon, only that in our cases the metric becomes degenerate at the singularities. So, we can now write an equation equivalent to Einstein's, valid even at the singularities of these black holes, or at more general black holes which change in time, for example by Hawking evaporation (arXiv:1111.4837, arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099). We can write field equations on such spacetimes, and the information can now pass through these singularities. Similarly, we can write this extend version of Einstein's equation through a FLRW singularity, without having problems with the infinities (arXiv:1112.4508).
About the problem of the wavefunction collapse. If it is discontinuous, it will lead to violations of the energy conservation. It will also imply (never observed) violation of the conservation of other quantities like spin or electric charge. So, maybe the wavefunction remains all the way unitary. How can this be possible, when the projection postulate seems to tell that it is discontinuous? A possibility is described here, here, and at arxiv:1309.2309.
I don't want by this to claim that General Relativity, and semi-classical gravity, have no problems, or that they are all solved by the solutions presented here. What I want to say is that, to justify other more radical theories, people frequently make claims about how General Relativity fails in one place or another. GR will most likely be replaced by a more complete theory, but it is important to know how much we can keep from it, and how much we should change.
Thursday, August 11, 2011
Local Hidden Variables Correlations
To generate random angular momentum variables, press "Generate Random Data". In this case the correlations will be the classical ones (along the blue line), as it is predicted.
To test it on random generated orientations and plot the result, press "Plot Data".
Alternatively, if anybody believes that a local hidden variable theory can provide a set of data which gives the correlation of Quantum Mechanics, he or she can paste the data instead of randomly generating it, and then plot it.
For example, Joy Christian claims in arxiv:0806.3078v2, page 4, that an experiment he describes can provide a list of angular momenta which gives correlations = -cos of the angle between the two orientations chosen by Alice and Bob (the green curve). He claims by this that his local hidden variables can reproduce the outcomes of the EPR-Bohm experiment.
More exactly, Joy saids that after an experiment involving balls which explode in halves which have total angular momentum 0, a list of angular momenta can be collected. The second part of his experiment is to randomly generate on a computer pairs of directions a and b in space, and calculate the result using equation (16) from his paper, page 4. He then claims that the result will be -cos of the angle between the two orientations (the green curve), rather than the linear function represented in blue.
My application does exactly the second part of Joy's experiment. If Joy Christian or anybody else can produce this kind of data, they he can test it in this application. I already provided a mathematical proof that the only possible correlation depending only on the angle between a and b is the linear one, but there are people who don't trust the mathematical proof. Therefore, I challenge them to produce the data which will contradict my proof by counterexample.
Given that the output of the first part of Joy's experiment is just a list of angular momenta, you can produce it by performing the first part of Joy's experiment. But I will not require anybody to get the data only by actually making the experiment. Joy can produce the list by any means he wants, I will not constrain him to make the experiment. Just to provide a list of angular momenta which give his prediction.
I used JavaScript, so that anyone can easily verify the source code.
P.S. For the moment, there is a problem viewing the results in Internet Explorer, so please use Mozilla, Chrome or Opera instead.
P.P.S. Thanks to Florin Moldoveanu for the uniformization of the random generator.
P.P.P.S. This test may look similar to the Quantum Randi Challenge of Sascha Vongehr, but its purpose was different, as it can be seen from the text. I fully agree with Vongehr, just that my script has a different purpose, although it can be used for the Quantum Randi Challenge too. Vongehr's challenge appeared few months earlier than mine, in connection to J.C.'s claimed disproof of Bell's theorem. Mine refers only to J.C.'s proposed experiment in which he claimed that classical mechanics reproduces Bell's correlations, and appeared from a discussion I had with J.C. on FQXi's blog, where I also gave a proof that the kind of correlations he proposed can't be obtained in that way. The discussion started on Florin Moldoveanu's post, before he turned completely against J.C., and continued here, here, here, and here. For the "disproof", I shown there that J.C.'s proposal works only for singlet state, and it is in fact the quaternion formulation from Peres's book, hence no locality involved; I gave then many other arguments, but I will not repeat the discussion here. The only reason I made the script in this post was to show that the classical proposed experiment of J.C. (also inspired from the same book by Peres) can't give the same correlations as Bell's. He rejected the math proof (both Peres's and mine), so I made this script and asked him to introduce at that time, or after the experiment will be performed, the data he claimed will show the same correlations as Bell's. My point was that no data can give the Bell correlations by using the data as generated in his proposed experiment. I maintain this page only because I said I will wait for him to come with the data, but I am no longer interested in that topic.
Thursday, May 19, 2011
On Singular Semi-Riemannian Manifolds
Abstract
On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this article we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.
On Singular Semi-Riemannian Manifolds on arXiv
On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this article we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.
On Singular Semi-Riemannian Manifolds on arXiv
Wednesday, February 16, 2011
"Bit from It" vs. "It from Bit"
Julian Barbour presented his essay "Bit from It" at the FQXi essay contest Is Reality Digital or Analog?.
The essay is beautiful and I agree with the conclusion "Bit from It", in a way I will try to make clear. But I disagree with the way the conclusion was reached - it seems to me that the central part of Wheeler's 'ontology' "It from Bit" was overlooked, and this makes it look naive, while it is in fact very profound.
In a classical world, Wheeler's "It from Bit" would be obviously silly. When we measure something, we can write down the outcome as a string of digits, and by collecting all these digits we can determine the state. In such a world, "bit" would indeed originate from "it".
But Wheeler is discussing the quantum world. And for Wheeler, the quantum world is not just "classical world" plus "probability". Julian Barbour said: "Crucially, even if individual quantum outcomes are unpredictable, the probabilities for them are beautifully determined by a theory based on 'its'", but this is not the whole story. If this would be all, then he would indeed be right to say "I see nothing in Wheeler's arguments to suggest that we should reverse the mode of explanation that has so far served science so well". Julian Barbour tries to understand how Wheeler could do so trivial mistakes: "Wheeler's thesis mistakes abstraction for reality", and "A 'bit' has no meaning except in the context of the universe". Yet, there is no such a gross mistake.
Wheeler's "It from Bit" can be understood in the context of the "delayed choice experiment". He realizes that it is not enough to specify the outcome, but also what we measure - for example "which way" or "both ways" in the Mach-Zehnder experiment. But he realizes that our choice of what to measure determines how the state was (yes, in the past). This is the key problem of quantum mechanics, and this is the fundamental obstacle of all realistic interpretations of quantum mechanics: we choose "now" what to measure, and our present choice dictates how the state was, long time before we made our choice. We can think that there is an ontology behind the outcomes of our measurements, as in the classical world. But the "delayed choice experiment" shows that the "elements of reality" depend of the future choice of our measurements. And the outcomes depend of these choices too. So, it is in fact "the choice of what to measure" (Hermitian operator) plus "the outcome" (eigenvalue) that forms the "Bit" from Wheeler's "It from Bit". And the "It" is in fact the eigenstate corresponding to the obtained eigenvalue, given that the observable was that particular Hermitian operator. Wheeler was not that naive to think that eigenvalues determine eigenstates by themselves, without considering the Hermitian operator, so he accounted well for the prescription "A 'bit' has no meaning except in the context of the universe".
The central point of Wheeler's "It from Bit" is that the reality of today depends on the choices we make tomorrow, when we decide what to observe, and of the outcomes of the observations. He compares this with the game of 20 questions, when we try to guess a word by asking 20 yes/no questions, under the prescription that the choice of the word is not done at the beginning. The person who "knows" the word changes it by wish, so long as it remains consistent with the answer she already gave to our question. Wheeler wants to emphasize by this the similarity with the quantum state we try to determine, but which depends on what we choose to observe. This is why he was led to the idea that the state of the universe (it) results from the observations (bit).
I give more credit than Julian Barbour to the "It from Bit" philosophy - I view it as a way to present a central problem of quantum mechanics. I think, nevertheless, that it is exaggerated to conclude from this, as many do, that the world is digital. It may be or it may be not, but we should not force the conclusion. After all, the "It from Bit" philosophy is intended to clarify some points of a theory based on continuum - Quantum Mechanics.
My viewpoint on "It from Bit" is that we should regard the outcomes of measurements as "delayed initial conditions" for the Schrödinger's equation. I presented my view in this article, and this video. A solution of a partial differential equation like Schrödinger's is determined by a set of initial conditions. Classically, the initial conditions can be determined from future observations. In Quantum Mechanics, the future observations determine the state in the two meanings of the word "determine": passive - "find out what it is" (by the selection of an eigenvalue of the observable), and active - "choose what it is" (by the choice of that observable). Another central problem is that two consecutive observations of the same quantum system are incompatible, if the observables do not commute. That is, they impose incompatible initial conditions to the wavefunction. But, the second measurement is not, in fact, a measurement of the same system. The system interacted with the first measurement device, and this measurement device has many degrees of freedom which are not determined yet. So, the second observation measures in fact the composed system - the observed system plus the apparatuses used for the previous observations, and all the past interactions of the observed system. This may offer enough degrees of freedom to maintain the unitary evolution and to avoid a discontinuous collapse of the wavefunction.
My interpretation comes with a realistic wavefunction, which is not yet determined among the possible wavefunctions, but whose "delayed initial conditions" are determined by all future and past observations. I think that we cannot avoid the idea of "delayed initial conditions", no matter what "It" we choose to consider as the underlying ontology.
My view is therefore that "It from Bit" and "Bit from It" are reciprocal: a set of possible "It"s (solutions to the Schrödinger's equation), a set of possible "Bit"s (observations, delayed initial conditions) and the Universe is a pair (It, Bit), so that the "It" and the "Bit"s are compatible.
On the other hand, the "Bit" itself is part of the solution of the Schrödinger's equation, that is, of the "It". This is why I said at the beginning that I agree with "Bit from It". But if we have some "delayed initial conditions" - the "Bit"s - the "It" that satisfies to them is not necessarily unique. So, in fact, what we have is not a pair (It, Bit), but a pair ("It"s that satisfy to the observed "Bit", the observed "Bit"). There is a relation "one-to-many" between the "Bit" and the "It"s. The "Bit" appears to be discrete, but the "It" may very well be continuous. So, although "It from Bit" reflects an important aspect of Quantum Mechanics, it should not be taken too far.
The essay is beautiful and I agree with the conclusion "Bit from It", in a way I will try to make clear. But I disagree with the way the conclusion was reached - it seems to me that the central part of Wheeler's 'ontology' "It from Bit" was overlooked, and this makes it look naive, while it is in fact very profound.
In a classical world, Wheeler's "It from Bit" would be obviously silly. When we measure something, we can write down the outcome as a string of digits, and by collecting all these digits we can determine the state. In such a world, "bit" would indeed originate from "it".
But Wheeler is discussing the quantum world. And for Wheeler, the quantum world is not just "classical world" plus "probability". Julian Barbour said: "Crucially, even if individual quantum outcomes are unpredictable, the probabilities for them are beautifully determined by a theory based on 'its'", but this is not the whole story. If this would be all, then he would indeed be right to say "I see nothing in Wheeler's arguments to suggest that we should reverse the mode of explanation that has so far served science so well". Julian Barbour tries to understand how Wheeler could do so trivial mistakes: "Wheeler's thesis mistakes abstraction for reality", and "A 'bit' has no meaning except in the context of the universe". Yet, there is no such a gross mistake.
Wheeler's "It from Bit" can be understood in the context of the "delayed choice experiment". He realizes that it is not enough to specify the outcome, but also what we measure - for example "which way" or "both ways" in the Mach-Zehnder experiment. But he realizes that our choice of what to measure determines how the state was (yes, in the past). This is the key problem of quantum mechanics, and this is the fundamental obstacle of all realistic interpretations of quantum mechanics: we choose "now" what to measure, and our present choice dictates how the state was, long time before we made our choice. We can think that there is an ontology behind the outcomes of our measurements, as in the classical world. But the "delayed choice experiment" shows that the "elements of reality" depend of the future choice of our measurements. And the outcomes depend of these choices too. So, it is in fact "the choice of what to measure" (Hermitian operator) plus "the outcome" (eigenvalue) that forms the "Bit" from Wheeler's "It from Bit". And the "It" is in fact the eigenstate corresponding to the obtained eigenvalue, given that the observable was that particular Hermitian operator. Wheeler was not that naive to think that eigenvalues determine eigenstates by themselves, without considering the Hermitian operator, so he accounted well for the prescription "A 'bit' has no meaning except in the context of the universe".
The central point of Wheeler's "It from Bit" is that the reality of today depends on the choices we make tomorrow, when we decide what to observe, and of the outcomes of the observations. He compares this with the game of 20 questions, when we try to guess a word by asking 20 yes/no questions, under the prescription that the choice of the word is not done at the beginning. The person who "knows" the word changes it by wish, so long as it remains consistent with the answer she already gave to our question. Wheeler wants to emphasize by this the similarity with the quantum state we try to determine, but which depends on what we choose to observe. This is why he was led to the idea that the state of the universe (it) results from the observations (bit).
I give more credit than Julian Barbour to the "It from Bit" philosophy - I view it as a way to present a central problem of quantum mechanics. I think, nevertheless, that it is exaggerated to conclude from this, as many do, that the world is digital. It may be or it may be not, but we should not force the conclusion. After all, the "It from Bit" philosophy is intended to clarify some points of a theory based on continuum - Quantum Mechanics.
My viewpoint on "It from Bit" is that we should regard the outcomes of measurements as "delayed initial conditions" for the Schrödinger's equation. I presented my view in this article, and this video. A solution of a partial differential equation like Schrödinger's is determined by a set of initial conditions. Classically, the initial conditions can be determined from future observations. In Quantum Mechanics, the future observations determine the state in the two meanings of the word "determine": passive - "find out what it is" (by the selection of an eigenvalue of the observable), and active - "choose what it is" (by the choice of that observable). Another central problem is that two consecutive observations of the same quantum system are incompatible, if the observables do not commute. That is, they impose incompatible initial conditions to the wavefunction. But, the second measurement is not, in fact, a measurement of the same system. The system interacted with the first measurement device, and this measurement device has many degrees of freedom which are not determined yet. So, the second observation measures in fact the composed system - the observed system plus the apparatuses used for the previous observations, and all the past interactions of the observed system. This may offer enough degrees of freedom to maintain the unitary evolution and to avoid a discontinuous collapse of the wavefunction.
My interpretation comes with a realistic wavefunction, which is not yet determined among the possible wavefunctions, but whose "delayed initial conditions" are determined by all future and past observations. I think that we cannot avoid the idea of "delayed initial conditions", no matter what "It" we choose to consider as the underlying ontology.
My view is therefore that "It from Bit" and "Bit from It" are reciprocal: a set of possible "It"s (solutions to the Schrödinger's equation), a set of possible "Bit"s (observations, delayed initial conditions) and the Universe is a pair (It, Bit), so that the "It" and the "Bit"s are compatible.
On the other hand, the "Bit" itself is part of the solution of the Schrödinger's equation, that is, of the "It". This is why I said at the beginning that I agree with "Bit from It". But if we have some "delayed initial conditions" - the "Bit"s - the "It" that satisfies to them is not necessarily unique. So, in fact, what we have is not a pair (It, Bit), but a pair ("It"s that satisfy to the observed "Bit", the observed "Bit"). There is a relation "one-to-many" between the "Bit" and the "It"s. The "Bit" appears to be discrete, but the "It" may very well be continuous. So, although "It from Bit" reflects an important aspect of Quantum Mechanics, it should not be taken too far.
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