# Foliation of Space Time Hypersurfaces

by Alfrez
Tags: foliation, hypersurfaces, space, time
 P: 127 Hello, Anyone has figure or web site which illustrates how foliation of space time hypersurfaces look like or what they meant in the context of the following statement? "We have presented a means of embedding quantum non-locality within a background classical space-time (flat or curved), by introducing an absolute simultaneity associated with a preferred foliation by space-like hypersurfaces (where the preferred foliation defines a preferred local state of rest)."
 P: 2,284 Ummmm... what that paragraph says isn't the case... you know that right? "A preferred rest frame" read" 'New Aether'. Anyway, I can't imagine how you'd graphically represent spacetime-foliation... although Demystifier or Zenith8 probably do.
P: 8,470
 Quote by nismaratwork Ummmm... what that paragraph says isn't the case... you know that right? "A preferred rest frame" read" 'New Aether'. Anyway, I can't imagine how you'd graphically represent spacetime-foliation... although Demystifier or Zenith8 probably do.
Well, foliation just means dividing 4D spacetime into a stack of 3D slices--so by analogy, if you think of spacetime as a 3D chunk of salami, then put it through a salami slicer to divide it into a stack of thin-sliced 2D pieces, that's the basic idea of a foliation. Of course with spacetime you have to make sure that each slice is "spacelike", so any worldline confined to a single slice will be spacelike as well...if the OP doesn't know about the difference between timelike, spacelike and lightlike worldlines, he/she can ask (there's a basic intro on the wiki Spacetime article).

P: 2,284
Foliation of Space Time Hypersurfaces

 Quote by JesseM Well, foliation just means dividing 4D spacetime into a stack of 3D slices--so by analogy, if you think of spacetime as a 3D chunk of salami, then put it through a salami slicer to divide it into a stack of thin-sliced 2D pieces, that's the basic idea of a foliation. Of course with spacetime you have to make sure that each slice is "spacelike", so any worldline confined to a single slice will be spacelike as well...if the OP doesn't know about the difference between timelike, spacelike and lightlike worldlines, he/she can ask (there's a basic intro on the wiki Spacetime article).
True... and thanks... that makes perfect sense when put that way. I wish someone had used the salami analogy with me when I first tried to learn that one.

I can understand the necessity for space-like 'slices', of course, it would be rather useless to have a bunch of slices that no longer correspond to the original salami. edit: for worldlines. Also, thanks very much for the link, which has other good links. I have so much to learn, and I love it.
P: 127
 Quote by JesseM Well, foliation just means dividing 4D spacetime into a stack of 3D slices--so by analogy, if you think of spacetime as a 3D chunk of salami, then put it through a salami slicer to divide it into a stack of thin-sliced 2D pieces, that's the basic idea of a foliation. Of course with spacetime you have to make sure that each slice is "spacelike", so any worldline confined to a single slice will be spacelike as well...if the OP doesn't know about the difference between timelike, spacelike and lightlike worldlines, he/she can ask (there's a basic intro on the wiki Spacetime article).

I know what spacelike, lightlike, worldlines meant. But how do you model a preferred foliation in spacetime such that you can accomodate quantum non-locality in Bell's Theorem into spacetime? The idea is this. The original SR metric is lorentz invariant. Now what they do is to add a preferred foliation such that it can connect two spacelike points instantaneously. How do you do that? Illustrations by graphs or words, thanks.
P: 8,470
 Quote by Alfrez I know what spacelike, lightlike, worldlines meant. But how do you model a preferred foliation in spacetime such that you can accomodate quantum non-locality in Bell's Theorem into spacetime? The idea is this. The original SR metric is lorentz invariant. Now what they do is to add a preferred foliation such that it can connect two spacelike points instantaneously. How do you do that? Illustrations by graphs or words, thanks.
There is no preferred foliation in relativity, and likewise no preferred definition of simultaneity so no unique meaning to "connect two spacelike points instantaneously" (each foliation defines a separate definition of simultaneity, and the equations of GR work the same in any coordinate system regardless of the definition of simultaneity used). Searching for the text you quoted, it apparently comes from a chapter of the book "Einstein, Relativity, and Absolute Simultaneity" which seems to be mostly discussing philosophical ideas about how relativity could be compatible with the presentist philosophy...the quote comes from p. 148 which can be viewed on google books. I haven't read all the sections of this chapter that are available online, but just skimming some parts I don't think they are asserting there is any physical experiment that would define absolute simultaneity, they are just imagining that there is some "metaphysical" truth about absolute simultaneity and talking about how you'd express the laws of physics if you knew it (if God came down and told us which sets of events were 'really' simultaneous, say). Basically it can be seen as a different way of thinking about what's "really going on" without changing any empirical results, like the different "interpretations" of quantum mechanics (the article seems to start out with a discussion of the Bohmian interpretation of QM in fact).
P: 2,284
 Quote by JesseM There is no preferred foliation in relativity, and likewise no preferred definition of simultaneity so no unique meaning to "connect two spacelike points instantaneously" (each foliation defines a separate definition of simultaneity, and the equations of GR work the same in any coordinate system regardless of the definition of simultaneity used). Searching for the text you quoted, it apparently comes from a chapter of the book "Einstein, Relativity, and Absolute Simultaneity" which seems to be mostly discussing philosophical ideas about how relativity could be compatible with the presentist philosophy...the quote comes from p. 148 which can be viewed on google books. I haven't read all the sections of this chapter that are available online, but just skimming some parts I don't think they are asserting there is any physical experiment that would define absolute simultaneity, they are just imagining that there is some "metaphysical" truth about absolute simultaneity and talking about how you'd express the laws of physics if you knew it (if God came down and told us which sets of events were 'really' simultaneous, say). Basically it can be seen as a different way of thinking about what's "really going on" without changing any empirical results, like the different "interpretations" of quantum mechanics (the article seems to start out with a discussion of the Bohmian interpretation of QM in fact).
With human luck, god would probably tell us that there is no preferred frame for him either. Wouldn't that be a hilarious mixture of religion and science... heh.

Bohmian, that's 'Schrodinger Trajectories" and trying to keep up with the Bell ansatz leaving QM alone... with deBB... I think. I might be totally off here, but it seems that when your theory is based on interpretation, not unique results... we're no longer talking about meaningful research.

Unless the Bohmians are right... then they were unsung heroes. That would be known as, "International Crow-Eating Day"... may I not see it come to pass.
P: 127
 Quote by JesseM There is no preferred foliation in relativity, and likewise no preferred definition of simultaneity so no unique meaning to "connect two spacelike points instantaneously" (each foliation defines a separate definition of simultaneity, and the equations of GR work the same in any coordinate system regardless of the definition of simultaneity used). Searching for the text you quoted, it apparently comes from a chapter of the book "Einstein, Relativity, and Absolute Simultaneity" which seems to be mostly discussing philosophical ideas about how relativity could be compatible with the presentist philosophy...the quote comes from p. 148 which can be viewed on google books. I haven't read all the sections of this chapter that are available online, but just skimming some parts I don't think they are asserting there is any physical experiment that would define absolute simultaneity, they are just imagining that there is some "metaphysical" truth about absolute simultaneity and talking about how you'd express the laws of physics if you knew it (if God came down and told us which sets of events were 'really' simultaneous, say). Basically it can be seen as a different way of thinking about what's "really going on" without changing any empirical results, like the different "interpretations" of quantum mechanics (the article seems to start out with a discussion of the Bohmian interpretation of QM in fact).

No. That is not talking Bohmian or metaphysics. It is about explaining Bell's Theorem non-locality in the context of Special Relativity. Bell test Experiment like Aspect's clearly show there was non-local correlation that occur instantaneously. How do you explain that in SR? Even though no signal is transmitted. It still violates the spirit of Special Relativity because there is a prefered privileged frame of the non-local correlations. So the book is about how to relate Bell's Theorem to Special Relativity. Only Valentin and Maudlin still believe in SR and speculate that the only thing to make SR be compatible with it is to add preferred foliations in SR. While the other authors simply want to go back to absolute space and time. Bell himself stated the following:

"I think it’s a deep dilemma, and the resolution of it will not be trivial; it will require a substantial change in the way we look at things. But I would say that the cheapest resolution is something like going back to relativity as it was before Einstein, when people like Lorentz and Poincare ´ thought that there was an aether – a preferred frame of reference – but that our measuring instruments were distorted by motion in such a way that we could not detect motion through the aether. . . . that is certainly the cheapest solution. Behind the apparent Lorentz invariance of the phenomena, there is a deeper level which is not Lorentz invariant. . . . what is not sufficiently emphasized in textbooks, in my opinion, is that the pre-Einstein position of Lorentz and Poincare´, Larmor and Fitzgerald was perfectly coherent, and is not inconsistent with relativity theory. The idea that there is an aether, and these Fitzgerald contractions and Larmor dilations occur, and that as a result the
instruments do not detect motion through the aether – that is a perfectly coherent point of view. . . . The reason I want to go back to the idea of an aether here is because in these EPR experiments there is the suggestion that behind the scenes something is going faster than light.
Now if all Lorentz frames are equivalent, that also means that things can go backward in time. . . . [this] introduces great problems, paradoxes of causality, and so on. And so it is precisely to avoid these that I want to say there is a real causal sequence which is defined in the aether. (‘‘John Bell,’’ (1986) interview in Davies and Brown; cf. Bell 1987: 279; see also Bell 1984: 66–76)"

*back to me Alfrez*

Now since we don't like any Aether or Absolute Space and Time, then the only way to make Bell's Theorem compatible with SR is to add an artifical structure called preferred foliations and many physicists actually do this as you will see in many papers in arvix.

I just want to imagine how one do a preferred foliation of spacetime hypersurfaces. Any illustration or pointer or drwaings would be greatly appreciated.
P: 8,470
 Quote by Alfrez No. That is not talking Bohmian or metaphysics. It is about explaining Bell's Theorem non-locality in the context of Special Relativity. Bell test Experiment like Aspect's clearly show there was non-local correlation that occur instantaneously.
No they don't, they just rule out local hidden variables theories, there's no way to use these experiments to define a preferred definition of simultaneity (the equations of relativistic QM look the same in every frame). You may believe that there are nonlocal hidden variables and that the equations governing them do involve a preferred definition of simultaneity, but this is a belief about entities that cannot be probed experimentally in any way, only God could tell you which events are "really" simultaneous or what values the hidden variables take. Besides, although Bell's theorem does rule out local realist interpretations of QM it doesn't rule out all local interpretations of QM, for example advocates of the many-worlds interpretation say it is compatible with locality--see my post here along with the simple toy model of how a MWI-like interpretation can explain Bell inequality violations in a local way in this post.
 Quote by Alfrez So the book is about how to relate Bell's Theorem to Special Relativity. Only Valentin and Maudlin still believe in SR and speculate that the only thing to make SR be compatible with it is to add preferred foliations in SR. While the other authors simply want to go back to absolute space and time.
A preferred foliation is absolute time in the sense of a preferred definition of simultaneity. Do you see some distinction between these concepts?
 Quote by Alfrez Now since we don't like any Aether or Absolute Space and Time, then the only way to make Bell's Theorem compatible with SR is to add an artifical structure called preferred foliations and many physicists actually do this as you will see in many papers in arvix.
Can you give some examples of these papers? Again I would imagine they are dealing with interpretations of QM that postulate hidden variables not accessible to experiment, or else speculations that a theory of quantum gravity might contradict the "no preferred definition of simultaneity" rule of relativity. But certainly according to known physics, there can be no experiment that will determine a physically preferred definition of simultaneity, and thus no physically preferred foliation.
P: 127
 Quote by JesseM No they don't, they just rule out local hidden variables theories, there's no way to use these experiments to define a preferred definition of simultaneity (the equations of relativistic QM look the same in every frame). You may believe that there are nonlocal hidden variables and that the equations governing them do involve a preferred definition of simultaneity, but this is a belief about entities that cannot be probed experimentally in any way, only God could tell you which events are "really" simultaneous or what values the hidden variables take. Besides, although Bell's theorem does rule out local realist interpretations of QM it doesn't rule out all local interpretations of QM, for example advocates of the many-worlds interpretation say it is compatible with locality--see my post here along with the simple toy model of how a MWI-like interpretation can explain Bell inequality violations in a local way in this post. A preferred foliation is absolute time in the sense of a preferred definition of simultaneity. Do you see some distinction between these concepts? Can you give some examples of these papers? Again I would imagine they are dealing with interpretations of QM that postulate hidden variables not accessible to experiment, or else speculations that a theory of quantum gravity might contradict the "no preferred definition of simultaneity" rule of relativity. But certainly according to known physics, there can be no experiment that will determine a physically preferred definition of simultaneity, and thus no physically preferred foliation.

Are you saying that only Bohmian models use the concept of adding preferred foliations in spacetime hypersurfaces?

Let's say we don't add any foliations. But the fact is that non-local correlations really happen in Bell's experiments. You explained that Many World Intepretation can explain them. But what if MWI is not correct? Then how do you make sense SR and Bell's Theorem?

Let's assume the Copenhagen is right that there are no position properties before measurement, in other words, the particle doesn't exist in position B before it is measured. This means no signal is being exchanged and realism doesn't hold. But the fact that there are correlations in the wave function means the wave function itself has instantaneous capability. This would still violate the spirit of Relativity.

Also maybe you are saying the correlations is just random. No signal is being interchanged. So since causality is not violated, SR remains. But the fact is that the wave function itself can correlate itself faster than light outside of spacetime. That still violate SR, no??

About the papers at arvix, you may be right that only Bohmians use preferred foliations, but I'll search more.
 P: 1,781 I dont know why this upsets people. There may be nothing in relativity that calls out a special frame. But if there was a special frame for a particular physical phenomenon that is outside of relativity theory that doesn't mean relativity is broken. Personally I think it's an elegant solution to the problem. You can always find a frame in which two widely separated events are simultaneous. It seems like the perfect frame to use with correlated observations. The paradoxes go away in that frame. Is this that much different than the relativistic limo that fits into the short garage in one frame but not another? Relativity may not prefer the short car over the short garage but maybe QM requires they be the same length to have an expiation that doesn't seem paradoxical.
P: 2,284
 Quote by Antiphon I dont know why this upsets people. There may be nothing in relativity that calls out a special frame. But if there was a special frame for a particular physical phenomenon that is outside of relativity theory that doesn't mean relativity is broken. Personally I think it's an elegant solution to the problem. You can always find a frame in which two widely separated events are simultaneous. It seems like the perfect frame to use with correlated observations. The paradoxes go away in that frame. Is this that much different than the relativistic limo that fits into the short garage in one frame but not another? Relativity may not prefer the short car over the short garage but maybe QM requires they be the same length to have an expiation that doesn't seem paradoxical.
Well said, and why I always thought Relativity was intuitive in many ways.
P: 127
 Quote by JesseM No they don't, they just rule out local hidden variables theories, there's no way to use these experiments to define a preferred definition of simultaneity (the equations of relativistic QM look the same in every frame). You may believe that there are nonlocal hidden variables and that the equations governing them do involve a preferred definition of simultaneity, but this is a belief about entities that cannot be probed experimentally in any way, only God could tell you which events are "really" simultaneous or what values the hidden variables take. Besides, although Bell's theorem does rule out local realist interpretations of QM it doesn't rule out all local interpretations of QM, for example advocates of the many-worlds interpretation say it is compatible with locality--see my post here along with the simple toy model of how a MWI-like interpretation can explain Bell inequality violations in a local way in this post. A preferred foliation is absolute time in the sense of a preferred definition of simultaneity. Do you see some distinction between these concepts? Can you give some examples of these papers? Again I would imagine they are dealing with interpretations of QM that postulate hidden variables not accessible to experiment, or else speculations that a theory of quantum gravity might contradict the "no preferred definition of simultaneity" rule of relativity. But certainly according to known physics, there can be no experiment that will determine a physically preferred definition of simultaneity, and thus no physically preferred foliation.
 Quote by Antiphon I dont know why this upsets people. There may be nothing in relativity that calls out a special frame. But if there was a special frame for a particular physical phenomenon that is outside of relativity theory that doesn't mean relativity is broken. Personally I think it's an elegant solution to the problem. You can always find a frame in which two widely separated events are simultaneous. It seems like the perfect frame to use with correlated observations. The paradoxes go away in that frame. Is this that much different than the relativistic limo that fits into the short garage in one frame but not another? Relativity may not prefer the short car over the short garage but maybe QM requires they be the same length to have an expiation that doesn't seem paradoxical.
Craig argued (pls give your objections, clarifications, etc.) that suppose, on the other hand, that we reject the existence of causal connections between events occurring at A and B in favor of some interpretation according to which the photons at A and B are somehow correlated, but not causally connected. On this interpretation, the composite state consisting of the two photons with their respective measuring devices constitutes a single system, which is in a definite state (see Bohr 1964: 145–51). To affect the behavior of one photon via measurement is to disturb the whole system. When the wave function of a photon at A collapses, there is an immediate and correlated collapse of the wave function of its counterpart at B. But clearly, even though this interpretation denies the superluminal causal influence from A to B posited by the de Broglie-Bohm theory, it still just as effectively abrogates the relativity of simultaneity, since the collapse of the paired wave functions is simultaneous. Therefore, asseverations that such an interpretation would not run contrary to the received interpretation of SR because that theory prohibits only signals of superluminal velocities, not instantaneous correlations, are quite beside the
point.23 The point is that such correlations furnish the means of establishing relations of absolute simultaneity and a fundamental frame.24 As Maudlin points out

In Minkowski spacetime this theory of wave collapse no longer makes sense. The collapse can be instantaneous in at most one reference frame, leading to two possibilities: either some feature of the situation picks out a preferred reference frame, with respect to which the collapse is instantaneous, or the collapse is not instantaneous at all. (Maudlin 1994: 196; cf. pp. 137–38, 144)

The problem posed by instantaneous collapse of the wave function for relativity theory can be clarified by realizing that since, according to SR, simultaneity is relative to reference frames, the collapse of the wave function for spatially separated photons will itself become relative to a reference frame. The problem is that properties like polarization and spin are not relational,
but intrinsic properties of photons (Ryff 1992: 249). If the universe contained a single photon, it would have a definite polarization. Therefore the possession of such properties should not be made hyper-plane dependent. After surveying various attempts to integrate Bell’s Inequalities with relativity theory, Maudlin concludes that these attempts ‘‘entail such severe dislocations of our physical view that one must seriously consider whether our grounds for adhering to Relativity are really strong enough to justify such extreme measures’’ (Maudlin 1994: 239; cf. Eberhard 1978: 392–419). ‘‘Indeed, the cost exacted by those theories which retain Lorentz invariance is so high that one might rationally prefer to reject Relativity as the ultimate account of spacetime
structure’’ (Maudlin 1994: 220).
P: 2,284
 Quote by Alfrez Craig argued (pls give your objections, clarifications, etc.) that suppose, on the other hand, that we reject the existence of causal connections between events occurring at A and B in favor of some interpretation according to which the photons at A and B are somehow correlated, but not causally connected. On this interpretation, the composite state consisting of the two photons with their respective measuring devices constitutes a single system, which is in a definite state (see Bohr 1964: 145–51). To affect the behavior of one photon via measurement is to disturb the whole system. When the wave function of a photon at A collapses, there is an immediate and correlated collapse of the wave function of its counterpart at B. But clearly, even though this interpretation denies the superluminal causal influence from A to B posited by the de Broglie-Bohm theory, it still just as effectively abrogates the relativity of simultaneity, since the collapse of the paired wave functions is simultaneous. Therefore, asseverations that such an interpretation would not run contrary to the received interpretation of SR because that theory prohibits only signals of superluminal velocities, not instantaneous correlations, are quite beside the point.23 The point is that such correlations furnish the means of establishing relations of absolute simultaneity and a fundamental frame.24 As Maudlin points out In Minkowski spacetime this theory of wave collapse no longer makes sense. The collapse can be instantaneous in at most one reference frame, leading to two possibilities: either some feature of the situation picks out a preferred reference frame, with respect to which the collapse is instantaneous, or the collapse is not instantaneous at all. (Maudlin 1994: 196; cf. pp. 137–38, 144) The problem posed by instantaneous collapse of the wave function for relativity theory can be clarified by realizing that since, according to SR, simultaneity is relative to reference frames, the collapse of the wave function for spatially separated photons will itself become relative to a reference frame. The problem is that properties like polarization and spin are not relational, but intrinsic properties of photons (Ryff 1992: 249). If the universe contained a single photon, it would have a definite polarization. Therefore the possession of such properties should not be made hyper-plane dependent. After surveying various attempts to integrate Bell’s Inequalities with relativity theory, Maudlin concludes that these attempts ‘‘entail such severe dislocations of our physical view that one must seriously consider whether our grounds for adhering to Relativity are really strong enough to justify such extreme measures’’ (Maudlin 1994: 239; cf. Eberhard 1978: 392–419). ‘‘Indeed, the cost exacted by those theories which retain Lorentz invariance is so high that one might rationally prefer to reject Relativity as the ultimate account of spacetime structure’’ (Maudlin 1994: 220).
If I may sum this up...

You still believe that collapse is REAL?
P: 8,470
 Quote by Antiphon I dont know why this upsets people. There may be nothing in relativity that calls out a special frame. But if there was a special frame for a particular physical phenomenon that is outside of relativity theory that doesn't mean relativity is broken.
Yes it does, because relativity claims that all physical laws locally obey the first postulate of SR (work the same way in all inertial frames). Also, it would open up the possibility that the metric only predicts proper time for a restricted class of clocks, while other types of clocks could follow the same worldline but elapse a different amount of time (so in that sense the metric would no longer describe the structure of 'spacetime' in general but only certain types of clocks).
P: 8,470
 Quote by Alfrez Craig argued (pls give your objections, clarifications, etc.) that suppose, on the other hand, that we reject the existence of causal connections between events occurring at A and B in favor of some interpretation according to which the photons at A and B are somehow correlated, but not causally connected. On this interpretation, the composite state consisting of the two photons with their respective measuring devices constitutes a single system, which is in a definite state (see Bohr 1964: 145–51). To affect the behavior of one photon via measurement is to disturb the whole system. When the wave function of a photon at A collapses, there is an immediate and correlated collapse of the wave function of its counterpart at B.
Are you imputing some objective physical reality to wave collapse, as opposed to treating it as just part of our technique for modeling events in order to produce predictions about empirical results? If so this does not really even fit the standard version of the Copenhagen interpretation, and it is certainly disputed by other interpretations like Bohmian mechanics and the many-worlds interpretation. Also, note that if two observers at different locations are measuring members of an entangled pair, each is free to model things by assuming that their measurement collapsed the wavefunction of both particles, they will get consistent predictions about the probabilities of different possible combinations of measurements.

In any case, whatever you believe about what is "really happening" behind the scenes, do you dispute my basic point that in orthodox QM and relativity, regardless of which interpretation you choose, there is no experimental method that will pick out any physically preferred definition of simultaneity?
 Quote by Alfrez But clearly, even though this interpretation denies the superluminal causal influence from A to B posited by the de Broglie-Bohm theory
If you treat the collapse as having some objective physical reality that seems to me like a superluminal causal influence, as a measurement here is causing a change to an entangled wavefunction which also determines probabilities of different outcomes over there.
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 Quote by JesseM Are you imputing some objective physical reality to wave collapse, as opposed to treating it as just part of our technique for modeling events in order to produce predictions about empirical results? If so this does not really even fit the standard version of the Copenhagen interpretation, and it is certainly disputed by other interpretations like Bohmian mechanics and the many-worlds interpretation. Also, note that if two observers at different locations are measuring members of an entangled pair, each is free to model things by assuming that their measurement collapsed the wavefunction of both particles, they will get consistent predictions about the probabilities of different possible combinations of measurements. In any case, whatever you believe about what is "really happening" behind the scenes, do you dispute my basic point that in orthodox QM and relativity, regardless of which interpretation you choose, there is no experimental method that will pick out any physically preferred definition of simultaneity? If you treat the collapse as having some objective physical reality that seems to me like a superluminal causal influence, as a measurement here is causing a change to an entangled wavefunction which also determines probabilities of different outcomes over there.
Even if there is no experimental method that will pick out any physically preferred definition of simultaneity. You can't deny the fact that position A and B results are correlated in say Aspect experiment which violated Bell's Inequality. How can you explain the correlations supposing the Many World Interpretation and Bohm Mechanics are both incorrect??

Bottom line is, there are correlations. How can you explain them using local realism??