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Notions of simultaneity in strongly curved spacetime

  1. Nov 16, 2012 #1

    PAllen

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    While simultaneity conventions for inertial frames in flat spacetime (SR) are non-controversial, numerous questions, discussions, and debates in this forum indicate how confusing and controversial notions of simultaneity can be for more general cases. A couple of formal and true answers are generally unsatisfying to many:

    - Simultaneity is undefinable, in any preferred way, in general. It is never observable or measurable anyway.

    - You can pick any any event not in your past or future light cone to be a simultaneous event to your now. Except locally, there is no preference. (Sufficiently locally, one can argue for a preference for the Fermi-Normal simultaneity).

    I thought of a possibly useful way to classify simultaneity notions for fairly general spacetimes and observers (I assume an orientable spacetime). Of critical importance is that any sensible implementation of these notions for inertial observers in flat spacetime produce the same result. However, they may differ wildly in curved spacetimes and/or for non-inertial observers. I assume, in what follows, that any observer can be considered past/future eternal unless their world line encounters a singularity.

    1) It is reasonable to expect that any event in your causal past (on or inside your past light cone) is simultaneous to some event in your past.

    2) It is reasonable to expect that any event in your causal future (on or inside your future light cone) is simultaneous to some event in your future.


    Executing (1) in various ways produces a foliation (family of simultaneity surfaces) that at least covers the union of all of your past light cones (all events that are ever in your causal past). I propose to call such foliations past inclusive if they at least cover your total causal past, but may cover more; and past only if they cover nothing except your total causal past.

    Similarly, notion (2) leads to future inclusive and future only simultaneity conventions.

    Finally, one may require that simultaneity be designed to cover any event in your total causal past or future. Call these causal inclusive and causal only

    As mentioned, for inertial observers in flat spacetime, these are all the same, and the obvious implementation is Minkowski frames.

    Now consider these for the Oppenheimer-Snyder spacetime (asymptotically flat; collapsing space time region; interior and exterior SC regions eventually). I choose this for qualitative plausibility and to avoid the white hole region (the notions certainly apply to full SC geometry).

    A) Consider a distant, hovering, eternal, observer. Exterior SC type time slices represent an implementation of past-only simultaneity. No events on or inside the EH are covered. On the other hand, any future-only simultaneity implementation covers the interior, and indeed, is also a causal inclusive simultaneity. There are infinite such choices which can agree with local Fermi-Normal simultaneity.

    B) Consider an observer that is distant and hovering into eternal past, but at some moment free falls into the BH (late enough so they hit the singularity). For this observer, both past-only and future-only conventions include both interior and exterior events. However, past only covers only a portion of spacetime - ending with the past of the termination of free fall world line on the singularity. A future only simultaneity covers all of space time, and is thus also a causal inclusive simultaneity.

    In my opinion, it seems clearly desirable to favor causal inclusive simultaneity; and thus it is unfortunate that so much attention is paid to SC time slice simultaneity, which is exclusively a past-only simultaneity.
     
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  3. Nov 17, 2012 #2
    Without claiming to have understood all of the above, especially the different theories/interpretations mentioned, I would like to add the question on how far we can analyze such things as 'simultaneity' based on existing GR theory when considering 'strongly curved spacetime'.

    The reason for this thought stems from some statements of Einstein in his book 'The Meaning of Relativity' (6th Ed, 1955):
    "In this connexion the following should be noted: The present theory of relativity is based on a division of physical reality into a metric field (gravitation) on the one hand, and into an electromagnetic field and matter on the other hand. In reality space will probably be of a uniform character and the present theory be valid only as a limiting case. For large densities of field and of matter, the field equations and even the field variables which enter into them will have no real significance. One may not therefore assume the validity of the equations for very high density of field and of matter, and one may not conclude that the ‘beginning of the expansion’ must mean a singularity in the mathematical sense. All we have to realize is that the equations may not be continued over such regions."
     
  4. Nov 17, 2012 #3

    PeterDonis

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    In general this is more or less the present understanding of GR; it is an "effective field theory" that is a low-energy approximation to some more fundamental theory.

    However, none of that affects the predictions of GR about event horizons and black holes, at least not for BHs of sufficiently large mass (certainly any BH of stellar mass or more), because at the horizon of any such BH, and even far into its interior, there are no "large densities of field and of matter"; spacetime curvature for a hole of that size does not become large until you get close to the singularity at r = 0.
     
  5. Nov 17, 2012 #4
    Actually, this is a question which bugs me all the time. What do we consider as 'large' in terms of matter and field density? What is the cut-off point?

    I realize there is no hard answer to this question, and we have to go with certain heuristics. My understanding has been that matter and field density near or within a stellar mass black hole can be considered large enough.

    This may not apply to supermassive black holes, as the average matter density does tend to get closer to that of ordinary matter, but any black holes even a few order of magnitude smaller should qualify as having large matter and field densities?
     
  6. Nov 17, 2012 #5

    PAllen

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    If you want to consider quantum theories or 'what really happens in our universe', those are very different questions from what classical GR predicts.

    Because of the strength of evidence for supermassive collapsed object, any quantum + gravity theory must address the facts:

    - during collapse, average matter density is not large at time of crossing EH
    - curvature = tidal gravity is mild.

    Be that is it may, in the context of beyond classical GR, the question is wide open. There are, for example, several approaches where a true horizon never forms, even for a super massive BH (fuzz balls approach from string theory is just one of half dozen such approaches). Any approach that preserves unitarity would seem (IMO) to require some relaxation of true horizon behavior.
     
  7. Nov 17, 2012 #6

    PeterDonis

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    As you say, there is no "hard" answer to the question, because we don't know for sure what more fundamental theory classical GR is the low energy limit of. However, the best current belief, AFAIK, is that "large" means "approaching a value of 1 in Planck units", since Planck units are the natural units of quantum gravity. In other words, curvature becomes "large" when the radius of curvature becomes small enough to be of the same order as the Planck length. This was the criterion I was using when I said that the curvature at the horizon, and even deep into the interior, of a BH of stellar mass or larger is not "large"--the radius of curvature is many, *many* orders of magnitude larger than the Planck length.

    "Field density" means radius of curvature; see above for why it's not "large" near or within a stellar mass BH. For matter density, the corresponding criterion would be the Planck density (one Planck mass per Planck length cubed). The density of collapsing matter in an idealized spherically symmetric collapse is far smaller than the Planck density until the matter has collapsed almost to r = 0 (i.e., it is not "large" at the horizon and well inside it).

    Not by the Planck criterion. By that criterion what we consider "ordinary matter" has a density of something like 10^-93. Even neutron star matter has a density of something like 10^-80 in Planck units. It takes a *lot* more than a few orders of magnitude to get from "ordinary" densities, or even neutron star densities, to "large" densities in Planck units.
     
  8. Nov 17, 2012 #7
    OK, I wasn't aware of this. If that is the case, then I suppose nowhere in the Universe is matter and field density large, except very close to singularities within black holes.

    I was going by references I have come across stating 'curvature of space is large near a big star' or a neutron star. In fact, thinking about it, that may not be quite the same thing as matter/field density (rather its rate of variation perhaps), or may be those are also relative statements.
     
  9. Nov 17, 2012 #8

    PeterDonis

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    And very close to the Big Bang.

    Those statements are using a different criterion for "large", basically comparing the matter/field density to that of "ordinary matter". Which criterion you use depends on what you want to use it for. If you want to determine at what point classical GR, as a low-energy effective field theory, starts breaking down (i.e., stops being a good approximation), the Planck unit criterion is the right one to use (at least, according to our best current understanding).
     
  10. Nov 17, 2012 #9
    Yes, there is that. Perhaps another case would be at velocities close to c? Not at all sure that is correct, just a random thought...

    I can accept that as a good answer, to the best of our current knowledge as you mentioned.
     
  11. Nov 17, 2012 #10

    PeterDonis

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    No. The criterion can't be frame-dependent, and "velocity close to c" is frame-dependent.
     
  12. Nov 18, 2012 #11
    OK. I thought it might be a wrong idea even when I posted it. The clarification helps.
     
  13. Nov 18, 2012 #12
    I somehow missed this post earlier. Yes, I can see that logic.
     
  14. Nov 19, 2012 #13
    That looks very interesting. Can you translate the above into normal English, with which I mean the kind of physicists English that Einstein and Feynman used? Then likely more people will understand what you mean and participate. :tongue2:
     
    Last edited: Nov 19, 2012
  15. Nov 19, 2012 #14

    PAllen

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    If you ask a specific question, maybe I can help. I put a lot of time into writing that up, and it is as clear and simple as I know how to make it without writing a 'book'. IMO, Einstein and Feynman would understand it perfectly and be able to discuss it.

    Are you familiar with backward and forward going light cones, and their use to define causal structure of spacetime?
     
  16. Nov 19, 2012 #15
    ?? I suppose that they would be able to understand it and translate your mathematical English into plain English. Einstein would perhaps talk of rods and clocks, and Feynman would give colourful examples.
    Light cones, yes; their use to define "causal structure of spacetime", no. And I don't believe in "structure of spacetime" as a physical entity. Of course, I do believe in space-time events as physical occurrences.
     
  17. Nov 19, 2012 #16

    PAllen

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    Though we have had little luck understanding each other, I will try one tack (it would be really helpful if you asked a specific question).

    Do you think it is plausible to expect that if I compute that a physical detector somewhere in the history of the universe receives a signal from me, that I would want to assign a time coordinate to this predicted physical event?

    Background: I can compute, purely using SC coordinates (exterior + interior, with limiting process over SC radius), that a physical detector with its own clock falling with (but above, in vacuum) a collapsing body will receive a signal from me at a specific finite reading on its own clock, when it has fallen through an event horizon to near the singularity. Is there some 'hand of god' that prohibits me from assigning a time coordinate to this predicted physical event, because I will never detect this event?
     
  18. Nov 19, 2012 #17

    zonde

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    So basically you are saying that simultaneity is a convention, right?
    But it does not mean that you can use different conventions at the same time.

    With SC type time slicing there is no EH and no interior region for collapsing mass. All you can get is "frozen star". EH appears at infinite future i.e. never.

    In order to have EH and interior region with SC type time slicing you have to have eternal BH.
     
  19. Nov 19, 2012 #18
    Regretfully it would require specific questions about nearly ALL of the cited text - and I really think that this is why there was little feedback on your first post. However, it seems that we won't need it, see next:
    In fact you are continuing the discussion that I started earlier about the theoretical possibility of assigning distant time, even putting physical clocks at distant places, thus making the discussion very concrete and physical. Evidently this is what we agree on.

    However, that brings us immediately to the real sticking point that has all the time been lurking over the discussions of the last weeks:
    If you use a valid coordinate system, then there is nothing against it. The issue is about what kind of coordinate systems are valid in GR, and if perhaps contradictory mapping models can be made that match the mathematics of GR (but perhaps not all equally well matching the foundations), thus resulting in contradictory predictions.

    We know that this can happen with earth maps; however that is without consequence, as it's easily verified (I can give a simple example). It appears that the same problem occurred in GR, but without the possibility for a direct "reality check".

    On a side note there is a somewhat similar case in SR, with tachyons. Are tachyons really SR? Must they exist if one can "fix" the math to contain their mathematical possibility?
     
    Last edited: Nov 19, 2012
  20. Nov 19, 2012 #19

    PAllen

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    For the case of earth maps, do you claim there is case of conflicting prediction for maps as used in differential geometry:

    - associated with each map is a metric expression, such that each map expressed the same geometry
    - if one is talking about the same sphere using different maps, and one map doesn't cover all of the sphere, you use other maps to cover the rest, such that you are always describing the same complete sphere.

    Who decides what is a valid coordinate system? Differential geometry has a well defined, precise, answer to this (see any definition of topological manifold, refined further to become a pseudo-riemannian manifold).

    If you think this is wrong, then what is your precise criteria for a valid coordinate system? If it is different from above, you have a new theory, not GR as understood by everyone else. And in this new theory, general covariance is rejected, because that requires that any coordinates allowed by the criteria in the prior paragraph or good.

    One possible analogy for our disagreement is:

    - Imagine a 2-sphere in polar coordinates. Bob doesn't like what happens at or near the poles. So Bob decides to analyze only different object: a sphere missing a little disk around each pole. This is a valid, different geometric object. It is easy to demonstrate that you have holes using only polar coordinates with metric.

    Now in the case of O-S collapse, the hole you are proposing 'must' be accepted as the correct prediction of GR is rather strange. A clock in the middle of collapsing dust ball stops for no reason. It stops in a strange sense - locally everything proceeds at a normal rate until it is declared to stop.

    Note that for Krauss, et. all, assuming their quantum simulation is correct, they have good physical justification for this - this central clock is not acatually stopping; it evaporates in finite local time. Then it makes sense to talk about chopping a classical model at similar point.

    If, instead, you accept the the interior clock proceeds normally, there is no escaping (using any coordinates), that the clock is proceeding for some time after an event horizon has formed around it. Any signals it sends will not escape, but it can readily receive signals from an external observer.
     
  21. Nov 19, 2012 #20

    PeterDonis

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    No, this is not correct. The correct statement is: SC type time slicing cannot *cover* the EH and interior region.

    This is not correct either. SC type time slicing cannot cover the EH and interior region for *any* black hole spacetime; it only covers the exterior. But in both cases you cite (collapsing mass and eternal BH), the EH and interior region are part of the spacetime; they are just not covered by the SC type time slicing.
     
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