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Standing waves in GR

  1. Oct 4, 2007 #1
    I would appreciate if some learned folk could answer these questions to increase my understanding.

    1. Does GR support standing wave solutions to the equations?

    2. Do physicists study these standing wave solutions?

    3. Is it correct to describe GR equations as wave equations? (e.g. as Maxwell's equations and Schroedinger equations are)

    4. Do Black holes have vibrational modes that depend mainly on the event horizon radius R or 2*pi*R?

    5. Would such vibrational modes have periods calculated as R/c and 2*pi*R/c or are relativistic adjustments needed?

    6. Are periods related to these vibrational modes observed in galactic core black holes? I know that we cannot observe the gravitational waves, I am referring to associated emissions, such as light, that we might observe.

    7. Are GR standing waves ever considered as possibly relevant to large scale structure in the Universe?

    Any other comments along similar lines are most welcome.

    Thanks
    Ray Tomes
     
  2. jcsd
  3. Oct 5, 2007 #2
    Wave equations are linear. This is crucial, as it means the sum of two separate solutions (i.e. two waves) is also a solution (this is how we get interference effects in phenomena governed by wave equations).

    The Einstein Field equations that describe the curvature of spacetime are non-linear. That means simple sums of solutions are not also solutions. These equations are not wave equations.

    A lot of your other questions do not apply, since GR is not primarily a theory of waves.
     
  4. Oct 5, 2007 #3
    Masudr, thanks for your reply.

    I understand what linear wave equations are and that GR equations are not linear wave equations, but it does seem to me that there are such things as non-linear wave equations. Indeed a search for such finds such sites as these:
    http://en.wikipedia.org/wiki/Wave_equation which states that "Another common correction is that, in realistic systems, the speed also can depend on the amplitude of the wave, leading to a nonlinear wave equation"...
    http://tosio.math.toronto.edu/wiki/index.php/Wave_equations which states that "Non-linear wave equations: Nonlinear wave equations arise in physics from two major sources: relativity and elasticity. All relativistic field equations in (classical) physics are variants of the free wave equation or Klein-Gordon equation on Minkowski space."

    So while understanding that any wave solutions in GR will not in general be able to be added together to get other solutions, it does seem that there are such solutions in GR. My question is really whether these are studied and how much is known about them.

    I had been assuming for a long time that such was the case, until someone questioned me on that assumption and I find that it may not be so.
     
  5. Oct 5, 2007 #4
    Here's my take on it:
    1,3. Both GRT and Maxwell's equations obviously do support waves (and hence standing waves), but neither should be called a wave equation.
    2. There is no reason to expect such standing waves to actually occur, I imagine soliton-type waves are a much more interesting topic.
    7. No. The large scale structure doesn't exactly appear periodic like a standing wave, does it? (The relationship between GR and fundamental particles might be a better question.)
    4,5,6. Look up quasinormal modes. (No, we'd of heard if they'd been detected, and is it not obvious that you can't expect the correct answers about black holes from classical physics?)
     
  6. Oct 6, 2007 #5
    Well I am confused by this. Wikipedia http://en.wikipedia.org/wiki/Wave_equation states:
    "is an important second-order linear partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics."
    And http://en.wikipedia.org/wiki/Maxwell's_equations states:
    "A Dynamical Theory of the Electromagnetic Field Maxwell's modified version of Ampère's circuital law enabled him to derive the electromagnetic wave equation, hence demonstrating that light is an electromagnetic wave."
    Furthermore, http://en.wikipedia.org/wiki/Electromagnetic_wave_equation states:
    "The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum."
    Perhaps you can explain what you mean by your statement?
    Is there a reason to expect them not to occur?
    Well, yes it does. There is a periodic pattern of galactic clusters on a scale of 128/h Mpc that is pretty consistently detected.

    OK, I did and it talked about dissipation of energy. I suppose that is your point.
    Exactly. That is why I asked about standing waves in GR. Certainly I understand that in a non-linear system any standing waves will dissipate energy. The questions then are how fast will it do so and does that mean there cannot be such systems with substantial energy still in existence? I am wondering if these are solved problems or too difficult problems or if people have just assumed that there will be nothing going on?
     
    Last edited by a moderator: Oct 9, 2007
  7. Oct 6, 2007 #6
    When you write the Einstein equations in 3+1 (so called ADM) form, you get a hyperbolic-elliptic equation and hyperbolic equations are sometimes called wave equations. People actualy use this form to prove existence and uniqueness of solutions to and computationally simulate the Einstein equations. Also you have to remember that if you linearize the equations you get exactly the linear plane wave equations. But in case the 3+1 splitting does not work, I do not know what the situation is. About the standing wave I have no idea (I am just starting to study GR).
     
  8. Oct 9, 2007 #7
  9. Nov 8, 2007 #8
    Last edited by a moderator: Nov 8, 2007
  10. Nov 8, 2007 #9
  11. Nov 8, 2007 #10

    ZapperZ

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    I would strongly advise that you ease up on the use of purely ArXiv references unless you have the exact references to where these were published.

    Furthermore, we require reputable, peer-reviewed journals as valid sources, not "livingreviews.org". Please refer to the PF Guidelines that you have agreed to.

    Zz.
     
  12. Nov 8, 2007 #11
    Hey.. what's wrong with living reviews?
     
  13. Nov 8, 2007 #12
    Zapper, read this FAQ. My supervisor specifically encouraged my citing living reviews for a thesis. And as for the two arXiv links, only the first one fails to immediately cite it's publication reference (and it's sufficiently recent, what do you expect?). For articles published in actual journals, I still think it is preferable here to link to the arXiv abstract (which normally cites a print journal in addition to giving those without subscription access a choice of full text formats).
     
    Last edited: Nov 8, 2007
  14. Nov 8, 2007 #13

    ZapperZ

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    I have. And I would hope that other than the Particle physics and BTSM forum, we can at least go easy on using websites and arxiv articles when we're dealing with research-front issues in this and the rest of the physics forums.

    I shouldn't have deleted that link, so if someone who has a buffer of it, please either PM me or repost it back in here.

    Zz.

    Edit: Yes, in cases where the Arxiv article actually cite the print journal reference, that is perfectly fine. I know that at least one of the Arxiv article being referred to here has such a citation.
     
    Last edited: Nov 8, 2007
  15. Nov 8, 2007 #14
    Thank you so much for reminding me that the ArXiv is obviously a crank and crackpot site. Please forgive me for linking to a paper by one of Kip Thorne's students - God knows those people in Southern California have gotten way too much sun.
     
  16. Nov 8, 2007 #15

    ZapperZ

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    I ASKED that you go easy on those, rather than delete them outright. I have all of my papers on ArXiv as well, and based on my experience, at least 2 of them are quite different than the versions that finally got published. So just because they are uploaded on there still do not make them the source that one wants to cite, no matter who wrote them.

    I fully understand its use - High energy/particle physics/String/etc. all make use of them extensively. I only REQUEST that we use them sparingly, rather than predominantly, in the other physics sub-forums on here since most of these other subject areas are still strongly based on peer-reviewed sources.

    The other issue here is that in cases of disagreement and rebuttals, citation indexes for peer-reviewed journals are a lot easier to look up. One needs to know who have cited such papers, and how they are cited. In cases where there are still disagreement, this is a valuable tool.


    Zz.
     
    Last edited: Nov 8, 2007
  17. Nov 8, 2007 #16

    Stingray

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    FYI, relativity is fairly close to particle physics in how it uses the arXiv. I'm not really familiar with the process in condensed matter, but I get the impression that it's rather different (I assume that's the viewpoint you're coming from).

    Also, Living Reviews is a very reputable source. Despite the cheesy name, it's essentially a relativity-specific version of RMP.
     
  18. Nov 8, 2007 #17
    ZapperZ -

    I apologize for not understanding the protocol concerning links, although I must admit that all I was trying to do was point someone in a general direction to answer their questions (which clearly is a research problem).

    But now that I have your attention, perhaps you could explain to me what happens when two gravitational waves interact - without using sources.
     
  19. Nov 8, 2007 #18

    pervect

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  20. Nov 9, 2007 #19

    robphy

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