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Curvatures in space-time: actual reality or mathematical concept?

  1. Oct 31, 2012 #1
    According to GR is space actually supposed to be bending around mass, or is the concept of "curvature" just abstract and purely mathematical idea?
  2. jcsd
  3. Oct 31, 2012 #2


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    You could ask that question for basically every physical concept. Physics just desribes reality as we perceive it and beyond, without referring to the "actual reality" (whatever that means).

    Perhaps you like this story of Feynman:

  4. Oct 31, 2012 #3


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    It's space-time that's curved, not just space. But with the usual slicing of space-time into space + time, the spatail slices are curved as well.

    As far as it being "real" or not - I think of it as "real". But I'm not sure what sort of experiment (if any), would convince you that "curvature" or anything else for that matter was "real" or "not real".

    Typically, "reality" is addressed by philosophy and isn't experimentally testable.
  5. Oct 31, 2012 #4
    Can you rephrase that as a specific statement?

    a.) "curvatures" are only abstract mathematical idea

    b.) "curvatures" are abstract idea with a possibility to be actual reality

    c.) "curvatures" are ...?

    I'm not sure I agree with what you're saying. When physics say there is such thing as electron, do we consider it actual reality or just abstract idea? And when physics say there is such thing as electric or magnetic field, do we consider it actual reality or just abstract idea as well?
  6. Oct 31, 2012 #5
    I think "curvatures" can not really exist in some 3D spatial volume as actual geometrical feature without reference frame against which it would be curved against. Actual curves would require actual reference frame and "empty space" contains nothing, so for it to contain some actual topology seem to be direct contradiction.

    If you can explain some phenomena by supposing space is actually curved that I can not explain with the concept of potentials, gradients or fields, then I will submit space is actually curved as the best explanation.
  7. Oct 31, 2012 #6


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    That is one of the best criteria I have seen for "actual reality". Most people asking similar questions simply avoid trying to define what they mean by it. I congratulate you for making the effort.

    However, if you have even the slightest mathematical background then you should be aware that it is always possible to write the same equation multiple ways. That is basically what algebra is all about. Therefore, it is always possible to come up with a different but mathematically equivalent expression. Meaning that there isn't anything in physics which meets this standard.
  8. Oct 31, 2012 #7


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    Particles and fields are human ideas. They are useful in predicting phenomena in the observed reality.
  9. Oct 31, 2012 #8


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    You seem to be saying that in order to detect \ define curvature in any possible way you would need your hyper - surface to be embedded in some ambient space. This isn't true; even when dealing with regular surfaces in R^3 you will see that Gauss curvature is an intrinsic property of the regular surface. Also, on manifolds endowed with a riemannian metric, the curvature endomorphism is given in terms of the levi - civita connection which make sense without reference to some embedding. I don't understand your statement about "actual" topologies: you don't need to refer to curvature or the likes to even endow a set with a topology.
  10. Oct 31, 2012 #9
    That's a great story! Where is it from?
  11. Oct 31, 2012 #10
    On what basis do you claim that curvature is less-than-real, while fields and gradients are really-real? What does really-real mean anyway? How do you determine when something is really-real?
  12. Oct 31, 2012 #11
    I agree. The thing is people in cosmology field take the idea seriously and so we have these 'shape of the universe' models where the space is actually curved and can be closed and finite. I thought that was not logical assumption and is unreasonable because it could not be proven even if it was true, but people I spoke to felt differently and were firmly believing the space actually bends and curves and pointed me to 'Wilkinson Microwave Anisotropy Probe' measurements for the curvature of the whole universe. They measured zero curvature, as I naturally expected, but apparently NASA and our astronomer friends are far less agnostic about the whole "curved-space" idea.
  13. Oct 31, 2012 #12
    I would assume that, but I would not claim it without trying it out myself. I think there is a difference when you call abstract things abstract names like "field" or "potential gradient", because even if they are curvatures in reality they would still conform to this abstract definition, relations to distance and mass would stay the same. But if you give abstract things specific names, then you kind of insist that what it actually is. So I understand how there are people who consider 'curvatures' are actual reality rather than abstract idea, I just wondered if that's a bad thing or whether I lack some understanding.
  14. Oct 31, 2012 #13
    tris_d - just to be clear on this matter, GR's geometrical curved spacetime basis does provide a very successful predictive model - which may however fail at a certain level yet to be tested. Current measurements suggests perfectly flat or very close to perfectly flat overall spacial curvature but this is a prediction of inflationary model of cosmology rather than GR itself. Real problem imo is this conceptual dichotomy where in GR 'gravitational field' has totally different connotation to say electric field. One glaring problem for me is that gravitational field is allowed to have an ambiguously defined energy content but not allowed to act as it's own further source, despite the insistence that all other forms of stress-energy must contribute. Ask an expert why and good luck getting a sensible answer.

    The fact of gravitational waves as per binary pulsar data imo screams out one of two things - curvature of an actual physical medium on geometrical formulation of gravity, or physical field propagation through flat spacetime on field formulation of gravity. But definitely not just curvature of spacial and temporal relationships.
  15. Oct 31, 2012 #14


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    Well, in Newton's days one could say that gravity is actually some sort of invisible arm pulling masses together instantaneously. Einstein says it's because energy curves spacetime. Someone like Verlinde claims gravity is some sort of entropic force. From a string point of view it's because energy is exchanging closed strings with vibrational modes we call a "graviton". Some of these ideas overlap, some don't.

    What is "actual reality"? That's rather an abstract notion for a physicist, I think. Perhaps it's some fixed point in "theory-space" of which we hope our theories are converging to. Perhaps it's not fixed at all. Perhaps there are more points. I would say that every theory which describes nature as accurately, consistently and completely as is possible in a certain paradigm is as close as you can get. Personally I don't like concepts as "actual reality" or "absolute truth". It makes you forget that physics is all about description and perception. A wise man once said that description and perception come with deception.

    Perhaps I've missed it, but I didn't see a definition of "actual reality".
    Last edited: Oct 31, 2012
  16. Oct 31, 2012 #15


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    I'm sure it is in one of its books, but I've forgotten which one and google isn't helping :P
  17. Oct 31, 2012 #16


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    That is fine, as long as they understand that EVERYTHING in physics is, according to your classification, an "abstract idea" and not "actual reality". Force, energy, mass, time, potential, momentum, and everything else all fail to meet your criterion for being "actual reality", so it shouldn't be surprising that curvature does also.
  18. Oct 31, 2012 #17


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  19. Oct 31, 2012 #18


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    This is because gravity is different from electromagnetism. (Specifically, gravity obeys the equivalence principle while electromagnetism does not.) Why is that a problem? The two things are physically different, so the theories that best describe them are different as well.

    If it's the word "field" that you're getting hung up on, then drop it; or at least, don't apply it to gravity. "Field" is just a word.

    We've been here before, of course. :wink: You didn't like the answers you got before, but that doesn't mean they're not sensible (or correct, for that matter). But how about the answer I gave above: gravity is different from other things. Does that help?

    I really don't understand why people think it's a big problem that gravity is different from other things. What's so hard about that? I agree it would make some theoretical tasks easier if gravity shared more properties with other things, but the fact is that it doesn't.

    What is the difference, in your view, between "curvature of an actual physical medium" and "curvature of spatial and temporal relationships"?

    Btw, you are correct that, in the particular case of the binary pulsar data, you can also interpret the result as "physical field propagation", but not through a flat background spacetime--through a curved background spacetime. The propagating gravitational waves do not constitute the entire curvature; they only constitute the rapidly varying portion of the curvature.

    However, this "physical field propagation" interpretation breaks down in other cases, such as cosmology; I'm not aware of any workable interpretation of the expansion of the universe in terms of "physical field propagation".
  20. Oct 31, 2012 #19
    Yes, everything is human idea, it's all just words. We give things names, and we name their properties as well. We then relate this information to other such things we know, and that's our knowledge, it's all relative. The problem here I see is the question whether this thing we call "curvature" is a property of "space", or property of "mass", or both, whatever those things actually are.
  21. Oct 31, 2012 #20


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    I don't see the problem there. For our purposes, the curvature of a pseudo riemannian 4 - manifold is well - defined and computed on local coordinate charts. The EFEs then give the relation between the curvature and the stress - energy tensor.
  22. Oct 31, 2012 #21
    Curvature is here a mathematical characteristic of the "world of events" that is called space-time.

    Compare: http://www.bartleby.com/173/17.html
    http://en.wikipedia.org/wiki/Spacetime (see the intro + "Spacetime in general relativity")
    As a matter of fact, Einstein and others also used the expression "gravitational field". But "field" is a very vague, abstract concept; its original meaning is merely "area" or "zone".
    Last edited: Oct 31, 2012
  23. Oct 31, 2012 #22
    By 'actual' I mean just "measurable". But as long as the same measurement can be explained with some other concept where the space can stay flat I will believe that explanation is better. Basically, if there is some experiment that strongly suggests gravity indeed curves space rather than being some field or something in flat space, I'd like to hear about it.
  24. Oct 31, 2012 #23
    Hello Peter - I just had this sixth-sense feeling we would meet up again about here :rolleyes:. Sure the two (gravity vs EM) are different, and yes it's true I didn't like the answers given before because as I recollect there was no proper dealing with the question of gravity being excluded as source term in the SET. As I further recall you cited ADM definition as evidence gravity (but only as GW's) sort of contributed. But that begs the question - what is fundamentally different about static vs propagating field curvature that one can sort of contribute but absolutely not the other? This is what I mean about consistency issues in GR. There is no such divide with EM as source for instance - static or wave, they both carry energy and both act as contributors in SET. Hocus without the pocus?
    Gravitational waves for one!
    I understand that source - binary pulsars - as gravitating objects necessarily generates a static/quasi-static curvature through which GW's must propagate, but that's in geometric interpretation. You are well aware there exist other formulations of gravity where everything is owing to fields existing in an unobservable flat background spacetime. That's what I meant there.
    I'm no expert, but there are alternate descriptions/formulations using field concept - Yuri Baryshev comes to mind but there are others. In any case, sticking to geometric concept, how do you explain energy-momentum transport in GW's without it directly implying a physically 'stressed' medium of some sort?
  25. Oct 31, 2012 #24
    As far as I know, experiment cannot distinguish such things. Note that also "flat space" is a mathematical term; it has no meaning that I know of in a physical sense (how can empty space be literally "flat"?; that's like saying that it is "green"!).
  26. Oct 31, 2012 #25
    Sum of angles in a triangle! Greater than 180 degrees = +ve curvature, less than 180 degrees = -ve curvature. Note that this can be explained also using a non-geometric field theory = 'effective' curvature.
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