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Basic questions about general relativity concepts

  1. May 28, 2010 #1
    I have a few things I don't quite understand in general relativity. In my course we have gone through a lot of mathematical stuff that I have blindly followed without really appreciating its significance and I feel bad about that.

    The first thing is, where did this idea of the curvature of spacetime come from? I know that the relativistic view of gravitation is based on the idea that the more massive an object is, the more it warps spacetime around it. I'm not sure where this idea came from though - as I understand it, Einstein started by forgetting about two massive bodies attracting each other by a force, and just said that the presence of a massive object affects the motion of other stuff (either massive or massless) around it.
    What made Einstein make the leap from that to a curved spacetime?
    How is this curvature actually measured?
    What is it about mass that causes spacetime curvature?
    Does an object moving in a curved spacetime have any perception of it?

    Also, when it comes to measuring distances (i should really say "intervals", right?) in a curved spacetime, do we still think of this as being measured with, say, a straight metre ruler, or does the metre ruler itself become curved to conform to the spacetime it is in?
    Does a metre in a curved spacetime still mean a metre in a flat one?
    When we measure the distance between two points in a curved spacetime, do we do the straight line distance as if the curvature wasnt there, or do we have to curve our metre ruler along the shortest distance between the points in this curved space (eg. surface of a sphere)?

    Clocks are another thing that get me. Say we are talking about two clocks, one in free fall and one fixed at some radius from a massive object. The one in free fall has no perception of a gravitational field, am I right? (If it was enclosed in a box, that box would freefall with it, and so it would not feel any resistive force.) It ticks "normally". But then say someone freefalling along with that clock sees another clock at some fixed radius from the large mass. That one does notice itself accelerating.
    The free falling clock-observer compares his clock to the fixed-radius one, and sees that in 1second, his clock hand has ticked further than the other clock. Well, what about that other clock - would an observer sitting on that fixed-radius clock be aware that his time was running slower? would he be sitting around getting bored thinking, bloody hell, this is taking ages?

    Also, spacetime expansion is my other problem, again about distances. Say I measure the distance to a star, I place down my metre ruler X number of times until I get from Earth to the star. Then I allow time to pass, the universe "expands". I repeat the measurement again.
    Do I measure that it takes (X + some more) metre rulers to get to the star, or has the length of my metre ruler expanded with the universe so that I still need X metre rulers to get there?

    A major thing that I didn't understand is the way that light supposedly slows down as a photon approaches the Schwarzschild radius of a massive object. How come this is allowed?

    Finally, the cosmological principle. "The universe looks the same to all observers". How can that be true?
    We could go to mars and set up a telescope, and we could identify all the same stars we can already see from Earth, but they would be from a new angle. How can two observers see the same thing?
    There could be an asteroid coming towards us, but an observer on a planet on the far side of that asteroid, we would see it moving away from us. How does the cosmological principle work there?

    Can you enlighten me on any of these issues?

    Last edited: May 28, 2010
  2. jcsd
  3. May 28, 2010 #2


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    Hi, don't feel too bad ;)

    Take a look at chapter 4 of Carroll's notes. One reason is because of the equivalence principle: locally in spacetime you can use special relativity and gravity is not detectable, just like locally on a manifold curvature is not "detectable" because a manifold looks flat locally per definition.

    It's not mass, but the energy momentum tensor! So every form of energy in general curves spacetime. For instance, electromagnetic fields cause curvature, unlike what Newton tries to tell you.

    The metric becomes a function of the coordinates and you need integration.

    If you parametrize a line with a parameter tau, the distance becomes

    s = \int \sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}d\tau

    Maybe I'll come to the rest tomorrow ;)
  4. May 28, 2010 #3
    ah thanks man, I'm looking forward to part 2.

    there was one other thing I was thinking - if a really strong gravitational wave passed by me, would I see everything distort like when you look at yourself in the back of a spoon?
  5. May 28, 2010 #4


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    GR doesn't answer that. It just describes the relationship between how things are distributed and how things move.

    Not if it only makes measurements in a small enough region of spacetime.

    This is pretty tricky. I think that both special and general relativity needs an axiom that says that we measure infinitesimal lengths with radar devices. (See e.g. this post). Then we should be able use the theory to determine the circumstances when a meter stick can make reliable measurements. (I'm guessing that it's reliable only when every part of it is doing geodesic motion). It really bugs me that books don't bother to discuss these things.

    A length measurement determines the proper length of a spacelike curve, so "length" is appropriate and "interval" is not.

    He can of course become aware of it the same way that your falling observer became aware of it, by a direct comparison of their ticking rates when they're really close to each other. You shouldn't think of this as one clock ticking "normally" and the other not. Clocks always do what they're supposed to, which is to measure the proper time of the curves in spacetime that represent their motion.

    The stick isn't expanding. And neither is the solar system. This post might help you understand why.

    It doesn't, and it isn't.

    It's just an approximate statement about the distribution of matter on very large scales, much larger than galaxies. It doesn't apply to the solar system any more than it does to your bathroom. But you will see about the same number of galaxies in every direction, and you can expect an alien in a galaxy a billion light-years away to do the same.
  6. May 28, 2010 #5
    You might find MTW's Gravitation worth the read, at least in the first portions.

    For you last question, no you would not perceive any distortions as you would be distorting yourself. You are in part however, describing a real effect that doesn't relate to G-Waves, called gravitational lensing. http://en.wikipedia.org/wiki/Gravitational_lens

    More of a fishbowl than a spoon, but you seem to have the grasp of it.

    The "Universe appears roughly uniform" doesn't mean that with a frame of reference we could not identity our position in a coordinate system. We are speaking in terms of the larger scheme of things: we appear to live in an Isotropic and Homogeneous Universe, not that the stars are in some fixed grid that is unchanging. This is a matter of what scale we are talking about. "the neighbors' lawn is homogeneous" does not extend to the individual height of each blade of grass when viewed on one's hands and knees.

    http://curious.astro.cornell.edu/question.php?number=453 [Broken]


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  7. May 28, 2010 #6
    Wow, some good questions!

    Where does the idea of curvature come from?
    Two things really:

    1. When two objects in space accelerate with respect to each other without a need for any propulsion (proper acceleration) one could postulate that space and time are curved.
    2. The theory predicts that space and time get deformed (e.g. Weyl and Ricci curvature), for instance the volume of a ball does no longer show the relation: volume = 4/3 *pi * r3

    How is this curvature actually measured?
    Using measurement of light from natural phenomena. Think for instance of the famous experiment taken in Principe.

    What is it about mass that causes spacetime curvature?
    Actually it is the total energy and momentum for a given volume.
    Why? I do not think anyone can explain, it is more of a philosophical question.

    Does an object moving in a curved spacetime have any perception of it?
    In principle, with any object except for a zero dimensional point one would be able to test it but remember that gravity is extremely weak. So it only would be practical in strong gravity situations. Unfortunately (or rather fortunately from our personal welfare perspective) we do not have any strong gravity situations nearby.

    Distances in GR
    Measuring distances in GR are very problematic. Especially in spacetimes that are non stationary, and any realistic spacetime is non stationary. One of the problems is that the "distance" is not fixed during the measurement. In some cases measuring A to B gives a different result than measuring B to A. One can for instance use a radar distance or distance measurement by slow movement. But each method could give a different distance.

    Clocks in GR
    Consider two small boxes with a clock in it, at a different heights above the planet free falling towards the center of a planet. Both would experience no difference in the perception of time but the clock falling faster towards the earth will run slower compared to the other clock.

    Spacetime expansion, same number of meter sticks?
    Yes all lengths get longer in the case of space expansions. However you have to see this in perspective, an expansion of the length of a meter obviously would not practically be measurable. But, in principle, you would need the same number of rulers as they also would have expanded. However the time it takes for a light signal would increase during expansion, and this is usually the way we measure large distances. Instead of saying space expands over time one could also say that light slows down over time. Distance expansion or light speed reduction in GR are completely equivalent, it just depends on what coordinate chart you pick.

    Slowing down of light at the Schwarzschild radius
    Because measuring distance is problematic in GR so is measuring speed.
    However, locally measured, the speed of light is always c.

    Everything expands including a meter stick. Obviously it would be not measurable since expansion is only a real measurable factor on the scales of distances between galaxies. You even state that in your own posting that you are referencing here.
    Last edited: May 28, 2010
  8. May 29, 2010 #7


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    I have skimmed a few articles about this, and they all agree that meter sticks and the solar system aren't expanding. The reasons they give are usually different than mine. I don't claim that my explanation is a complete answer, but I have never seen anything that I would consider a complete answer to this question. So I keep referencing mine because even though it's incomplete, I think it's essentially correct and much easier to understand than those articles.
  9. May 29, 2010 #8
    The expansion is explained using a GR dust solution. It is a metric expansion, one that applies for the whole universe, so everything in this universe will expand. Of the scale of a meter this expansion is obviously unmeasurable.

    The complete answer is not answerable at this stage because in GR you cannot simply plug and play various solutions at different scales and say it is so and so. GR is a non-linear theory and voodoo mathematics won't do.
  10. May 29, 2010 #9


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    There's no way that a solution derived from the assumptions of exact homogeneity and isotropy can describe the metric of a region of spacetime that contains our solar system and not much more. There's no reason to expect it to be even approximately correct.

    I find it quite bizarre that you dismiss my claim that the metric in such a region will have some features in common with a Schwarzschild metric and some features in common with a FLRW metric as "voodoo mathematics", when your claim is that one feature of the FLRW metric is completely intact in spite of the presence of a star. How do you justify that?

    You have also failed to realize that if we and our measuring devices are expanding with the cosmological expansion, we wouldn't see light from distant galaxies as redshifted. If the light was emitted with a wavelength of 600 nm, it will arrive with a wavelength of 600 nm. ((5.3.6) in Wald (on page 104) and the comment just below it confirms that the redshift factor is the same as the expansion factor).
  11. May 29, 2010 #10
    I agree with that.

    Well you have a point. :approve:
    Best answer would probably be not to speculate about meter sticks using a model that is totally no suitable for answering such questions.

    Not really
  12. May 29, 2010 #11
    with regards to that last bit, doesn't gravitational redshift come from a photon climing out of a massive object's potential well which costs it energy, rather than because of the motion of the source?

    also, lets forget about "metre rulers" when we are talking about expanding spacetimes. to be more clear, does our definition of how far a metre is change when expansion occurs?
  13. May 29, 2010 #12
    how can we ever hope to measure a gravitational wave at all then, if the measuring tool becomes distorted with the space?
  14. May 29, 2010 #13

    Sounds a bit like universal forces proposed by Reichenbach and others many years ago which affect all spatial distances similarly. To be "realistic" they are set to zero in the equations in which they appear and so, if they existed, would be undetectable at all scales even in theory and so a bit pointless. I believe even Reichenbach discarded the idea later. As I understand it, or perhaps misunderstand it, universal expansion is not of the same nature but has some non universal component, that is, its effect is not the same at all scales.

  15. May 29, 2010 #14


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    Yes (close enough), but we're talking about cosmological redshift, which is caused by the expansion of space.

    It would if our measuring devices are expanding too. That's why we can't just forget about them.
  16. May 29, 2010 #15
    It is the very fact of the distortion of the instrument in question that allows such a measurement. You, a person, would not see what the OP describes, but that does not mean that the event cannot be compared with the state "before", "during" and back to "before". If your instrument did not have a baseline, it would not be able to measure a change, but we are talking about a passing wave, and the measurement device (LISA, LIGO) records the minute difference from "before" and "during".
  17. May 29, 2010 #16


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    I'm no expert on this, but the impression I've picked up from what I've read is as follows.

    If rulers were made out of idealised "dust", i.e. particles that don't interact with each other via gravity, electromagnetism or any other mechanism, you would be right to say that dust-rulers expand with the Universe.

    But real rulers are made up of molecules that are held together in a semi-rigid lattice by electromagnetic forces (which are a huge order of magnitude higher that than the ultra-weak "repulsive force" of expansion), so they don't expand. Universe-expansion will add a very very tiny level of stress to the ruler, attempting to pull the two ends away from each other, but that force will be successfully resisted by the ruler's internal forces.

    A similar argument applies to the electromagnetic forces holding an atom together, the nuclear forces holding an atomic nucleus together, or the gravitational "forces" holding solar systems or galaxies together.
  18. May 29, 2010 #17
    It's fun that we don't fly apart like test particles, isn't it? :)
  19. May 29, 2010 #18
    I understand the argument.

    However, counter to this, and for the sake of argument: conform GR the metric expansion is equivalent to a universal slowing down of the speed of light, as a result the binding of those rulers will simply be a little less.

    Note that the measured distances in space would not increase when one would apply expanding rulers but the distance measured by radar methods would.
  20. May 29, 2010 #19
    does this not go against what someone has mentioned earlier (that if space expands, then the definition of a metre also expands?) because, say if the electrostatic force F=kQq/(r^2), then the value of r wouldn't change due to the expansion?

    personally I'm not happy with that idea of the definition of a metre growing as space expands, because then how would we ever be aware of any expansion going on at all?

    am I talking $hite?
  21. May 29, 2010 #20
    The definition of a meter is different than a rod of a given length. One is abstract, the other is material, and no, a meter is still a meter. There is more to measure with that meter, but if expansion were overcoming the electrostatic force, we'd be torn apart.
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