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I Einstein's elevator: gravity without curvature?

  1. Jul 1, 2017 #1

    pervect

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    This is a rather old issue, but one that has recently been on my mind.

    We often say that gravity is the curvature of space-time, with good reason. At the same time, we also talk about the "gravity" in Einstein's elevator, as an example of the equivalence principle. This is also with good reason, and is historically important. But Einstein's elevator is set in flat space-time. There is no curvature. So if we talk about gravity in Einstein's elevator, we're talking about gravity without curvature. But we have perhaps just previously said - or some other poster has previously said - that gravity is due to the curvature of space-time. Thus we are left with what appears to be a confused message.

    What's the best way of resolving this issue? Are we forced to talk about connections and Christoffel symbols to adequately define what we mean by gravity in Einstein's elevator? Is there a better way of doing this, preferably one that is accessible to B and I level readers? I don't really regard either connections or Christoffel symbols as being I level, and definitely not B level.

    On the flips side, can we modify or qualify saying that gravity is due to the curvature of space-time , in a way that will not confuse the presentation, but also allow us to talk about the "gravity" in Einstein's elevator?
     
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  3. Jul 1, 2017 #2
    In his thought experiment, by "gravity", Einstein meant Newtonian gravity, not GR gravity.
     
  4. Jul 1, 2017 #3

    Dale

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    @pervect

    I agree with your point. In Einstein's historical papers he seemed to (inconsistently but usually) use "gravity" to refer to the Christoffel symbols. That is also what is closest to the Newtonian force of gravity.

    Perhaps we should use "gravitation" to refer to the general phenomenon which is modeled using curved spacetime in GR and use "gravity" to refer specifically to the Christoffel symbols.
     
  5. Jul 1, 2017 #4

    Ibix

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    I'd be wary of that particular choice leading to the gravity wave/gravitational wave thing again.
     
  6. Jul 1, 2017 #5

    Nugatory

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    That's always felt backwards to me. The equivalence principle doesn't quite say that acceleration IS gravity, so it doesn't say that what's going on in the elevator is gravity. Instead, it says that (the local effects that we ascribe to) gravity can be understood as acceleration effects, like those we observe in the elevator. The elevator tells us a lot about how gravity works, but gravity tells us nothing about the elevator.
     
  7. Jul 1, 2017 #6

    PeterDonis

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    To be clear that the term "gravity" does not have a single meaning; it can refer to different things, so for clarity, if you use it, you have to make clear which meaning you are using. For example, if I want to use "gravity" to mean "spacetime curvature", for clarity I will say specifically "tidal gravity". Or if I want to use "gravity" to mean what is observed in an "Einstein elevator" scenario, I will say "acceleration due to gravity".
     
  8. Jul 1, 2017 #7

    Ibix

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    Agreed. Newton says that there's a fundamental difference between the elevator under thrust and sitting on a planet, but there's no way to tell which is which. The elevator is Einstein saying "here are two things that have webbed feet and go quack - maybe we should stop calling one of them a cat".

    So I think the point is that if we're trapped in a box and the floor is pushing up on our feet, we really ought to limit ourselves to observing that we're using an accelerating frame of reference. Only when we've detected tidal effects, or managed to spot some stars with changing redshift or something should we come down one way or another and say "this is gravity" or "this is acceleration".
     
  9. Jul 2, 2017 #8
    Perhaps I'm just illustrating the confusion you're talking about, but
    The clock at the top of the elevator run faster than the clock at its bottom, don't they? So we do have a curved spacetime here, don't we?
    Maybe it's not "curved" in the mathematical sense, but it certainly isn't Minkowski spacetime.
     
  10. Jul 2, 2017 #9

    PeterDonis

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    Yes.

    No. The difference in clock rates from bottom to top of the elevator does not, in and of itself, mean that spacetime is curved.

    Yes, it is. Google "Rindler coordinates". These are the coordinates in which observers at rest in the elevator are at rest. You will find that in these coordinates, the "rate of time flow" of an observer at rest depends on their "height" (position in the direction from bottom to top of the elevator), and that these coordinates are just a different set of coordinates on Minkowski spacetime.
     
  11. Jul 2, 2017 #10

    Ibix

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    It is Minkowski spacetime. How could me sitting on top of a rocket change the shape of spacetime? It's just that the natural coordinates to use in an accelerating elevator are curved ones that don't have the same notion of simultaneity as the usual inertial coordinates. And therefore objects that are at rest in this coordinate system are not following inertial paths and don't have the same simultaneity relationship that you would expect from inertial clocks.
     
  12. Jul 2, 2017 #11

    DrGreg

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    The coordinates in which the elevator is at rest are not Minkowski coordinates (and so non-inertial coordinates), but the spacetime itself is independent of your choice of coordinates and is still flat and is still Minkowski spacetime.
     
  13. Jul 2, 2017 #12
    So maybe the curvature isn't the most important part of gravity?
    Or, are you saying that gravity is the difference between an accelerating elevator, and a stationary elevator near a massive object?

    That would be quite an important problem with terminology, because to most people, gravity is the thing pulling everyone down, not the minuscule effects that can't be directly measured until you actually orbit the Earth in a rocket.
     
  14. Jul 2, 2017 #13
    Well I never quite understood how you can "transform away" acceleration. No matter how hard I'm imagining space and time axes around me, I still can't fly. So something must be wrong with the spacetime around here.
     
  15. Jul 2, 2017 #14

    PeterDonis

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    No, it means that curvature is not the same as gravitational time dilation. Spacetime curvature is tidal gravity.

    The difference in the geometry of spacetime, and therefore in the global "shape" of the path through spacetime of the elevator, yes. See below.

    That's why, when we want to be precise, we say that spacetime curvature is tidal gravity, specifically, instead of just "gravity".

    You can't. The acceleration (meaning proper acceleration--the acceleration that is felt by the person inside the elevator) is exactly what remains the same in the flat spacetime vs. the curved spacetime cases. What is different is the global "shape" of the path through spacetime of the person inside the elevator; but there is no way for that person, purely on the basis of measurements made inside the elevator, to know what the global "shape" of his path through spacetime is. To know that he has to look outside, at distant objects.
     
  16. Jul 2, 2017 #15

    Ibix

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    You can transform away coordinate acceleration. Just drop a set-square, and use it to define your coordinate system while it is free-falling, rather than while it is sitting on the surface of the Earth. In this description of the world the freely moving set-square is at rest, and the floor is accelerating upwards towards it.
    No, but you can free-fall. And you can choose whether to treat the floor as coming up to meet you or yourself as falling towards the floor. The former is the natural description in Newtonian physics; the latter in general relativity (at least at the local level).

    The point of the elevator/surface of the earth is that, in both cases, the floor is not moving inertially and a dropped ball is moving inertially. So the explanation for the ball falling is, in both cases, better phrased as "the floor is coming up to meet it". What's different is why the floor is non-inertial. In the elevator it's because the spacetime is flat and the floor is being accelerated through it. In the room it's because spacetime is curved but the "bottom" of the curved region is filled with matter which pushes up on the layer above it, which pushes up on the layer above it until you get to the surface, which pushes upwards on your feet, but not the ball because it's not in contact with it.
     
  17. Jul 2, 2017 #16
    AFAIK Einstein claims that there is no physical way to tell apart the effects of gravity from the effects of acceleration in the elevator. And not just locally but for as long as the elevator accelerates. And as DrGreg says the use of noinertial coordinates shouldn't affect the fact the elevator is in flat Minkowski spacetime. So I can se pervect's point that this thought experiment seems to blur the distinction between flat and curved spacetime as pertains to gravity physical effects. On the other hand there's the fact that the gravity effects that are usually associated to curvature in the curved spacetime setting are tidal effects that seem different from the effect obtained from acceleration in flat spacetime, but then one has to wonder how exactly then the principle of equivalence links acceleration and "actual" gravitational tidal effects.
     
  18. Jul 2, 2017 #17

    PeterDonis

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    I realized after posting this that this does raise a question about an argument that appears in the literature. I have posted a separate thread about it (note that this thread is an "A" level thread, since it requires considerable technical background to fully understand the argument in the literature that I am referring to):

    https://www.physicsforums.com/threa...me-dilation-imply-spacetime-curvature.919181/
     
  19. Jul 2, 2017 #18

    PeterDonis

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    No, this is not the case. Einstein's argument is limited to a short enough interval of time as well as a small enough elevator in spatial extent.

    It doesn't. Investigating whether, and to what extent, tidal gravity (spacetime curvature) is present goes beyond the EP.
     
  20. Jul 2, 2017 #19
    Sure, I was referring to the "gravity" effect, feeling weight.

    Yes. But then to what purely gravitational effect does the equivalence principle link acceleration in flat spacetime? Or you mean the EP is not related to GR?
     
  21. Jul 2, 2017 #20
    (Feel free to move this to the other thread)
    So if there is an infinitely wide and finitely deep flat slab, is the spacetime above it curved or not? Do I feel gravity or not? What is attracting me to it? And most importantly, how is this different from the accelerating elevator?
     
  22. Jul 2, 2017 #21

    PeterDonis

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    None. It links proper acceleration in flat spacetime to proper acceleration in curved spacetime.
     
  23. Jul 2, 2017 #22

    PeterDonis

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    It doesn't belong there. If anything, it belongs in a separate thread of its own, at least if you want to dig into it deeply enough. But I'll give a brief response here.

    It turns out that there is not a single consistent solution to the Einstein Field Equation that meets this description. (There is in Newtonian gravity, but we're not talking about Newtonian gravity here.) There are different solutions that come somewhat close, but each one has properties that are different from the Newtonian solution, and which will create intuitive difficulties if you try to interpret it the way the Newtonian solution is interpreted.

    You never feel gravity. You feel proper acceleration. If you put yourself in a rocket and turn on its engine, you will feel proper acceleration, which you can adjust to any desired value (such as 1 g) by adjusting the thrust of the engine.

    What your global path through spacetime looks like when you do that will depend on the global spacetime geometry--it will be different in flat spacetime vs. Schwarzschild spacetime vs. whatever "infinitely wide flat slab" solution you adopt for purposes of discussion--but none of that has anything to do with the equivalence principle or Einstein's "elevator" argument. As far as you can tell from measurements made inside the elevator, all of those cases will be indistinguishable.

    Um, the stress-energy of the slab, and its effect on the spacetime geometry? Remember that in GR, gravity is not a force, so thinking of yourself as being "attracted" to the slab isn't the best way to think of it--that's Newtonian thinking, and we're not talking about Newtonian gravity.

    As far as measurements made inside the elevator are concerned, it isn't. See above.
     
  24. Jul 2, 2017 #23
    Really? What about a globally (horizontally) closed space, so that the total mass would be finite?
    My point is, the space above the slab will likely have no curvature, since it is the same everywhere in space, and one dimension (time) is not enough to create a curvature.
    So we have a flat spacetime, but I can still walk on the slab. And there is a "gravitational" time dilation. Sounds like gravity to me.

    Edit:
    What effect really? If it's not curvature, then there must be some intermediate spacetime that fits between a flat and a curved one?
    Hmmm... (thinks a lot)... So what is "accelerating" me towards the slab?
     
    Last edited: Jul 2, 2017
  25. Jul 2, 2017 #24

    pervect

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    I'd tend to agree that most people think of gravity as a "thing that pulls everyone down". Henceforth "TTPED". And this does raises some important issues of terminology, which is one of the issues I wanted to discuss - what is a good terminology to use, one that ideally will reach a broad audience, from the most to the least sophisticated?

    In Newtonian physics, we can consider two cases. One is "real gravity" due to some nearby massive body. In Newtonian physics, "TTTPED" is in this case considered to be a real force. Another case we can consider is an accelerating elevator, Einstein's Elevator. In this case, "TTTPED" is not a real force at all, but a pseudo-force, or an inertial force. The easiest way to see this is to analyze EInstein's elevator in an inertial frame, where we can directly apply Newton's laws, and note that the only real force on an object on the floor of the elevator is the floor pushing up on the object, causing it to accelerate. Most I (intermediate) level and above readers will have been exposed to this concept, and perhaps some B-level readers as well, so in many cases it should (hopefully) just be a matter of refreshing their memory.

    Where do we get the idea that there is a "TTPED" in the elevator? This happens when we switch to a non-inertial frame of reference. To do this, we have to modify Newton's laws, they no longer directly apply in a non-inertial frame of reference. This is perhaps not a terribly hard thing to do, but the details of doing it are somewhat involved, and I think there are a lot of readers who just accept the result, because they've heard it before, without going through the calculations, or knowing the details of the calculations. Also, calculations of this sort would be at I-level, I think, and doing them would leave out all the B-level readers. But the result is what's important, in Newtonian physics, when we switch to an accelerated frame of rerference, we use laws that are formally similar to Newton's laws, but we add in these "fictitious forces" that are not real forces. And it is these fictitious forces that we feel on our feet when we stand in Einstein's elevator. (add). To be a bit pendantic, and try to be as precise as possible, I would say that what we actually feel on the soles of our feet is pressure, and that when we integrate the normal component of the pressure over the area of the "footprint", we get the total force, or weight.

    I'll touch lightly on another issue here that is that we expect there to be a lot of similarities between the elevator rider and "real gravity". It appears though that this point occasionally needs more discussion - perhaps not in this thread, though. We can be a bit more specific here - we can use a scale to measure the weight (as a force) of an object on the elevator, via the technique mentioned above (integrating the normal component of the pressure over the footprint), and that an identical scale in the "real" gravitational field will yield an identical reading for the weight (force) on the scale. In Newtonian theory this is a happy accident, GR provides a theoretical basis for the two scales to read the same number.

    The above discussion was Newtonian, things start to get even more interesting when we try to extend the calculations for special relativity. What happens is that in an accelerated frame, we not only have a "TTPED", but we also have other effects. One of the best-known of these effects is what can be loosely termed "gravitational time dilation". When we look at how we express the laws of physics in an accelerating frame in special relativity, we wind up having to do more than just add in a fictitious force, as we used to be able to do in Newtonian physics.

    I would say at the A-level that the best mathematical representation of the "TTPED" s the Christoffel symbol. (At least it's the best I can think of). Specifically we consider a free particle, and write the geodesic equations of motion for said free particle via the Christoffel symbols. We compare the resulting equations of motoins to those we'd get via "Newton's laws" to gain insight into how the particle moves. And we can write out the metric for the accelerated frame, and discuss the aspects of "gravitational time dilation". Unfortunately, I don't see these observations as being too helpful to the vast majority of readers at the B and I levels But I hope I can get across the point that going from an inertial frame to an accelerated frame is no longer a matter of adding in a fictitious force - we need to do more. And that this starts us down the path (though it doesn't take us all the way to the end of the path) in thinking that gravity in GR isn't just a force, at least not a "real" force.

    I don't think this is all that hard to understand, personally. If you take a ride on the "Vomit comet", or imagine doing so, you have a physical example of what it means to "transform away acceleration". You can fly - you just need a plane. When you do fly, you see that spacetime isn't so very different. Specifically, riding in a plane doesn't change the nature of space-time, but it can (with the right trajectory) get rid of the "TTPED", at least for a while.
     
    Last edited: Jul 2, 2017
  26. Jul 2, 2017 #25

    A.T.

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    As Peter said, distinguish between "attraction" and "tidal effects". DrGreg made a great diagram (see below). Explain that attraction needs no curvature locally, but is globally the only way to combine all those local non-inertial coordinate charts, that are at rest relative to each other.

     
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