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A Better Equivalence Principle

  1. Feb 3, 2015 #1
    I am new to relativity, so bare with me. I do not want to argue against it; however, one must admit it is a rather awesomely unwieldy theory. In my self education on the subject, I am bothered by the Equivalence Principle, mainly this:

    An elevator accelerating through space is no different from an elevator at rest in a gravitational field. I understand the principle, but it bothers me, because unless relativity is applied to it, the two situations are not strictly equivalent. (An accelerometer will register a uniform gravitational force at both 'top' and 'bottom' with the first elevator, but less at the top than the bottom of the second elevator.

    Only within an accelerated frame of reference will an accelerometer register any readings. We can think of this as the sensing element of the accelerometer traveling at a lesser velocity than its surroundings, or that it is traveling at the velocity that its surroundings were traveling a measurable amount of time ago. Dividing the difference by this time shows us the acceleration of the elevator.

    [Remainder of post deleted, personal theory.]
     
    Last edited by a moderator: Feb 3, 2015
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  3. Feb 3, 2015 #2

    phinds

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    Yes, this is a well-known caveat on the equivalence principle
     
  4. Feb 3, 2015 #3

    collinsmark

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    That's not necessarily true for a rigid elevator.

    But before continuing, let's eliminate the tidal forces. Tidal forces are a major impediment for the math simplifying in GR, so let's model the situation differently for now so we can ignore tidal forces, and compare apples to apples.

    Consider a hypothetical, massive slab. The slab is infinite in both length and width, but has a finite thickness. The elevator is at rest relative to the slab, and is held by the gravitational field of the slab.

    This is the situation you want to use for the equivalence principle, comparing it to an elevator undergoing acceleration in empty space. In both situations you will find that clocks at the top of the elevator tick faster than clocks at the bottom, and accelerometers will measure different values depending on their height within the rigid elevator.

    No, I wouldn't go that route. Empty space doesn't flow through objects in any measurable sort of way. That's the idea of relativity: it doesn't rely on moving space. Everything is relative.
     
    Last edited: Feb 3, 2015
  5. Feb 3, 2015 #4

    PeterDonis

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    Only if the second elevator is so tall that tidal gravity is detectable over its height. If it's short enough (how short it has to be depends on the precision of measurement), it will be indistinguishable from the first elevator.

    The "caveat" being the clarification I gave above.

    I assume you mean "not necessarily true" in the sense I describe above, correct?

    Not really, no. The situation you describe is indistinguishable from an accelerating elevator in flat spacetime regardless of the height of the elevator; but it is still not indistinguishable from, for example, the gravitational field of the Earth.

    The equivalence principle says that, in the limit as the height of the elevator goes to zero, being at rest in any gravitational field is indistinguishable from being at rest in the accelerating elevator in flat spacetime (provided the proper accelerations in both cases are the same). That is both broader (it applies to any field) and narrower (it only applies in a sufficiently short elevator) than what you propose.
     
  6. Feb 3, 2015 #5

    PeterDonis

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    Basically, this says that an accelerometer will only register a reading if it is accelerated; in other words, it's a tautology.

    Unless you can present a consistent mathematical model of this and what you present in the rest of your post (and strictly speaking, by PF rules, you should provide a peer-reviewed reference that does this), we can't really discuss it here, because we don't have a well-defined model to discuss. I would strongly advise you to first become familiar with the model provided by SR and GR, before trying to construct one of your own.
     
  7. Feb 3, 2015 #6

    collinsmark

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    I also want to emphasize that the accelerometers in a rigid elevator rocketing/accelerating through empty space will not measure identical acceleration from the top to bottom of the elevator. Accelerometers placed near the bottom of the rigid elevator will measure more acceleration than those at the top. Similarly, clocks at the bottom of the elevator will tick slower. Shine a light from the bottom of the elevator to the top, and measured by somebody at the top, and it will be red-shifted; light shining from the top to the bottom will be blue-shifted.

    It might aid understanding to consider the elevator being observed from a non-accelerating inertial frame. As the elevator picks up speed, it undergoes Lorentz contraction as observed by the inertial observer; it gets squished up -- the top and bottom of the elevator become closer together when measured in this inertial frame. The bottom of the elevator has traveled a greater distance than the top of the elevator within the same amount of time. Thus it stands to reason that the bottom of the elevator has a greater acceleration.
     
    Last edited: Feb 3, 2015
  8. Feb 3, 2015 #7

    collinsmark

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    And this is equivalent to the acceleration measured when the elevator is sitting on the infinite slab.

    And that presents an important difference between Newtonian gravity and GR. In Newtonian gravity, the gravitational field of such an infinite slab is constant, independent of the height above the slab. However in GR, the measured gravitational acceleration is function of the height above the slab; it falls off as the distance to the slab increases.

    And it gets weirder too. In GR, whether we speak of an elevator accelerating through empty space, or an elevator sitting on the infinite slab, a virtual horizon will form below the elevator. This places a strong limit on the height of the elevator, if the top of the elevator is to undergo some given, finite proper acceleration. If the elevator is any taller than this limit, it would be ripped asunder by the horizon.
     
    Last edited: Feb 3, 2015
  9. Feb 3, 2015 #8

    PeterDonis

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    Strictly speaking, yes, this is correct; but for any finite precision of measurement, there will be some height such that, if the elevator is shorter than that height, the difference will not be detectable. Similarly for the redshift.

    It's worth noting, btw, that the presence of the redshift can be derived (using the basic Doppler shift argument that Einstein used) even if we assume that the elevator is short enough that the accelerations at the top and bottom are the same.

    Yes, but the nature of this horizon is different in the two cases (which makes the word "virtual" problematic, at least in the second case). In the case of the elevator accelerating in flat spacetime, the horizon is relative to the elevator and is not present at all for an inertially moving object. In the case of the infinite slab, the horizon is a real thing which is present for all observers; in fact it turns out to be similar to a "domain wall", a surface on which the stress-energy is, strictly speaking, infinite.

    No, it doesn't. The elevator can be as tall as desired. What must be finite is the distance from the bottom of the elevator to the horizon; that distance must be positive, and the shorter it is, the higher the acceleration at the bottom of the elevator (so for any finite acceleration at the bottom, there will be a specific finite distance to the horizon). But given that the bottom is at a particular height above the horizon, the top can be as high as desired.
     
  10. Feb 3, 2015 #9

    collinsmark

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    What I mean is that there is a height limit if the top of the rigid elevator is to experience a given, non-zero proper acceleration.

    If the top of the elevator is to experience an acceleration of g, then the elevator cannot be taller than c2/g, since that would place the horizon above the bottom of the elevator.
     
  11. Feb 3, 2015 #10
    Is this because of length contraction? What if its propulsion was mounted to the top of the elevator's frame and "dragged" the rest of the rigid elevator behind? Would that still be true? And what if infinite propulsion systems where mounted to the elevator's frame at every point ensuring that every point of the elevator moved with respect to every other point and independent of the atomic forces that might otherwise cause such a length contraction?

    I am curious about that.

    I don't really see how light red/blue shifting helps to ascertain what is happening locally at the top and bottom of the elevator...
    At best it shows that the elevator is moving
     
  12. Feb 3, 2015 #11

    PeterDonis

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    Ah, I see. Yes, this is true. Sorry for the confusion on my part.
     
  13. Feb 3, 2015 #12

    collinsmark

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    That is one valid way of looking at it, yes.

    As PeterDonis points out, the difference is usually negligible for short elevators and low rates of acceleration. But since we're free to discuss extremes in our thought experiments, it in fact becomes quite significant for very tall elevators and high rates of acceleration. These extremes also highlight some of the important differences between Newtonian gravity and GR.

    It matters not where the propulsion mechanism is mounted. The only assumption I'm making is simply that the elevator is rigid: The height of the floor to the ceiling of the elevator, as measured by the occupants of the elevator, doesn't change as the elevator accelerates.

    Not just moving, but accelerating. If the elevator was moving at a constant velocity there would be no blueshifting or redshifting, as measured by the occupants of the elevator, by lights shining within the elevator. But in an accelerating elevator, the occupants at the bottom will measure blueshifts and those at the top redshifts (as well as differences in the rates that their clocks tick, and differences in their accelerometers).
     
    Last edited: Feb 3, 2015
  14. Feb 3, 2015 #13
    My main purpose with posting this is that I cannot find any reference for such a theory. (not even a wack-job one)

    [Moderator note: remainder of post deleted, personal theory.]
     
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  15. Feb 3, 2015 #14

    PeterDonis

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    Then you should not be posting about it here. PF is not for discussion of personal theories. It is for discussion of mainstream science. Thread closed.
     
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