Does Gravity Bend Light or Curve Space-Time?

[pervect]

Pervect, I was simply asking a question on what you wrote:


Obviously I am talking about the top and the bottom of the elevator in your example.
If the elevator accelerates exactly as in a gravitational field does the distance between the top and the bottom change?
If so, would it not be a logical assumption that since top and bottom are in relative motion with each other that there must be a Lorentz factor involved?

OK, if we imagine the Rindler metric as an elevator...

In this elevator, the acceleration goes on, without change, for infinity, so there are no dynamical issues. The elevator never starts accelerating or stops accelerating - it is, and always has been accelerating. This makes the math a lot simpler and avoids a lot of "rigid body" related issues.

Both the distance (integrated Lorentz interval) and the light-travel time between the top and the bottom of the elevator are constant.

 

[MeJennifer]

OK, if we imagine the Rindler metric as an elevator...

In this elevator, the acceleration goes on, without change, for infinity, so there are no dynamical issues. The elevator never starts accelerating or stops accelerating - it is, and always has been accelerating. This makes the math a lot simpler and avoids a lot of "rigid body" related issues.

Both the distance (integrated Lorentz interval) and the light-travel time between the top and the bottom of the elevator are constant.

If that were true then would there not be only one kind of accleration?
But we can distinguish between proper acceleration and acceleration outside the accelerating frame, would that not indicate a Lorentz factor to you?

 

[notknowing]

OK, if we imagine the Rindler metric as an elevator...

In this elevator, the acceleration goes on, without change, for infinity, so there are no dynamical issues. The elevator never starts accelerating or stops accelerating - it is, and always has been accelerating. This makes the math a lot simpler and avoids a lot of "rigid body" related issues.

Both the distance (integrated Lorentz interval) and the light-travel time between the top and the bottom of the elevator are constant.

Did you maybe find the time to take a short look at my little derivation in post 29 ? I'm afraid it was drown into the other messages ...
I'm not so sure there are no dynamical issues at constant acceleration. At every small time step, the bottom of the elevator is increased in speed relative to to the top of the elevator ...

 

[pervect]

If that were true then would there not be only one kind of accleration?
But we can distinguish between proper acceleration and acceleration outside the accelerating frame, would that not indicate a Lorentz factor to you?

I'm not sure I'm getting your point. This thread is getting a bit overloaded, so I'll start another thread about distance in accelerated frames.

 

[Chris Hillman]

Clarifying my position

Hi all,

there is apparently more to this question. See for instance

http://www.math.ucr.edu/home/baez/PUB/deser

I also believe that Jheriko was probably referring to the work by Deser et al. on an infinite sequence of corrections to the naive linearized theory, which eventually "yields general relativity". (This is one of the most difficult, but most intriguing, routes to "deriving gtr from first principle".)

If we assume (for the sake of argument) that there is a self-consistent theory that enforces a particular topology on space-time but is somehow locally equivalent to GR,

This is an essential point (unfortunately probably too sophisticated for PF, since it might take years of graduate level study to appreciate "local" versus "global" issues, and most readers here do not possesses this kind of background). There were one or two threads in sci.physics.research on this point long ago, where I and others pointed out that newcomers to the literature can easily misunderstand claims from string theory proponents, for example.

If I understand Chris Hillman's position on this issue correctly, he believes that there isn't even a self-consistent theory of this nature (?).

I insist that the "local versus global distinction" is absolutely critical when examining claims that some theory constitutes a "reinterpretation" or "reformulation" of gtr. In particular, I believe that the work of Deser et al. (which is solidly mainstream) need to be carefully interpreted in this light. That is, if I am not mistaken, Deser et al. show that under their assumptions, in any sufficiently small neighborhood one must obtain something indistinguishable from gtr. I would add that it shouldn't be surprising that a classical field theory of gravitation, which is a metric theory, might have difficulty in unambiguously determining a unique topology, or that for many "initial values", solutions in such a theory might develop Cauchy horizons, so such difficulties appear to be common to a large class of theories.

As far as I tell, it is not yet known whether some well-defined theory of gravitation exists which is "locally equivalent" to gtr, but which in some sense excludes solutions which are spacetimes with nontrivial topology. Although there are many claims to this effect in the literature, as far as I can recall, I consider the ones I have studied unconvincing or even incorrect. And I think we must expect that obtaining the required "topological filter" in a convincing fashion might be very difficult. It appears to me that this would require exiting the domain of classical field theories.

One can also ask whether or not there is yet rock-solid evidence for nontrivial topological features of the universe in which we live. Or perhaps better put: one can ask whether or not there is rock-solid evidence that no model in gtr (Lorentzian four-manifold plus any additional mathematical structure required to describe nongravitational physics in the model) which fails to feature nontrivial topology can be consistent with all the available evidence. As far as I know, a reasonable answer would be "not yet, but astrophysics seems to be generally headed in that direction".

Note that nontrivial topological features could arise in many ways:

1. It might turn out that the "best-fit" FRW models are actually quotient manifolds of an FRW lambdadust model, having nontrivial topology (c.f. Cornish and Weeks),

2. Of those (lamentably rare!) known exact solutions in gtr which have clear and unobjectional physical interpretations, including many models of black holes, many do feature nontrivial topology. (For example, the Kerr vacuum is homotopic to the real line with circles attached to each integer, and the deSitter lambdavacuum is homeomorphic to $${\bold R} \times S^3$$.) However, "idealized but realistic models" would presumably be (at best) nonlinear perturbations of exact solutions with nontrivial symmetries, so to tell whether or not gtr firmly predicts nontrivial topological features in realistic scenarios, one would have to characterize a local neighborhood (in the solution space) of one of these solutions. At present, the only rigorous results appear to concern models like Minkowski vacuum (small nonlinear perturbations of Minkowski vacuum are indeed homeomorphic to $${\bold R}^4$$ and de Sitter lambdavacuum (small nonlinear perturbations of de Sitter lambdavacuum are indeed homeomorphic to $${\bold R} \times S^3$$). Caution: these results are actually a bit weaker than we would really want, even in the case of these particular neighborhoods, which are unfortunately not the ones we really want.

 

[pervect]

I did not find it in literature. I made a simple derivation myself, though I must admit that I still miss a factor two somewhere .
Suppose you have a rocket (or elevator if you like) in free space. What I'm interested in is not the length Lorentz contraction, but a contraction due to acceleration itself. Therefore, to separate both effects, I assume that the rocket is initially at rest. If the effect is due to acceleration only, this assumption should not influence the result. Next, the rockets are switched on, which induce a push on the "back" of the rocket. An external observer is located at some distance in a line perpendicular to the velocity of the rocket.
At the moment the acceleration starts, the back of the rocket is set into motion, while the front of the rocket is absolutely still, since it takes a time L/c for the "push" to reach the top of the spacecraft (L is length of rocket). This is purely a consequence of the finite speed of light, to be distinguished from an elastic compression. Then, one just calculates the length the bottom has moved in the time T=L/c using Dx=a*t^2/2 (where a is the acceleration). The new length is then L-a(L/c)^2/2. Dividing this by L gives the contraction factor :
1-aL/(2*c^2). This can be considered as a Taylor expansion of SQRT(1-aL/c^2). Now, the corresponding term for aL in a gravitational field is GM/R, such that one obtains the contraction factor SQRT(1-GM/c^2 R). This is very similar (except for the factor 2) to the true length contraction SQRT(1-2*GM/c^2 R) in a gravitational field.

Rudi

Here are my comments.

The time taken for the "push" to reach the front of the rocket is going to be equal to L/(speed of sound in rocket material), not L/c.

The front of the rocket is then going to undergo some complicated dynamic oscillations.

Eventually it will reach some "steady state" length. Your analysis doesn't address what this steady state length is.

I'm not sure what you mean by the "contraction due to the acceleration itself". If you look at the steady state length, this would be due to the Young's modulus of the rocket material. Looking at an non-equilbrium "length" isn't going to tell us anything meaningful, in my opinion - but that's what your analysis is looking at.