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Limit of Rindler coordinates 
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#1
Nov2712, 11:43 PM

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It seems that acceleration at some point in Rindler coordinates completely determines it's distance from rindler horizon, right?
If we have two rockets with equal hight and experiencing equal acceleration at the bottom there are no other parameters we can vary to get different results for two cases. So that means that time dilation at the top of the rocket is the same for both rockets. That contrasts with gravitational field where we have two parameters determining acceleration (r and r_{s}) so we don't know time dilation at the top of the rocket given acceleration at the bottom (and given hight) if it stand on the surface of some gravitating body. So it seems that there is no limiting case where proper acceleration and gravitational acceleration would tend to become equal. 


#2
Nov2812, 12:25 AM

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#3
Nov2812, 03:35 AM

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It seems to me there isn't any huge difference between the two cases. The time dilation for a small height h will always be 1+gh/c^2, you can't really change this fundamental fact.
There are some second order differences, but to first order you can say that the time dilation is what I said above. 


#4
Nov2812, 11:11 AM

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Limit of Rindler coordinates
I'm getting the impression that Zonde has figured out that the strong principle of equivalence doesn't apply to spatially varying gravitational fields, or, equivalently, that it only applies in the limit of uniform gravitational fields or infinitesimal displacements.



#5
Nov2812, 11:06 PM

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But your proposal that we can find some situation where the equivalence works is not quite satisfactory either. If there is a limit then where is this limit? Then from different side. proper acceleration in Rindler coordinates is proportional to 1/x Newtonian gravitational acceleration is proportional to 1/r^2 Is this right? Maybe that 1/x acceleration is actually (Rindler) coordinate acceleration and you have to take into account time dilation to get proper acceleration? And the same about Newtonian gravitational acceleration. From GR perspective does it approximate coordinate acceleration (as seen by far away observer) or proper acceleration? 


#6
Nov2812, 11:15 PM

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I calculated with example values that gravitational time dilation seems to give the same value. But I don't understand that. Gravitational acceleration seems to vary differently than Rindler acceleration (see replay to PeterDonis) and than there is still mass parameter. 


#7
Nov2812, 11:19 PM

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None of this contradicts what I said above. Show me an accelerated observer in flat spacetime, feeling proper acceleration 1/x, and I will show you an accelerated observer in Schwarzschild spacetime who feels the same proper acceleration; it's just a matter of setting the two formulas equal: [tex]a = \frac{1}{x} = \frac{m}{r^2 \sqrt{1  2m / r}}[/tex] You should be able to convince yourself that for any x > 0, we can pick any m > 0 that we like, and then find some r > 2m for which the equality above is satisfied. That means that for any Rindler observer (accelerated in flat spacetime), we can find some corresponding Schwarzschild observer (hovering over a black hole in curved spacetime) who feels the same proper acceleration. That's all I was trying to say. 


#8
Nov2812, 11:31 PM

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But I doubt that strong principle of equivalence applies in the limit of infinitesimal displacements. And I want to either confirm my doubt or get over it. And what I just figured out is that strong equivalence principle in GR is the same as relativity principle in SR i.e. it is the statement that gives physical content to GR. So I want good understanding of things around it. And as a physical statement it is subject to experimental tests. So in perspective I want to understand how (tested quantitative) predictions of GR follow from strong principle of equivalence. 


#9
Nov2812, 11:47 PM

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Also, are you really intending to talk about the *strong* EP, as opposed to the weak EP or the Einstein EP? I'm using the terminology that's used on the Wikipedia page, which gives a brief overview of the different versions of the EP: http://en.wikipedia.org/wiki/Equivalence_principle I ask because the statement you appear to be concerned about is, more or less, "proper acceleration in flat spacetime is equivalent to being at rest in a static gravitational field". In the Wikipedia page terminology, this is the weak EP, not the strong EP. 


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Nov2912, 12:19 AM

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#11
Nov2912, 01:29 AM

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There seems to be more argument about the EEP than one would expec, even in the literature. BUt my view is that the EEP says that gravity and acceleration are the same to the first order. At higher orders, gravity has tidal effects, and acceleration doesn't really (not in the sense of geodesic deviation at least). But that's not really the point, the point is that it's the same at lower orders.



#12
Nov2912, 11:23 PM

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I would like to add that when speak about tidal effects I usually think about radial effects and not convergence of different angular directions. PeterDonis, this answers your question too "Why do you doubt this?". My doubts are that acceleration and tidal effects might be at the same level of orders. And in that case we can't speak about the limit where EP tends to hold better. 


#13
Nov3012, 12:06 AM

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#14
Nov3012, 12:10 AM

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(1) Spacetime is a geometric object. Gravity is curvature of this geometric object. (2) Physics is contained in geometric invariants. 


#15
Nov3012, 03:51 AM

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I'm using "order" in the calculus sense, first order effects are proportional to the derivative, second order effects are proportional to the second derivative. As you take the limit as dx>0, the first order terms will dominate the second. If we expand U in a taylor series we'd get U = some constant + force terms * dx + (1/2) tidal force terms * dx^2 (Expanding U in a taylor series is the same as expanding the "gravitational time dilation" in a series, the two are proportional). By choosing a small enough dx, i.e. by limiting the size of your box, you can always guarantee that the second order effects are low enough to ignore. This is the sense in which the EEP says acceleration is the same as gravity. It doesn't mean that tidal forces don't exist, it just means that by taking your box small enough you can ignore them. 


#16
Nov3012, 05:31 AM

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The EEP says that we can treat an uniformly accelerated reference system in space that is free from gravitation, as being "at rest" in a homogeneous gravitational field. That is quite different from "gravity and acceleration are the same to the first order". 


#17
Nov3012, 09:48 AM

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#18
Nov3012, 12:12 PM

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http://dx.doi.org/10.1007/BF02450447 "Covariance, invariance, and equivalence: A viewpoint" But how small is "small enough?" I think it's sufficient to say that it works in the limit as the box size approaches zero. The remark about the "orders" is simply a helpful way to help understand why the equivalence is exact in the limit, and nonexact over larger regions. It was not my intention to represet is aspart of any formal statement or defintion of the equivalence principle. Other proferred statements of the equivalence principle as "The Universality of Free Fall" seem to me to be clearer formulation of the equivalence principle than the original formulation about trying to determine whether or not you're in an elevator or on a planet by the means of physical experiments  as there are known means of building reasonably compact gravity gradient meters (such as the Forward mass detector, variants of which are actually used in prospecting for oil) [add] Another reference: http://dx.doi.org/10.1007/BF00763538 


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