Question about 'uniform' pseudo-field felt by accelerating observer

[JesseM]

From some previous discussions in this forum I had gotten the idea that under the equivalence principle, observations inside a small room sitting on the surface of a planet should be seen as equivalent, not just to any ol' accelerating room in flat spacetime, but to a room undergoing Born rigid acceleration; I thought, for example, that the assumption of Born rigid acceleration was needed to explain why clocks at the top and bottom of the room would have the same difference in ticking rates for both the room on the planet and the accelerating room. I've read that an object experiencing Born rigid acceleration actually measures G-forces at different points along it, so I had the idea that when people talked about "uniform" gravitational pseudo-fields in flat spacetime in the context of the equivalence principle, "uniform" just meant it would be experienced by an observer undergoing uniform acceleration in the Born rigid sense, not that the G-forces measured by accelerometers would actually be the same everywhere. But someone pointed out to me that in the Equivalence Principle Analysis on the Twin Paradox page, it says:

The Equivalence Principle analysis of the twin paradox simply views the scenario from the frame in which Stella is at rest the whole time. This is not an inertial frame; it's accelerated, so the mathematics is harder. But it can certainly be done. When the mathematics is described fully, what results is that we can treat a uniformly accelerated frame as if it were an inertial frame with the addition of a "uniform pseudo gravitational field". By a "pseudo gravitational field", we mean an apparent field (not a real gravitational field) that acts on all objects proportionately to their mass; by "uniform" we mean that the force felt by each object is independent of its position. This is the basic content of the Equivalence Principle.

So, I guess I was probably wrong in some aspect of my understanding above, unless talk of a "uniform gravitational (pseudo) field" can have multiple meanings in different contexts...if I did go wrong somewhere though, can someone point out where?

 

[DrGreg]

I think you have actually uncovered an error in the usually very reliable Physics FAQ.

You are quite right that in Born rigid acceleration, the proper acceleration varies with height. And it is my understanding that this is, by convention, described as "uniform acceleration".

If a rocket were accelerating in such a way that the proper acceleration was equal everywhere, that would be the "Bell's paradox" scenario, and the rocket's proper length would have to stretch to achieve this (breaking any strings holding it together). That's hardly an appropriate scenario for the equivalence principle.

Of course, the equivalence principle is only a local principle, so in a "small enough" spaceship over a "short enough" period of time you wouldn't really be able to tell the difference between a fixed-size room with varying proper acceleration and an expanding room with unvarying proper acceleration, so maybe the difference doesn't really matter in this context?

However in my timezone it is now way past bedtime. I'll have another think about this tomorrow.

 

[atyy]

In Newtonian gravity, the EP can be thought of in terms of a uniform gravitational field, which can exist according to the theory. But a uniform field can also be thought of as an approximate description of a small region of a non-uniform field. The EP is that we can make gravity disappear by an appropriate choice of coordinates over some small region of spacetime, and suggests a geometric formulation - which does exist as Newton-Cartan theory.

In SR, there is no EP since there is no gravity.

In GR, the EP has little to do with uniform fields, and applies to any (vacuum) field, no matter how wildly curved, as long as it's not singular. Like in Newtonian gravity, the EP has to do with making gravity disappear which means in any small region of spacetime (i) Minkowski coordinates can be found (ii) the laws of special relativity apply.

 

[atyy]

I may have garbled the GR EP conditions, take a look at Rindler or Carroll's texts, but the rough idea is that the EP holds for any small region of non-uniformly curved spacetime.

 

[JesseM]

I may have garbled the GR EP conditions, take a look at Rindler or Carroll's texts, but the rough idea is that the EP holds for any small region of non-uniformly curved spacetime.

Right, but I'm not really talking about the nature of the gravitational field in curved spacetime, I'm talking about what the experiences of a non-freefalling observer in this curved spacetime would be equivalent to in flat spacetime, i.e. what kind of acceleration the small room in flat spacetime would need to undergo in order for observations in this room to be indistinguishable from observations of the non-freefalling observer in curved spacetime. Perhaps as atyy said the answer is that since we're talking about a limiting case where the time-interval in which the observations are made goes to zero along with the volume of the room, it doesn't actually matter whether the accelerating room in flat spacetime is undergoing Born rigid acceleration or some other form of acceleration where the proper distance between the top and bottom of the room isn't constant. And it also seems possible that the "best" form of acceleration to use in flat spacetime would depend on what sort of non-freefalling motion the corresponding room in curved spacetime was undergoing, although if this is the case I was implicitly thinking of a room at rest on the surface of a planet, so we can think of a room in a Schwarzschild spacetime (or more precisely a Schwarzschild spacetime outside the radius of the planet's surface as per this thread) where the position of the top and bottom of the room are both constant in Schwarzschild coordinates.

 

[Dale]

Hi JesseM,

My understanding is similar to yours (uniform -> Born rigid -> different proper acceleration), so if you are wrong at least you are wrong in good company! The way that I reconciled the difference was the idea that for any given sensitivity of your accelerometers you could always find a small enough region of spacetime where they were equal to within the sensitivity. So you still must constrain things to be a small region.

 

[atyy]

And it also seems possible that the "best" form of acceleration to use in flat spacetime would depend on what sort of non-freefalling motion the corresponding room in curved spacetime was undergoing, although if this is the case I was implicitly thinking of a room at rest on the surface of a planet, so we can think of a room in a Schwarzschild spacetime (or more precisely a Schwarzschild spacetime outside the radius of the planet's surface as per this thread) where the position of the top and bottom of the room are both constant in Schwarzschild coordinates.

The hovering observer frame in Schwarzschild spacetime is locally like the Rindler frame (p27, Eq 2.36)
http://books.google.com/books?id=T94hD5rR8oYC&dq=schwarzschild+rindler&source=gbs_navlinks_s

I don't know whether every 4-accelerated observer in curved spacetime is locally Rindlerian. Blandford and Thorne comment that all 4-accelerations in curve spacetime produce first order deviations from Minkowski, but that's a slightly weaker statement. Comments following Eq 24.16 at http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html

 

[atyy]

For an arbitrarily accelerating particle in curved spacetime, one can construct Fermi-Walker coordinates, which are locally Minkowskian. So maybe the frame of an accelerated particle in curved spacetime should be equivalent locally to the frame of the unaccelerated particle in flat spacetime. Incidentally, that would make more sense to me as "uniform", like the gravitational field inside a spherical shell which is locally Minkowski.

BTW, is the EP resolution of the twin paradox really correct - ie. can one use the EP to calculate the time difference exactly equal to that of difference in integrated proper times for arbitrary accelarations for the spaceship twin?

 

[Ironhead]

Clocks at equal separations on the planet and in Born rigid acceleration may have equal differences in ticking rates by choosing the parameters in each experiment carefully. This does not imply that Born rigid acceleration is functionally equivalent to a room on a planet surface. The planet is a three dimensional field with acceleration vs height changing according to one function, while Born rigid acceleration is basically a one dimensional experiment with a different function.

Simplifying the Equivalence Principle to the point where these two experiments become the same defeats the purpose of having an EP. You want to know what the difference in gravitational and inertial experimences is (as well as the similarities).

 

[Al68]

BTW, is the EP resolution of the twin paradox really correct - ie. can one use the EP to calculate the time difference exactly equal to that of difference in integrated proper times for arbitrary accelarations for the spaceship twin?

That would not be a coincidence, since using the EP and accelerated observers is how the gravitational time dilation equations were derived originally. Gravitational time dilation is just SR time dilation for an accelerated observer plus EP.

 

[atyy]

That would not be a coincidence, since using the EP and accelerated observers is how the gravitational time dilation equations were derived originally. Gravitational time dilation is just SR time dilation for an accelerated observer plus EP.

How does one calculate the ageing with a sharp turn-around? I naively think the gravitational time dilation would be infinite corresponding to infinite acceleration.

 

[DrGreg]

How does one calculate the ageing with a sharp turn-around? I naively think the gravitational time dilation would be infinite corresponding to infinite acceleration.

I haven't actually done it, but I imagine you could calculate the result of a short period of high acceleration using Rindler coordinates. Then let the time t go to zero and the acceleration a will go to infinity but at will remain finite and so will the final answer.

 

[Al68]

How does one calculate the ageing with a sharp turn-around? I naively think the gravitational time dilation would be infinite corresponding to infinite acceleration.

The simple way would be to just use the SR simultaneity equation for the instantaneous change in velocity. If you were to do this for realistic acceleration, breaking up the acceleration into infinitesimally small changes in velocity, you would end up with the gravitational time dilation equation anyway.

 

[atyy]

I haven't actually done it, but I imagine you could calculate the result of a short period of high acceleration using Rindler coordinates. Then let the time t go to zero and the acceleration a will go to infinity but at will remain finite and so will the final answer.

The simple way would be to just use the SR simultaneity equation for the instantaneous change in velocity. If you were to do this for realistic acceleration, breaking up the acceleration into infinitesimally small changes in velocity, you would end up with the gravitational time dilation equation anyway.

But doesn't the gravitational time dilation equation (dtau~exp(gdr)) only have time curvature? The metric for the accelerated twin (eg. dss=exp(2gr)(dtt-drr) following Fig 8 of http://arxiv.org/abs/gr-qc/0104077) has both time and space curvature. Will it really work?

 

[George Jones]

But doesn't the gravitational time dilation equation (dtau~exp(gdr)) only have time curvature? The metric for the accelerated twin (eg. dss=exp(2gr)(dtt-drr) following Fig 8 of http://arxiv.org/abs/gr-qc/0104077) has both time and space curvature. Will it really work?

There is no gravity here, and invoking the equivalence principle, without taking a lot of care, might not be useful. Forget about gravity, and just do the SR calculation!

 

[atyy]

There is no gravity here, and invoking the equivalence principle, without taking a lot of care, might not be useful. Forget about gravity, and just do the SR calculation!

Yeah, that's what I normally do, but the discussion here is whether the EP can be made to work in principle with lots of care. I suspect not, but am not sure.

 

[Al68]

But doesn't the gravitational time dilation equation (dtau~exp(gdr)) only have time curvature? The metric for the accelerated twin (eg. dss=exp(2gr)(dtt-drr) following Fig 8 of http://arxiv.org/abs/gr-qc/0104077) has both time and space curvature. Will it really work?

It should if done correctly. But obviously the gravitational time dilation equation for the accelerated twin is different from one for the accelerated reference frame near a spherical gravitational mass.

Here's a link that I only glanced at, but might help: http://en.wikipedia.org/wiki/Gravitational_time_dilation

 

[George Jones]

Yeah, that's what I normally do, but the discussion here is whether the EP can be made to work in principle with lots of care. I suspect not, but am not sure.

I deleted my post because I didn't read the whole thread. Since you have responded to my post, I have undeleted my post.

My take on the equivalence principle is as follows. The equivalence principle, in the hands of Albert Einstein, was an amazing conceptual insight that led from special relativity to general relativity. Mere mortals like us who want to know what the now mature theory of relativity says about a given situation should grab hold of the metric appropriate for the situation and calculate. Use of the equivalence principle is not a substitute for calculation and can even be misleading. See comments 11, 12, 18 and 20 in

http://blogs.discovermagazine.com/cosmicvariance/2009/06/02/susskind-lectures-on-general-relativity/.

Sean is Sean Carroll.

 

[Al68]

My take on the equivalence principle is as follows. The equivalence principle, in the hands of Albert Einstein, was an amazing conceptual insight that led from special relativity to general relativity. Mere mortals like us who want to know what the now mature theory of relativity says about a given situation should grab hold of the metric appropriate for the situation and calculate. Use of the equivalence principle is not a substitute for calculation and can even be misleading.

That's a good point, especially for the twins. It would be especially misleading to say that the gravitational time dilation for the accelerated twin was a consequence of the EP. It's the time dilation predicted for clocks in actual gravitational fields that is a consequence of the EP.