Equivalence of Clocks in Gravitational Fields: A Thought Experiment

[pervect]

If you put a gamma radiation source at the nose of an accelerating rocket, gamma rays emitted from a source at the nose which propagate to the tail would slightly change in frequency, detectable by the Mossbauer effect. So the answer is that yes, a Pound-Rebka type experiment would find evidence of "gravitational redshift" as seen in the frame-field of the rocket.

An observer in an inertial frame would also see the same amount of blueshift of the photons, but would attribute this to a simple doppler effect.

 

[Voltage]

Thanks for the input, pervect. Can I ask a few questions to get this clear:

Does that mean that I can't perform a Pound-Rebka experiment to determine that the (windowless) rocket I'm in is accelerating at 1g rather than just sitting on the ground?

Did you mean inertial frame in your second paragraph? If I and my rocket were in free-fall, could I perform a Pound-Rebka experiment to determine that I was in free fall rather than floating quietly in space?

Is there some other experiment or experiments I can perform to distinguish these situations?

edit: yogi, sorry if I've hijacked your thread a little here.

 

[Ich]

Lest I go completely unnoted, I hijack back.
There is no experiment to determine whether I'm accelerating or in a grav field, as both situations are the same.
There is no experiment to determine whether I'm in free fall or floating quietly, as both situations are the same.
Small regions of spacetime presumed.

 

[pervect]

Let's talk about the actual experimental arrangement since this seems to be the only way to make sure the answer to the question is understood, I think there may be some semantic issues here (i.e. issues related to the words rather than the physics).

What you have for the experiment is a gamma emitter source at the top of some tower.

The tower is at a "fixed height". Experimentally a fixed height for a non-time varying "gravitational field" due to either a planet or an accelerating rocket can be defined by a constant number of wavelengths of some fixed frequency signal (such as the cesium standard) between the top and bottom, or by a constant round-trip propagation delay for a light signal. We will avoid talking about any time-varying case and assume we have such a non-time varying situation so we can use this simple defintion of "fixed height".

You can also use a "rigid rod" to keep the height constant. The light beam is simply the most rigid possible rod that is able to be physically constructed. The higher the velcocity of sound in a material, the more rigid it is, and ultimate rigidity occurs when the velocity of sound is equal to 'c'. Using light directly essentially insures ultimate rigidity, because light travels at 'c'.

The Mossbauer effect will detect small variations in frequency, so it can be used as a very precise frequency measurement device.

When you set up the experiment, you find that the frequency of the gamma rays at the floor has been blueshifted as they "fall down". This happens in both the spaceship and on a planet. So the emitted gamma rays are not absorbed by an identical "reciever".

One can argue the semantics of exactly what causes the blueshift, but it is important to realize that the blueshift is what the experiment is measuring, and that everyone agrees that there is a blueshift, no matter what they attribute its cause to.

That's why I talked about a blueshift in my last post, to try and emphasize the experimental aspects and avoid any semantic issues.

Note that non-absorption actually just illustrates that some frequency shift has occured. To determine that it actually a blueshift, a refinement of the method is required.

If you read the details of the experiment, for instance http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html
the amount of blueshift is measured by a doppler method, i.e. you move either the source or the reciever so that you get absorption of the gamma rays. This technique is needed because you can't tune the frequency of emission or absoprtion of the gamma rays, they are set by the nuclear energy levels, but you can add an additional doppler shift so that absorption occurs.

So let's recap:

Gamma rays emitted at the top of a tower of "fixed height", (see the previous defintion for what this means) are blueshifted when they fall. This means that with two identical Fe-57 bar at the top and bottom of the tower, the bottom bar won't absorb the gamma rays emitted from the top bar, because the frequency has shifted outside the absorption band.

This blueshift occurs both on an accelerating spaceship, or due to the gravitational field of a large mass (such as a planet).

 

[meopemuk]

Let's talk about the actual experimental arrangement since this seems to be the only way to make sure the answer to the question is understood, I think there may be some semantic issues here (i.e. issues related to the words rather than the physics).

What you have for the experiment is a gamma emitter source at the top of some tower.

The tower is at a "fixed height". Experimentally a fixed height for a non-time varying "gravitational field" due to either a planet or an accelerating rocket can be defined by a constant number of wavelengths of some fixed frequency signal (such as the cesium standard) between the top and bottom, or by a constant round-trip propagation delay for a light signal. We will avoid talking about any time-varying case and assume we have such a non-time varying situation so we can use this simple defintion of "fixed height".

You can also use a "rigid rod" to keep the height constant. The light beam is simply the most rigid possible rod that is able to be physically constructed. The higher the velcocity of sound in a material, the more rigid it is, and ultimate rigidity occurs when the velocity of sound is equal to 'c'. Using light directly essentially insures ultimate rigidity, because light travels at 'c'.

The Mossbauer effect will detect small variations in frequency, so it can be used as a very precise frequency measurement device.

When you set up the experiment, you find that the frequency of the gamma rays at the floor has been blueshifted as they "fall down". This happens in both the spaceship and on a planet. So the emitted gamma rays are not absorbed by an identical "reciever".

One can argue the semantics of exactly what causes the blueshift, but it is important to realize that the blueshift is what the experiment is measuring, and that everyone agrees that there is a blueshift, no matter what they attribute its cause to.

That's why I talked about a blueshift in my last post, to try and emphasize the experimental aspects and avoid any semantic issues.

Note that non-absorption actually just illustrates that some frequency shift has occured. To determine that it actually a blueshift, a refinement of the method is required.

If you read the details of the experiment, for instance http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html
the amount of blueshift is measured by a doppler method, i.e. you move either the source or the reciever so that you get absorption of the gamma rays. This technique is needed because you can't tune the frequency of emission or absoprtion of the gamma rays, they are set by the nuclear energy levels, but you can add an additional doppler shift so that absorption occurs.

So let's recap:

Gamma rays emitted at the top of a tower of "fixed height", (see the previous defintion for what this means) are blueshifted when they fall. This means that with two identical Fe-57 bar at the top and bottom of the tower, the bottom bar won't absorb the gamma rays emitted from the top bar, because the frequency has shifted outside the absorption band.

This blueshift occurs both on an accelerating spaceship, or due to the gravitational field of a large mass (such as a planet).

This is a fine description of the Pound-Rebka frequency-shift experiment in an accelerated elevator cabin. Observer on the ceiling sees that gamma quanta emitted by the source on the floor are red-shifted. Observer on the floor sees that gamma quanta emitted by the source on the ceiling are blue-shifted. All this occurs due to the Doppler-shift-type effect. Check.

Now, suppose that instead of gamma sources we used identical atomic clocks on the ceiling and on the floor. I don't think there is any doubt that observer on the ceiling will see that the clock on the floor runs slower than his own clock. Check.

Now, the question is: is this difference in clock rates an "apparent" and "relative" difference? Or it is an "absolute" difference? In other words, if we allow these ceiling and floor clocks to run for a prolonged time and then bring them together, will we find their readings the same or different? It is known experimentally, that such a side-by-side clock comparison in the gravity field will definitely show that the ceiling clock shows later time. This means that the difference in clocks' rates is "absolute" for all observers.

If you believe in the principle of equivalence, then you should conclude that in an accelerated cabin the ceiling clock ticks faster than the floor clock in an "absolute" sense. This difference in clock rates should be visible to everyone, including observers outside the elevator cabin.

Now, let us take the point of view of such an outside observer. We concluded that she should see the ceiling clock ticking faster than the floor clock. But what is the physical reason for such a difference? We cannot invoke the Doppler-shift-type arguments, because both ceiling and floor clocks move with the same velocity and acceleration with respect to the outside observer. Isn't it a logical contradiction?

Eugene.

 

[yogi]

ich - the Briatore-Leschiutta experiment referred too is cited in post 10

This is not a SR problem - To arrive at actual age differences between two clocks, some acceleration takes place somewhere. What is curious is exactly how it affects outcomes.

meopemuk: What seems to be at issue is whether there is a physically different operative in the case of G fields and free space accelerations. In Newtonian physics, G fields are divergent - in GR we attribute the clock differences to spacetime curvature. In the case of a free space accelerating rocket, there is neither curvature nor divergence - nor is there a net global reaction since the accleration of the rocket is always balanced by an equal and opposite momentum communicated to the universe in the opposite direction.
Whatever the mechanism that brings about permanent time dilation, it does not seem to have a corresponding counterpart in the case of free space acceleration

Thanks for the reference to the Briatore-Leschiutta experiment in post 10

 

[meopemuk]

meopemuk: What seems to be at issue is whether there is a physically different operative in the case of G fields and free space accelerations. In Newtonian physics, G fields are divergent - in GR we attribute the clock differences to spacetime curvature. In the case of a free space accelerating rocket, there is neither curvature nor divergence - nor is there a net global reaction since the accleration of the rocket is always balanced by an equal and opposite momentum communicated to the universe in the opposite direction.
Whatever the mechanism that brings about permanent time dilation, it does not seem to have a corresponding counterpart in the case of free space acceleration

After some thinking I tend to conclude that behaviors of clocks in the gravity field and in an accelerated rocket are, indeed, different. However, this difference cannot be used to disprove the principle of equivalence. This is because the principle of equivalence (as usually stated) refers to infinitesimally small regions of spacetime. If you allow clocks to run for a long period of time, you are not talking about an infinitesimally small spacetime region. The "time dimension" is now finite.

Thanks for the reference to the Briatore-Leschiutta experiment in post 10

You are welcome.

 

[yogi]

Now, the question is: is this difference in clock rates an "apparent" and "relative" difference? Or it is an "absolute" difference? In other words, if we allow these ceiling and floor clocks to run for a prolonged time and then bring them together, will we find their readings the same or different? It is known experimentally, that such a side-by-side clock comparison in the gravity field will definitely show that the ceiling clock shows later time. This means that the difference in clocks' rates is "absolute" for all observers.

If you believe in the principle of equivalence, then you should conclude that in an accelerated cabin the ceiling clock ticks faster than the floor clock in an "absolute" sense. This difference in clock rates should be visible to everyone, including observers outside the elevator cabin.

Now, let us take the point of view of such an outside observer. We concluded that she should see the ceiling clock ticking faster than the floor clock. But what is the physical reason for such a difference? We cannot invoke the Doppler-shift-type arguments, because both ceiling and floor clocks move with the same velocity and acceleration with respect to the outside observer. Isn't it a logical contradiction?

Eugene.

Well Put - and that was provocation for this thread - either the two clocks accumulate time at the same rate or different rates - and if they accumulate time at different rates, what mechanism is involved?

 

[pervect]

There isn't really such a thing as the "absolute rate" at which a clock ticks, at least not other than the trivial fact that all clocks tick at one second per second.

What we have here is a situation with a time-translation symmetry. Methods of clock comparison that respect this time translation symmetry always show one clock as ticking faster. But it is still a (subtle) mistake to conclude from this that one clock ticks faster than the other in an absolute sense. One still needs to compare two different clocks to determine if one is "faster" or "slower" than another. There is no sensible notion of the "absolute rate" at which a clock ticks (at least none that I'm aware of).

The comparison process is still the key, and while it is quite natural to take advantage of the time-translation symmetry of the problem in the comparison process, one could use other methods. For instance, one might consider an inertial clock that is intially moving faster than the accelerating clocks, and use this clock (or a pair of such clocks) to perform the comparison process.

Thus there is still no meaning for the "absolute rate" at which a clock ticks, one still needs to think about multiple clocks, relative rates which are defined by comparing one clock to another clock, and in general one needs to consider the comparison means as well (though it is natural to use a comparsion means that matches the symmetry of the problem)

 

[yogi]

Pervect: "One still needs to compare two different clocks to determine if one is "faster" or "slower" than another"

Ok - no one seems to be claiming absolutes - nor preferred frames, at least I am not. We actually have a two clock situation - a floor clock and a ceiling clock -So for an extended free space acceleration, what would you find when you brought the two clocks together?

 

[Ich]

After some thinking I tend to conclude that behaviors of clocks in the gravity field and in an accelerated rocket are, indeed, different.

So for an extended free space acceleration, what would you find when you brought the two clocks together?

Acceleration and gravity are exactly equivalent [...] The clocks in the elevator would read different times, just as clocks on Earth would.

The shifting of simultaneitiy which produces the effect in question is not more than basic SR.

Different curvature is not necessary. Even in GR, time dilation is not a local property of spacetime (but curvature is). It is defined only as a relation between two points, its magnitude (in small fields) is proportional to the difference in gravitational potential, not to its first or second derivative (gravitational acceleration or tidal acceleration respectively, where tidal acceleration corresponds to curvature). A "difference in potential" is also present in flat spacetime when you change to accelerating frames.

So the answer is that yes, a Pound-Rebka type experiment would find evidence of "gravitational redshift" as seen in the frame-field of the rocket.

This blueshift occurs both on an accelerating spaceship, or due to the gravitational field of a large mass (such as a planet).

Is there some threshold number of answers needed to get noticed?

 

[HallsofIvy]

Thanks for that, Halls of Ivy. Can you clarify this point: If I was in the rocket conducting a Pound-Rebka experiment, would I be able to tell whether the rocket was sitting on the ground rather than accelerating through space?

Remember that the equivalence of gravity and acceleration is purely local.

 

[Voltage]

You've been noticed, ich. If you think you haven't, maybe it's because we're not quite getting to the heart of the matter. We all agree that the principle of equivalence means the accelerating rocket situation is equivalent to being in the rocket sitting on the ground. But the question is this: is it exactly equivalent? As confirmed by HallsofIvy, the former situation exhibits a "uniform gravitational field", and the latter does not, because no real gravitational fields are ever uniform. As far as I can tell this says they are not exactly equivalent. As meopemuk says, they can only be equivalent when you consider your local frame to be an infinitesimaly small region. This means your local frame no longer has any extent. It's not there any more, so it doesn't feel like a sound basis for a rationale.

Edit: noted, HallsofIvy. We overlapped, see the last portion of the paragraph above.

Pervect, thanks for the lengthy response above. I can appreciate that a Pound-Rebka experiment would not distinguish between the accelerating rocket and the rocket on the ground. Sorry to be a pain, but I've been thinking it would distinguish between a free-falling rocket and a rocket floating in space. Can you confirm or refute this?

 

[pervect]

Tidal forces in a free-falling rocket near a large mass would produce very small shifts for a Pound-Rebka experiment. Such effects would be quadratic in the distance, however, and not linear, i.e. the fractional frequency shift with a gravitational acceleration g is of the order gh/c^2 (see for instance http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html), the effect due to tidal forces would be of the order kh^2 / 2 c^2, where k is some spring constant representing the tidal force. Thus for small enough distances, the quadratic effect will be ignorable.

I don't think such small frequency shifts could be measured with current experimental apparatus, however the tidal forces themselves can be measured by other means (the Forward mass detector comes into mind as a device designed for this purpose).

The principle of equivalence should not be interpreted to mean that the tidal forces in an accelerating elevator are the same as those on a planet, nor should it be interpreted to mean that a Forward mass detector "can't work".

 

[Voltage]

Thanks pervect.

 

[yogi]

If we are treating the issue as one of measurable differences - then it seems we do not have the accuracy needed - at least that is the way i interpret pervect's answer, in part. The interesting issue as to whether the two experiments involve a difference in the modus operandi is left unanswered - did Einstein intend to define equivalence in terms of the same dynamic? It seems as we try to focus on what is actually occurring, the G field time dilation involves a change in some property of space (I hesitate to suggest it on this forum, but perhaps something akin to the ideas of Lorentz). I don't think Einstein would have objected to this interpretation in the case of gravitational acceleration as he stated frequently that "matter conditions space."

This leaves the free space rocket. I fail to see why two free space accelerating clocks would be affected differently - this would seem to violate the principle that one location is as good as another from the standpoint of making physical experiments. I suggested earlier that two clocks might be attached to a spinning disk at different radial distances. From the experiments that have been performed, there seems to be no evidence that acceleration per se will result in clocks accumulating different quantites of time other than that which can be directly correlated to their SR velocity profiles.

 

[Ich]

Some more comments:

As confirmed by HallsofIvy, the former situation exhibits a "uniform gravitational field", and the latter does not, because no real gravitational fields are ever uniform. As far as I can tell this says they are not exactly equivalent. As meopemuk says, they can only be equivalent when you consider your local frame to be an infinitesimaly small region. This means your local frame no longer has any extent.

First: I disagree with HallsofIvy's post#28. There is no first order difference in the dilation effects.
Second: While it is certainly true that the equivalence is exact only in an infinitesimally small region, this is no reason to state a qualitative difference.
For example, experiments which measure time-dilation (in one dimension!) cannot distinguish monopole gravitation from an accelerating Rindler frame, to first or second order in x. This is already quite extended. Further, I see no reason why one could not set up a mass distribution which mimicks a Rindler frame to arbitrary precision in an extended region of space. Again, no point in insisting on a qualitative difference.

Sorry to be a pain, but I've been thinking it would distinguish between a free-falling rocket and a rocket floating in space.

Neglecting tidal forces, there is no difference.

It seems as we try to focus on what is actually occurring, the G field time dilation involves a change in some property of space (I hesitate to suggest it on this forum, but perhaps something akin to the ideas of Lorentz). I don't think Einstein would have objected to this interpretation in the case of gravitational acceleration as he stated frequently that "matter conditions space."

Einstein surely would have objected, as he did not attribute time dilation to local properies of spacetime. GR doesn't work this way.

I fail to see why two free space accelerating clocks would be affected differently - this would seem to violate the principle that one location is as good as another from the standpoint of making physical experiments.

It occurs to me that you still accept only local properties as physically significiant. In this example, both locations are indistinguishable locally. But time dilation is relative, and the relation of leading and trailing end is clearly asymmetric: the trailing end is down the potential well as seen from the leading end, and the leading end is up the well.

I suggested earlier that two clocks might be attached to a spinning disk at different radial distances. From the experiments that have been performed, there seems to be no evidence that acceleration per se will result in clocks accumulating different quantites of time other than that which can be directly correlated to their SR velocity profiles.

That is true, acceleration does not make clocks run faster or slower.
In flat spacetime, you have two possibilities: you analyze the problem in an inertial frame, and you get the standard SR time dilations. Or you switch to accelerated frames where the objects are at rest, and you recover the very same time dilation, but this time in terms of gravitational potential, as there is no more motion. It is just a different point of view, not a different physical effect.
The equivalence principle gives you the power to extend this point of view to gravitation. You get the description of gravity as spacetime curvature. The Einstein equation describes the connection between curvature and matter, now you have the complete theory of gravitation.

 

[Voltage]

Some more comments: First: I disagree with HallsofIvy's post#28. There is no first order difference in the dilation effects.

You're wrong to do so. See my post #9, where I said:

You'll never obtain experimental evidence for this, yogi, because it's based upon a misconception. The principle of equivalence does not confer absolute equivalence. In the accelerating rocket, your two clocks experience the same acceleration. In the rocket standing on the surface of the earth, they do not. They can only experience the same acceleration if they're in what's called a uniform gravitational field, and in the real world, gravitational fields are not uniform.

Second: While it is certainly true that the equivalence is exact only in an infinitesimally small region, this is no reason to state a qualitative difference. For example, experiments which measure time-dilation (in one dimension!) cannot distinguish monopole gravitation from an accelerating Rindler frame, to first or second order in x. This is already quite extended. Further, I see no reason why one could not set up a mass distribution which mimicks a Rindler frame to arbitrary precision in an extended region of space. Again, no point in insisting on a qualitative difference. Neglecting tidal forces, there is no difference.

Oh yes it is. You're missing the point. The "tidal forces" are a sign of a very vital difference, and it's utterly wrong to neglect them. If you neglect them, everything within your frame is the same. When you then move across the extent of your local frame to occupy a new local frame, everything is still the same. And that means your uniform gravitational field, which doesn't exist in nature, makes as much sense as a flat hill.

Einstein surely would have objected, as he did not attribute time dilation to local properies of spacetime. GR doesn't work this way.

That's your misinterpretation. Check with pmb about the way the modern interpretation of General Relativity has shifted away from Einstein's interpretation.

It occurs to me that you still accept only local properties as physically significiant. In this example, both locations are indistinguishable locally. But time dilation is relative, and the relation of leading and trailing end is clearly asymmetric: the trailing end is down the potential well as seen from the leading end, and the leading end is up the well.

No, in a gravity situation, the time dilation is absolute. Both observers agree that the observer in the gravity field experiences time dilation.

That is true, acceleration does not make clocks run faster or slower. In flat spacetime, you have two possibilities: you analyze the problem in an inertial frame, and you get the standard SR time dilations. Or you switch to accelerated frames where the objects are at rest, and you recover the very same time dilation, but this time in terms of gravitational potential, as there is no more motion. It is just a different point of view, not a different physical effect. The equivalence principle gives you the power to extend this point of view to gravitation. You get the description of gravity as spacetime curvature. The Einstein equation describes the connection between curvature and matter, now you have the complete theory of gravitation.

I agree that accleration is not responsible for time dilation. I also agree that the SR time dilation is equivalent to GR time dilation. But go and find the original GR translation. You will not find spacetime curvature mentioned anywhere. There's plenty of talk of curvature, but it's associated with the Weyl, Ricci, and Reimann tensors. And it's not a complete theory of gravitation anyhow, because it doesn't actually explain what gravity is.

 

[yogi]

Ditto Voltage - GR is not a complete theory - it does not explain why mass distorts spacetime and it does not predict the value of the gravitational constant.

Voltage - in a previous post you vowed to introduce some argument as to the applicability of Sagnac to this thread - will that be forthcoming?

 

[Hurkyl]

Ditto Voltage - GR is not a complete theory - it does not explain why mass distorts spacetime?

It's not supposed to. Questions like "why?" don't make sense unless you've assumed a foundation upon which things can be explained. GR is a fundamental theory; it's meant to provide the foundation.

 

[Ich]

You're wrong to do so. See my post #9, where I said:
...
You'll never obtain experimental evidence for this, yogi, because it's based upon a misconception. The principle of equivalence does not confer absolute equivalence. In the accelerating rocket, your two clocks experience the same acceleration. In the rocket standing on the surface of the earth, they do not. They can only experience the same acceleration if they're in what's called a uniform gravitational field, and in the real world, gravitational fields are not uniform.
...
Oh yes it is. You're missing the point. The "tidal forces" are a sign of a very vital difference, and it's utterly wrong to neglect them. If you neglect them, everything within your frame is the same. When you then move across the extent of your local frame to occupy a new local frame, everything is still the same. And that means your uniform gravitational field, which doesn't exist in nature, makes as much sense as a flat hill.
...
That's your misinterpretation. Check with pmb about the way the modern interpretation of General Relativity has shifted away from Einstein's interpretation.
...
No, in a gravity situation, the time dilation is absolute. Both observers agree that the observer in the gravity field experiences time dilation.

I merged you answers because they all seem to be based on the same misunderstanding of gravitational potential, field, and tidal force. I don't think I can convince you in this discussion, so I recommend you read http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html" for an introductory explanation of how even uniform fields produce time dilation. We can come back to the fine points and remaining objections to my post afterwards.

 

[Voltage]

ich: Baez opens with same misconception as you suffer. He talks about a "uniform" pseudoforce field where the lower clock goes slower. This is a contradiction in terms. There is something there that isn't uniform, that's why the pseudoforce is there, and that's why the lower clock goes slower. You are talking about a uniform non-uniformity. Note however that he highlights the shifting interpretation of GR, and also note that the GR explanation for the Twins Paradox does not account for "passing clocks".

yogi: no, the Sagnac matter is a little tangential, and IMHO the article I was thinking of goes too far with its conclusions.

 

[Ich]

ich: Baez opens with same misconception as you suffer.

I'm glad to hear this. Instead of relying on his authority, I will try to explain the uniform non-uniformity:
The pseudo-force (or gravitational acceleration) is described by a uniform vector field: same size and direction everywhere in space.
But the relevant parameter is the potential that is assigned to this vector field; you get it by integrating over x. This potential is linearly increasing with position, and so is time dilation.

 

[K.J.Healey]

I'm glad to hear this. Instead of relying on his authority, I will try to explain the uniform non-uniformity:
The pseudo-force (or gravitational acceleration) is described by a uniform vector field: same size and direction everywhere in space.
But the relevant parameter is the potential that is assigned to this vector field; you get it by integrating over x. This potential is linearly increasing with position, and so is time dilation.

But in a gravitational field potential is proportional to 1/r^2 ? How is that linear?

 

[Ich]

But in a gravitational field potential is proportional to 1/r^2 ? How is that linear?

I was talking about uniform acceleration, as at Baez's site. Acceleration in an "accelerating frame" isn't constant, too. That's why you can match it even in an extended region with true gravitation, concerning time dilation.

 

[K.J.Healey]

How is acceleration of a particle not constant(if we define it so)? Are you talking about a change in acceleration reflecting the effects that would imitate gravity(on time)?
I must be confusing the terms "uniform" and "constant" when talking about acceleration of a point particle here? What do you mean by each?
I thought "uniform acceleration" was dv/dt = constant?
In which case it WOULDN'T represent the same effect as a gravitational potential well, right?

 

[yogi]

It's not supposed to. Questions like "why?" don't make sense unless you've assumed a foundation upon which things can be explained. GR is a fundamental theory; it's meant to provide the foundation.

That is a subjective opinion - If it ultimately turns out that G is not some God Given factor, but is in fact relatable to cosmological properties, then GR is not foundational. In this area, Einstein played with different ideas to account for the apparent stability of a closed gravitationally dominated positively curved space - he first tried the cosmological constant, then threw it out when the cosmological red shift was discovered. He himself called the right side of the equation a "house of straw" ...Just as in SR, Einstein converted the problem into a postulate...add up the energy on one side and proclaim it would produce the needed spacetime curvature.

It is great as an interum, but I think it will ultimately be viewed as a constructive consequence of a something more fundamental.

 

[yogi]

But in a gravitational field potential is proportional to 1/r^2 ? How is that linear?

Do you mean force or potential?

 

[Ich]

How is acceleration of a particle not constant(if we define it so)? Are you talking about a change in acceleration reflecting the effects that would imitate gravity(on time)?
I must be confusing the terms "uniform" and "constant" when talking about acceleration of a point particle here?

No, I confused the terms. I did not consider single particles, but an "accelerated frame". That is the famous elevator extended to significant height in the direction of acceleration.
One finds that the proper acceleration decreases in fact from bottom to top, while being constant in time. So it is non-uniform but constant.
As a consequence, you can always match this acceleration distribution with the distribution due to gravitation of a point mass up to first order in x. That means that the time dilation effects will match to second order.
That exercise was meant only to show that the equivalence principle does not become worthless in extended regions of space; it does not hold exactly, but to arbitrary precision.

 

[Ich]

That is a subjective opinion - If it ultimately turns out that G is not some God Given factor, but is in fact relatable to cosmological properties, then GR is not foundational.

Well, c=1, G=1, and the sun has a mass of 3 km. How would you derive that from cosmological properties?