Is the Equivalence Principle Contradictory in General Relativity?

[Jonathan Scott]

In my experience, the principle of equivalence is very robust, and only fails when the zone in which the observation is being made is large enough to detect significant tidal effects or variations in the direction of the acceleration.

In the case of the book on the table, this is locally just like a book on a table in a spaceship accelerating upwards relative to the table at 1g. A free-falling observer in that spaceship will see the kinetic energy of the book increasing simply because it is being pushed by the spaceship engines, but an observer sitting beside the book will see the falling observer being accelerated by a fictitious force which looks just like gravity.

When this is converted to the Earth gravity picture (for example by landing the spaceship), locally there will be no difference (and I think this applies even if electric charges are involved). From the falling observer's point of view, the book will appear to acquire kinetic energy. My opinion is that the question about the ultimate source of the apparent energy in that case cannot be answered under the Principle of Equivalence because we cannot extend our local inertial frame to include the whole Earth.

 

[JustinLevy]

atyy,
Thanks for the links, but when I search published literature there appear to be contradictory views on the charged particle problem. Even you disagree with some of the things you linked.

As stated I do not have the GR skills to pare good published resources vs. probably incorrect publications (yes, incorrect papers do get published). So links to archiv papers just makes me even more wary. I don't know enough to figure out what 'authority' is correct. That is the problem.

Is the following two sentences a fair summary of a response to the charged particle question?

There is still scientific debate on these topics, so the question cannot be definitively answered.
Current consensus is that the orbitting electric charge will spiral inwards, but the concept of radiation is not agreed on with an inertial observer claiming only charges with non-zero proper acceleration radiate and observer's having non-zero proper acceleration can claim otherwise.

Yes. That is correct. The equivalence principle is, roughly speaking, that anytime you have a flat spacetime (no tidal effects) you cannot experimentally distinguish between "gravity" and "acceleration". Whenever you see the word "local" in a GR context, it is talking about a region of spacetime that is small enough to be considered flat (tidal effects within the region are too small to be measured).

Then the equivalence principle is not a princple of GR, but a principle of SR. Since while a spacetime point can always be given the -1,1,1,1 metric, the curvature is intrinsic and even at zero size the point region isn't equivalent to the flat spacetime of a Rindler observer in an accelerating rocket.

That doesn't sound right to me. I thought the equivalence principle referring to acceleration is a GR principle.

Can someone state mathematically what the principle is?
I think this would cut through any mis-communication that may be due to word choice issues.
This is also probably the easiest way for me to correct my understanding.

No it's not. If a rocket in free fall switches on its engines, the velocity and the acceleration will be in the same direction. The engine is clearly doing work in this case.

The velocity and acceleration according to who?
The velocity of the rocket in an instantaneously co-moving inertial frame is still zero. Hence why it is a vacuous truth. If you are going to argue there isn't work being done on the book because of this argument, then the engine is not doing work on the rocket by the same argument. You can't have it both ways.

Different frames are not going to agree on energy. But they will still agree energy is conserved. According to a freefalling frame there is work being done on the book. Where is this energy coming from?

Any metric in GR has the property of local Lorentz invariance. That means that the components of the metric in a local inertial frame are approximately \eta_{\mu\nu} in a small region of spacetime that contains the origin. The approximation becomes exact in the limit where the size of that region goes to zero. When you use the local inertial frame to describe a finite region of spacetime (i.e. when you don't take that limit), things won't look the same as in SR.

Consider a coordinate system with the metric diagonal -1,1,1,1 right at the point of the book (make the book a point particle if you want) and in which the book has a non-zero velocity. Sure, I realize that due to spacetime curvature the metric can't be exactly that of flat spacetime in a finite region, but I only need to talk about a region of size limiting to zero. What is the velocity and acceleration of the book right at that point where the metric is -1,1,1,1 ? It can, according to an inertial observer, give results showing that work is being done on the book. Where is this energy coming from?

 

[Dale]

Then the equivalence principle is not a princple of GR, but a principle of SR.

That's fine. Classify it however you will, any principle of SR is also a principle of GR.

Since while a spacetime point can always be given the -1,1,1,1 metric, the curvature is intrinsic and even at zero size the point region isn't equivalent to the flat spacetime of a Rindler observer in an accelerating rocket.

Let's say you have a measurement technique that is sensitive to curvature (tidal effects) greater than a finite magnitude k, then it is always possible to define a region of finite size s such that the tidal effects within that hypervolume are less than k. Such a region can be treated as flat.

 

[atyy]

atyy,
Thanks for the links, but when I search published literature there appear to be contradictory views on the charged particle problem. Even you disagree with some of the things you linked.

As stated I do not have the GR skills to pare good published resources vs. probably incorrect publications (yes, incorrect papers do get published). So links to archiv papers just makes me even more wary. I don't know enough to figure out what 'authority' is correct. That is the problem.

All the papers I referenced were from excellent sources, not merely posted on arXiv, but in peer-reviewed journals with good reputations (eg. Foundations of Physics, American Journal of Physics, Physical Review; except the Chiao article, which is published in a crackpot volume edited by Physics Nobel laureates). So I agree there is no consensus, at least not at the level for Maxwell's equations or Taylor and Hulse's observations and analyses.

Is the following two sentences a fair summary of a response to the charged particle question?

There is still scientific debate on these topics, so the question cannot be definitively answered.
Current consensus is that the orbitting electric charge will spiral inwards, but the concept of radiation is not agreed on with an inertial observer claiming only charges with non-zero proper acceleration radiate and observer's having non-zero proper acceleration can claim otherwise.

I believe there is little disagreement remaining over the physics (predictions of experimental outcomes), but there is substantial disagreement over when the "equivalence principle" applies (because the predictions can be made using Einstein's and Maxwell's field equations, without the "equivalence principle"). I would like to suggest this modification of your summary:

"There is still scientific debate on these topics, so the question cannot be definitively answered. There is consensus that: the orbitting electric charge will spiral inwards; the ability to detect radiation is observer dependent; a comoving observer will not detect radiation in at least some situations; some non-comoving observers will detect radiation. There is no consensus on whether this may be derived from or is consistent with various forms of the equivalence principle".

OK, now some points which I don't expect agreement on. The orbitting charge is not falling freely, hence the comoving observer is not inertial. There is a loose, "heuristic", "historical" form of the equivalence principle stated from outside GR. There is a "mathematical" equivalence principle stated from within GR that (i) in curved Lorentzian spacetime at any point (or sufficiently close) one may set up a coordinate system in which the metric is exactly flat Lorentzian at the point, and deviates from flatness away from that point only at or above second order in Taylor series (ii) the known fundamental laws of physics (Maxwell's equations), but possibly not the derived laws involving second derivatives, in curved Lorentzian spacetime at that point have the same form as those in flat Lorentzian spacetime.

 

[Fredrik]

The velocity and acceleration according to who?
The velocity of the rocket in an instantaneously co-moving inertial frame is still zero. Hence why it is a vacuous truth. If you are going to argue there isn't work being done on the book because of this argument, then the engine is not doing work on the rocket by the same argument. You can't have it both ways.

You're definitely right about this. I don't know how I was able to confuse myself enough to make such a claim.

Consider a coordinate system with the metric diagonal -1,1,1,1 right at the point of the book (make the book a point particle if you want) and in which the book has a non-zero velocity. Sure, I realize that due to spacetime curvature the metric can't be exactly that of flat spacetime in a finite region, but I only need to talk about a region of size limiting to zero. What is the velocity and acceleration of the book right at that point where the metric is -1,1,1,1 ? It can, according to an inertial observer, give results showing that work is being done on the book. Where is this energy coming from?

Man, I feel dumb right now. The equivalence principle, as I stated it in my previous post, applies to any local inertial frame, not just the ones that have a time axis that's tangent to the world line of the object that concerns us. And, there's obviously motion in the direction of the acceleration in some of them. D'oh.

 

[Phrak]

"Will a charge in a gravitational orbit radiate?"

Under simple considerations it's fairly obvious that two charged bodies, freely falling in independent orbits, will be coupled electromagnetically. Each will percieve the other as accelerating. But in which directions does the energy and momentum move, and according to whom?

 

[JustinLevy]

Then the equivalence principle is not a princple of GR, but a principle of SR.

That's fine. Classify it however you will, any principle of SR is also a principle of GR.

That is not fine. There is no principle of SR that refers to gravity. You missed the point I was trying to make with that statement.

Let's say you have a measurement technique that is sensitive to curvature (tidal effects) greater than a finite magnitude k, then it is always possible to define a region of finite size s such that the tidal effects within that hypervolume are less than k. Such a region can be treated as flat.

Look, consider a spacetime that for a finite region (or all of it) has a constant and positive Ricci scalar curvature. This is an intrinsitc geometric quantity. Sure, in GR we can still choose the metric to be diagonal -1,1,1,1 at any point, but that is NOT enough to garauntee that we can't measure the curvature.

Using your 'version' of the equivalence principle, we could have a force that depends on the Ricci scalar since in the scalar->0 limit we still get back SR. So it could be possible to measure the scalar at any point. So your argument above is assuming the equivalence principle says more.

I'm starting to agree with atyy. It sounds like the major problem here is just a precise definition of the equivalence principle, so that it is clear exactly what it says, and exactly when it applies.

Please, can you give me a mathematical definition of the equivalence principle?

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So I agree there is no consensus, at least not at the level for Maxwell's equations or Taylor and Hulse's observations and analyses.

I believe there is little disagreement remaining over the physics (predictions of experimental outcomes), but there is substantial disagreement over when the "equivalence principle" applies (because the predictions can be made using Einstein's and Maxwell's field equations, without the "equivalence principle").

I may be misunderstanding you, but these two back-to-back sentences seem to contradict each other. So there is disagreement on precisely what the "equivalence principle" is (in particular its region of applicability), but is there or isn't there consensus on application of Maxwell's equations?

I would like to suggest this modification of your summary:

"There is still scientific debate on these topics, so the question cannot be definitively answered. There is consensus that: the orbitting electric charge will spiral inwards; the ability to detect radiation is observer dependent; a comoving observer will not detect radiation in at least some situations; some non-comoving observers will detect radiation. There is no consensus on whether this may be derived from or is consistent with various forms of the equivalence principle".

Okay, sounds good. Thank you for your help on this.

There is a loose, "heuristic", "historical" form of the equivalence principle stated from outside GR. There is a "mathematical" equivalence principle stated from within GR that (i) in curved Lorentzian spacetime at any point (or sufficiently close) one may set up a coordinate system in which the metric is exactly flat Lorentzian at the point, and deviates from flatness away from that point only at or above second order in Taylor series (ii) the known fundamental laws of physics (Maxwell's equations), but possibly not the derived laws involving second derivatives, in curved Lorentzian spacetime at that point have the same form as those in flat Lorentzian spacetime.

Ah, okay. I was not aware of the portions I underlined.
Is there a non "heuristic" statement of the equivalence principle? Or was this just a rough feature that Einstein wanted that he used to give insight into the development of GR, but not a precise final feature of the theory?

-----------------------

"Will a charge in a gravitational orbit radiate?"

Under simple considerations it's fairly obvious that two charged bodies, freely falling in independent orbits, will be coupled electromagnetically. Each will percieve the other as accelerating. But in which directions does the energy and momentum move, and according to whom?

Just in case there is confusion (I can't tell if you are trying to ask a new question), the original question involved only one charged body, as the other was neutral.

 

[Phrak]

Just in case there is confusion (I can't tell if you are trying to ask a new question), the original question involved only one charged body, as the other was neutral.

Charge is measured by charge. If you want, one body is charged and the other is neutral, and contains charges doing the measuring.

But, you are right, it's something of a new question---maybe. But as you might recall from what Dale said, where energy resides is not well defined in general relativity, so the answer is anything but trivial--unless someone can come up with a clever thought experiment.

 

[JustinLevy]

Charge is measured by charge. If you want, one body is charged and the other is neutral, and contains charges doing the measuring.

Measure the charge, then setup the experiment. You don't need to measure any EM stuff again, only mechanics ... like distance between the two objects as the charged one orbits (does the orbit decay?), and like momentum (is there a force on the charged object causing the orbit to decay... does it have a non-zero proper acceleration?).

But, you are right, it's something of a new question---maybe. But as you might recall from what Dale said, where energy resides is not well defined in general relativity, so the answer is anything but trivial--unless someone can come up with a clever thought experiment.

This one asks about proper distances, and proper acceleration. We can answer whether the orbit decays without referring to energy if we must.

Furthermore, consider in the two problems (book problem or orbitting charge problem) the book and the charge to be test particles (mass small enough to have negligible impact on the spacetime for this problem). In this case the spacetime is static, and as already noted we can apply energy conservation quite easily.

 

[Dale]

That is not fine. There is no principle of SR that refers to gravity. You missed the point I was trying to make with that statement.

A flat spacetime can be considered a special case of a gravity field, one without tidal effects. Again, any principle of SR is a principle of GR. I didn't miss your point, I was just trying to open your mind a little to see the connection between the two theories.

Please, can you give me a mathematical definition of the equivalence principle?

Sorry, I can't.

 

[JustinLevy]

A flat spacetime can be considered a special case of a gravity field, one without tidal effects. Again, any principle of SR is a principle of GR. I didn't miss your point, I was just trying to open your mind a little to see the connection between the two theories.

My understanding of the connection of the two theories are:
- the equivalence principle (whatever that turns out to be) and
- the poincare symmetry of SR is retained as a local symmetry in GR
If that is incorrect, please help me, for I really thought the equivalence principle was something relating a GR situation involving gravity to an SR situation (and therefore without gravity) in some limit.

In particular, I don't understand the first sentence in your quote there. Remember that in GR, even an infinite planar mass results in curved space and thus tidal effects (unlike in Newtonian gravity). What exactly is it that you are referring to as 'a gavity field' in that sentence?

I'm worried I'm missing some important points because I'm not understanding some of the terminology precisely enough.

 

[atyy]

Ah, okay. I was not aware of the portions I underlined.
Is there a non "heuristic" statement of the equivalence principle? Or was this just a rough feature that Einstein wanted that he used to give insight into the development of GR, but not a precise final feature of the theory?

I'm not exactly sure. Try Blandford and Thorne's discussion of their Eq. 24.15 and 24.16, and section 24.7 http://www.pma.caltech.edu/Courses/ph136/yr2006/0424.1.K.pdf. There's also the interesting comment here on "Nordström's second theory" that is a consistent relativistic theory of gravity, incorporates the "equivalence principle", but there's no global bending of light http://www.einstein-online.info/en/spotlights/equivalence_deflection/index.html .

 

[Naty1]

Well, in reading this thread for yet another time this morning, I believe some of my own confusion is finally cleared...funny how subtle some of this is.

Can someone help me with the following:

from the original posted question:

Where is the energy coming from that allows the Earth to continually accelerate the book?

My prior post:

I see three questions here: If the book is accelerated, does it take energy? Does the gravitational field have energy? If so, where does it come from. Yes,yes,unsure:

F = -dU/dr = -GMm/r^2 (Halliday and Resnick) so I say yes, acceleration requires a force and energy .

The gravitational field clearly has potential energy ...

I ask because I saw elsewhere, another thread, a knolwedgeable person had posted that acceleration does NOT require energy...

 

[Phrak]

Is there some confusion on what is the Einstein Equivalence Principle (EEP)? The Einstein equivalence principle is to general relativity what the equivalence of inerial frames is to special relativity. The EEP is an element of general relativity, but not special relativity.

It was the genius of Einstein to take the Weak Equivalence Principle (WEP); that the inertial mass and gravitational mass of an object are equal, and obtain general relativity.
The WEP is restated as the EEP: "In small enough regions of spacetime the laws of physics reduce to those of special relativity; it is impossible to detect the presence of a gravitational field."

The physical differences between the WEP and EEP are hard to imagine. It's more a change in perspective; inerial mass and gravitational mass are not just equal, but the same stuff. So that no matter how sophisticated or clever the experiment, that two will always be found to be exactly the same (in principle ).

 

[JustinLevy]

Is there some confusion on what is the Einstein Equivalence Principle (EEP)?

Yes, that essentially is the problem. Can you state the equivalence principle mathematically?

The WEP is restated as the EEP: "In small enough regions of spacetime the laws of physics reduce to those of special relativity; it is impossible to detect the presence of a gravitational field."

That cannot be the equivalence principle for as shown by calculations in the journal papers atyy listed, a charged particle orbitting a neutral mass will feel a proper acceleration. This does not reduce to SR even in the region size -> 0 limit.

 

[Jonathan Scott]

That cannot be the equivalence principle for as shown by calculations in the journal papers atyy listed, a charged particle orbitting a neutral mass will feel a proper acceleration. This does not reduce to SR even in the region size -> 0 limit.

Sorry I've not been able to keep up with this thread, but I'd just like to point out that mentioning an "orbit" automatically means you are outside the scope of the Einstein Equivalence Principle, as an orbit requires a region in which the acceleration field is not even approximately uniform.

 

[JustinLevy]

Sorry I've not been able to keep up with this thread, but I'd just like to point out that mentioning an "orbit" automatically means you are outside the scope of the Einstein Equivalence Principle, as an orbit requires a region in which the acceleration field is not even approximately uniform.

Are you saying I can't apply the EEP here even in the limit the region goes to zero size?
That is like claiming there is no Equivalence Principle at all, because it would be restricted to flat spacetime, which would make it a principle of SR not GR.

 

[Jonathan Scott]

Are you saying I can't apply the EEP here even in the limit the region goes to zero size?
That is like claiming there is no Equivalence Principle at all, because it would be restricted to flat spacetime, which would make it a principle of SR not GR.

You can't use it for an orbit (and obviously there's no such thing as a zero-sized orbit). You can use it for a segment of an orbit which is small enough that the change in direction of the field is negligible.

 

[JustinLevy]

You can't use it for an orbit (and obviously there's no such thing as a zero-sized orbit). You can use it for a segment of an orbit which is small enough that the change in direction of the field is negligible.

I don't need to use it for the whole orbit to see that there is a non-zero proper acceleration. I only need to look at a portion of the motion in a region of size limiting to zero. I only need to ask: what is the proper acceleration of the charged particle at this one point on its worldline.

Since the proper acceleration is non-zero, a freefalling observer at the particle will see some kind of anomalous force that can't be accounted for in SR... despite the region being of size limiting to zero.

So we need a better, more precise statement of the EEP. So far no one has been able to present a rigorous mathematical definition. I think we need that to figure out the subtlety that is going on here.

 

[Jonathan Scott]

[Sorry - somehow I got duplicate copies of the same post, with only a tiny difference, so I've deleted the first; I think the first was intended to be a preview, and I don't know how I managed to post it twice]

 

[Jonathan Scott]

I don't need to use it for the whole orbit to see that there is a non-zero proper acceleration. I only need to look at a portion of the motion in a region of size limiting to zero. I only need to ask: what is the proper acceleration of the charged particle at this one point on its worldline.

Since the proper acceleration is non-zero, a freefalling observer at the particle will see some kind of anomalous force that can't be accounted for in SR... despite the region being of size limiting to zero.

So we need a better, more precise statement of the EEP. So far no one has been able to present a rigorous mathematical definition. I think we need that to figure out the subtlety that is going on here.

The EEP simply applies to any region which can be approximately described using an inertial frame of reference. The limit of the approximation mainly depends on how accurately you want your results.

You can't necessarily expand the results of the EEP to cover a larger region, as the approximations involved in the EEP may well be of the same order as the approximations involved in the result.

That means for example that although calculations suggest that charge does not radiate in a uniform gravitational field, you cannot extend that to cover a whole orbit, as that is not uniform.

I'd also guess that if you have a charge sitting on the Earth, it may not radiate due to its gravitational acceleration, but it might well radiate due to the rotation of the Earth (effectively causing a slowing torque on the Earth's rotation) and the Earth's orbit around the sun. Obviously, such effects are many orders of magnitude smaller and negligible in practice.

I think that a charge in a uniform gravitational field would fall just like any other matter, with no proper acceleration relative to a falling inertial frame of reference. However, in orbit, where the field isn't uniform, it would presumably in theory radiate and get a slight back-reaction which would mean that its motion would not be exactly the same as free fall, but these effects would be immeasurably tiny.

 

[atyy]

However, in orbit, where the field isn't uniform, it would presumably in theory radiate and get a slight back-reaction which would mean that its motion would not be exactly the same as free fall, but these effects would be immeasurably tiny.

The backreaction is electromagnetic, which is non-gravitational, so we do not expect it to fall freely. An interesting point is, what if we consider a point mass? Would its "gravitational backreaction" cause it not to fall freely? It turns out that it doesn't fall freely on the "background" metric, but it does fall freely on the "background + gravitational backreaction" metric. http://relativity.livingreviews.org/Articles/lrr-2004-6/

 

[Phrak]

Yes, that essentially is the problem. Can you state the equivalence principle mathematically?

I am behind, in this thread, aren't I? Unfortunately I haven't the time nor wherewithall to keep up, but your probing questions have been well appreciated all around, I'm sure. I'm far more interested in the implications a charged body (or bodies) in orbit, anyway.

That cannot be the equivalence principle for as shown by calculations in the journal papers atyy listed, a charged particle orbitting a neutral mass will feel a proper acceleration. This does not reduce to SR even in the region size -> 0 limit.

That was one of the few ways the equivalence principle is usually quoted. To be sure, I'd forgotten that it bothered me, as well. After all, the stress-energy tensor consists of first and second derivatives of the metric.

I think the above quote can be considered an oversimplification in need of several conditionals. After all, gravity waves are a vacuum solution.

 

[atyy]

The Relation between Physical and Gravitational Geometry
Jacob D. Bekenstein
http://arxiv.org/abs/gr-qc/9211017

Has interesting comments on the weak and strong equivalence principle at the start (rest of paper not relevant to this thread). I suspect most of this is in MTW, which unfortunately is not on arXiv.