(Alpha) General relativity dissatisfies the equivalence principle

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This is at least the ninth time you’ve blatantly violated the Alpha rules. Four times you’ve attacked people not the idea, and five times you’ve ignored my direct relevant questions. That’s enough. All you’ve proven in this thread is that you think the rules apply only to others, who are free to take over your arguments.

 

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Hi Zanket,

I assume that what Ben meant by "purely mathematical theory" is a theory based on consistent axioms and the rules of logic. I don't wish to assert that this is strictly speaking the only possible sort of "purely mathematical theory," only that this is probably what he was talking about. At least it was what I thought he was talking about, so with this understanding, he was basically saying what I had in mind which is why I regarded his response as adequate.

I assure you there are no ill feelings on my part. I'm happy to be specific if you like. First, I'm not sure why you mentioned Birkhoff's theorem since it has nothing to do with local flatness. Now, to the relevant theorems. Within the confines of Riemannian geometry it is possible to prove that every \( C^\infty \) Riemannian manifold may be imbedded isometrically in a Euclidean space of sufficiently high dimension. This theorem is quite important and it does have a name: Nash's imbedding theorem. Now this isn't quite what we want since GR is a kind of pseudo-Riemannian geometry, but it has been proven by Sokolov that the same sort of isometric imbedding of psedo-Riemannian manifolds in pseudo-Euclidean spaces is possible. This basically means that the manifolds in question are as consistent as Euclidean spaces which I think we all agree are consistent (though I freely admit I know of no proof of this). Accepting the consistency of Euclidean spaces we conclude that the axioms of (pseudo-)Riemannian geometry, including the local flatness axiom, are consistent.

One can also be more concrete with a number of constructive theorems. Schild's ladder can be used to construct geodesics crossing the horizon, and some sort of orthonormal transport, like Fermi-Walker transport, can be used to carry good local coordinate systems across the horizon. Of course, all this is based on local flatness which is, as I have repeatedly emphasized, a fundamental feature of GR.

Hope this helps.

 

I'll defer to you

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Zanket will neither understand nor accept these maths. 10 to 1.

 

And aparently a consensus is not enough to show that your argument is logically inconsistent, which also violates your hallowed Alpha rules. I'm glad you're keeping count of MY violations.

True or false: you are proposing that one of the most experimentally well-verified theories that man has ever conceived is wrong. I have already linked you to a publication which proves this.

True or false: you have studied physics in any detail, beyond an introductory high school or college course.

True or false: you have shown a robust knowledge of the theory of general relativity.

True or false: you have been accused by others of violating the same rules.

They are free to take over my arguments as I am free to take over theirs, I'm sure.

But I am here for the physics and not the company (Physics Monkey and Trippy excluded of course), so I will re-list my main objections here. I know that you are responding to many objections, so I will pick one and focus on it. I will drop the debate about Lorentz trnasforms because you are not understanding the point, whether it be some failure on my part to explain it coherently or otherwise. I will drop this argument with the caveat that it will probably be used again in the future, because, as it is with physics, many things are tied together. I will focus, at least in the short term, on attacking your definition of reference frame, which is used out of the context of Taylor and Wheeler's original intent, I am sure.

I also don't want to debate about the difference between "negligible" and "non-existant" anymore, unless it is necessary. The two must be the same if you are to do a calculation. Your arguments about "different negligible tidal forces" are amusing, but are not really relevant to the experiment, I feel. If you can show otherwise, we can continue bickering about this.

I have often not answered some of the questions that you have asked because I felt that they had been adequately addressed in other posts, or because I felt that they should be clear to someone who had some knowledge of GR. I will assume neither in the future. If you ask a question, I will either cut and paste a past response from me or someone else, or give an explicit answer.

Your definition of reference frame leads to a contradiction:

Trippy has pointed out that 4 doesn't follow from 3, for some communications that he has had with you in the past, namely that the rocket in Y is behaving in a non-inertial manner. If this is so, then the SEP is trivially violated, as expected. The rocket is non-inertial, and cannot be considered under the SEP, as only inertial reference frames can be compared. The rocket must be moving with a constant velocity. In essence you are comparing an inertial and a non-inertial reference frame, that is, X and the rocket's frame in Y. (Note that the rocket has a reference frame whether you want it to or not.)

If both X and Y are inertial then I respectfully disagree with Trippy---that is, if the rocket is moving with a constant velocity. If X and Y are inertial then you must accept that your definition of inertial frame implies that some larger frame S contains both X and Y.

 

This is the way I understand it, too. Like Newtonian mechanics is completely consistent mathematically and only happens to apply to the real world or something.

The only thing that Zanket can possibly argue is that GR violates the SEP in some special limit---like the limit where GR fails to apply anymore. This is like Newtonian mechanics being violated when masses and Compton wavelengths are of similar size. We already have an upper bound for this, called the Planck energy (or legth).

What I understood from Space Monkey's reply is that the existence of a smooth Riemannian manifold is a necessary and sufficient ingredient for GR. (Riemannian means it is smooth and admits a metric.) So Zanket would have to proove that the geometry at the horizon is not described by a Riemannian manifold---that is, that it is either not smooth or doesn't admit a metric. But this contradicts his own statements that the horizon can have a flat geometry, described by the Schwarzchild solution. In other words, the SEP follows from GR, it is not required for GR to be consistent.

Is this correct?

A minor point---Riemannian manifolds contain the set of Lorentzian manifolds? Is this true?

 

Oh. I meant Physics Monkey's post.

 

Yes I have checked. This is true... Lorentzian manifolds just have a metric which has a single negative eigenvalue. The Riemannian manifolds have metrics which have real eigenvalues.

 

If it is non-inertial how can you use the SEP????

 

I think some are forgetting the original example by Zanket:

Neither frame X nor frame Y is the rocket's reference frame. Zanket is using two different inertial frames X and Y, and an accelerating rocket in each, separate, inertial frame. 'Y' inertial frame has within it an accelerating rocket dragging a rope in flat spacetime. 'X' inertial frame has within it an accelerating rocket dragging a rope across an event horizon, not flat spacetime. In both cases, the rocket's frame is non-inertial, while the observer is located in an inertial frame, either X or Y. The two rockets are located in two different non-inertial frames in which the spacetime curvature is different in each frame. The observer in frame Y will not see the rope break unless the acceleration is sufficient to place a great enough stress along the length of the rope to cause it to break. The observer in frame X will not see the rope break unless the tidal forces at the event horizon are great. The only 'problem' I see is if the observer in frame X decides to tell the rocket to 'reel in' a rope that has fallen through an event horizon without breaking when it fell through. As I understand GR, the observer in frame X would never actually see the rope 'fall through' the event horizon in the first place because of gravitational time dilation near the event horizon. Special Theory does not adequately address this scenario because it does not recognize gravitational time dilation. There is no problem with GR. Because of the gravitational time dilation, this thought experiment is outside the bounds of Special Theory.

 

So what? Just because you put a non-inertial experiment in an inertial frame doesn't mean you can suddenly apply special relativity. The experiment is non-inertial, and this is all that matters.

Zanket has said that the space-time across the horizon is approximately flat, as it must be for the SEP to apply. If the frame was inertial, and the experiment was also inertial, then the SEP could be applied.

Yes, so the observations are effectively non-inertial. The rocket's frame is the only one that matters.

If this is the case then the SEP definitely doesn't apply.

This is exactly correct. That is why the frame that sees the rope cross the horizon must be non-inertial. This has been pointed out to Zanket (by Physics Monkey, I think). (It does, of course, depend on where the observer is located.)

Good luck convincing Zanket of this!

 

We should make a list of Zanket quotes. This one is my favorite:

 

God forbid he give me another infraction.

 

It says “as she falls through”, as in “while she is falling across the horizon, straddling the horizon, her frame is inertial and SR works fine”.

You have not done that.

The results of the tests would be irrelevant using your logic. I claim that your logic precludes testing GR’s adherence to the SEP in regards to the lab’s location in spacetime, because for any set of experiments one could always say that the lab was at the same location, using your logic. By your logic “location” is something that can be a constant property of the lab, rather than something that changes when the lab is located differently relative to other objects. The lab can be considered stationary in some cases, but not in reference to the part of the SEP used in the OP (or else the SEP becomes untestable in that regard).

No, that’s a test of a part of the SEP that is irrelevant to the OP. More on that below.

SR deals with a subset that includes nongravitational acceleration (like the rocket’s acceleration in the OP), which you’d realize with a simple Google search.

Yes, this is true, with one exception: an object undergoing Born acceleration; that’s the only type of accelerating object that can avoid being continuously physically stretched. It doesn’t matter that one end of the rope (in X or Y) is free; the rope will still physically stretch because it is not undergoing Born acceleration. Bell’s spaceship paradox shows that the rope physically stretches: if a rocket pushing the “trailing” end of the rope is not enough to keep it from physically stretching, then it will certainly physically stretch when that end is not pushed. Texts on Born acceleration confirm that, by showing that the trailing end of an object must accelerate at a higher rate than the leading end, as measured by onboard accelerometers, or else the object will physically stretch.

False, as I showed above about ANDed statements. The OP need focus on only one of a set of ANDed statements, because any such statement shown to be false falsifies what the set says as a whole (e.g. the entire SEP is violated if any one of its ANDed statements is violated). You didn’t refute that, so you’re expressing only an unsupported opinion at this point.

I was polite. When I say “that should tell you something”, I mean “that should tell you that your point is invalid”. I now understand that you think “No part of the SEP is irrlevant to the OP”, but I had already refuted that by when I replied, so there was no reason for me to assume that’s what you meant. If I refute something and you don’t dispute it, then I assume that you agree that it’s refuted, and I assume that you will not carry on with the refuted point, using it in further arguments.

The OP implies that a horizon does have an effect on causality (i.e. affects the outcome of an experiment). Then it would make no sense for me to claim the opposite. Your point here essentially restates your belief that changing the lab’s location is tantamount to changing the experiment, invalidating the thought experiment; I’ve refuted that.

Here you’re essentially repeating your claim that an inertial frame cannot straddle a horizon. That’s already been refuted.

Here you’re essentially repeating your claim that any change to the outcome of the experiment that is due to changing the lab’s location in spacetime means that the forces changed, hence the thought experiment is invalid. And I have already noted that such logic implies that the SEP is untestable in regards to the lab’s location in spacetime, hence not falsifiable in that regard, and that is not science.

I reviewed your posts. Nowhere have you given an example that would refute my claim that, using your logic, the SEP is untestable in regards to the lab’s location in spacetime. You are indeed arguing that “Location in space time has no applicability”. Obviously it has no applicability when, by your logic, the lab’s location in spacetime can be the same no matter where in spacetime the lab is located, relative to other objects.

Nope. Maybe you’re talking about your claims regarding the other part of the SEP that I say is irrelevant to the OP? Not only have I shown that I am right that it’s irrelevant, but also if you were right then the SEP would contradict itself, as I have already mentioned.

I already covered this above. I disagree that the SEP is untestable in regards to the lab’s location in spacetime; that’s not science. If the forces on the rope change solely by changing the lab’s location in spacetime, that shows that GR does not adhere to the SEP, rather than show a problem with the thought experiment.

What you said was “[a Rindler horizon] predicts all of the behaviour (as near as I can tell) associated with the event horizon ...” and that’s what I addressed.

You didn’t say why, so I ignore this claim.

No matter how many times you say it, it’s still incorrect. Nothing in the relativistic definition of “event” specifies that it occurs in only one reference frame. And if you were right, other definitions would be nonsensical. For example, the definition of “proper distance” is “a distance measured between two events in a frame in which those events occur simultaneously”. That definition makes sense only when a given event can occur in multiple frames. Taylor and Wheeler give an example of an event: an exploding firecracker. A given such event can occur in multiple frames; the firecracker might be stationary in my frame and moving in yours.

I could not have quoted your answer, because you never answered it.

I drop it, but you have blatantly violated the Alpha rules, by refusing to answer the question, which for the record is:

Anyone can see that you did not answer the question. You say you answered it in what I quoted of yours here, but you ignore that I showed above that this answer is insufficient, by noting the moths are "in a different inertial environment to the person observing the moths" whenever they move relative to the experimenter, so by your own logic your answer must show how the measurement "needs to take into account the Lorentz transformations". You haven't done that. Then in fact you have not answered the question. Your "answer" translates to this: "In your very specific example, it wouldn't. However, it would, because if (as previously stated) the moths are ..."

I don’t deny that SR is self-consistent; that covers the Wikipedia quote. The second reference does not show that GR is self-consistent at the level referenced in the OP (a rocket hovering about the horizon of a black hole).

Minkowski did not demonstrate that there are no internal inconsistencies in GR. He did that only for SR.

Yes, the rocket’s frame is noninertial, and that’s irrelevant. The SEP says “any local experiment”. That includes experiments of objects whose frame is noninertial. For example, it includes experiments of meandering moths. Only the lab’s (the experimenter’s) frame need be inertial.

I have not obfuscated about that. I have clearly said that you are wrong and noted that you offer no support for your claim but your opinion. When the SEP says “any local experiment”, it is not up to me to show that some local experiments are excluded. The burden of proof is instead on you to show that.

I didn’t say your objection is irrelevant. I say that you have not supported the objection; you have not supported your claim that the experiment is not covered by “any local experiment” in the SEP.

You’ve offered nothing but your opinion to support this claim. Your claim contradicts the dictionary definition of “any”, as in “any local experiment”.

You aren’t specific here, but I gather that you think this because you think SR does not handle acceleration like the rocket’s. Even if you were right about that, it wouldn’t matter, because the experiment would still be in the domain of “any local experiment” to which the SEP refers. But I suggested above that you see for yourself that you’re wrong, by googling for “relativistic rocket”. You found that comment “insulting”, and continued to express your unsupported opinion that contradicts relativity. I can’t force you to google for text like “A noninertial frame can be described in special relativity using the proper mathematical forms for relativistic acceleration”, or “It is a common misconception that Special Relativity cannot handle accelerating objects or accelerating reference frames”.

You have again blatantly violated the Alpha rules here. This is all “attacking people”; you suggest that my points are laughable without refuting them scientifically. You also repeat claims which I refuted, while ignoring my refutations (for example, about ANDed statements, and about SR’s ability to handle accelerating objects). You also ignore my repeated questions. I specified in the OP that posters in this thread must adhere to the Alpha rules, and you have repeatedly and blatantly violated those rules. Thus my obligation to respond to your posts ends.