# (Alpha) General relativity dissatisfies the equivalence principle

First, I'm not sure why you mentioned Birkhoff's theorem since it has nothing to do with local flatness.
I mention it because many people have told me that the theorem proves GR is self-consistent.

Now, to the relevant theorems. ... Accepting the consistency of Euclidean spaces we conclude that the axioms of (pseudo-)Riemannian geometry, including the local flatness axiom, are consistent.
I don’t disagree with any of this.

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.
If these are based on local flatness, which I agree is a “fundamental feature of GR”, then they don’t prove that GR doesn’t contradict itself about that. Do you disagree? From what I can tell based on an extensive search, GR is thought to be self-consistent only because it has not been shown to be self-inconsistent, and not because it has been proven to be self-consistent. While a simple theory may be readily shown to be self-consistent, GR is not a simple theory. For example, it took years after GR was published before it was shown that a Schwarzschild radius is not a physical singularity.

Hope this helps.
It does, and thanks. I’ll be posting another thread about this topic, another attempt to show that GR violates the SEP, that doesn’t have the flaw pointed out by Pete on pg. 2. I hope you will give your input.

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.
Agreed to all.

'X' inertial frame has within it an accelerating rocket dragging a rope across an event horizon, not flat spacetime.
The spacetime is flat within X. That was amply shown above, with references, esp. on page 1 and 2. Spacetime curvature is synonymous with the tidal force. Then by the definition of an inertial frame given in the OP, the spacetime in X is negligibly curved (all but flat). Three references were given on page 1 and 2 to show that an inertial frame can straddle a horizon.

In both cases, the rocket's frame is non-inertial, while the observer is located in an inertial frame, either X or Y.
Agreed.

The two rockets are located in two different non-inertial frames in which the spacetime curvature is different in each frame.
I disagree that the spacetime curvature in X and Y significantly differ, as noted above.

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.
On page 2, Pete showed that the rope in Y will eventually break regardless of its tensile strength. That refuted the OP’s contention that the rope in Y need not break.

The observer in frame X will not see the rope break unless the tidal forces at the event horizon are great.
By the definition of an inertial frame, which X is, the tidal force throughout the frame is negligible. So a tidal force will not break the rope. Nevertheless, the rope must break because one end is falling below the horizon (GR’s prediction) whereas the other end is attached to the rocket, which hovers above the horizon.

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.
This would indeed cause the rope to break; I gave a link on page 1 or 2 to show that (from Wikipedia topic “event horizon”). But, as above, the rope in X will break regardless.

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.
The experimenter could be beside and at rest with respect to the rope as they both fall freely across the horizon. Then the experimenter can certainly see the rope fall across the horizon. It is an observer in the rocket who cannot see the rope cross the horizon, but there is no such observer mentioned in the OP; such observer is irrelevant to the conclusions in the OP.

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.
The OP is about GR’s adherence to the SEP, not SR’s. Noting that SR “does not recognize gravitational time dilation” neither shows that there “is no problem with GR” nor shows a problem with the thought experiment in the OP.

If these are based on local flatness, which I agree is a “fundamental feature of GR”, then they don’t prove that GR doesn’t contradict itself about that. Do you disagree? From what I can tell based on an extensive search, GR is thought to be self-consistent only because it has not been shown to be self-inconsistent, and not because it has been proven to be self-consistent. While a simple theory may be readily shown to be self-consistent, GR is not a simple theory. For example, it took years after GR was published before it was shown that a Schwarzschild radius is not a physical singularity.

The constructive theorems I've mentioned are a part of GR and so depend on various of its axioms including the local flatness axiom. In this precise sense, quoting them does nothing to "prove" the axiom, the axiom is already assumed. My point in mentioning them is that they provide an explicit way to set up nice local coordinate systems near the horizon. Nothing goes wrong with these constructions so the horizon is regular.

The real point of the post was the possibility of imbedding all nice (pseudo-)Riemannian manifolds into (pseudo-)Euclidean spaces. Since the spacetimes which can appear from the GR field equations constitute a subset of the nice pseudo-Riemannian manifolds, and since all such manifolds can be imbedded in pseudo-Euclidean spaces, we expect that all spacetimes in GR are as nice and consistent as pseudo-Euclidean spaces. Since proving a theory is truly self consistent is a very complicated business in mathematics (e.g. Godel's theorems), my argument is probably about the best one can do.

As for the confusing history of the Schwarzschild solution, much of the chaos and misunderstanding results from the over dependence on coordinates. In fact, this is the story in general for GR. Real progress began to be made when people stopped relying on coordinates so much. The Schwarzschild coordinates are bad at the horizon, but that tells you nothing about the actual geometry. I suspect, and this is only meant to be friendly advice, that your own issues with GR stem largely from the fact that you rely too much on coordinates. Coordinates are often extremely misleading; for example, you're using S coordinates when you should use EF or KS coordinates, both of which reveal the benign nature of the horizon.

zanket said:
It does, and thanks. I’ll be posting another thread about this topic, another attempt to show that GR violates the SEP, that doesn’t have the flaw pointed out by Pete on pg. 2. I hope you will give your input.

Well, I'm certain you won't find any violations, but I'll participate if I can.

That's an excellent post full of good info.

The OP relies on GR's predicted "benign nature of the horizon" for an inertial frame falling across it. That might imply a particular global coordinate system, but the OP doesn't need to explicitly mention one.

Physics Monkey: Thankyou for clarifying that. I don't suppose you could post some links to some info? Or send me a PM with some Info in it.

Zanket: I have apologized for any offense caused. I have even gone as far as voluntarily removing the contended post. How about showing a little good faith and addressing the points I raised in my last post? I will attempt to only address one point at a time.

OK, let's address one point at a time.

Trippy said:
So here, I have cited four independent sources ALL of which agree with the definition of an event that I have been using to refute your arguments.

You say a given event can occur in only one frame. I disagree; I say a given event can occur in multiple frames (i.e. a given point in spacetime exists in multiple frames). None of your four sources limits an event to only one frame. So how do these sources refute me?

Hi Zanket---

In the interest of coninuing in the discussion I will apologize to you. I should have adhered to the rules that you have set out in this discussion, as I would expect the same courtesy from you. I took a loose interpretation of the alpha rules, and apparently overestimated my ability to be sarcastic is a manner which was not interpretted as insulting. In attempting to prove you wrong I have learned a bit about GR that I have not known before, and I should thank you for this.

Either way, I would like to pick up where I left off, in proving you wrong.

Trippy has provided about four definitions which refute your interpretaiton of Wheeler and Taylor's definition of reference frame---namely that "physics" and "measurment" are local ideas. I have showed you that your interpretaiton leads to a logical fallacy, which I will not bother cutting and pasting here again.

Zanket. While I work on a reply (I work full time, and have a wedding to plan), I'm going to ask you to cite a reference that refutes me. If I'm as wrong as you seem to think I am, then this should not be too much to ask. I'm also going to suggest that refusing to do so would constitute (another) violation of the Alpha rules.

Okay.
So I'm willing to concede a point, that I may have been mis-using the term reference frame, but that does not invalidate my argument.
Let John be the Experimenter in the lab X.
Let Bill be the Experimenter in the lab Y.

We have established that an Event occurs at a unique set of space time co-ordinates.
John will measure the event to occur at some co-ordinate (relative to him) of (x,y,z,t).
Bill will measure the event to occur at some other co-ordinate (relative to him) of (x',y',z',t').
John and Bill will agree, once the take the appropriate lorentz transformations into account, that (x,y,z,t) = (x',y',z',t').
Therefore, in that respect it could be argued that the event that John and Bill are observing has occured in two different reference frames. John's reference frame, and Bill's reference frame.

HOWEVER.

Let Andrew be the pilot of the rocket ship in X.
Both Jon and Bill will also agree that the rocket is behaving non-inertialy. It is impossible to fully account for any observations made within the rockets reference frame without accounting for some outside force - in this case, gravity.
John and Bill will both agree that although the rocket ship exists within their reference frames, and that John's reference frame, and Bill's reference frame are both inertial, that Andrew's reference frame is not inertial. They would also agree that any experiment carried out in Andrew's reference frame, is carried out in a non-inertial reference frame, and that any observations of events that occured in Andrew's reference frame would need to account for the outside force being applied to Andrew's reference frame.

This is the point that I was attempting to communicate when I stated that an event can occur in only one reference frame, but is observed from many.

I hope that makes it clearer.

Either way, I would like to pick up where I left off, in proving you wrong.
OK, let’s discuss one point at a time, starting with this one:

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.)
The part of the SEP used in the OP refers to “any experiment” in a lab moving in an inertial frame. How did you infer that “any experiment” is limited to an experiment that measures something moving at a constant velocity? The vast majority of experiments conducted throughout recorded history have not been so limited.

Y is a copy of X, including the experiment within X, with the sole exception that Y’s location in spacetime differs from that of X. Both X and Y are inertial frames. Only the outcomes of the experiments in X and Y are compared in the OP. X and the rocket’s frame in Y would be incomparable even if the rocket in Y moved at a constant velocity; those frames are apples and oranges.

Therefore, in that respect it could be argued that the event that John and Bill are observing has occured in two different reference frames. John's reference frame, and Bill's reference frame.
Yes, and in relativity there are no preferred inertial frames, so the event belongs to John’s frame no more or less than it does to Bill’s frame.

John and Bill will both agree that although the rocket ship exists within their reference frames, and that John's reference frame, and Bill's reference frame are both inertial, that Andrew's reference frame is not inertial. They would also agree that any experiment carried out in Andrew's reference frame, is carried out in a non-inertial reference frame, and that any observations of events that occured in Andrew's reference frame would need to account for the outside force being applied to Andrew's reference frame.
Yes. However, in the OP the rocket + rope is the experiment, so no experiment is conducted in a noninertial frame. Observers in the labs that are in free fall in X and Y, both inertial frames, monitor the rope and record whether or not it breaks.

I hope that makes it clearer.
If you have no more issues with events, then please choose your next point of disagreement with the OP.

The part of the SEP used in the OP refers to “any experiment” in a lab moving in an inertial frame. How did you infer that “any experiment” is limited to an experiment that measures something moving at a constant velocity? The vast majority of experiments conducted throughout recorded history have not been so limited.

Using this logic, it is quite easy to construct experiments which violate the SEP! One can take some grossly noninertial experiment and stick it in an inertial reference frame, and be done with the matter.

Here is an example, I think. Let my mass be constant and nonzero (which, I assure you, it is). Now put me on a scale in an inertial frame, and my weight is 0. Now put me on a scale on the Earth, and my weight is mg. No problem, you say, because the Earth is a non-inertial reference frame.

But now, put me on a scale on Earth, but observe my weight from an inertial reference frame. I am behaving in a non-inertial manner, but the experiment is still being conducted in an inertial reference frame. So the contradiction---how can my weight be both zero and non-zero.

Zanket says "Ah ha, the SEP is violated", and I say "Sorry but you'll have to do better".

Let my mass be constant and nonzero (which, I assure you, it is). Now put me on a scale in an inertial frame, and my weight is 0.
This experiment is okay; it’s within the realm of “any local experiment in a lab moving in an inertial frame” to which the SEP refers.

But now, put me on a scale on Earth, but observe my weight from an inertial reference frame.
A local experiment measures something within the lab. You’d need to put the Earth within the lab, but then the curvature of spacetime (the tidal force) caused by the Earth makes the inertial status of the lab’s frame doubtful. A better example of such experiment would be to put you on a scale in a rocket that is noninertially accelerating (thrusting) within the lab.

Zanket says "Ah ha, the SEP is violated", and I say "Sorry but you'll have to do better".
When testing GR’s adherence to the SEP regarding the lab’s location in spacetime, the same experiment must be conducted in both (or all) inertial frames, like X and Y. Your two experiments differ, hence their outcomes are incomparable. You could show that GR violates the SEP only by showing that the outcome of one experiment or the other differs depending on the lab’s location in spacetime. But I don’t see how you could do that. Presumably the outcome of the first experiment is always the same (always zero) regardless of the lab’s location in spacetime. Presumably the outcome of the second experiment is always the same (always the same nonzero value) regardless of the lab’s location in spacetime.

Zanket said:
When testing GR’s adherence to the SEP regarding the lab’s location in spacetime, the same experiment must be conducted in both (or all) inertial frames, like X and Y. Your two experiments differ, hence their outcomes are incomparable. You could show that GR violates the SEP only by showing that the outcome of one experiment or the other differs depending on the lab’s location in spacetime. But I don’t see how you could do that. Presumably the outcome of the first experiment is always the same (always zero) regardless of the lab’s location in spacetime. Presumably the outcome of the second experiment is always the same (always the same nonzero value) regardless of the lab’s location in spacetime.

This exact same logic applies to the OP as I have tried to point out repeatedly, and yet you have refused to acknowledge it. Any attempt to pint this out has been met with claims about it's irrelevancy, and the claim that the person making the argument (in this case myself) is claiming that the GR is untestable in this regard.

By your own logic the two experiments outlined by Ben are equivalent.

By your own logic the lab which encompases X must encompase the entire blackhole (thus tidal forces could never be considered negligible).

And if you can't see it?
Replace 'Earth' with 'Event Horizon'
Replace 'sitting on' with 'hovering above'
Replace 'What is the mass as measured' with 'does the rope snap' actually, these are much the same question anyway.

By your own logic the two experiments outlined by Ben are equivalent.
By what logic? They are clearly different experiments. One measures the weight of an inertial observer, the other measures the weight of a noninertial observer. In the OP the experiments in X and Y are identical.

By your own logic the lab which encompases X must encompase the entire blackhole (thus tidal forces could never be considered negligible)
No, because the experiment in X is contained wholly within the lab in X. "Local" in "local experiment" in the SEP means "within the lab, doing good measurements". The experimenter cannot look outside a window of the lab and say "there's a planet out there I didn't see before, therefore the SEP is violated". BenTheMan's experiment involving a scale on Earth would be like that, because you couldn't even see the scale on the Earth unless you looked out the window. The Earth could be included within the lab, but then the lab's inertial status is doubtful.

The same type of experiment can be conducted within the lab without the tidal force created by the Earth being an issue, by putting the scale within a noninertially accelerating (thrusting) rocket within the lab. That would be a local experiment. But then the outcome of that same local experiment must differ in an inertial frame at a different location in spacetime before one can say that GR violates the SEP in regards to the lab's location in spacetime.

If the outcome of a local experiment (an experiment contained wholly within the lab, i.e. without looking out a window, doing good measurements) differs depending on the lab's location in spacetime, then GR violates the SEP, rather than (as you suggest) indicating that whatever caused the outcome to differ must be included within the lab.

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Zanket said:
One measures the weight of an inertial observer, the other measures the weight of a noninertial observer. In the OP the experiments in X and Y are identical.

No, instead one measures the weight of a non inertial rope in X, and compares it to the weight of an inertial rope in Y, because ultimately that is what breaks the rope.

How can X and Y be identical when you have already agreed that the rocket in X is behaving non-inertialy, and the rocket in Y is behaving inertialy?

Zanket said:
No, because the experiment in X is contained within the lab in X. "Local" in "local experiment" in the SEP means "within the lab, doing good measurements". The experiment cannot look outside the windows of the lab and say "there's a planet out there I didn't see before, therefore the SEP is violated". BenTheMan's experiment involving a scale on Earth would be like that. But the same type of experiment can be conducted within the lab, by putting the scale within a noninertially accelerating (thrusting) rocket within the lab. That would be a local experiment. But then the outcome of that same local experiment must differ in an inertial frame at a different location in spacetime before one can say that GR violates the SEP in regards to the lab's location in spacetime.

As I said in my post. The experimenter in X MUST take into account the presence of an outside force when observing the rocket.

By your own logic, the only thing that Ben has changed is the labs location in space time.
By your own logic, your now arguing that the SEP is untestable in relation to it's location in space time.

No, instead one measures the weight of a non inertial rope in X, and compares it to the weight of an inertial rope in Y, because ultimately that is what breaks the rope.
The experiment in both X and Y determines whether or not the rope breaks, that's all. The rope in both X and Y is noninertially accelerating, due to being attached to a thrusting rocket in both cases.

How can X and Y be identical when you have already agreed that the rocket in X is behaving non-inertialy, and the rocket in Y is behaving inertialy?
I didn't agree that "the rocket in Y is behaving inertialy". Quote me.

As I said in my post. The experimenter in X MUST take into account the presence of an outside force when observing the rocket.
I edited my post after yours to improve the explanation; take a look. The experimenter in either X or Y must not take into account anything outside of the lab, otherwise it's not a local experiment.

By your own logic, the only thing that Ben has changed is the labs location in space time.
By your own logic, your now arguing that the SEP is untestable in relation to it's location in space time.
No, the two experiments BenTheMan gave differ. They are apples and oranges.

“any local experiment in a lab moving in an inertial frame”

We should be clear---you have not defined your reference frames to be local. That is, they are not defined at one point in space time. Your experiments are non-local anyway. You are sure to run to Wikipedia and cut and paste a definition for me. I have checked Wikipedia and found that there isn't an acceptable definition there. So you'll have to try harder

Your experiment is non-local for several reasons. The principle of which is that, if the rope crosses the horizon, it is no longer in causal contact with the rest of the rope. The end of the rope cannot "talk" to the beginning of the rope---this much should be clear from the causal structure of the space-time in which the black hole is located.

A local experiment measures something within the lab.

Not as you've defined "lab". Your labs are not local. See the previous paragraph.

You’d need to put the Earth within the lab, but then the curvature of spacetime (the tidal force) caused by the Earth makes the inertial status of the lab’s frame doubtful.

And this is different from your lab containing an event horizon how?

A better example of such experiment would be to put you on a scale in a rocket that is noninertially accelerating (thrusting) within the lab.

Ah yes but GR tells us the two are the same. But ok---let's use your example.

When testing GR’s adherence to the SEP regarding the lab’s location in spacetime, the same experiment must be conducted in both (or all) inertial frames, like X and Y. Your two experiments differ, hence their outcomes are incomparable.

If my experiments differ then so do yours. If I define a reference frame to contain an accelerating rocket, and observe my weight from an inertial reference frame, do my measurements suddenly become valid? This is, I should stress, what you are proposing. All you have done is defined a frame and placed the rocket there. I have done the same thing. Aparently, the fact that the rocket is non-inertial doesn't matter.

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We should be clear---you have not defined your reference frames to be local. That is, they are not defined at one point in space time.
In the interest of time, I'll argue only one point of yours at a time.

You've made this argument before but you haven't backed it up. Whereas I gave multiple references from top relativity physicists to show that the inertial frame to which the SEP refers need not be point-sized. Even the SEP itself refers to a lab in an inertial frame. I don't see why you keep bringing this up. You’ve already agreed that the tidal force (spacetime curvature) can be neglected in a frame for the purposes of a given experiment. So why do you think the frame must be point-sized? What would be gained by that? More importantly, when are you going to support your point with some outside reference? I call yours the “not exactly flat” argument, as in “if the spacetime is not exactly flat then it’s not an inertial frame”. I once mentioned to a physics professor that people bring up this argument ad nauseam. He said “Why bother having discussions with people like that?” I don’t say that to be rude. I say it to impress upon you that you’re nitpicking. Einstein's “luckiest thought” that led to the idea of the equivalence principle was this: “For an observer in free fall from the roof of his house there exists no gravitational field—at least in his immediate proximity”. Einstein’s next thought was not “scratch that—it’s a useless idea because it’s true only at a point”.

Ok fair enough---maybe I should jump off a building because one physics professor somewhere in the world thinks I don't know what I am talking about

Look---the point is that your experiment isn't local. This was the point that I have made several times before, and you have refused to acknowledge. One cannot define a local experiment stradling the horizon of a black hole---it is not possible. Defining frames as little patches of space is fine, so long as the little patch of space has the same causal structure, as Phisics Monkey pointed out. Clearly the causal structure in one part of your frame is different from the other part of the frame because one end cannot send a message to the other end. The idea of locality ceases to exist across the horizon of a black hole. Once one part of your reference frame crosses a horizon, it is non-local---that is, a beam of light cannot go from one end of the frame to another. There is no way for the end of your rope to comunicate with the beginning of your rope. This is something that you cannot get around. And I don't know how many ways I can say this.

If you ever talk to that physics prof again, tell him I said that defining your reference frames at a single space-time point can ensure that this doesn't happen. You are automatically granted locality when you do this.

To be fair, I have talked with several physics professors about this conversation (even one of the world experts in black holes) and they have all said the same thing---"Did you tell him that you can't define a frame that straddles a black hole?"