Thanks Ben, your explanation and post was perfectly logical to me. I also liked the pion example.
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(Alpha) General relativity dissatisfies the equivalence principle
- Thread starter zanket
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No, I am talking about your statement by fiat about the Lorentz frames being curved. Or, have you edited your post by now? No matter, I will check my copy of your original thread start post on my hard drive.
So, does your statement by fiat that the Lorentz frames are curved agree with the definitions of Taylor, Thorne and Wheeler, all top physicists ( and first mentioned in this thread by me)?
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This subject has cropped up before, and I've never heard a satisfactory answer that explains a difference that has niggled me for a while.
My layman's understanding is that gravity exists not because of some magical action-at-a-distance force, but because of a "local tension gradient" that is exhibited as a varying c. This alters a particle's velocity, effectively steering it towards the centre of mass/energy. Without this gradient, which exists everywhere, there is no gravitational field. Reducing the situation to a study of points removes this gradient entirely. In other words the tidal forces are a crucial difference between an accelerating frame and a "gravitational field". If anybody could put me straight on this I'd be grateful, or point out any problems in this historic paper written by Peter M Brown:
http://xxx.lanl.gov/abs/physics/0204044
However one of the main tenets of general relativity is the Principle of equivalence: A uniform gravitational field is equivalent to a uniformly accelerating frame of reference. This implies that one can create a uniform gravitational field simply by changing one’s frame of reference from an inertial frame of reference to an accelerating frame, which is rather difficult idea to accept. A uniform gravitational field has, by definition, no tidal forces and thus no space-time curvature. Thus according to the interpretation of gravity as a curvature in spacetime a uniform gravitational field becomes a contradiction in terms (i.e. no tidal forces where there are tidal forces). This apparent contradiction is obviously quite confusing and can certainly be misleading. In A brief history of relativity published in the above mentioned issue of Time, Stephen Hawking writes...
My layman's understanding is that gravity exists not because of some magical action-at-a-distance force, but because of a "local tension gradient" that is exhibited as a varying c. This alters a particle's velocity, effectively steering it towards the centre of mass/energy. Without this gradient, which exists everywhere, there is no gravitational field. Reducing the situation to a study of points removes this gradient entirely. In other words the tidal forces are a crucial difference between an accelerating frame and a "gravitational field". If anybody could put me straight on this I'd be grateful, or point out any problems in this historic paper written by Peter M Brown:
http://xxx.lanl.gov/abs/physics/0204044
However one of the main tenets of general relativity is the Principle of equivalence: A uniform gravitational field is equivalent to a uniformly accelerating frame of reference. This implies that one can create a uniform gravitational field simply by changing one’s frame of reference from an inertial frame of reference to an accelerating frame, which is rather difficult idea to accept. A uniform gravitational field has, by definition, no tidal forces and thus no space-time curvature. Thus according to the interpretation of gravity as a curvature in spacetime a uniform gravitational field becomes a contradiction in terms (i.e. no tidal forces where there are tidal forces). This apparent contradiction is obviously quite confusing and can certainly be misleading. In A brief history of relativity published in the above mentioned issue of Time, Stephen Hawking writes...
I will take your argument and show how it negates your thought experiment first, and then point out the flaw in logic.
If the laws of physics in all inertial frames are the same, clearly there exists some (inertial frame) S that contains X and Y Because we have assumed the frames in X and Y to be different except for a horizon, the curvatures at X and Y should be the same, and S should be a good frame. If S is "self-compatible" (as we are calling it), then clearly the events in X and Y must be compatible. But this violates out initial conclusion (which I am beginning to doubt) that the events in X and Y are not the same.
(You are playing games with reference frames, as I just did above.)
The new frames are not compatible. Implicit in all of the thought experiments is that the events occur locally, at one point in space-time. If this were not the case, then one could not say that there is a frame with constant curvature, only a frame with approximately constant curvature. Then one would have to work to varying accuracies, as you mentioned in your previous post.
I think I am very glad that you have brought up this point, because I think it shows the flaw in your thought experiment without anymore business with Lorentz Transforms and covariant derivatives. You have defined a set of events to happen across a range of space-time, instead of locally at one point. (In fact, I would like to see one of Einstein's original thought experiments that was not comparing two isolated points in space-time, or at least that could be formulated that way.) The thing we are observing is the tension in the rope---if it is greater than the tensile strength, then the rope breaks. But "tension" required more than 0 dimensions (i.e. one cannot have "tension" in a point particle). So, perhaps, this is a backdoor way out of your conclusion.
To Farsight:
Be careful referencing documents posted on the ArXiV. One should not mistake this as peer-reviewed literature. This article was published in no journal, by an author not affiliated with any university or research institution. This is not to say his conclusions are wrong, it is to say that they have not been examined by the scientific community. (This always opens the door to complaints that the fraternal order of scientists doesn't take outsiders seriously, which I have found not to be the case. The most important math problem that was solved last year was done so by a Russian hermit who lived with his mother!)
Thanks Ben. I've had similar feedback elsewhere, but still seem unable to get much of an answer to what seems like a quite crucial point.
Re what you're saying above, if you can't have tension in a point you can't have negative tension (stress) in a point. And if like me you view energy as stress x volume you can't have energy/momentum in a point, then if you view mass as tied-up energy/momentum you can't have mass in a point. Everything just slips between your fingers, because points appear to be a mathematical convenience that is simply not apt.
Re what you're saying above, if you can't have tension in a point you can't have negative tension (stress) in a point. And if like me you view energy as stress x volume you can't have energy/momentum in a point, then if you view mass as tied-up energy/momentum you can't have mass in a point. Everything just slips between your fingers, because points appear to be a mathematical convenience that is simply not apt.
The fiat would be that negligibly curved spacetime is flat. And yes, that is what Taylor and Wheeler do.
Here is their definition for “flat spacetime” (from their book Exploring Black Holes): “Region of spacetime in which it is possible to set up a free-float (inertial) reference frame.”
Here is their definition for “free-float frame (inertial frame)”: “Generally, a reference frame in which a free test particle initially at rest remains at rest. More technically, a reference frame with respect to which relative (tidal) accelerations of test particles can be neglected for the purposes of a given experiment.”
In other words, an inertial frame is a frame in free fall and throughout which the tidal force is negligible. (That is the definition I gave in the original post.) And the spacetime within an inertial frame is deemed flat, even though it’s negligibly curved.
Thorne agrees with Taylor and Wheeler’s definition for an inertial frame. I haven’t seen a definition of flat spacetime from him.
Regarding the size of an inertial frame: Let a lab be a cube with sides each a proper length of one million light years. Is it possible in principle for this lab to fit wholly within an inertial frame, where the maximum difference among the local gravitational accelerations (g’s) throughout the lab is less than the difference in g between your head and your feet now? Yes it is possible. And it is possible in principle for the lab to remain in such an inertial frame as it crosses the horizon of a black hole, provided the black hole is sufficiently large.
If the new frames (X2 and Y2) are incomparable, then experiments at different r-coordinates in the lab in X are incomparable as well, in which case any experiment taking place at more than one r-coordinate (i.e. every experiment in practice) is inapplicable to the SEP. Do you agree?
It seems you are changing your argument here (which would be okay). Are you saying that an experiment to which the SEP refers must take place at a single point in spacetime? Can you list out fully and succinctly the criteria you think are required for two experiments to be comparable in regards to the SEP?
Hmm. Let me think. Yes I agree. Each event is occuring at a single space-time point. Or it should be, in order for the SEP to apply. One can have a dynamic system by observing the experiment at each point in time---this is ok if the metric itself is static (i.e. your black hole isn't changing size). We have assumed the metric to be static, so this is not an issue.
I am changing the way that I argue your point, not the argument itself. (I am actually meeting you on your turf, as this all makes much more sense to me when formulated in terms of quantum observables
I should be careful, though, because I don't fully understand QM on curved manifolds!) All of the stuff I said still applies. This argument (that frames should be defined locally at one space-time point) still follows from the link I posted earlier---I just haven't said it quite so explicitly. (The business about causal structures was me attempting to support this claim from another position, one which I think I will ultimately abandon. It is still true, I will just not try to argue from this point in the future.)The SEP says that the laws of physics hold locally at each point in space-time, provided we replace derivatives with covariant derivatives. The first part is what concerns us. We can take the Schwarzchild metric as before, with space otherwise empty. A measurement in two different frames, X and Y, will be the same if they are constructed such that r_1 = r_2. Otherwise, the Lorentz Transformations must be used to connect one frame with the other (example: pion decay). The Lorentz Transformation makes all frames "comparable", because it gives us a recipe of how to get from one frame to the other.
Perhaps we should draw a distinction between "comparable" and "the same". The two frames are always comparable, because one can use Lorentz Transformations to shift one frame to the other. The results from measurements in two "comparable" metrics can only be "compared" when one employs the Lorentz Transform. The two frames are "the same" if they are constructed such that they are equidistant from the singularity. (Here I've assumed a Schwarzchild metric.)
How does this affect a measurement of whether the rope will break?
To me, the only way the rope can break is if the tension is greater than the restoring force. So one should take a measurement of force (which is a Lorentzian four-vector), preformed at the horizon of the black hole, and Lorentz Transform it to some frame sufficiently far away from the horizon, and see what happens. This is probably tremendously complicated, as most GR calculations are. But this may be using the SEP to test the SEP, so Zanket will no doubt take issue here.
The other, quantum explanation that I will offer is this. (Zanket has already informed me that he hates QM---or at least refuses to acknowledge its applicability---so I will put it here for posterity, and any others who may be following along.)
The electromagnetic force is mediated by virtual photons---that is, two charged objects which exchange virtual photons are either attracted or repeled based on the signs of their charges. Now, suppose the rope is made of some (linear) molecule, called "rope-ite", which has the property that it is positive on one end and negative on the other. Then the tensile strnegth of the rope can be understood as the positive-negative attraction of "ropeite", or, quantum mechanically, as the exchange of virtual photons from one end of the ropeite molecule with the other. Now, suppose that we lay a rope composed of ropeite across the horizon of the black hole, such that the horizon lay between two molecules of ropeite:
+oooooo- +oooooo- +oooooo- H +oooooo- +oooooo-
where H is the horizon, and +oooooo- is a ropeite molecule. Now, the virtual photons cannot travel across the horizon, and the rope breaks because the electromagnetic bond just doesn't exist anymore. Does this make sense? So then one isn't surprised when they measure the rope and fine it broken---the electromagnetic force, which is effectively a way for the ropeite molecules to communicate, cannot propogate across the horizon.
The other example is if the rope-ite is cut by the horizon:
+oooooo- +oooooo- +oooooo- +oooHooo- +oooooo-.
Supposedly, rope-ite is composed of some atoms, which also communicate using virtual photons, so the molecule is cut in the middle. There's really no difference with this case.
So it seems to me that the laws of physics are safe because the rope-ite molecules cannot communicate across the horizon, as we expect. Also, if you detest the "rope-ite", you can use carbon nanotubes or something more physical.
To me, the only way the rope can break is if the tension is greater than the restoring force. So one should take a measurement of force (which is a Lorentzian four-vector), preformed at the horizon of the black hole, and Lorentz Transform it to some frame sufficiently far away from the horizon, and see what happens. This is probably tremendously complicated, as most GR calculations are. But this may be using the SEP to test the SEP, so Zanket will no doubt take issue here.
The other, quantum explanation that I will offer is this. (Zanket has already informed me that he hates QM---or at least refuses to acknowledge its applicability---so I will put it here for posterity, and any others who may be following along.)
The electromagnetic force is mediated by virtual photons---that is, two charged objects which exchange virtual photons are either attracted or repeled based on the signs of their charges. Now, suppose the rope is made of some (linear) molecule, called "rope-ite", which has the property that it is positive on one end and negative on the other. Then the tensile strnegth of the rope can be understood as the positive-negative attraction of "ropeite", or, quantum mechanically, as the exchange of virtual photons from one end of the ropeite molecule with the other. Now, suppose that we lay a rope composed of ropeite across the horizon of the black hole, such that the horizon lay between two molecules of ropeite:
+oooooo- +oooooo- +oooooo- H +oooooo- +oooooo-
where H is the horizon, and +oooooo- is a ropeite molecule. Now, the virtual photons cannot travel across the horizon, and the rope breaks because the electromagnetic bond just doesn't exist anymore. Does this make sense? So then one isn't surprised when they measure the rope and fine it broken---the electromagnetic force, which is effectively a way for the ropeite molecules to communicate, cannot propogate across the horizon.
The other example is if the rope-ite is cut by the horizon:
+oooooo- +oooooo- +oooooo- +oooHooo- +oooooo-.
Supposedly, rope-ite is composed of some atoms, which also communicate using virtual photons, so the molecule is cut in the middle. There's really no difference with this case.
So it seems to me that the laws of physics are safe because the rope-ite molecules cannot communicate across the horizon, as we expect. Also, if you detest the "rope-ite", you can use carbon nanotubes or something more physical.
Ben said:
I agree with this.
zanket defines a "lab frame" that covers a region of spacetime. If it is near an event horizon, then the spacetime curvature changes across the width of the "lab". And if that is the case, then the lab cannot constitute a single inertial reference frame any more.
If every experiment in practice is inapplicable to the SEP, then you agree that it cannot be experimentally tested in practice, right?
You’re talking about a measurement in principle, not one that can be done in practice, right? Because the measurement has to be done using a point-sized measuring tool. Otherwise the measuring equipment itself would be spread across multiple incomparable frames, invalidating the measurement, right? Or could the measurement be valid after a Lorentz transformation is applied between every possible pair of points that are at different r-coordinates in the spacetime filling the measuring equipment?
OK, let’s consider the muon experiment (same as the pion decay experiment). Let the muon do a local experiment at r_2: measure the speed of light. Let the ground-based Earth observer do the same local experiment at r_1. How would you apply a Lorentz transformation to make those frames comparable so you can compare the outcomes of the experiments? Let the muon’s velocity in the Earth’s frame (same as the Earth’s velocity in the muon’s frame) be denoted by v. Can you describe to me how you would input v into the Lorentz formula to make those frames comparable? The Lorentz transformation must be applied to the outcomes of the experiments in some way, otherwise the exercise is for naught, right?
I say that it’s irrelevant in a discussion about what GR, a different theory, predicts. This thread is about comparing GR’s predictions to the SEP. QM’s predictions or explanations are irrelevant to that.
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Then you believe an inertial frame cannot cross a horizon in GR, right?
Spacetime curvature is synonymous with tidal force. The tidal force is negligible in an inertial frame by definition. Do you agree to those?
Here’s an excerpt from the book Black Holes: A Traveler’s Guide, pg. 21: “If you were approaching a 10 solar masses black hole with a radius of 30 kilometers, you would be killed long before you reached the horizon, at an altitude of 400 kilometers. However, you could reach the horizon of a 1,000 solar masses black hole, and even be able to explore the interior of a 10 million solar masses black hole. The tidal forces at the horizon of this gigantic black hole would be weaker than those produced by Earth, which are already impossible for us to feel.”
An online reference: “In a supermassive black hole the tidal forces are weaker, and you could survive well inside the horizon of the black hole before being torn apart.”
If an inertial frame the size of a breadbox can cross the horizon of black hole J, then what size of an inertial frame can cross the horizon of black hole K that is 10^7000 times larger than J? A huge one no doubt!
Taylor, Thorne, and Wheeler think so. The following definitions are verbatim from the glossary of Thorne’s book Black Holes & Time Warps:
In the glossary of Exploring Black Holes, Taylor and Wheeler define “curvature of spacetime” as: “Property of spacetime evidenced by tidal accelerations (relative accelerations) of free test particles.”
It was given that X is an inertial frame. Then according to the above, the lab can constitute a single inertial frame. In principle there is no upper limit to the size of the lab.James R said:
Sigh. First, you are playing games with reference frames. You have chosen your reference frames in a manner which you have defined, as opposed to a manner that is consistent with GR. Second, the SEP can never be tested exactly, only approximately. This should not surprise you or anyone---experimenters can test the SEP, but cannot verify it exactly. This is how experimental physics works.
Please respond to this, that I posted several comments ago. It is based on your definition of coordinate frames:
Next:
I thought all of this was "in principle". If not, then I challenge you to conduct the experiments as you have described and prove that GR violates the SEP experimentally "in practice". If you can do this then I will concede this argument.
First, the Lorentz transformations here are easy---they are the the length contraction, time dialation, etc. Second, of course, there is no Lorentz transformation to be done, and I can tell you what the results of the measurements will yield. The muon and the Earth observer will both measure the speed of light to be the same. Third, I described how one would go about applying the Lorentz transforms to such an experiment when I described the pion measurements several posts ago. Finally, I have shown you the (general) forms of the Lorentz transformations several posts ago. I will freely admit that I cannot do the calculation in the GR case, but if you want me to show you how time dialation works I certainly can.
That's not at all the case, but ok. I gave two ways to show the outcome of your experiment, one GR calculation that I (and probably many physicists) can't do, and a very accessible explanation of the event on an atomic level. Because we are dealing with almost flat space-times, one can still apply quantum reasoning to these experiments.
Zanket---I have paid you the courtesy of responding to your points as they are presented, please respond in kind.
How so?
OK, so my understanding is that you are okay with real-life (i.e. not point-sized) measuring equipment being used. And X and Y can share a smaller space than any real-life measuring equipment takes up. Then you no longer have a problem with the original post, right? You think X and Y are in fact comparable, if only approximately, right? The closer they are to each other, the better the approximation to full comparability gets, right? So Y can just be put close enough to X that your issue goes away, right?
I didn’t see anything to respond to. You didn’t ask me any questions. It looks to me like a remake of mine:
Zanket said:
It is; “in principle” was not a good choice of words. It was a question as to whether you think point-sized measuring equipment must be used, as opposed to real-life (i.e. not point-sized) measuring equipment.
This seems to contradict what you said before:
In my example, r_1 (Earth observer) <> r_2 (muon). According to what you said before, “the Lorentz Transformations must be used” in that case. But now you say “there is no Lorentz transformation to be done” in that case. Which is it? Can you give me an example of a local experiment in the muon’s or Earth observer’s frame for which the Lorentz transforms must really be used in order for measurements of like experiments in two frames at different r-coordinates to be the same?
Why would “one would go about applying the Lorentz transforms to such an experiment” when “there is no Lorentz transformation to be done”, as you say? Or are there other local experiments to which the Lorentz transformation must really be done? If so, what makes the experiment I chose special? That is, what is the criterion for determining whether the Lorentz transformation must really be done?
Sure, but quantum reasoning will just muddy the thread. If GR predicts “red” whereas QM predicts “blue”, then “red” is all that could possibly matter here, because this thread is about whether GR’s predictions adhere to the SEP.
I believe I am responding in full. I will respond, and believe I have responded, to all of your relevant distinct questions. Other than that it makes sense for me to cherry-pick your responses for whatever I think makes my best case against you. I won’t play games though. (I really don’t get why you think I’m “playing games” with frames.)
I'm not sure if that's true.
I suspect that regardless of the presence of an event horizon, a small lab in freefall into a large black hole can still be considered inertial.
Interesting, this implies that the lab can't detect the presence of the event horizon as it passes.
So there's a test, I think... can a small lab in freefall determine when it crosses an event horizon?
Zanket's original post says "Yes". But, I don't think so... I think that if the same rocket and rope were accelerating at the same rate in the lab frame through an identical lab in freefall some distance outside the event horizon, the rope would behave in the same way.
First to James:
This is what Zanket was trying to point out---if the rope breaks then one knows that he is passing a horizon. Originally I agreed with the conclusion of Zanket's thought experiment, however, now I am beginnning to doubt that---that is, I am not sure the rope breaks, especially if the horizon is moving (relative to the room) near the speed of light.
On to Zanket.
Again, your reference frame is not defined at one point in space-time.
Zanket, you are splitting hairs here. X and Y must be defined at one point in space-time. Observations are made by co-moving observers in the same frame. Because we are not talking about preforming an actual experiment, one can assume that we are testing these things to infinite accuracy. That means that X and Y will never be the same unless one either sets them the same distance from the event horizon, or uses a Lorentz Transformation on the results of one measurement.
I assumed that you would respond to this because it specifically invalidates your results, based on your logic. In other words, it shows that your experiment is not consistent given your definition of reference frame. To paraphrase you, if I can show your argument is not consistent, then there is no need in continuing this discussion.
Zanket I said this because your experiment was to measure the speed of light. According to the postulates of Special Relativity, the speed of light is the same in all frames. There is no Lorentz Transformation to be done because I know what the answer will be---c. Do you understand? Any experiment other than measuring the speed of light will need to be related with Lorentz Transformations. Is this clear, or should we discuss the postulates of SR next?
Should we also discuss the Correspondence principle? The correspondence principle states that measurements must be consistent with both classical physics (GR) and quantum mechanics. QM never predicts "red" when GR predicts "blue". It is easier to understand these things in terms of quantum mechanics for me, so I have chosen to explain them as such. I can use the correpondence principle (which doesn't break down here) to relate the quantum results to the GR results. Thus these examples are very relevant.
If by "cerry-pick your responses" you mean "completely ignore some relevant parts", then I agree with this statement. From the "Alpha Rules" post:
You cannot continue to choose to selectivley ignore my responses, as per the rules that you set out in this discussion. I have refuted your results on several levels---I have presented you with evidence that you have not shown is wrong or inconsistent. If you wish to convince anyone, you must show that ALL of my responses are wrong, not just the parts that you "cherry-pick".
For example, using your definition of reference frame, I have already shown your experiment to not be valid. You will have had three chances to argue against this point, and you have passed on two of them.
I will cut and paste again. Here are the prerequisites: 1) a reference frame may be defined across a range of space-time if the curvature is negligible (this is your definition), and 2) X and Y are inertial reference frames.
If you need a question to answer, then answer this: Where is the flaw in logic? I will give you a hint, one of the initial assumptions was wrong. Near as I can figure, there are only two answers.
This is what Zanket was trying to point out---if the rope breaks then one knows that he is passing a horizon. Originally I agreed with the conclusion of Zanket's thought experiment, however, now I am beginnning to doubt that---that is, I am not sure the rope breaks, especially if the horizon is moving (relative to the room) near the speed of light.
On to Zanket.
Again, your reference frame is not defined at one point in space-time.
Zanket, you are splitting hairs here. X and Y must be defined at one point in space-time. Observations are made by co-moving observers in the same frame. Because we are not talking about preforming an actual experiment, one can assume that we are testing these things to infinite accuracy. That means that X and Y will never be the same unless one either sets them the same distance from the event horizon, or uses a Lorentz Transformation on the results of one measurement.
I assumed that you would respond to this because it specifically invalidates your results, based on your logic. In other words, it shows that your experiment is not consistent given your definition of reference frame. To paraphrase you, if I can show your argument is not consistent, then there is no need in continuing this discussion.
Zanket I said this because your experiment was to measure the speed of light. According to the postulates of Special Relativity, the speed of light is the same in all frames. There is no Lorentz Transformation to be done because I know what the answer will be---c. Do you understand? Any experiment other than measuring the speed of light will need to be related with Lorentz Transformations. Is this clear, or should we discuss the postulates of SR next?
Should we also discuss the Correspondence principle? The correspondence principle states that measurements must be consistent with both classical physics (GR) and quantum mechanics. QM never predicts "red" when GR predicts "blue". It is easier to understand these things in terms of quantum mechanics for me, so I have chosen to explain them as such. I can use the correpondence principle (which doesn't break down here) to relate the quantum results to the GR results. Thus these examples are very relevant.
If by "cerry-pick your responses" you mean "completely ignore some relevant parts", then I agree with this statement. From the "Alpha Rules" post:
You cannot continue to choose to selectivley ignore my responses, as per the rules that you set out in this discussion. I have refuted your results on several levels---I have presented you with evidence that you have not shown is wrong or inconsistent. If you wish to convince anyone, you must show that ALL of my responses are wrong, not just the parts that you "cherry-pick".
For example, using your definition of reference frame, I have already shown your experiment to not be valid. You will have had three chances to argue against this point, and you have passed on two of them.
I will cut and paste again. Here are the prerequisites: 1) a reference frame may be defined across a range of space-time if the curvature is negligible (this is your definition), and 2) X and Y are inertial reference frames.
If you need a question to answer, then answer this: Where is the flaw in logic? I will give you a hint, one of the initial assumptions was wrong. Near as I can figure, there are only two answers.
Not only that, I show above to JamesR that the tidal force on a person in the lab can be less than the tidal force on you right now. In fact it can less than that to any degree across the breadth of the whole lab, and regardless how large the lab is, for a sufficiently large black hole.
My favorite way of putting that: you could have crossed a horizon while reading this sentence.
You’re saying that the rope in Y must break too, right? If so, why must it break? We know the rope in X must break because of the horizon. But there’s no horizon in Y.
zanket:
Ok, you've convinced me that a sufficiently small inertial frame can straddle an event horizon, and that "sufficiently small" could be quite large, given a large enough black hole.
Now you need to convince me that the rope must break in the case with the event horizon, but that it need not break in the case without the event horizon.
Ok, you've convinced me that a sufficiently small inertial frame can straddle an event horizon, and that "sufficiently small" could be quite large, given a large enough black hole.
Now you need to convince me that the rope must break in the case with the event horizon, but that it need not break in the case without the event horizon.