On the nature of the infinite fall toward the EH

[stevendaryl]

About geodesic completeness, the theorems of Hawking and Penrose show that "quite often" there are geodesicly incomplete unextendable manifolds.

Isn't the usual Schwarzschild geometry geodesically incomplete? You can't extend geodesics beyond the singularity, can you?

Or does "geodesic completeness" only require that any geodesic that does not pass through a singularity must be complete?

 

[PeterDonis]

Or does "geodesic completeness" only require that any geodesic that does not pass through a singularity must be complete?

This is the definition of "geodesic completeness" that I was using, yes. The technically correct mathematical definition would call a geodesic that ends on the singularity incomplete (because the proper time to the singularity is finite), but physically that isn't interpreted the same way as the geodesic incompleteness of exterior Schwarzschild coordinates at the horizon. At the curvature singularity at r = 0, geometric invariants are infinite (more precisely, they increase without bound as r -> 0). That isn't true at r = 2m. The physical "requirement" of geodesic completeness only applies at boundaries where the geometric invariants are finite.

 

[PAllen]

That's a good point. That's a counter-argument to PeterDonis' claim that the EFE implies geodesic completeness. If the manifold is an open set, then the EFE would be satisfied at every point in the manifold, whether or not there is geodesic completeness. Similarly, the Bianchi identities would be satisfied at every point. So I don't think that anything would be violated by simply declaring that nothing exists outside the manifold.

I think the best way to approach this is as follows:

1) As you are presumably aware, you can derive inclusive coordinate systems directly from the EFE and see that e.g. exterior SC is simply a subset of one of these.

2) So to rule out 'regions you don't like' you must modify GR. One variant I proposed, that, I think, fully expresses the desired boundary condition (including the requirement that it is open) is:

-----
Consider what this modification might look like, classically, and assuming we want to keep the coordinate independent nature of the equations of GR.1) We must add a couple of new axioms the theory: Universes containing naked singularities are prohibited (as a corollary, closed universes are prohibited because event horizons cannot technically be defined for them; the required new law I give next cannot be stated for a closed universe). Eternal WH-BH are prohibited. (Much stronger than 'we think not physically plausible').

2) We supplement the EFE with a new universal boundary law: The universe is bounded such that the world line of any particle or fluid element always has null paths extending from it to null infinity.

-------

Aesthetically, why should we add this 'universal boundary law' to the EFE?

Physically, I think the sharpest problem is shown by a collapsing shell of matter. One may posit a shell that would one light year as it reached EH. Inside, we have a 1 ly region following exactly the laws of SR (exactly flat spacetime). Now add a solar system in this region (small local deviations from pure SR). On an 'earth' in this solar system, Alice has dropped a ball. All normal laws of physics in the region say this ball hits the floor at time t1. However, it happens that the Universal Boundary Law kicks in and says the the ball will approach but never reach the half way point in its fall (because if it reached the half way point, the shell would have reached a radius such that no null path from from the ball can reach null infinity; thus the ball would be assigned infinite SC type time coordinate on approach to the half way point).

Geodesic completeness is equivalent to excluding such physical absurdities (as well as ruling out my proposed 'Universal Boundary Law'). Any situation where geodesics end for no local reason (e.g. singularity) are equivalent to dropped ball stopping for no reason.

 

[PeterDonis]

We have a "patch" P, with a boundary B. We propose the (quite weird, I admit) rule that any geodesic that intersects B ceases to exist on the "far" side of B. How can that rule possibly violate a tensor identity?

It does if the manifold includes B, because derivatives can't be defined at B if the manifold "ceases to exist" on the far side of B. But it doesn't if, as DaleSpam pointed out, the manifold is an open set*, because then B would not be included in the manifold. In that case, no geodesic would intersect B; it would approach B closer and closer, without ever reaching it, because B itself is not in the manifold. This is true for both of the cases we have discussed in Schwarzschild spacetime: the exterior Schwarzschild coordinate patch does not include r = 2m (it is an open set with r -> 2m), and patches like the Painleve patch that include the horizon and the black hole interior do not include the singularity (they are open sets with r -> 0). In that case, you are correct that it is not an actual contradiction to suppose that B and the region beyond B "don't exist".

All this is mathematical, though, and doesn't address the question of whether a proposed manifold that is geodesically incomplete is physically reasonable.

* - I believe DaleSpam is right that the technical definition of "manifold" (at least the one that is used in GR) requires manifolds to be open sets, but I haven't looked it up to confirm.

 

[rjbeery]

Yes, it differs substantially. Coordinate velocities are frame variant quantities and can easily exceed c. A null tangent vector is frame invariant and is only possible for massless particles.

I can easily come up with a coordinate system where my coordinate velocity sitting here typing this response is c, but there is no coordinate system where my worldline is null. [EDIT: and why settle for c, I can make a coordinate system where my v>>c, woohoo FTL travel solved!]

That's interesting. How would you do so without involving rotating frames (or black holes )?

That depends entirely on the coordinate chart selected. However, note that you could not select Schwarzschild coordinates for this since they don't cover the EH. The closest you could do in Schwarzschild coordinates is the limit of Alice's velocity as she approached the EH.

I object to this. You're saying SC coordinates don't cover the EH but they do at the limit, and what you end up with is a coordinate velocity of the infalling object = c. This is equivalent to saying that the escape velocity at that point is (just above) c, hence impossible.

 

[PAllen]

I object to this. You're saying SC coordinates don't cover the EH but they do at the limit, and what you end up with is a coordinate velocity of the infalling object = c. This is equivalent to saying that the escape velocity at that point is (just above) c, hence impossible.

NO, as Peter pointed out, the limiting SC coordinate velocity of a radial infaller is zero not c, on approach to the EH. Showing just how meaningless it is to talk about coordinate velocity as a physical thing.

 

[Dale]

That's interesting. How would you do so without involving rotating frames (or black holes )?

I don't know why you would exclude rotating frames. But anyway, e.g. start with the usual Minkowski coordinates (t,x,y,z) for the rest frame of an object in flat spacetime. Then use the following coordinate transformation:
T=t
X=x+1000000ct
Y=y
Z=z

I object to this. You're saying SC coordinates don't cover the EH but they do at the limit, and what you end up with is a coordinate velocity of the infalling object = c. This is equivalent to saying that the escape velocity at that point is (just above) c, hence impossible.

This is related to the lecture notes I posted earlier, please read through them. Note how coordinate charts and manifolds are defined on open sets, meaning that they do not include the boundary. The EH is a boundary for SC coordinates, so they coordinates do not include the EH. There are some solid mathematical reasons for this, please read the notes.

Also, as PeterDonis pointed out earlier the coordinate velocity in SC coordinates goes to 0, not c.

 

[PeterDonis]

* - I believe DaleSpam is right that the technical definition of "manifold" (at least the one that is used in GR) requires manifolds to be open sets, but I haven't looked it up to confirm.

This is related to the lecture notes I posted earlier, please read through them. Note how coordinate charts and manifolds are defined on open sets, meaning that they do not include the boundary.

Easier to look it up there then to dig out my copy of MTW. Yes, I see that in Chapter 2 Carroll goes into detail about manifolds being defined on open sets.

 

[PAllen]

Something that someone has mention is completeness of geodesics. If there is a geodesic leading to a boundary, and nothing singular happens at that boundary, then we need to describe what happens on the other side of the boundary, to "complete" the geodesic. But is that just an aesthetic consideration, or is there some reason we must have geodesic completeness?

And, to relate this to my physical motivation for geodesic completeness, imagine the geodesics representing free fall of a lab towards a supermassive BH horizon. In the lab, a student is observing and timing a spring oscillator. They see that the next peak should occur at t1 (last one at t0). Instead, the oscillator approaches, but never reaches the half way point of its oscillation, for no local physical region. Any time you have incomplete geodesics for no local reason, you can set up a scenario of this type.

 

[martinbn]

* - I believe DaleSpam is right that the technical definition of "manifold" (at least the one that is used in GR) requires manifolds to be open sets, but I haven't looked it up to confirm.

Any topological space is open and closed, part of the definition of topology.

 

[rjbeery]

NO, as Peter pointed out, the limiting SC coordinate velocity of a radial infaller is zero not c, on approach to the EH. Showing just how meaningless it is to talk about coordinate velocity as a physical thing.

Of course I meant the velocity of the infaller is zero. They "freeze", and never appear to cross the EH. Locally, the escape velocity is c, remotely the coordinate velocity is zero and the escape velocity is anything > 0.

Philosophically, it's curious though: if we say that coordinate velocity is not a physical thing then can we even say that any sort of velocity is a physical thing? Velocity seems to "mean something" and have an a physicality to it, yes?

 

[PAllen]

Of course I meant the velocity of the infaller is zero. They "freeze", and never appear to cross the EH. Locally, the escape velocity is c, remotely the coordinate velocity is zero and the escape velocity is anything > 0.

Philosophically, it's curious though: if we say that coordinate velocity is not a physical thing then can we even say that any sort of velocity is a physical thing? Velocity seems to "mean something" and have an a physicality to it, yes?

Only velocity comparisons are meaningful in relativity. In GR, only local velocity comparisons have unambiguous definition. To compare velocities of distant world lines, you have to bring one tangent close to the other to compare - that is done via parallel transport. In flat spacetime, parallel transport is path independent, so there is a unique definition [given a choice of 'now', which is a whole other matter]. In GR, the result is path dependent so there is just no preferred way to compare velocities of distant objects.

Also, note, that despite the ambiguity, you can pick any of an infinite number of paths along which to transport the 4-velocity of an infaller at the EH, or up to the singularity, to some distant static world line. If you do this, no matter what choice you make, the comparison both for horizon crossing and up to the singularity, will always be < c.

 

[PeterDonis]

if we say that coordinate velocity is not a physical thing then can we even say that any sort of velocity is a physical thing? Velocity seems to "mean something" and have an a physicality to it, yes?

The 4-velocity of a timelike worldline can be defined in an invariant way (it's the worldline's tangent vector), so it qualifies as a "physical thing". But that's a 4-vector, not a 3-vector, so it doesn't directly tell you anything about "velocity" in the ordinary sense. (One can also define a null tangent 4-vector to the worldline of a light ray; that is not usually called a 4-velocity because "4-velocity" normally means a unit vector, i.e., one with length 1, or c in conventional units, not zero.)

The relative velocity of two worldlines, at least one of which is timelike, at a particular event where they cross can also be defined in an invariant way, by taking the inner product of their two tangent vectors. If both worldlines are timelike, this will always give a result less than c. If one is timelike and one is null, this will always give a result equal to c. This happens, for example, at the event where a timelike object crosses the horizon: the inner product of the worldline's timelike tangent vector with the null tangent vector to the horizon gives c. (This is often misinterpreted as saying that the object "moves at c" when it crosses the horizon; in fact it's the *horizon* that is "moving at c".) So relative velocity in this sense also qualifies as a "physical thing".

These are the two senses of "velocity" that have direct physical meanings. Note that neither of them corresponds to coordinate velocity. Note also that both of them are local: a worldline's tangent vector has to be evaluated at a particular event, and the inner product of two worldlines' tangent vectors has to be evaluated at the event where they cross.

 

[harrylin]

It's no more of a paradox than the twin "paradox". In fact, it's more or less an extreme version of said paradox - A thinks it takes an infinite amount of time for something to happen, B thinks its' finite.

If you mean the SR twin paradox: once more, that is very different as the (t, t') sets are finite and agree with each other. It's different however with Einstein's GR twin paradox which is much more interesting and relevant as background for this topic. It would distract too much from this topic to discuss it here, but I encourage you to study it.

Similar "paradoxes" occur outside relativity, Zeno's paradox is very similar, and the answer is very similar as well. Basically one can map a finite interval of the real numbers (say 0-1) to an infinite interval (0-infinity) with a 1:1 mapping. Thus having an infinite expanse of coordinate time means nothing. Having an infinite amount of proper time does have physical significance, but the proper time here is fnite.

I always considered Zeno's paradox as a joke - it may have been serious for philosophers, but not for physicists IMHO.

 

[PeterDonis]

Einstein's GR twin paradox

What are you referring to here? If you just mean the part of the Usenet Physics FAQ entry on the twin paradox that talks about the equivalence principle, that's not a different paradox, it's the same twin paradox analyzed from another viewpoint. If, OTOH, you mean something else, can you give a reference?

 

[stevendaryl]

I always considered Zeno's paradox as a joke - it may have been serious for philosophers, but not for physicists IMHO.

But it has the same "punch line" as the paradox of the infalling observer. Using one time coordinate, the runner starts running at time t=0 and crosses the finish line at time t=1 (say). But you can set up a different time coordinate, t', with the mappings:
t=0 → t'=0
t=1/2 → t'=1
t=3/4 → t'=2
etc.
(in general, t' = log₂(1/(1-t)))

Clearly, as t' runs from 0 to ∞, the runner never reaches the finish line.

That's simply an artifact of the choice of coordinates.

 

[harrylin]

Hi harrylin and rjbeery,

I would recommend that you read page 37 and 38 of Carroll's lecture notes on GR (it may be necessary to read earlier pages too if you don't understand some of the terminology used there, and of course I recommend reading the entire chapter 2). [..]

http://arxiv.org/abs/gr-qc/9712019

Hi Dalespam, I already commented on Carroll some 10 days ago, and what he discusses on those pages is similar to what was discussed in earlier threads, in fact I had started a similar sub topic as Caroll in order to clarify different philosophy. Patchwork is in my eyes not good physics. My earlier comment on his views hasn't changed:

"I looked it up (interesting, thanks!) and I note that he has a different opinion of reality than I have. In my experience, only opinions about verifiable facts can be argued in a convincing way for those who are of a contrary opinion. Do you disagree?"

Next I gave a little sample of different philosophy, but just for information as anything beyond that is a waste of time. From that point on (which has passed), it try to stick to the agreed physics, which means to discuss and compare the (t, τ, r) numbers of the models under discussion. I thought that I was clear but will be clearer: like Vachaspati I find no problem at all with Schwarzschild's system, it looks perfectly usable for say t=10¹⁰ years. I don't see how an object can cross the horizon without violating either laws of nature or the physical claims of GR about the nature of clock rate and light propagation in a gravitational field; so, once more, a numerical example would be helpful. Sorry but I don't have time for long-winding philosophical discussions and surely this forum is not meant for that.

 

[PeterDonis]

I don't see how an object can cross the horizon without violating either laws of nature or the physical claims of GR about the nature of clock rate and light propagation in a gravitational field

This isn't going to be fixed by numerical examples. Either you understand how GR translates its math into physical predictions or you don't. Evidently you don't.

We have repeated countless times that the proper time for an object to fall to the horizon is finite, and that all physical invariants are finite there, and that by the rules GR uses to translate math into physical predictions (only invariants count, coordinate quantities don't count), that means objects can cross the horizon. Are you saying that if we show you the actual calculations behind those claims I just made, you will change your mind?

 

[PeterDonis]

Patchwork is in my eyes not good physics.

Then evidently you think GR is not good physics, since it uses "patchwork". You're quite entitled to hold that opinion, but it doesn't entitle you to claim that GR's predictions are something other than they are. What you should be saying is simply that you don't think GR is good physics.

 

[Dale]

I always considered Zeno's paradox as a joke - it may have been serious for philosophers, but not for physicists IMHO.

That is interesting, since your position in this and other threads is the same as Zeno's as far as I can tell.

In both cases there is a mapping from points on an objects worldline to real numbers. In both cases the limit of the mapping goes to infinity as the object approaches some location. In both cases the claim is that therefore the object does not reach the location. In both cases, the proponent of the argument ignores the fact that the distance between successive points on the mapping is decreasing.

 

[PAllen]

Hi Dalespam, I already commented on Carroll some 10 days ago, and what he discusses on those pages is similar to what was discussed in earlier threads, in fact I had started a similar sub topic as Caroll in order to clarify different philosophy. Patchwork is in my eyes not good physics. My earlier comment on his views hasn't changed:

"I looked it up (interesting, thanks!) and I note that he has a different opinion of reality than I have. In my experience, only opinions about verifiable facts can be argued in a convincing way for those who are of a contrary opinion. Do you disagree?"

Next I gave a little sample of different philosophy, but just for information as anything beyond that is a waste of time. From that point on (which has passed), it try to stick to the agreed physics, which means to discuss and compare the (t, τ, r) numbers of the models under discussion. I thought that I was clear but will be clearer: like Vachaspati I find no problem at all with Schwarzschild's system, it looks perfectly usable for say t=10¹⁰ years. I don't see how an object can cross the horizon without violating either laws of nature or the physical claims of GR about the nature of clock rate and light propagation in a gravitational field; so, once more, a numerical example would be helpful. Sorry but I don't have time for long-winding philosophical discussions and surely this forum is not meant for that.

Come on! What does the clock on a an infaller's watch read as they approach an O-S collapse? Let's way it is 2 PM. They are also bouncing a yo-yo as they fall. The apply normal laws of physics, including GR, and predict that the yo-yo will reach their hand at 2:05 PM. Describe the new law of physics that says the yo-yo will instead never reach their hand because some other privileged observer won't see it?

 

[Dale]

Hi Dalespam, I already commented on Carroll some 10 days ago, and what he discusses on those pages is similar to what was discussed in earlier threads, in fact I had started a similar sub topic as Caroll in order to clarify different philosophy. Patchwork is in my eyes not good physics. My earlier comment on his views hasn't changed:

What is patchwork?

Btw, you are in dangerous territory. If patchwork is something mentioned in those lecture notes then it is part of mainstream physics and your claim that it isn't good physics would therefore be quite speculative.

Furthermore, all of the comments on manifolds and coordinates in that section apply in simple spacetimes too. Like flat or constant curvature.

 

[PeterDonis]

What is patchwork?

In my response to this a few posts ago, I assumed he meant using more than one coordinate patch to describe a spacetime.

If patchwork is something mentioned in those lecture notes then it is part of mainstream physics

Which it is if my assumption above is correct.

 

[harrylin]

We've already stipulated that he can, because he can detect tidal gravity (as can Eve'). But given that, why would he ever assume he was moving in a straight line in the first place?

Maybe I should expound a bit more on what I'm looking for here. The standard view of this scenario is that the two cases are exactly parallel: in both cases, the accelerated observer (Eve, Eve'), because of her proper acceleration, is unable to observe or explore a region of spacetime that the free-falling observer (Adam, Adam') can. The physical criterion that distinguishes them is clear, and is the same in both cases (zero vs. nonzero proper acceleration).

You are claiming that, contrary to the above, the cases are different: Adam is "privileged" in the first case, but Eve' is in the second. So I'm looking for some criterion that picks out Adam in the first case, but picks out Eve' in the second; in other words, something that applies to Adam but not Eve, and applies to Eve' but not Adam'. The only criterion I have so far is "moves in a straight line according to my chosen coordinates", but that only pushes the problem back a step: what is it that applies to the coordinates of Adam but not Eve, *and* to those of Eve' but not Adam'? I haven't seen an answer yet.

You made that view sufficiently clear; and I thought that I made the opposing view also sufficiently clear - but apparently not. I'll try to explain once more, but will subsequently let it rest - in case you forgot, my intended role here was just that of a curious but critical reporter, but suddenly people start to argue with the reporter and asking him questions.

The opposing view that I came up with is that Eve and Adam usually will be able to distinguish the two cases; the physical criterion that experimentally distinguishes the two cases is clear (tidal effect). I noted that in case that one or both are unable to do so (for example Adam only has a simple accelerometer and no windows), that could make them like bees that fly against a window. Surely you'll agree that nature can't care less if they did not predict the window, and the window is not "unphysical" if the bee didn't notice it before hitting it.

The criterion of that opposing view is just as "mainstream" as the one you presented: real gravitation can be distinguished from acceleration in flat space-time and is physically different from it. This weaker version of Einstein's equivalence principle remains, in the form cited in the first post of https://www.physicsforums.com/showthread.php?t=656240. Also, special Relativity is the theory of flat spacetime, without equivalence principle. That enables the use of universal ("global") descriptions such as Minkowski space-time for negligible effects of gravitation on "clocks and "rulers" (that's extremely handy for solving Langevin's original "twin" example!) and similarly universal descriptions such as Schwarzschild space-time for negligible effects of velocity; the two systems can be combined to globally account for both effects. That is de facto how the ECI frame is constructed.

From that point of view, the use of Rindler/EEP coordinates in flat spacetime is the use of pseudo coordinates like those of Zeno. Branding such fictitious gravitational fields as pseudo gravitational fields allows for a consistent physical interpretation, globally (as far as I can see).

That's enough philosophy. It will be most interesting to elaborate with numbers how a consistent physical interpretation is possible with what you call "the standard view".

 

[martinbn]

Patchwork is in my eyes not good physics.

Here you are either very confused or deliberately using the word "patchwork" out of the context of manifold charts.

 

[Dale]

the physical criterion that experimentally distinguishes the two cases is clear (tidal effect).

Note that tidal effects are arbitrarily small at the EH if the mass of the BH is sufficiently large. So the physical criterion can be made arbitrarily small.

 

[PeterDonis]

my intended role here was just that of a curious but critical reporter, but suddenly people start to argue with the reporter and asking him questions.

Because if the reporter is going to report about a theory, we want to make sure he reports accurately. He can add an editorial about how he doesn't really think certain aspects of the theory are "good physics", but that goes on the editorial page, not the news page. Nobody said the reporter's job was easy.

The opposing view that I came up with is that Eve and Adam usually will be able to distinguish the two cases; the physical criterion that experimentally distinguishes the two cases is clear (tidal effect).

That distinguishes the cases, but it doesn't help in deciding which observer should be privileged in each case. Eve and Adam both measure zero tidal gravity; Eve' and Adam' both measure nonzero tidal gravity. Nothing in that helps to pick out Adam vs. Eve, or Eve' vs. Adam.

The criterion of that opposing view is just as "mainstream" as the one you presented: real gravitation can be distinguished from acceleration in flat space-time and is physically different from it.

I agree that the view you state here is mainstream, but it's not the criterion we need, because it only helps to distinguish the two spacetimes (flat vs. curved); it doesn't help in picking out which observer is privileged in each (Adam vs. Eve, Eve' vs. Adam'). See above.

Schwarzschild space-time for negligible effects of velocity

I don't understand this; Schwarzschild spacetime can handle any velocity. Unless you mean that the central gravitating body is at rest?

the two systems can be combined to globally account for both effects.

No "combination" is necessary; Schwarzschild spacetime by itself can handle the regions with negligible gravity, since the metric coefficients go to the Minkowski values as r -> infinity.

That is de facto how the ECI frame is constructed.

See my note above about "combination"; the ECI frame doesn't have to combine a Minkowski spacetime and a Schwarzschild spacetime. It's just a Schwarzschild-type chart centered on the Earth whose time coordinate is rescaled to the rate of proper time on the geoid.

With the word "combination" you may be thinking of the fact that the ECI is also a sort of "local inertial frame" for the Earth in its orbit about the Sun. This is true (with some technicalities), but note the word "local"; it is certainly not any kind of "combination" of a global Minkowski frame with a global Schwarzschild frame. If we look at the Solar System as a whole, the global frame is a Schwarzschild frame centered on the Sun.

From that point of view, the use of Rindler/EEP coordinates in flat spacetime is the use of pseudo coordinates like those of Zeno.

But in so far as these coordinates are "pseudo", it's not because there is a "pseudo gravitational field" in them. It's because they go to infinity at the Rindler horizon, yet the spacetime itself is finite there. Exactly the same criticism applies to Schwarzschild coordinates for a black hole.

Branding such fictitious gravitational fields as pseudo gravitational fields allows for a consistent physical interpretation, globally (as far as I can see).

Sure, but that doesn't help with the issues you're having with the SC chart and black hole horizons, because it only differentiates between spacetimes, not between observers. See above.

 

[PeterDonis]

It will be most interesting to elaborate with numbers how a consistent physical interpretation is possible with what you call "the standard view".

Have you read my post #123 yet?

 

[stevendaryl]

The criterion of that opposing view is just as "mainstream" as the one you presented: real gravitation can be distinguished from acceleration in flat space-time and is physically different from it. This weaker version of Einstein's equivalence principle remains, in the form cited in the first post of https://www.physicsforums.com/showthread.php?t=656240. Also, special Relativity is the theory of flat spacetime, without equivalence principle. That enables the use of universal ("global") descriptions such as Minkowski space-time for negligible effects of gravitation on "clocks and "rulers" (that's extremely handy for solving Langevin's original "twin" example!) and similarly universal descriptions such as Schwarzschild space-time for negligible effects of velocity; the two systems can be combined to globally account for both effects. That is de facto how the ECI frame is constructed.

From that point of view, the use of Rindler/EEP coordinates in flat spacetime is the use of pseudo coordinates like those of Zeno. Branding such fictitious gravitational fields as pseudo gravitational fields allows for a consistent physical interpretation, globally (as far as I can see).

As I said earlier, "fictitious gravitational fields" have nothing to do with this topic, at all. It's simply a matter of coordinate systems, and whether or not they cover the entire manifold.

 

[stevendaryl]

Hi Dalespam, I already commented on Carroll some 10 days ago, and what he discusses on those pages is similar to what was discussed in earlier threads, in fact I had started a similar sub topic as Caroll in order to clarify different philosophy. Patchwork is in my eyes not good physics.

What do you mean by that? On the contrary, all physics involves splitting up the world into pieces that can be analyzed in (approximate) isolation.

 

[pervect]

If you mean the SR twin paradox: once more, that is very different as the (t, t') sets are finite and agree with each other. It's different however with Einstein's GR twin paradox which is much more interesting and relevant as background for this topic. It would distract too much from this topic to discuss it here, but I encourage you to study it.

I always considered Zeno's paradox as a joke - it may have been serious for philosophers, but not for physicists IMHO.

The funny thing is, you're basically using the same reasoning that Zeno did - without apparently even realizing this fact, not even when it's pointed out!

Let's compare what happens issue by issue.

In Zeno's paradox, we have two times. Let's let the "real" time be represented by tau, and zeno time by Z.

In the infalling black hole case, we have proper time tau, and Schwarzschild time t

The mapping from t to tau that we worked out previously for the Schwarzschild case in great detail is:

$$t = \tau-4\,\left(-3 \, \tau \right)^{\frac{1}{3}}+4\,\ln \left[ \left(-3 \, \tau \right)^{\frac{1}{3}}+2 \right] -4\,\ln \left[\left(-3 \, \tau \right)^{\frac{1}{3}}-2 \right]$$

(You can probably find this in a textbook if you want to check my math).

The characteristic features of this mapping is that t increases monotonically with tau, and that infinite range of tau only covers a finite range of t.

This is due specifically to the term

$$-4 \ln \left[\left(-3 \, \tau \right)^{\frac{1}{3}}-2 \right]$$

This is rather complicated, the argument will be clearest if we assume this term, which is the one that approaches infinity, is the dominant term near the event horizon, in which case we can solve for $\tau$ assming that this is the only term that matters.

$$\tau \approx \frac{1}{3} \, \left[\exp^{-t/4} - 2\right]^3$$

and we see that $\tau$ approaches -8/3 as t-> infinity (which is when the horizon is reached).

In the Zeno paradox, the mapping is something like

$$\tau = a(1-\exp^{z/b})$$
where a and b are some constants

where $\tau$ approaches some constant a as Z approaches infinity

And we see the issue -in both cases, even though t (in one case), Z (in the other case) cover infnite ranges, $\tau$ does not.

So, essentially Zeno never assigns a label, Z to some events. And he concludes from this that these events don't exist.

And you assume the same thing - because you never assign a coordinate "t" to some events, you assume they don't happen.

And this conclusion is just as unjustified when you do it, as when Zeno does it.

So the not-very-complicated moral of the story is that because you can choose ANY coordinates you want, you need to be careful in your interpretation of the results. Specifically, it's possible to choose coordinates like Zeno did, that exclude important regions from analysis, because the coordinates don't label physically significant events. However, not giving something a label doesn't make it not exist, any more than closing your eyes does. At least not for most defitnitions of "existence".

 

[Mentz114]

Pervect, that's a showstopping reply !

Harrylin, pay heed.

 

[stevendaryl]

The opposing view that I came up with is that Eve and Adam usually will be able to distinguish the two cases; the physical criterion that experimentally distinguishes the two cases is clear (tidal effect).

The issue is not whether you can distinguish flat spacetime from curved spactime; of course, you can, by looking at the curvature tensor. The issue is that you seem to think that because an event is not covered by the Schwarzschild coordinate system, then that event never happens. What reason is there for believing that, as opposed to realizing that the Schwarzschild coordinates don't cover the entire manifold?

It is easy to demonstrate that it is possible to choose coordinates that leave out part of the manifold. What reason do you have for thinking that's NOT the case with Schwarzschild coordinates? (It PROVABLY is the case, so what I'm really asking you is why you seem to believe something that is provably false.)

I noted that in case that one or both are unable to do so (for example Adam only has a simple accelerometer and no windows), that could make them like bees that fly against a window. Surely you'll agree that nature can't care less if they did not predict the window, and the window is not "unphysical" if the bee didn't notice it before hitting it.

I don't understand the point of your story. What you seem to be thinking is that one of the observers is "correct" and the other is "wrong", and you have to look to clues such as accelerometers to figure out which is which. That is a completely wrong way to think about it. ANY coordinate system can be used to describe events within a chart. There is no "correct" coordinate system or "incorrect" coordinate system. But a coordinate system only works within a chart. It can't possibly describe events that are NOT in its chart.

So in Rindler coordinates, someone sees a dropped object asymptotically approach the location X=0 as time T → ∞. The correct interpretation of this situation isn't: "Rindler coordinates are wrong. Cartesian coordinates are right." The correct interpretation is "The event of the object crossing the 'event horizon' at X=0 is not an event covered by the Rindler coordinates". Rindler coordinates are perfectly fine for describing any events taking place within its chart, but it can't possibly describe events outside that chart.

The same thing is true of an object crossing the event horizon in Schwarzschild coordinates. That event is not covered by Schwarzschild coordinates. Schwarzschild coordinates are perfectly good for describing events within its chart, but can't be used to describe events outside its chart. It's not a question of whether the "hovering observer" is correct and the "infalling observer" is wrong, or vice-verse. The only issue is whether the event of crossing the horizon is in fact covered by this coordinate system or that coordinate system.

The criterion of that opposing view is just as "mainstream" as the one you presented: real gravitation can be distinguished from acceleration in flat space-time and is physically different from it.

This doesn't have anything to do with "real" versus "pseudo" gravitation! It has to do with whether a coordinate system covers the entire manifold, or not. An easy way to prove that it does not is to show that there is a second coordinate system that has an overlapping chart with the first coordinate system, yet includes points that are not covered by the first. That's been done, with Schwarzschild coordinates.

This weaker version of Einstein's equivalence principle remains...

This doesn't really have anything to do with the equivalence principle.

From that point of view, the use of Rindler/EEP coordinates in flat spacetime is the use of pseudo coordinates like those of Zeno.

There is no such thing as "pseudo coordinates". The only issue about coordinates is what region of spacetime do they cover, and are there regions that are not covered by them.

Branding such fictitious gravitational fields as pseudo gravitational fields allows for a consistent physical interpretation, globally (as far as I can see).

It doesn't have anything to do with gravitational fields, pseudo or otherwise.

That's enough philosophy. It will be most interesting to elaborate with numbers how a consistent physical interpretation is possible with what you call "the standard view".

If you look at KS coordinates, the metric looks like this:
$\dfrac{4 R_s^3}{r} e^{-r/R_s}(dV^2 - dU^2)$
where $R_s$ is the Schwarzschild radius, and $r$ is the Schwarzschild radial coordinate. This metric is defined everywhere, except at the singularity $r=0$. The "time" coordinate is $V$. The event horizon in these coordinates consists of all points with $V^2 = U^2$. So an object can certainly cross the event horizon at a finite value for the time coordinate $V$.

Now, to see that this is describing the SAME situation as the Schwarzschild black hole, you note that the "patch" with $U > 0$ and $-U < V < +U$ describes exactly the same region of spacetime as the Schwarzschild patch $r > R_s$ and $-\infty < t < +\infty$, and the "patch" with $1 > V > 0$ and $-V < U < +V$ describes exactly the Schwarzschild patch with $0 < r < R_s$ and $-\infty < t < \infty$. But the KS coordinates also covers the boundary between these two regions, the event horizon.

 

[PeterDonis]

If you look at KS coordinates, the metric looks like this:
$\dfrac{32 R_s^3}{r} e^{-r/R_s}(dV^2 - dU^2)$
where $R_s$ is the Schwarzschild radius

Quick pedantic note: if you write the K-S line element this way, in terms of $R_s$, then the coefficient in front is $4 R_s^3 / r$. The 32 is there if you write it in terms of M:

$$\dfrac{32 M^3}{r} e^{-r/2M}(dV^2 - dU^2)$$

 

[A.T.]

The funny thing is, you're basically using the same reasoning that Zeno did - without apparently even realizing this fact, not even when it's pointed out!

Let's compare what happens issue by issue.

In Zeno's paradox, we have two times. Let's let the "real" time be represented by tau, and zeno time by Z.

That is an interesting comparison. One difference seems to be that "Zeno time" is the result of an arbitrary mathematical mapping, that doesn't have any physical significance. The coordinate time in Schwarzshild coordinates on the other hand, can be interpreted as the proper time of a clock at infinity, which is a observable physical quantity.