Notions of simultaneity in strongly curved spacetime

[harrylin]

[...] It is absolutely possible for a distant observer to assign remote times in a consistent way such that they consider the object to have crossed the horizon in finite time. They can also consistently assign remote times so that never happens. It will never be possible to verify one assignment over another precisely because event horizon crossing will never be seen.

This is an essential clarification for me; it is exactly the kind of disagreement that I tried to illustrate with Earth maps. It doesn't solve the issue, but at least we agree that maps are used that disagree about events - which is a thing that was for me unheard of.

 

[harrylin]

another detail:

[..] for the infalling observer the thus predicted effect will be very dramatic, with starlight in front of him reaching nearly infinite intensity as the universe speeds up around him and his observations come to a halt when this universe ends. [..]

[..] This is not what SC or O-S geometry predicts. They predict that an infaller will see the external universe going at a relatively normal rate, with no extreme red or blueshift. [..]

Either I made a calculation error, or you made an interpretation error, or both.

I estimated the intensity of starlight that hits the eye of the infaller when looking forward - thus what he literally will see; the prediction of an event. In contrast, the "see" in your sentence is probably a prediction of what the infaller will calculate.

 

[PeterDonis]

So here's my thought process: Schwartzschild's solution is the one that I heard about in the literature, and it happens to be the one that I happened to stumble on in the first papers that I read about this topic, dating from 2007 and 1939.

A clarification about terminology: there are several concepts/objects that can be referred to as "Schwarzschild's solution", or "the Schwarzschild metric":

(1) The full, maximally extended vacuum, spherically symmetric solution to the Einstein Field Equation, considered as a geometric object independent of coordinates.

(2) Some portion of #1, such as the exterior region (region I on the Kruskal chart), or that plus the future black hole region (region II on the Kruskal chart), again considered as a geometric object or objects independent of coordinates.

(3) The Schwarzschild coordinate chart--strictly speaking, the *exterior* Schwarzschild coordinate chart--that covers region I only, considered as a particular way of describing the geometric object which is region I of #1 or #2 above.

(4) And for good measure, one can also use a portion of the Schwarzschild exterior chart to cover the vacuum spacetime outside of a massive body like the Earth or the Sun.

My reason for making these distinctions will be evident in a moment.

It is obviously a valid reference system according to GR, and it is "privileged" in the same sense as inertial frames and centre of mass systems are "privileged": it allows for the most simple mathematics, so simple that one doesn't need to be an expert to understand it. Thus it is a natural choice in a public discussion about predictions of GR. And the way I understand the first paper that I read about this, it's probably all I will ever need to understand this topic.

Here you are talking about #3 above; but you have to bear in mind a key point about GR: all of the actual physics in the theory is independent of which coordinate chart you decide to express it in; i.e., all of the physics must be capable of being expressed in terms of invariants, things that don't change when you change coordinate charts. You can use a particular chart as a starting point, so to speak, to get you to the invariants; but if you're not talking about invariants, at the end of the day, you're not talking about the actual physics.

For example, when you say that Schwarzschild coordinate time goes to infinity at the horizon, you are not talking about an invariant; there is no invariant physical quantity that goes to infinity at the horizon. So this is not a statement about any actual physics; it's only a statement about a particular coordinate chart, with no real physical content. In order to make this a statement about the actual physics, you would have to be able to translate it into a statement about invariants, and you can't; all invariant quantities are finite at the horizon.

Also, the converse of the statement I made above is *not* true: there is no requirement that any physical invariant must be expressible in *every* coordinate chart. It is perfectly possible to have physical quantities (such as, for example, the proper time on an infalling observer's worldline at an event inside the horizon) that can't be expressed in some coordinate charts, because those charts don't cover that portion of spacetime.

In view of the above, I would have to disagree that understanding things in terms of the Schwarzschild exterior coordinate chart is "all you will ever need to understand about this topic". By limiting yourself to understanding things in terms of that chart, you are limiting yourself to understanding things outside the horizon only. You can't understand what happens at or inside the horizon if the only tool you have is the Schwarzschild exterior chart, because the relevant physical quantities simply can't be expressed in that chart.

Finally, a few words about the way, if any, in which the Schwarzschild exterior chart is "privileged". This chart is indeed a "valid reference system" in that region, and it has the attractive property, as I said before, of having surfaces of simultaneity that match up exactly with the surfaces of simultaneity of observers who are static--i.e., who "hover" at a constant radius above the horizon. This allows you to simplify your view of the physics, so that the connection with your intuitive understanding of Newtonian gravity is evident (for example, concepts like "potential energy" can be defined). So the math in terms of this chart does "look simple"; but the price you pay for that is, as I just said, only being able to express physical quantities in the region outside the horizon.

It's tempting to think that, since things look so nice and intuitive outside the horizon when expressed in the Schwarzschild exterior chart, it must be sufficient to express *all* of the physics everywhere in the spacetime, and therefore the region outside the horizon, since it's the region where the Schwarzschild exterior chart works, must *be* the entire spacetime. But it isn't; when you actually work through the solution of the Einstein Field Equation, in either case #1 above (mathematically simple, because the entire spacetime is vacuum, but physically unreasonable) or case #2 above (more complex because there is collapsing matter present in a portion of the spacetime, but physically more reasonable, though still highly idealized), you find that any solution that only includes the region outside the horizon is incomplete; the EFE itself tells you that that region cannot be the entire spacetime. That's why you can't get a complete understanding by just using the Schwarzschild exterior chart.

This is an essential clarification for me; it is exactly the kind of disagreement that I tried to illustrate with Earth maps. It doesn't solve the issue, but at least we agree that maps are used that disagree about events - which is a thing that was for me unheard of.

The only sense in which the maps "disagree about events" is that one map (SC coordinates) can't assign coordinates to some events (those on or inside the horizon), while another map (e.g., Painleve coordinates) can. But this is only a "disagreement" in the same sense as a Mercator projection "disagrees" with, say, a polar projection of the Earth's surface; the former can't assign coordinates to the North Pole, while the latter can. Yet both charts can assign coordinates to, say, Big Ben in London. So someone in London could choose either chart to map his surroundings, but one choice would allow him to map the North Pole on the same chart, while the other wouldn't.

These all seem like innocuous statements about coordinate charts on geometric manifolds to me. Apparently they seem like huge issues to you; I'm not sure why. Why is it such a big deal that some charts can cover points or regions that others can't?

 

[PAllen]

This is an essential clarification for me; it is exactly the kind of disagreement that I tried to illustrate with Earth maps. It doesn't solve the issue, but at least we agree that maps are used that disagree about events - which is a thing that was for me unheard of.

Actually they don't disagree about events. With one convention, assign remote times ranging to infinity for all the events I will ever see. I still compute that physical law says there are other events I will never actually see. So I assign an independent time range (a different chart) to these events which are still part of the universe (if the laws are true). So one choice is to use two charts to cover the universe, one for events I will see, eventually, one for those I will never see.

The other choice, equally consistent, is to construct a map which covers the whole universe in the one map, assigning coordinates both to events I never see and those I will see.

If you believe the physical laws, there is no disagreement at all about what events happen, or exist; or about what any instrument will measure, including a hypothesized instrument I can't communicate with.

 

[PAllen]

another detail:

Either I made a calculation error, or you made an interpretation error, or both.

I estimated the intensity of starlight that hits the eye of the infaller when looking forward - thus what he literally will see; the prediction of an event. In contrast, the "see" in your sentence is probably a prediction of what the infaller will calculate.

You made a calculation error. The way I described it is a textbook calculation you can look up. I am talking about the directly observed doppler for an infaller who has crossed the horizon and is looking out. I suspect you applied the gravitational redshift for a static observer (which cannot even exist inside). But there is a huge difference between what an infaller (from a good distance) sees near and beyond the horizon, versus what a static, near horizon observer sees (note, the static observer is experiencing acceleration measurable with an accelerometer approaching infinite near the horizon; the free faller is experiencing no proper (measurable) acceleration).

An infaller from a good distance away is passing a near horizon hovering observer at a speed approaching c. If the hovering observer is seeing extreme blue shift (correct), the passing infaller is seeing the hovering clock extremely slow, and almost all the blueshift disappears. They continue to see the outside quite normally (in time rate and frequency; there are optical distortions), until they hit the singularity. [Actually, I believe, if the infaller falls from far enough away, they see moderate redshift at and beyond the horizon.]

 

[harrylin]

You made a calculation error. The way I described it is a textbook calculation you can look up. I am talking about the directly observed doppler for an infaller who has crossed the horizon and is looking out. [..]

Sorry, you even doubly misunderstood my description! At Peter's request I attempted a description for what an infaller experiences who is looking forward towards starlight as he is going towards the horizon, according to the prediction of a Schwartzschild "distant observer".

Of course, I could still be mistaken. I simply multiplied the gravitational time dilation with "SR Doppler", which is itself Doppler times SR time dilation :
- gravitational time dilation f->∞ for r->r₀
- SR Doppler f->∞ for v->c

 

[PAllen]

Sorry, you even doubly misunderstood my description! At Peter's request I attempted a description for what an infaller experiences who is looking forward towards starlight as he is going towards the horizon, according to the prediction of a Schwartzschild "distant observer".

Of course, I could still be mistaken. I simply multiplied the gravitational time dilation with "SR Doppler", which is itself Doppler times SR time dilation :
- gravitational time dilation f->∞ for r->r₀
- SR Doppler f->∞ for v->c

Ok, our track record for failing to describe things in a mutually meaningful way continues. To me, as you are close the BH, most of the stars are behind you. Of course, beaming will shift things toward the front. In the very front, you are staring at the growing BH. For free fall from a great distance, you will see stars behind you moderately redshifted and stars in front of you highly blueshifted. The blue shift in front will never be infinite, because more and more of it is transverse + beaming (because the BH blocks the very front). Once inside the horizon, you can only see the outside in back of you, and it looks moderately red shifted. You will also see Einstein ring in front of you (before you get too close).

 

[harrylin]

I think that it is time to enquire more about observable events, in relation to possible mappings of simultaneity.

[...] All the maps agrees on every computation of an observable, for the events they have in common. [..] Kruskal, GP, Lemaitre, etc. are simply maps that cover more events. Every computed measurement in them agrees with SC for the events included in both. SC assigns infinite coordinate values at a boundary of its coverage, the others do not, but all measurements right up to this edge agree in all coordinates (that infaller's clocks pass finite time reaching the edge; that distant observers never see/detect anything reaching the edge = EH). [..]

Please elaborate that with one or two simple examples.

Voyager 35 is sent to a black hole, which for simplicity we assume to be eternal static and in rest wrt the solar system. And of course, the Voyager is indestructible and always in operation.

1. A time code is emitted from Earth that can be received by Voyager. A Schwartzschild observer calculates that for t->∞, Voyager's clock goes to τ=42. He predicts that no signal can be received by Voyager at τ>=42.

2. Voyager also emits a time code. A Schwartzschild observer predicts that the Earth will never loose its signal.

What will a Kruskal observer predict for those cases, and why?

 

[PeterDonis]

1. A time code is emitted from Earth that can be received by Voyager. A Schwartzschild observer calculates that for t->∞, Voyager's clock goes to τ=42. He predicts that no signal can be received by Voyager at τ>=42.

2. Voyager also emits a time code. A Schwartzschild observer predicts that the Earth will never lose its signal.

This is not what the Schwarzschild observer will predict if he is using standard classical GR. (Note: technically the Schwarzschild observer has to change charts to make some of the predictions I'm going to give; but there is nothing stopping him from doing that. See the end of this post, however, for some further comments on that.) What he will predict is:

0. Voyager crosses the horizon at τ=42. At some later proper time by Voyager's clock, let's say τ=48, Voyager hits the singularity and is destroyed. But no value of t can be assigned to any event on Voyager's worldline with τ>=42, assuming the Schwarzschild observer is using the simultaneity convention for "t" of the exterior Schwarzschild coordinate chart. (As PAllen has pointed out, he could choose other simultaneity conventions; but we'll focus on this one because it's the one you are considering "privileged", so it's good to illustrate its limitations.)

1. Time codes emitted from Earth are received by Voyager just fine at τ=42, and indeed all the way up to τ=48. Voyager is destroyed at τ=48, so obviously it can't receive any signals after that; its worldline stops at τ=48. Signals sent from Earth towards Voyager that don't reach Voyager before τ=48 will hit the singularity instead.

2. No signal sent *from* Voyager at τ>=42 can ever be received by Earth; and signals sent at τ<42, but closer and closer to τ=42, will be received by Earth at later and later times t->∞. In other words, as the Voyager time codes Earth receives get closer and closer to τ=42, the signals carrying those codes will be received at Earth times t->∞. Earth will never receive any Voyager time code with τ>=42.

What will a Kruskal observer predict for those cases, and why?

All of the predictions above are about invariant, physical observables, and so they are the same regardless of which chart you use. If I have time soon I'll try and draw an illustration of what the worldlines of Earth, Voyager, and the light signals going back and forth look like in the Kruskal chart.

Now for the further comments I promised. The Schwarzschild observer could frame predictions similar to the ones you gave, like this:

1. A time code is emitted from Earth that can be received by Voyager. A Schwartzschild observer calculates that for t->∞, Voyager's clock goes to τ=42. He predicts that no signal can be received by Voyager at τ>=42 if Voyager's worldline does not continue past τ=42.

2. Voyager also emits a time code. A Schwartzschild observer predicts that the Earth will never lose its signal if Voyager's worldline does not continue past τ=42.

However, you have not shown that the clause in bold is actually true; and as I've said before, if you look at the solution to the EFE that applies here, it tells you that the clause in bold is *false*. But even without looking at the gory details of the EFE, it should be obvious that Voyager's worldline can't just stop at τ=42. No physical quantity is singular there. Saying that Voyager's worldline suddenly stops there, for no apparent reason, would be like saying your worldline suddenly stops 42 minutes from now, for no apparent reason. Did you get hit by something and destroyed? No. Did you get torn apart by tidal forces increasing without bound? No. Then why does your worldline stop? No reason.

That's not physically reasonable. Voyager's worldline has a finite length up to τ=42, and worldlines don't just stop at a finite length unless some physical quantity, some invariant, is singular there. That doesn't happen at τ=42, so the only physically reasonable conclusion is that Voyager's worldline continues on *past* τ=42.

But then where does it go? It can't rise back up again to a larger radius; to do that it would have to move faster than light. It can't even *stay* at the same radius; to do that it would have to move at the speed of light, and it's an ordinary object and can't do that. The only possibility is for it to continue *inward*, and that means there has to be a region of spacetime inside the horizon, where the portion of Voyager's worldline with τ>=42 goes. Only when Voyager reaches the curvature singularity at r = 0, at τ=48, will its worldline actually stop, because there, a physical quantity, the curvature, *does* become singular.

 

[PAllen]

I think that it is time to enquire more about observable events, in relation to possible mappings of simultaneity.
Please elaborate that with one or two simple examples.

Voyager 35 is sent to a black hole, which for simplicity we assume to be eternal static and in rest wrt the solar system. And of course, the Voyager is indestructible and always in operation.

1. A time code is emitted from Earth that can be received by Voyager. A Schwartzschild observer calculates that for t->∞, Voyager's clock goes to τ=42. He predicts that no signal can be received by Voyager at τ>=42.

Not quite. After the clock reaches 42, you need to switch to interior SC coordinates. Then the trajectory continues for another short finite period of proper time before the voyager reaches the singularity. Voyager continues to get signals from outside until reaching the singularity. All of this can be calculated in pure SC coordinates (note the interior is the same coordinates and metric as the exterior; you just have to use limiting operations to 'step over' the horizon.

2. Voyager also emits a time code. A Schwartzschild observer predicts that the Earth will never loose its signal.

The last signal sent by voyager infinitesimally before crossing the horizon will be received from Earth in the infinite future. No signal voyager sends from past the horizon will reach earth. Not sure how much this agrees with what you wrote - as usual, I am not quite sure how to interpret your phrasing.

What will a Kruskal observer predict for those cases, and why?

All other coordinates predict exactly the same thing. By construction, computing observables (invariants) in different coordinates is as tightly guaranteed to produce the same result as (1/2)*(1/2) = (1/4)*(1). This is simply because the metric is transformed in such a way along with coordinate transform of e.g. world lines as to make this a pure mathematical identity.

 

[harrylin]

These surprising clarifications were very helpful for me to understand what the two of you were telling me in concert - it brought to light an important point.

For my account of what "Schwartzschild" would calculate, I used the Oppenheimer-Snyder "map" and directions (the O-S map of O-S, as explained by O-S). I held it for quite possible that I made a calculation or interpretation error. However, it now looks to me that you use a force-fitted "Schwartzschild" map that has been made to comply with other maps - perhaps based on a textbook treatment (is it perhaps de facto a MTW map?!).

BTW, the question "is this really GR?" is of course completely independent of "do I find this reasonable?".

I have in mind to dig deeper, but as this is very much the topic of O-S, I will do so (later) in the appropriate thread - https://www.physicsforums.com/showthread.php?t=651362

Thanks again!

 

[PAllen]

These surprising clarifications were very helpful for me to understand what the two of you were telling me in concert - it brought to light an important point.

For my account of what "Schwartzschild" would calculate, I used the Oppenheimer-Snyder "map" and directions (the O-S map of O-S, as explained by O-S). I held it for quite possible that I made a calculation or interpretation error. However, it now looks to me that you use a force-fitted "Schwartzschild" map that has been made to comply with other maps - perhaps based on a textbook treatment (is it perhaps de facto a MTW map?!).

BTW, the question "is this really GR?" is of course completely independent of "do I find this reasonable?".

I have in mind to dig deeper, but as this is very much the topic of O-S, I will do so (later) in the appropriate thread - https://www.physicsforums.com/showthread.php?t=651362

Thanks again!

Just a quick comment here - the SC metric you've seen (as derived from the EFE) applies inside the horizon. That is, if you look at it, it works just fine for r < Rs; it only doesn't work (without limits for the finite invariants) on Rs itself. So it is not added later - it is the same solution, and EFE are telling us it applies everywhere - the derivation applies up to r=0.

So, if you look back at my (u,v) versus (x,y) example, it is as if you got the full (u,v) solution; you just have to deal with technical problems at x and y-axis coordinate (but not invariant) discontinuity.

 

[PeterDonis]

For my account of what "Schwartzschild" would calculate, I used the Oppenheimer-Snyder "map" and directions (the O-S map of O-S, as explained by O-S).

Yes, I understand that. My rewrite of those predictions, with the key clause added in bold, was to make the point that the map and directions provided by O-S are incomplete; they tell you how to calculate things as long as tau < 42, but they do *not* say that Voyager stops existing at tau = 42. Nor do they say that it continues to exist at tau >= 42, but in a region of spacetime that needs a different chart to map it. They do not address that question either way. They do say that t -> infinity as tau -> 42; but they do not show (nor, I think, do they claim to show) that t -> infinity represents a *physical* limitation; they only show that it represents a limitation of SC exterior coordinates. (Showing that it represents a physical limitation would require showing that some invariant quantity goes to infinity there, and O-S certainly don't do that; and in the light of further knowledge since then, we know there isn't one.)

However, it now looks to me that you use a force-fitted "Schwartzschild" map that has been made to comply with other maps - perhaps based on a textbook treatment (is it perhaps de facto a MTW map?!).

No, we are just addressing the question that the O-S map does not address: what happens to Voyager at tau = 42? Based on the answer to that question given by the Einstein Field Equation, we are taking the incomplete O-S map and adding a new region, and an expanded set of directions, onto it to make it physically complete.

BTW, the question "is this really GR?" is of course completely independent of "do I find this reasonable?".

Understood; and the answer is yes, it is "really GR".

 

[PeterDonis]

I have in mind to dig deeper, but as this is very much the topic of O-S, I will do so (later) in the appropriate thread - https://www.physicsforums.com/showthread.php?t=651362

I would find this helpful; I was having trouble remembering which thread I was posting in.

 

[PAllen]

Having come this far, I want to emphasize a very physical reason for not being satisfied with an object's history stopping at an arbitrary time in its history. This is that a key physical foundation of GR is the principle of equivalence; one aspect of this, is that sufficiently locally, GR = SR, everywhere, every when. So we have an infall clock whose mechanics is following normal SR physics locally at all times; tidal gravity is irrelevant due to its small size (esp. for a supermassive BH); the relation between its time rate and some distant clock when sending messages is irrelevant locally. So you posit at 2:59:59 pm, it is operating the same as any similarly constructed clock; but at 3 pm it stops dead for no reason explicable with SR physics. This is a gross violation of the principle of equivalence: a free fall clock near the horizon has a behavior completely different than free fall clocks everywhere else.

---

A related mathematical point is that the EFE are system of 10 equations that satisfy 4 identities. They are thus sufficient only to determine 6 independent functions. But 10 are needed to specify a metric expressed in some coordinates. This means you need to pose 4 'coordinate conditions' to fully determine the metric expression. Given the same boundary and initial conditions, these coordinate conditions don't change the physics of the solution - they just determine its coordinate expression. So you arrive as SC geometry by saying you want a vacuum solution that is spherically symmetric. This is enough to uniquely determine the geometry. One type of coordinate conditions gives you SC coordinates; another can lead directly to Kruskal. You can verify that SC coordinates compute all the same physics and geometry as Kruskal except that they have coordinate discontinuity at the horizon. This mathematical structure leave little alternative but to see the SC coordinates as two patches and a boundary problem that correspond to two parts of the Kruskal coordinates that cover the whole geometry without coordinate difficulties.

Historically, I think all of the following being well understood did not occur until the mid 1960s:

- spherical symmetry + vacuum uniquely determine a solution (this part was known, in various forms a long time)

- using right coordinate conditions, you can directly get the Kruskal coordinates from
the EFE; all other known coordinates are subsets of these. Historically, Kruskal coordinates arose by geodesic extension of incomplete charts; but sometime in the mid 60s (I think) it was realized they actually follow directly from the EFE as the unique complete spherically symmetric solution if you impose the right coordinate conditions.

 

[zonde]

The path of an outer edge infall particle has finite proper time integrated to the SC radius. If you declare it stops there, you have a hole in spacetime. You have a geodesic ending with finite 'interval', where curvature is finite.

PAllen, do you realize that in Minkowski geometry zero space-time distance (zero proper time) between two points does not mean it's the same point?
This is very different from traditional geometry where zero distance between two points does mean it's the same point.

If you imagine the surface of such particles infalling, and you don't allow it to proceed to Sc radius, you have, geometrically, a hole: proper time on this surface is finite, area is finite, but if you stop it from continuing, it is geometrically a hole. Geometry is defined by invariants, not coordinate quantities. Consider the example I gave to harrylin several posts back of a horizontal geodesic in the plane in (u,v) coordinates. u coordinate goes to infinity on both sides of coordinate singularity, but it is still nothing but a geodesic in the flat plane.

I found your example. I just have to think what I have to say about it. It is just mock transformation when you undo all the consequences of transformation using transformed metric.

Nope. Einstein was very clear that simultaneity is purely a convention, not an observable. There is no observation or measurement in SR that changes if you use a different one than the standard one (but you have to change the metric as well; it is no longer eg. diag(+1,-1,-1,-1) if you use a funky convention.

There is one statement in SR that gives it physical content - it is principle of relativity.
But principle of relativity applies to certain class of inertial coordinate systems. This class of inertial coordinate systems is defined using particular simultaneity convention.
So you can't really speak about SR with different simultaneity convention as this particular simultaneity convention is integral part of the theory (and it's predictions).

If you want you can say that relativity principle gives physical content to particular simultaneity convention.

 

[harrylin]

Just a quick comment here - the SC metric you've seen (as derived from the EFE) applies inside the horizon. That is, if you look at it, it works just fine for r < Rs; it only doesn't work (without limits for the finite invariants) on Rs itself. So it is not added later - it is the same solution, and EFE are telling us it applies everywhere - the derivation applies up to r=0.

So, if you look back at my (u,v) versus (x,y) example, it is as if you got the full (u,v) solution; you just have to deal with technical problems at x and y-axis coordinate (but not invariant) discontinuity.

And a quick comment on that quick comment: I don't see the qualitative difference with "The light speed limit doesn't exist; the tachyon space works just fine, it is the same solution. You just have to deal with technical problems to get through c".

 

[martinbn]

Having come this far, I want to emphasize a very physical reason for not being satisfied with an object's history stopping at an arbitrary time in its history. This is that a key physical foundation of GR is the principle of equivalence; one aspect of this, is that sufficiently locally, GR = SR, everywhere, every when. So we have an infall clock whose mechanics is following normal SR physics locally at all times; tidal gravity is irrelevant due to its small size (esp. for a supermassive BH); the relation between its time rate and some distant clock when sending messages is irrelevant locally. So you posit at 2:59:59 pm, it is operating the same as any similarly constructed clock; but at 3 pm it stops dead for no reason explicable with SR physics. This is a gross violation of the principle of equivalence: a free fall clock near the horizon has a behavior completely different than free fall clocks everywhere else.

So far I was following your explanations, but here I am I little confused. Why do you say that the clock will stop? Surely passing the horizon will not stop the clock and someone with the clock will see it ticking after 3:00pm, no?

A related mathematical point is that the EFE are system of 10 equations that satisfy 4 identities.

What do you mean by this? Equations that satisfy identities!

 

[harrylin]

A clarification about terminology: there are several concepts/objects that can be referred to as "Schwarzschild's solution", or "the Schwarzschild metric" [..] My reason for making these distinctions will be evident in a moment.

As I elaborate in a parallel thread, I make a similar distinction between different "flavours" of GR.

[..] this is not a statement about any actual physics; it's only a statement about a particular coordinate chart, with no real physical content.

In fact that chart is an equation. You next suggest that another equation has more physical content than the equation which was derived from it, using reasonable physical assumptions. I suspect that the one who derived that equation would disagree with you for reasons that I will briefly* mention.

[..] when you actually work through the solution of the Einstein Field Equation, in either case #1 above ([..] physically unreasonable) or case #2 above (more complex [..] but physically more reasonable, though still highly idealized), you find that any solution that only includes the region outside the horizon is incomplete; the EFE itself tells you that that region cannot be the entire spacetime. [..] These all seem like innocuous statements about coordinate charts on geometric manifolds to me. Apparently they seem like huge issues to you; I'm not sure why. Why is it such a big deal that some charts can cover points or regions that others can't?

[..] technically the Schwarzschild observer has to change charts to make some of the predictions I'm going to give; but there is nothing stopping him from doing that. [..] why does your worldline stop? No reason.
That's not physically reasonable. [..]

[..] Based on the answer to that question given by the Einstein Field Equation, we are taking the incomplete O-S map and adding a new region, and an expanded set of directions, onto it to make it physically complete. [..]
Understood; and the answer is yes, it is "really GR".

If I correctly understood the explanations, those equations lead to white holes when blindly followed through without physical concerns; following your arguments, white holes are "really GR". Is it a big deal to you?

However, and as we discussed earlier, we agree that considerations of what makes physical sense should play a role. Most theories do contain more than mere equations, and GR is no exception. For me a theory consists of its physical foundations (both those postulated and those clearly mentioned). Those foundations are all kept except if they lead to contradictions; and an equation is only physically valid insofar as those physical foundations are not violated. An obvious one is the Einstein equivalence principle, but there are also the physical reality of gravitational fields and the requirement that theorems must describe the relation between measurable bodies and clocks.

Consequently I expect that Einstein would reject Peter's argument and say that Peter denies the physical reality of gravitational fields. Einstein would argue that the clock's worldline never really stops, but does not reach beyond 42 due to the physical reality of the gravitational field. I think that Kruskal's white holes and his inside solution for black holes are not compatible with Einstein's GR.

Having come this far, I want to emphasize a very physical reason for not being satisfied with an object's history stopping at an arbitrary time in its history. This is that a key physical foundation of GR is the principle of equivalence; one aspect of this, is that sufficiently locally, GR = SR, everywhere, every when. So we have an infall clock whose mechanics is following normal SR physics locally at all times; tidal gravity is irrelevant due to its small size (esp. for a supermassive BH); the relation between its time rate and some distant clock when sending messages is irrelevant locally. So you posit at 2:59:59 pm, it is operating the same as any similarly constructed clock; but at 3 pm it stops dead for no reason explicable with SR physics.
This is a gross violation of the principle of equivalence: a free fall clock near the horizon has a behavior completely different than free fall clocks everywhere else.

PAllen, it looks to me that you are mixing up reference frames. As far as I can see, in no valid GR reference system is the clock suddenly stopping dead.
As described from S, the clock never stops ticking. I guess that for such an extreme case the validity of SR probably shrinks to nothing. And as described from S', dramatic things happen upto 3 pm but no stopping of clocks is observed.

[ADDENDUM: It may look a little weird if you believe that the universe is eternal. But in case you believe that the universe is not eternal, as is commonly thought, then the universe ends at for example 2:59:58.]

And don't you think that Einstein would have exclaimed the same, if it was really a "gross violation of the principle of equivalence"? Instead he commented that "there arises the question whether it is possible to build up a field containing such singularities". (E. 1939.)

The Einstein principle of equivalence:
"K' [..] has a uniformly accelerated motion relative to K [..] [This] can be explained in as good a manner in the following way. The reference-system K' has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to K'."
- https://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity

*Regretfully this forum has been stripped from philosophy on the grounds that the mentors don't want to spend time on monitoring such discussions; I will respect that by not elaborating much on philosophy of science.

 

[harrylin]

[..] Surely passing the horizon will not stop the clock and someone with the clock will see it ticking after 3:00pm, no? [..]

According to the Schwartzschild equations as used by Einstein and Oppenheimer, that clock will only reach the horizon (and indicate 3:00pm) at t=∞. In common physics that means "never"; however some smart person (in fact, who?) invented a different interpretation!

 

[PeterDonis]

In fact that chart is an equation. You next suggest that another equation has more physical content than the equation which was derived from it, using reasonable physical assumptions.

No, I don't. I suggest that the first chart/equation (exterior Schwarzschild) does not cover a particular portion of the spacetime that the second chart/equation (Kruskal) does.

However, underlying all of this is just one equation, the EFE. That equation is what's really at issue here. See below.

If I correctly understood the explanations, those equations lead to white holes when blindly followed through without physical concerns; following your arguments, white holes are "really GR".

You didn't correctly understand the explanations. The EFE leads to white holes only if we assume the spacetime is vacuum everywhere (and spherically symmetric, but that's a minor point for this discussion). Nobody thinks that assumption is physically reasonable. If the spacetime is not vacuum everywhere--for example, if there is collapsing matter present--then the EFE does *not* predict white holes. So white holes are part of the set of all possible mathematical solutions of the EFE, but they are not part of the set of physically reasonable solutions of the EFE.

Just an "equation" isn't enough; you have to add constraints--initial/boundary conditions--to get a particular solution. Which solution of the equation you get--i.e., which spacetime geometry models the physical situation you're interested in--depends on the constraints.

However, and as we discussed earlier, we agree that considerations of what makes physical sense should play a role. Most theories do contain more than mere equations, and GR is no exception. For me a theory consists of its physical foundations (both those postulated and those clearly mentioned).

Of course. See above.

Those foundations are all kept except if they lead to contradictions; and an equation is only physically valid insofar as those physical foundations are not violated. An obvious one is the Einstein equivalence principle, but there are also the physical reality of gravitational fields and the requirement that theorems must describe the relation between measurable bodies and clocks.

Sure.

Consequently I expect that Einstein would reject Peter's argument and say that Peter denies the physical reality of gravitational fields.

Einstein *did* reject arguments of this type. Einstein was wrong.

Einstein would argue that the clock's worldline never really stops, but does not reach beyond 42 due to the physical reality of the gravitational field.

What is "the gravitational field"? What mathematical object in the theory does it correspond to? Before we can even evaluate this claim, we have to know what it refers to. But let's try it with some examples:

(1) The "gravitational field" is the metric. The metric (the coordinate-free geometric object, not its expression in particular coordinates) is perfectly finite and continuous at the horizon, and for reasons that both PAllen and I have explained, it can't "just stop" at the horizon without violating the EFE.

(2) The "gravitational field" is the Riemann curvature tensor. Like the metric, this is perfectly finite and continuous at the horizon.

(3) The "gravitational field" is the proper acceleration experienced by a "hovering" observer (an observer who stays at the same radius and does not move at all in a tangential direction). This *does* increase without bound as you get closer and closer to the horizon. However, there is *no* "hovering" observer *at* the horizon, because the horizon is a null surface: i.e., a line with constant r = 2M and constant theta, phi is not a timelike line; it's a null line (the path of a light ray--a radially outgoing light ray). So there is no observer who experiences infinite proper acceleration, and this definition of "gravitational field" simply doesn't apply at or inside the horizon.

As far as I can see, the only possible basis you could have for claiming that "the physical reality of the gravitational field" means that the clock's worldline stops as tau->42, would be #3. However, #3 doesn't apply to infalling observers; it only applies to accelerated, "hovering" observers. Infalling observers don't feel any acceleration, so there's nothing stopping them from falling through the horizon. The "gravitational field" in the sense of #3 is simply not felt by them at all.

Note that in all these cases, the physical "field" has to correspond to something invariant in the mathematical model, *not* something that only exists in a particular coordinate chart. That is something Einstein would have *agreed* with. Note also that none of the definitions of "gravitational field" I gave above used Schwarzschild coordinate time, or the fact that t->infinity as you approach the horizon. Einstein simply didn't understand that claims about t->infinity as you approach the horizon were claims about something that only exists in a particular coordinate chart.

I think that Kruskal's white holes and his inside solution for black holes are not compatible with Einstein's GR.

See above. You are equivocating on different meanings of "Einstein's GR". White holes are mathematically compatible, but not physically reasonable. Black hole interiors are both mathematically compatible *and* physically reasonable.

And don't you think that Einstein would have exclaimed the same, if it was really a "gross violation of the principle of equivalence"? Instead he commented that "there arises the question whether it is possible to build up a field containing such singularities". (E. 1939.)

As I've said before, Einstein's paper only considered the stationary case--i.e., he only considered systems of matter in stable equilibrium. All his paper proves is that *if* a system has a radius less than 9/8 of the Schwarzschild radius corresponding to its mass, the matter can't be in stable equilibrium. A collapsing object that forms a black hole meets this criterion: the collapsing matter is not in stable equilibrium. So Einstein's conclusion doesn't apply to it.

 

[PAllen]

I found your example. I just have to think what I have to say about it. It is just mock transformation when you undo all the consequences of transformation using transformed metric.

I only have time for one quick comment:

But that is the way all coordinate transforms work in differential geometry! That is the whole point! The the metric is transformed along with coordinates so all geometric properties (lengths, angle, intervals, etc.) are preserved as mathematical identities.

 

[martinbn]

According to the Schwartzschild equations as used by Einstein and Oppenheimer, that clock will only reach the horizon (and indicate 3:00pm) at t=∞. In common physics that means "never"; however some smart person (in fact, who?) invented a different interpretation!

That is not true, the clock will pass the horizon, and its proper time will go after 3:00pm.

 

[harrylin]

[..] Einstein *did* reject arguments of this type. Einstein was wrong.

Just give me the physics paper that proves that your philosophy is right, and Einstein's was wrong.

What is "the gravitational field"? [..]

Perhaps your beef with Einstein could be summarized as follows:

Peter: What is "the gravitational field"? It is not a real mathematical object
Einstein: What is a "region of spacetime"? It is not a real physical object.

In my experience it can be interesting to poll opinions, and to inform onlookers about different points of view; however discussions of that type are useless.

 

[harrylin]

That is not true, the clock will pass the horizon, and its proper time will go after 3:00pm.

Not in "Schwartzschild" time t of the Schwartzschild equation. Did you calculate it?
Once more: According to the Schwartzschild equations as used by Einstein and Oppenheimer, that clock will only reach the horizon (and indicate 3:00pm) at t=∞. In common physics that means "never"; however some smart person (in fact, who?) invented a different interpretation. In that different interpretation, which I still don't fully understand, the clock will pass the horizon despite Schwartzschild's t=∞.

For details, see the ongoing discussion: https://www.physicsforums.com/showthread.php?t=651362
incl. an extract of Oppenheimer-Snyder: https://www.physicsforums.com/showpost.php?p=4162425&postcount=50

 

[martinbn]

Not in "Schwartzschild" time t of the Schwartzschild equation. Did you calculate it?
For details, see the ongoing discussion: https://www.physicsforums.com/showthread.php?t=651362

This makes no sense. The clock shows its proper time, nothing else, and it will not stop when it reaches 3:00pm.

 

[harrylin]

This makes no sense. The clock shows its proper time, nothing else, and it will not stop when it reaches 3:00pm.

A so-called "asymptotic observer" predicts that it will slow down so much that it will not reach 3:00pm before the end of this universe. However, a "Kruskal observer" says that that is true from the viewpoint of the asymptotic observer but predicts that the clock will nevertheless continue to tick beyond 3:00pm.

PeterDonis and PAllen say that these predictions do not contradict each other. And that still makes no sense to me, despite lengthy efforts of them to explain this to me.

 

[pervect]

And a quick comment on that quick comment: I don't see the qualitative difference with "The light speed limit doesn't exist; the tachyon space works just fine, it is the same solution. You just have to deal with technical problems to get through c".

It does require closer inspection to see if the apparent singularity in the equations of motion is removable or not.

What do I mean by a removable singularity?

http://en.wikipedia.org/w/index.php?title=Removable_singularity&oldid=507006469

n complex analysis, a removable singularity (sometimes called a cosmetic singularity) of a holomorphic function is a point at which the function is undefined, but it is possible to define the function at that point in such a way that the function is regular in a neighbourhood of that point.

For instance, the function

$$f(z) = \frac{\sin z}{z}$$

has a singularity at z = 0. This singularity can be removed by defining f(0) := 1, which is the limit of f as z tends to 0. The resulting function is holomorphic.

It's been known for a very long time that in the black hole case that the singularity is removable.

IT does takes a bit of work to decide if the apparent singularity is the result of a poor coordinate choice , or is an inherent feature of the equations.

It might be helpful to give a quick example of how this happens. Consider the equations for spatial geodesics on the surface of the Earth. (Why geodesics? Because that's how GR determines equation of motion. So this is an easy-to-understand application of the issues involved in finding geodesics).

If you let lattitude be represented by $\psi$ and longitude by $\phi$, then you can write the metrc $ds^2 = R^2 (d \psi^2 + cos^2(\psi) d\phi^2)$ and come up with the equations for the geodesic (which we know SHOULD be a great circle) for $\psi(t)$ and $\phi(t)$

$$\frac{d^2 \psi}{dt^2} + \frac{1}{2} \sin 2 \psi \left( \frac{d \phi}{dt} \right)^2$$$$\frac{d^2 \phi}{dt^2} - 2 \tan \psi \left(\frac{d\phi}{dt}\right) \left( \frac{d\psi}{dt} \right) = 0$$

Now, one solution of these equations is $\phi$ = constant. It makes both equations zero. It's also half of a great circle. But, if we look more closely, we see that we have a term of the form 0*infinity in the second equation as we approach the north pole, because of the presence of $\tan \psi$ when $\psi$ reaches 90 degrees.

THis apparent singularity is mathematical, not physical. If you're drawing a great circle around a sphere, there's no physical reason to stop at the north pole.

Of course we already know what the answer is - we need to join two half circles together. In particular, we know we need to splice together two half circles, 180 degrees apart in lattitude, though as far as I know all the solution techniques (change of variable, etc) are equivalent to not using lattitude and longitude coordinates at the north pole, because the coordinates are ill-behaved there.

The same is in the black hole case, though to justify it you need to either do the math yourself, or read a textbook where someone else has. Note that you probably won't find this sort of thing in papers so much, it's assumed everyone knows it in the literature. Where you're more likely to find an explanation in a textbook or lecture notes.

Which brings me to the next point.

We don't have textooks online, but we've got several good sets of lecture notes.

What does Carroll's lecture notes have to say on the topic?
He defines the geodesic equation of motion - they're pretty complex looking, and I wouldn't be surprised if you didn't want to solve them yourself. But what does Caroll have to say about solving them?

I'll give you a link http://preposterousuniverse.com/grnotes/grnotes-seven.pdf , and a page reference (pg 182) in that link.

Then I'll give you some question

1) Does Carroll support your thesis? Or does he disagree with it?
2) What do other textbooks and online lecture notes have to say?

And for my own information
3) Do you think you know the difference between "absolute time" and "non-absolute time"
4) Do you think your argument about "time slowing down at the event horizon" depends on the existence of "absolute" time?

 

[Nugatory]

A so-called "asymptotic observer" predicts that it will slow down so much that it will not reach 3:00pm before the end of this universe. However, a "Kruskal observer" says that that is true from the viewpoint of the asymptotic observer but predicts that the clock will nevertheless continue to tick beyond 3:00pm.

PeterDonis and PAllen say that these predictions do not contradict each other. And that still makes no sense to me, despite lengthy efforts of them to explain this to me.

Here's an analogy that may help it make a bit more sense.
Consider an ordinary boring constant-velocity special relativity problem: You are rest and you watch me passing by at some reasonable fraction of the speed of light, so you observe that my clock is ticking more slowly than yours. If the universe has a a finite age, it is certainly possible for me to observe a time on my clock that you will claim will never be reached - all that necessary is that:
1) I get to read my clock on my worldline before it terminates at the end of the universe.
2) Your worldline terminates at the end of the universe before it intersects the line of (your) simultaneity through the event of me reading my clock.

But, you will say, I'm cheating by introducing this arbitrary "end of the universe" to cut off your worldline (actually, you introduced it - I'm just abusing it )before it can intersect the relevant line of simultaneity. If I didn't do that, then no matter how much of my time passes before I read my clock, you'd be able to extend your worldline to intersect the line of simultaneity. That is true enough, but then again the entire concept of "line of simultaneity" only really makes sense in flat space.

The bit about a "Kruskal observer" is a red herring. The geometry around a static non-charged non-rotating mass is the Schwarzschild geometry, no matter what coordinates we use, and the only meaningful notion of time that we have is proper time along a time-like worldline. The Kruskal coordinates allow us to calculate the proper time along the infalling clock's worldline as it crosses the Schwarzschild radius, whereas the the Schwarzschild coordinates (as opposed to geometry) do not. So it's not that the "Schwarzschild observer" and the "Kruskal observer" are producing conflicting observations, it is that the Kruskal coordinates are producing a prediction for the infalling observer's worldline and the Schwarzschild coordinates are not.

 

[zonde]

I only have time for one quick comment:

But that is the way all coordinate transforms work in differential geometry! That is the whole point! The the metric is transformed along with coordinates so all geometric properties (lengths, angle, intervals, etc.) are preserved as mathematical identities.

Okay, I have kind of working hypothesis about how this works.
We have global coordinate system where we know how to get from one place to another i.e. it provides connection, but this global coordinate system does not tell anything useful about distances and angles and such. And then we have another coordinate system that tells us distances and angles but it works only locally, meaning that if we have two adjacent patches with local coordinate systems we don't know how to glue them together.
So we take take global coordinate system with metric that will give us geometric values in accord with local coordinate systems.

Something like that. Only I don't know how to check if this is right.

 

[Austin0]

Since you're so insistent on doing calculations in Schwarzschild coordinates, try this one: write down the equation defining the proper time of an object freely falling radially inward from a finite radius r = R > 2M, to radius r = 2M. Write it so that the proper time is a function of r only (this is straightforward because it's easy to derive an equation relating r and the Schwarzschild coordinate time t, so you can eliminate t from the equation). This equation will be a definite integral of some function of r, from r = R to r = 2M. Evaluate the integral; you will see that it gives a finite answer. Therefore, the proper time elapsed for an infalling object is finite, even according to Schwarzschild coordinates.

.

Correct me if I am wrong but it appears to me that the integration of proper falling time does not have a finite value..

Yes, it appears that way, if you just try to intuitively guess the answer without deriving it. But when you actually derive it, you find that it *does* give a finite answer, despite your intuition.

It asymptotically approaches a finite limit.

This is equivalent to saying the proper time integral *does* have a finite value. If you try to evaluate the integral in the most "naively obvious" way in Schwarzschild coordinates, you have to take a limit as r -> 2m, since the metric is singular at r = 2m; but the limit, when you take it, is finite..

From the statement the limit "does" have a finite value can I assume you are basing this on a mathematical theorem "proving" that such limits at 0 or infinity resolve to definite values? While I understand the truth of such a theorem within the tautological structure of mathematics and also it's practical truth as far as, for most applications in the real world, the difference becomes vanishingly small (effectively vanishes) this does not imply that it necessarily has physical truth.

Example: Unbounded coordinate acceleration of a system under constant proper acceleration as t ---->∞

Mathematically you can say this resolves to c but in this universe as we know it or believe it to be, this is not the case.

What you are doing here seems to me to be equivalent to integrating proper time of such a system to the limit as v --->c to derive a finite value. Thus demonstrating that such a system could reach c in finite time even if it never happens according to external clocks..

The analogy is particularly apt as by assuming the free faller reaches the horizon this is also equivalent to reaching c relative to the distant static observer yes??

What difference do you see between the two cases?

In both cases it is equivalent to directly assuming reaching c or the horizon independent of determining whether they could actually arrive there. And then determining a temporal value for your assumption. Just MHO

However, even if you insist on doing the integral in Schwarzschild coordinates, you can still write it in a way that doesn't even require taking a limit; as I said in the previous post you quoted, you can eliminate the t coordinate altogether and obtain an integrand that is solely a function of r and is nonsingular at r = 2m, so you can evaluate the integral directly. .

The comments above apply to any method of integration but if freefall proper time is derived from the metric how does the additional dilation factor from velocity enter into this integration??
If you are directly integrating the metric without reference to coordinate time isn't this actually integrating an infinitesimal series of static clocks between infinity and 2M?

It seems to me that either the Schwarzschild metric accurately corresponds to reality outside the hole or it doesn't. But the idea that its perfectly true up to some indeterminate pathological point "somewhere" in the vicinity of the horizon seems very shakey.
Actually the idea of a horizon as a third sector of reality between inside and outside seems like a pure abstraction. Is there a surface between air and water?

 

[Austin0]

And for my own information
3) Do you think you know the difference between "absolute time" and "non-absolute time"
4) Do you think your argument about "time slowing down at the event horizon" depends on the existence of "absolute" time?

Does the returning twins age difference depend on a concept of absolute time?

What if the traveling twin hangs out close to the horizon for a time before traveling back to his distant outside brother. Does his younger age indicate time slowing down at the horizon?
Does it depend on an absolute time? Is it a coordinate effect?

 

[Nugatory]

It seems to me that either the Schwarzschild metric accurately corresponds to reality outside the hole or it doesn't. But the idea that its perfectly true up to some indeterminate pathological point "somewhere" in the vicinity of the horizon seems very shakey.

Be careful with that term "Schwarzschild metric"...

There's the metric that Schwarzschild discovered as a solution of the Einstein field equations. It corresponds to reality (assuming spherical symmetry, no charge, no rotation, static - the conditions under which the SW metric is solution of the EFE) inside the event horizon, outside the event horizon, and at the event horizon itself.

Then there are Schwarzschild coordinates, which we often use when we want to write that metric down in a particular coordinate system. These coordinates do not work well at the event horizon. That doesn't mean that there's anything wrong there with the spacetime described by the Schwarzschild solution to the EFE; it just means that we should use some other coordinates to describe the metric there.

 

[PeterDonis]

Just give me the physics paper that proves that your philosophy is right, and Einstein's was wrong.

How about every paper published on black holes since the 1960's, and every major GR textbook since then?

Perhaps your beef with Einstein could be summarized as follows:

My "beef" isn't with Einstein; last I checked he doesn't post on PF.

Peter: What is "the gravitational field"? It is not a real mathematical object

Huh? I gave several examples of mathematical objects that could be reasonably associated with the term "gravitational field".

Einstein: What is a "region of spacetime"? It is not a real physical object.

Einstein thought spacetime *was* physically real; since a "region" of spacetime is just a portion of it, it should be real as well, since a portion of a real object would also presumably be real.

In my experience it can be interesting to poll opinions, and to inform onlookers about different points of view; however discussions of that type are useless.

I agree, but that's not the discussion we're having. You are stating your understanding of a physical model, and I am saying your understanding is mistaken. You are then quoting Einstein as an authority supporting your understanding, and I am repeating that your understanding is mistaken, and also that, in so far as Einstein's understanding was the same as yours, his was mistaken too.

You might well say that discussions of that type are useless too; I agree to the extent that I think quoting authorities is useless if the objective is to talk about the physics. We should be able to talk about the physics without caring what Einstein, Oppenheimer, Schwarzschild, or anyone else thought; we can talk about the mathematical model and its physical interpretation directly. You're having trouble understanding how the things PAllen and I and others have been saying about the mathematical model can all be consistent with each other; fine, I understand that. But it does no good to quote Einstein or anyone else; either you are able to construct the model yourself, or you're not. If you're not, IMO you need to learn how to do so before criticizing it--or else you should be able to show your partial construction of the model and exactly where you are hitting a stumbling block.

It seems to me that your current stumbling block is the fact that t->infinity as tau->42; you appear to think that this requires the infalling object to never reach tau>=42. What is your argument for this? By which I mean, what are the specific logical steps that get you from "t->infinity as tau->42" to "tau can't be >=42", and what assumptions do they depend on? I know it seems obvious to you, but it's not obvious to me, because I have a consistent mathematical model that shows how tau>=42 is possible despite the fact that t->infinity as tau->42. So one or the other of us must have a mistaken assumption somewhere. Let's see if we can find it.

If it will help, I can post *my* logical argument; but that will have to wait for a separate post.

 

[PeterDonis]

A so-called "asymptotic observer" predicts that it will slow down so much that it will not reach 3:00pm before the end of this universe.

That's *not* what the asymptotic observer predicts. What he predicts is that he will never see a light signal from the infalling object that says "my clock reads 3:00 pm", and light signals saying "my clock reads 2:59 pm", "my clock reads 2:59:30", "my clock reads 2:59:45", etc., etc. will reach him at times on his clock (the asymptotic observer's clock) that increase without bound.

The asymptotic observer may try to *interpret* this prediction as showing that the infalling observer's clock will slow down so much that it will not reach 3:00 pm before the end of this universe. But that interpretation depends on additional assumptions, such as the adoption of a particular simultaneity convention for distant events. As PAllen has pointed out repeatedly, simultaneity conventions are just that: conventions. They can't be used as the basis for making direct physical claims like those you are trying to make.

However, a "Kruskal observer" says that that is true from the viewpoint of the asymptotic observer but predicts that the clock will nevertheless continue to tick beyond 3:00pm.

No, a "Kruskal observer" says that the asymptotic observer is claiming too much (see above).

Btw, all this talk about different "observers" making different predictions is mistaken as well. Predictions of physical observables are the same regardless of which coordinate chart you adopt. Also, which coordinate chart you adopt is not dictated by which worldline in spacetime you follow; there is nothing preventing the "asymptotic observer" from adopting Kruskal coordinates to do calculations.