On the nature of the infinite fall toward the EH

[harrylin]

In other words, you have no interest in actually defending the statements you've made? Why post anything, if you don't want people to respond to your statements? [..]

"Making statements" is not the question; explanations can be helpful, but it's weird to have to explain things to those who are supposed to provide the answers - I still have questions similar to the OP here concerning black holes. To my surprise, when I first asked about black holes I found myself drawn into philosophical discussions. And I noticed that people started debating their philosophical views. That is a waste of time for me; I don't want to waste time but to increase knowledge. Similarly, the philosophy forum has now been closed because it consumed too much time of the mentors.

[..] Schwarzschild coordinates are chosen for CONVENIENCE. [...]

Yes of course - I made a similar remark in an earlier thread (probably it was in the "simultaneity" thread).

 

[stevendaryl]

"Making statements" is not the question; explanations can be helpful, but it's weird to have to explain things to those who are supposed to provide the answers

I don't think you've asked any very specific questions. Maybe I missed them. It seemed to me that you were making the claim that an infalling observer never reaches the event horizon. You were making another claim that there was a contradiction between the description of the situation as described by the coordinate system of the distant observer and the coordinate system of the infalling observer. Those seemed to be claims, not questions.

 

[stevendaryl]

I still have questions similar to the OP here concerning black holes.

I looked back through your posts, and I didn't see a single question. So what are your questions about black holes?

 

[stevendaryl]

To my surprise, when I first asked about black holes I found myself drawn into philosophical discussions.

You think that the discussions of Rindler coordinates or Zeno paradox are philosophical? I don't consider them philosophical, they are pointing out a mathematical error that you seem to be making, which is assuming that any event that does not have a "t" label must never happen. That's an invalid way to reason, and the Rindler and Zeno examples show explicitly why it is invalid. The conclusion "Since there is never a time t that the infalling observer crosses the event horizon, then he must never cross the event horizon" is invalid for exactly the same reason as the Rindler case or the Zeno case.

The discussions become philosophical because you respond to a mathematical claim as if it were a philosophical claim.

 

[jartsa]

A laser beam shot into a black hole may return back after an arbitrarily long time, after having been reflected from some object that was dropped into the black hole earlier.

Half of the arbitrarily long time the laser beam was traveling down.

So we can see that it takes an arbitrarily long time for a laser beam to reach the event horizon.

 

[PAllen]

A laser beam shot into a black hole may return back after an arbitrarily long time, after having been reflected from some object that was dropped into the black hole earlier.

Half of the arbitrarily long time the laser beam was traveling down.

So we can see that it takes an arbitrarily long time for a laser beam to reach the event horizon.

No, all you can see is that light can't get back. That doesn't necessarily say anything about time. If there were a perfect black body that absorbed all light, would you conclude light never reaches it? Well a black hole is blackest of black - any light crossing its horizon will never escape. You can compute that light will pass an infalling clock that will read finite time on approach to singularity. (Discussion concerns classical GR).

 

[Dale]

I think jartsa meant "towards a black hole" rather than "into a black hole"

 

[PAllen]

I think jartsa meant "towards a black hole" rather than "into a black hole"

But the rest of the argument was: "So we can see that it takes an arbitrarily long time for a laser beam to reach the event horizon. "

I disagree this follows. It follows if, and only if, you adopt a simultaneity convention that requires getting light back. However, such a simultaneity convention is at least dubious for any situation where light is trapped.

 

[Dale]

But the rest of the argument was: "So we can see that it takes an arbitrarily long time for a laser beam to reach the event horizon. "

I disagree this follows. It follows if, and only if, you adopt a simultaneity convention that requires getting light back. However, such a simultaneity convention is at least dubious for any situation where light is trapped.

I agree with you on that point. I think that the key problem is jartsa's claim "Half of the arbitrarily long time the laser beam was traveling down."

First, that is an arbitrary convention and second in curved spacetimes it isn't necessarily the best convention since it fails for some events.

 

[Mike Holland]

Moving off the white hole topic for a moment, we seem to have established that according to Schwarzschild coordinates and the O-S calculations, collapse of a massive body will take place very quickly in its own proper time, but will be infinitely delayed as far as a remote observer is concerned - it will not become a black hole while there is a finite reading on his clock.

But it also seems to be agreed that using an alternative coordinate system, the collapse is found to take place in a finite time on this remote observer's clock.

Does this mean that time dilation is just a coordinate thing? Can those clocks up in orbit be synchronised with ours here on Earth simply by changing coordinate systems? The people responsible for maintaining GPS systems would be delighted to hear this!

The different coordinate systems provide different ways of assigning times to remote events. But when we send a clock up into orbit and bring it back, it is in advance of our dilated earth-bound clock. This is a direct result of the time dilation as is predicted by GR and has been verified. In principle we could lower a clock close to a neutron star and then bring it back (or send our twin down to orbit it), and we would find a much higher degree of time dilation. Then we could repeat this close to an event horizon, and find tremendous time dilation. At what point do these coordinate systems give different results, and which will be in accord with our measurements?

 

[PeterDonis]

Does this mean that time dilation is just a coordinate thing?

It depends on what you mean by "time dilation". Sometimes there is a direct physical observable that allows you to compare "rates of time flow" for two clocks; the simplest case is where the clocks separate and then later meet up again, but there are other possibilities as well.

Can those clocks up in orbit be synchronised with ours here on Earth simply by changing coordinate systems?

No. The GPS satellites are in periodic orbits, so there is a common reference we can use to compare "rates of time flow" (basically, how many clock ticks elapse on the satellite per orbit, vs. on the ground per orbit).

But when we send a clock up into orbit and bring it back, it is in advance of our dilated earth-bound clock.

This is a case where two clocks separate and then meet up again, so a direct comparison can be made between their readings.

In principle we could lower a clock close to a neutron star and then bring it back (or send our twin down to orbit it), and we would find a much higher degree of time dilation.

Yes, when it came back up to meet up with your clock again, your clock would have much more elapsed time. But again, there's a direct comparison you can make to establish this.

At what point do these coordinate systems give different results

They never give different results when predicting actual observables, provided both coordinate systems cover the events in question. The fact that no finite coordinate time for the distant observer can be assigned to events on the infalling observer's worldline at or below the horizon is not a "result"; it's just a limitation of the distant observer's coordinate system.

and which will be in accord with our measurements?

Once an object reaches the horizon, it can never come back up again, and there are no stable orbits or static observers at or inside the horizon, so there are no periodic phenomena that could be use as a common reference for comparing "rates of time flow". So the comparison methods that work in the cases above, don't work in the case where an object falls past the horizon, and there are no other methods that give an invariant answer.

So there is *no* measurement that can tell you how an infalling clock below the horizon is "really" synchronized with a distant observer's clock. There is no physical observable corresponding to this.

 

[Mike Holland]

This is a case where two clocks separate and then meet up again, so a direct comparison can be made between their readings.

Yes, when it came back up to meet up with your clock again, your clock would have much more elapsed time. But again, there's a direct comparison you can make to establish this.

They never give different results when predicting actual observables, provided both coordinate systems cover the events in question. The fact that no finite coordinate time for the distant observer can be assigned to events on the infalling observer's worldline at or below the horizon is not a "result"; it's just a limitation of the distant observer's coordinate system.

Once an object reaches the horizon, it can never come back up again, and there are no stable orbits or static observers at or inside the horizon, so there are no periodic phenomena that could be use as a common reference for comparing "rates of time flow". So the comparison methods that work in the cases above, don't work in the case where an object falls past the horizon, and there are no other methods that give an invariant answer.

So there is *no* measurement that can tell you how an infalling clock below the horizon is "really" synchronized with a distant observer's clock. There is no physical observable corresponding to this.

I never said anything about what happens at or below the horizon. I was only discussing events very near the horizon.

So if I suspend my clock very close to the horizon and bring it back, I can make a direct comparison, and will find that an extremely high degree of time dilation has taken place. In principle, I can hang my clock closer, or for longer, and get as much dilation as I like. And you tell me all the coordinate systems will agree. So then how can some coordinate systems say that an object will fall through the horizon, or that an event horizon has formed, as far as a remote observer's clock is concerned, when time dilation approaches infinity there?

 

[PeterDonis]

I never said anything about what happens at or below the horizon. I was only discussing events very near the horizon.

Ok, but then you're leaving out the part that causes all the arguments.

So if I suspend my clock very close to the horizon and bring it back, I can make a direct comparison, and will find that an extremely high degree of time dilation has taken place.

Yes.

In principle, I can hang my clock closer, or for longer, and get as much dilation as I like. And you tell me all the coordinate systems will agree.

Yes, because you've specified a direct observable: when the two clocks meet up again, what are their respective readings? That can't depend on which coordinate system you use.

So then how can some coordinate systems say that an object will fall through the horizon, or that an event horizon has formed

Coordinate systems don't "say" that physical events can or cannot happen. You have to look at invariants to determine that, i.e., you have to look at things that don't depend on which coordinate system you use. Some coordinate systems can't be used to determine all of those invariants because they don't cover the necessary events. SC exterior coordinates don't cover events at or below the horizon, so they can't be used to determine invariants there, such as the proper time on a clock falling inward that is at or below the horizon. But other coordinate systems *do* cover those events, so they can be used to determine invariants there. That's really all there is to it; the confusion and argument comes from people who can't let go of the idea that coordinate systems can "say" things about physical events.

as far as a remote observer's clock is concerned

As others have noted, this is not an invariant, because it depends on the coordinates; more precisely, it depends on the simultaneity convention you adopt, and simultaneity conventions are just like coordinates, they can't "say" things directly about the physics.

when time dilation approaches infinity there?

Again, the statement "time dilation approaches infinity" by itself is not sufficient, because it doesn't express an invariant. The invariant is something like "if I lower a clock down close to the horizon, let it hang static there for a while, and then bring it back up, it shows much less elapsed time than a clock that stayed distant from the hole, and the time gets shorter the closer the clock is brought to the horizon." There is no way to express an invariant like "the time dilation of a clock falling into the hole that hasn't quite reached the horizon" that makes it behave the same as a clock that goes down, is static for a while, and then comes back up.

 

[Mike Holland]

Peter, I think PAllen summed up the problem in this post from the beginning of this topic:

The confusion all starts with asking what clock readings on Bob's clock (world line) should Bob treat as corresponding various clock readings on Alice's world line. This is where you go beyond even computed physics to pure convention. If Bob uses a convention which requires getting a signal from an event in order to assign a 'Bob' time to it, Bob cannot assign any times to a portion of Alice's world line. If, on the other hand, Bob uses a different convention, allowing assignment of Bob times to events Bob can send a signal to, then Bob can assign specific, finite times to all event's on Alice's world line. SC coordinate time just happens to be an instance of the first class of simultaneity convention.

I think we agreed that the events in the manifold between our past and future light cones were candidates for "now", depending on the coordinate system we chose. But PAllen's Bob goes a bit too far in assigning his proper time to events on his future light cone, such as his photons going through an event horizon, because it assumes that the speed of light is infinite.

I know that the events I see are in the past, and I know that anyone observing me at my now will be in the future. How much in the past or future depends on relative motion and gravity fields present. The only coordinate frame that satisfys Bob's second choice is that of the emitted photon itself, where the distance traveled is Lorentz-contracted to zero and all points on the path are "now". In other words, he must be falling into the BH at the speed of light.

 

[PAllen]

Peter, I think PAllen summed up the problem in this post from the beginning of this topic:



I think we agreed that the events in the manifold between our past and future light cones were candidates for "now", depending on the coordinate system we chose. But PAllen's Bob goes a bit too far in assigning his proper time to events on his future light cone, such as his photons going through an event horizon, because it assumes that the speed of light is infinite.

Nonsense. Let's say Bob (remaining at r0) sends a signal to Alice at t0. Suppose Alice receives it at r1 < Rx.l Bob assigns a time to this reception event of t0 + (r0-r1)/C. Where is the infinite speed?

I know that the events I see are in the past, and I know that anyone observing me at my now will be in the future. How much in the past or future depends on relative motion and gravity fields present. The only coordinate frame that satisfys Bob's second choice is that of the emitted photon itself, where the distance traveled is Lorentz-contracted to zero and all points on the path are "now". In other words, he must be falling into the BH at the speed of light.

I can't even figure out what this means. Try to explain more. Note, there is no frame of a photon, so this is just nonsense. Lorentz contractions is meaningless in GR except locally.l This whole paragraph has no meaning that I can discern.

 

[PAllen]

Does this mean that time dilation is just a coordinate thing? Can those clocks up in orbit be synchronised with ours here on Earth simply by changing coordinate systems? The people responsible for maintaining GPS systems would be delighted to hear this!

Yes, definitely. Time dilation is the ratio of a particular clock's reading to a particular coordinate time assignment. In contrast, a physical measurement is something like two clocks are compared at event e1 and e2. Or one clock sends signals to another, which records reception times. These measurements are completely independent of coordinates, unlike time dilation.

 

[Mike Holland]

PAllen:
"If, on the other hand, Bob uses a different convention, allowing assignment of Bob times to events Bob can send a signal to, then Bob can assign specific, finite times to all event's on Alice's world line."
OK, I understood this to mean that if he sent a photon at 4pm then he would assign a time of 4pm to the event when it arrived there.

PAllen:
"Let's say Bob (remaining at r0) sends a signal to Alice at t0. Suppose Alice receives it at r1 < Rx.l Bob assigns a time to this reception event of t0 + (r0-r1)/C.?"

With this correction there is no infinite speed, but Bob has assigned a time to an imaginary event about which he knows nothing, and he has made unwarranted assumptions about the (constant?) speed at which his signal will travel. He has assumed there is no gravitational time dilation anywhere, which begs the whole question. So he is just assigning a time to some event in his Euclidian imagination.

 

[PeterDonis]

Bob has assigned a time to an imaginary event about which he knows nothing

He has assigned a time to an event that he calculates will happen, according to the information he has. Of course he can't guarantee that that event actually will happen just as he calculates, but so what? Any scheme that assigns time coordinates to events not in your past light cone has the same problem, if you think it's a problem.

he has made unwarranted assumptions about the (constant?) speed at which his signal will travel.

He hasn't made any assumptions about the speed of light. He has just adopted a particular rule for assigning times to events, that happens to use the speed of light as part of the assignment process. As long as such a rule doesn't assign the same time to events that could be causally connected, and doesn't assign multiple times to the same event, it will work.

He has assumed there is no gravitational time dilation anywhere

He hasn't assumed anything about "time dilation", gravitational or otherwise. Once again, he's just adopted a particular rule for assigning times to events, that meets certain minimal requirements.

which begs the whole question.

Begs what question? "Time dilation" isn't a direct observable, so there's no question to be begged.

 

[PeterDonis]

I think we agreed that the events in the manifold between our past and future light cones were candidates for "now", depending on the coordinate system we chose.

Yes, that's true. But apparently I didn't make clear enough what that tells us: it tells us that "now" is not a physically meaningful concept, because there is no coordinate-independent way to specify it.

I know that the events I see are in the past

Yes.

and I know that anyone observing me at my now will be in the future

More precisely, you know that the events at which anyone observes you at your now (which really should be your "here and now" to make it clear that we are talking about a single event, not a "now" surface of simultaneity) are in your future.

How much in the past or future depends on relative motion and gravity fields present.

How do you define "how much" in a coordinate-independent way? You can't. The only coordinate-independent notion of "how much in the past or future" we have is proper time along some particular worldline. So you can say that some particular event that you experienced directly was, say, exactly 24 hours before your "here and now" according to your clock, or you can predict that some particular event you will experience directly in the future will occur, say, exactly 24 hours after your "here and now" by your clock.

But if you are looking at light from the Sun, say, you can't make any coordinate-independent claims about "how long ago" that light was emitted. If you say it was emitted 500 seconds ago, what you really mean, whether you realize it or not, is 500 seconds ago in some convenient set of coordinates in which the Sun is 500 light seconds away from you. There's no way to eliminate a coordinate choice from such statements.

 

[harrylin]

[..] what you're saying makes no sense.
[..]
You think that the discussions of Rindler coordinates or Zeno paradox are philosophical? I don't consider them philosophical, they are pointing out a mathematical error that you seem to be making, which is assuming that any event that does not have a "t" label must never happen. [..] The discussions become philosophical because you respond to a mathematical claim as if it were a philosophical claim.

Your misinterpretation of what I said makes no sense indeed; and in earlier discussions I clarified that I don't assume that any event that does not have a "t" label must never happen. Of course mathematics, physics and philosophy have overlap but here I focus on the physics:

The claims are not contradictory because they are referring to different things. The first claim refers to the fact that the limit of the Schwarzschild t coordinate goes to infinity as the object crosses the horizon. The second claim refers to the fact that the coordinate time in other systems is finite as the object crosses the horizon. Since they are referring to coordinates of different coordinate systems there is no contradiction. [..]

I don't think that anyone has an issue with different coordinate systems! I got drawn into this topic of black holes because of what appears to be an issue about physical interpretation of Schwarzschild's coordinate system, which is obviously also the topic of this thread. The nature of the "infinite" fall according to that system is nicely animated with http://www.compadre.org/osp/items/detail.cfm?ID=7232 and I also have no problem with that (move the red dot to get it into a fall).

[..] assuming that we have a valid time coordinate and assuming that the coordinate system is well defined along some timelike worldline, then it is always possible to transform to a closely related coordinate system where the coordinate time matches the proper time along that worldline, but the hypersurfaces of simultaneity are unchanged. [..] whether or not an object falls across the EH according to a distant observer is simply a matter of convention.

That sounds reasonable to me, and then no informed person would object if someone else says that a crossing of a black hole horizon never happened; if I'm not mistaken that's simply according to our standard time convention on Earth (perhaps the ECI frame is based on Schwarzschild). However, it remains paradoxical to me, in the sense that in my experience such things always led to contradictions. Different from SR's relativity of simultaneity it seems to imply a possible disagreement if events will occur or not; however it turned out that the simplest examples lead to philosophical discussions, so that a more specific example would be needed. And in the context of this topic, there may be an issue of consistent physical interpretation: there has been talk of an "internal Schwarzschild solution" according to which an object crosses the horizon, and which is combined with it as a single solution. Is that correct? So far I don't understand how such a combined solution can give a consistent physical interpretation of events.

 

[Mentz114]

That sounds reasonable to me, and then no informed person would object if someone else says that a crossing of a black hole horizon never happened; if I'm not mistaken that's simply according to our standard time convention on Earth (perhaps the ECI frame is based on Schwarzschild). However, it remains paradoxical to me, in the sense that in my experience such things always led to contradictions. Different from SR's relativity of simultaneity it seems to imply a possible disagreement if events will occur or not; however it turned out that the simplest examples lead to philosophical discussions, so that a more specific example would be needed. And in the context of this topic, there may be an issue of consistent physical interpretation: there has been talk of an "internal Schwarzschild solution" according to which an object crosses the horizon, and which is combined with it as a single solution. Is that correct? So far I don't understand how such a combined solution can give a consistent physical interpretation of events.

Maybe it will help you to think about worldlines. Do you agree that the worldline of a body is a coordinate independent fact ? It obviously is so, because if something goes from a to b then this cannot be undone by any coordinate change. We can easily find worldlines which intersect the EH of a BH, with no discontinuities in their local clock time. The fact that some observer, not present at an event assigns a different ( possibly infinite) time to this event, or is unable to assign one, does not lead to any causal paradoxes.

As for coordinates that cover all the SC regions, the KS coordinates have been explained to you already.

 

[harrylin]

[..] it doesn't help in picking out which observer is privileged in each (Adam vs. Eve, Eve' vs. Adam') [..] [rearrange] But [..]

I don't understand that question as we discussed that right at the start, and neither of us had a problem there... And sorry, it's really useless to repeat disagreeing viewpoints. However, it appears that there is something unclear to you about identifying Adam from Eve according to the other viewpoint, although I don't understand what that may be. Please send me an email if you want to discuss that further.

[..] 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.

That rescaling is done accounting for SR time dilation which is not included in Schwarzschild. However little correction is necessary for the ECI frame itself; indeed I was more thinking of how GPS operates.

If we look at the Solar System as a whole, the global frame is a Schwarzschild frame centered on the Sun. [..]

Obviously.
Addendum - As you asked about another post of yours:

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.

You did not understand my question which was not about the kind of predictions that you mention next:

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?

I made those same calculations which are not an issue. Different people preach different interpretations of GR on this forum (and that merely reflects the literature) - for example, according to Dalespam (post #150) it is a matter of convention if an object falls across the EH while according to you objects can cross the horizon. Similar PAllen in "notions of simultaneity" thread (post #55). Their version sounds to me more in line with Einstein's own notion of GR. It would be counter productive to debate differing views; what may be useful is to discuss the effects of models on physical description.

 

[stevendaryl]

Your misinterpretation of what I said makes no sense indeed; and in earlier discussions I clarified that I don't assume that any event that does not have a "t" label must never happen.

Well, can you give an unambiguous claim or question? I thought you were saying that in the case of a black hole, there was a contradiction between the description given by a distant observer and the description given by an infalling observer. There is no contradiction UNLESS you assume that "any event that does not have a t label must never happen".

I don't think that anyone has an issue with different coordinate systems! I got drawn into this topic of black holes because of what appears to be an issue about physical interpretation of Schwarzschild's coordinate system, which is obviously also the topic of this thread. The nature of the "infinite" fall according to that system is nicely animated with http://www.compadre.org/osp/items/detail.cfm?ID=7232 and I also have no problem with that (move the red dot to get it into a fall).

That sounds reasonable to me, and then no informed person would object if someone else says that a crossing of a black hole horizon never happened;

No informed person would make that claim about GR. Now, there could be other theories that reduce to GR in most cases, that make different predictions about black holes. That's very likely to be the case when and if we develop a quantum theory of gravity.

if I'm not mistaken that's simply according to our standard time convention on Earth (perhaps the ECI frame is based on Schwarzschild). However, it remains paradoxical to me, in the sense that in my experience such things always led to contradictions.

What is it that you think is paradoxical? You say that you don't want to talk about philosophy, that you want to talk about physics, but your questions and comments never are specific enough to be physics questions.

Different from SR's relativity of simultaneity it seems to imply a possible disagreement if events will occur or not;

There is no disagreement UNLESS you make the (incorrect) assumption that if an event doesn't have a "t" label, then it does not happen. You're denying making that assumption, but without that assumption, why do you think that the event of falling through the horizon does not occur?

however it turned out that the simplest examples lead to philosophical discussions,

There would be no need for a philosophical discussion if the question was one of physics and mathematics. Whether the Schwarzschild coordinates cover the entire manifold is not a philosophical question, it's a mathematical question. And the answer is no.

so that a more specific example would be needed. And in the context of this topic, there may be an issue of consistent physical interpretation:

It would be helpful for you to say what you think the inconsistency is. Only then is it possible to worry about how to make things consistent. You haven't said what you think the inconsistency is. I THOUGHT you were saying that the inconsistency was:

• According to Schwarzschild coordinates, the infalling observer never crosses the event horizon.
• According to freefall coordinates, the infalling observer does cross the event horizon.

But the first claim is NOT true. Schwarzschild coordinates don't say "the infalling observer never crosses the event horizon"; they say something subtler: "The event of the infalling observer crossing the event horizon is not covered by the chart." Those two are NOT the same thing.

there has been talk of an "internal Schwarzschild solution" according to which an object crosses the horizon, and which is combined with it as a single solution. Is that correct? So far I don't understand how such a combined solution can give a consistent physical interpretation of events.

What is inconsistent about the description in terms of KS coordinates? The KS coordinates in turn agree completely with Schwarzschild coordinates in the region r &gt; 0 and -\infty &lt; t &lt; +\infty. So where do you think there is an inconsistency?

 

[Dale]

Does this mean that time dilation is just a coordinate thing?

Yes, time dilation itself is a coordinate thing. It is the ratio between a clock's proper time and the local coordinate time. Since one part of the ratio is a coordinate time and the other part is an invariant the ratio itself is coordinate dependent. This is true in SR as well as GR.

As Peter Donis mentioned, there are some closely related invariants that can be calculated from time dilation, but they are not themselves time dilation. E.g. the ratio of the proper times along two paths (twin paradox) can be calculated by the ratio of the time dilation along the paths.

Can those clocks up in orbit be synchronised with ours here on Earth simply by changing coordinate systems? The people responsible for maintaining GPS systems would be delighted to hear this!

Yes, and they already know it. That is what they do. They transform the coordinate system of the satellite clocks from their own MCIRF to the ECIRF.

The different coordinate systems provide different ways of assigning times to remote events. But when we send a clock up into orbit and bring it back, it is in advance of our dilated earth-bound clock. This is a direct result of the time dilation as is predicted by GR and has been verified.

It is not a direct result of time dilation, it is the ratio of two time dilations (the orbit clock and the Earth clock). That ratio is invariant.

At what point do these coordinate systems give different results, and which will be in accord with our measurements?

Different coordinate systems never give different results for any measurement. That is the essence of the principle of general covariance. Mathematically, any measurement is an invariant quantity, not a coordinate-dependent one.

 

[stevendaryl]

Different people preach different interpretations of GR on this forum (and that merely reflects the literature) - for example, according to Dalespam (post #150) it is a matter of convention if an object falls across the EH while according to you objects can cross the horizon.

I think you misinterpreted what Dalespam said. I think he was talking about how things look from the point of view of the distant observer. If the distant observer uses one coordinate system, then the crossing takes an infinite amount of time, and if he uses a different coordinate system, the crossing takes a finite amount of time. "Takes an infinite amount of time" doesn't mean "Never happens". Rindler coordinates clearly show that these are not the same. I don't know why you accept the conclusion for Rindler coordinates (or for the Zeno coordinates), but reject it for Schwarzschild coordinates.

 

[Dale]

I got drawn into this topic of black holes because of what appears to be an issue about physical interpretation of Schwarzschild's coordinate system

That is easy. There is NO physical interpretation of ANY coordinate system (incl. SC and all of the other coordinate systems that we have discussed); what has physical interpretations are the invariants.

The purpose of any coordinate system is simply to make calculations possible or even easy. In some coordinate systems the calculation of specific invariants becomes particularly easy, but even then it is the physical invariants which are easily calculated from the coordinates which have a physical interpretation, not the coordinates themselves.

That sounds reasonable to me, and then no informed person would object if someone else says that a crossing of a black hole horizon never happened; if I'm not mistaken that's simply according to our standard time convention on Earth (perhaps the ECI frame is based on Schwarzschild).

Yes, as long as it is clear that "never" is a coordinate-dependent statement meaning "not at any finite SC coordinate time". Unfortunately, that is rarely clear.

However, it remains paradoxical to me, in the sense that in my experience such things always led to contradictions. Different from SR's relativity of simultaneity it seems to imply a possible disagreement if events will occur or not

This is why I recommend that you read the Carroll's lecture notes. It seems to me that you don't yet understand the basic relationship between coordinate charts and manifolds. Essentially, you seem to not get the fact that a coordinate chart need not cover the entire manifold, and indeed, some simple manifolds are impossible to cover in a single chart (see p. 38). Whether or not a specific chart covers a given point has nothing to do with whether or not that point is in the manifold. The numerous examples of Rindler and Zeno coordinates in flat spacetime should make that abundantly clear.

And in the context of this topic, there may be an issue of consistent physical interpretation: there has been talk of an "internal Schwarzschild solution" according to which an object crosses the horizon, and which is combined with it as a single solution. Is that correct? So far I don't understand how such a combined solution can give a consistent physical interpretation of events.

Please read the Carroll notes again. You cannot have understood that chapter and not understand this concept.

 

[Dale]

This implies that whether or not an object falls across the EH according to a distant observer is simply a matter of convention.

for example, according to Dalespam (post #150) it is a matter of convention if an object falls across the EH while according to you objects can cross the horizon.

I feel like my name is being taken in vain

Obviously, if it is a matter of convention whether or not it does fall through then there are conventions in which it does fall through. If there is a convention where it does fall through, then obviously it can fall through. So the two statements are not contradictory.

Btw, I had hoped from the context that it was clear that I was talking about whether or not it happened in a finite amount of coordinate time for the different conventions. I do apologize for any lack of clarity. This highlights the importance of the math over the English.

Again, the difficulty lies in the translation of the English word "never" into math. If by "never" you mean "not in any finite SC time" then it is true that it never crosses. If by "never" you mean "not at any event in the manifold" then it is false that it never crosses.

I think you misinterpreted what Dalespam said. I think he was talking about how things look from the point of view of the distant observer. If the distant observer uses one coordinate system, then the crossing takes an infinite amount of time, and if he uses a different coordinate system, the crossing takes a finite amount of time. "Takes an infinite amount of time" doesn't mean "Never happens". Rindler coordinates clearly show that these are not the same. I don't know why you accept the conclusion for Rindler coordinates (or for the Zeno coordinates), but reject it for Schwarzschild coordinates.

Yes, a better way to say it.

 

[harrylin]

I think you misinterpreted what Dalespam said. I think he was talking about how things look from the point of view of the distant observer. If the distant observer uses one coordinate system, then the crossing takes an infinite amount of time, and if he uses a different coordinate system, the crossing takes a finite amount of time. [...]

Sorry for the ambiguity, but that is indeed how I understood it - this whole discussion 'On the nature of the "infinite" fall toward the EH' is obviously meant from the distant Schwarzschild observer's point of view.
[addendum

[..] Obviously, if it is a matter of convention whether or not it does fall through then there are conventions in which it does fall through. If there is a convention where it does fall through, then obviously it can fall through. So the two statements are not contradictory.

I understood him to mean that it will fall through according to us - he did not write that, but that is what he seemed to be arguing for several threads. If that is not the case, then some of my discussion with him was based on miscommunication.

Btw, I had hoped from the context that it was clear that I was talking about whether or not it happened in a finite amount of coordinate time for the different conventions. I do apologize for any lack of clarity. This highlights the importance of the math over the English.

That was, as I stated earlier, very clear (at least to me!) ]

Again, the difficulty lies in the translation of the English word "never" into math. [..]

Yes, exactly - as I also remarked.

That is easy. There is NO physical interpretation of ANY coordinate system (incl. SC and all of the other coordinate systems that we have discussed); what has physical interpretations are the invariants.

The purpose of any coordinate system is simply to make calculations possible or even easy. In some coordinate systems the calculation of specific invariants becomes particularly easy, but even then it is the physical invariants which are easily calculated from the coordinates which have a physical interpretation, not the coordinates themselves.

Of course coordinates are not absolutes, but isn't that easy solution perhaps a little too easy? For example, gravitational time dilation follows from Schwarzschild's system, and that is normal is it already appeared during the development of GR. Thus for example Moller finds dτ/dt= √(1+2g/c² - v²/c²) and such findings are what Einstein called a "physical interpretations". Similarly the Schwarzschild simulation to which I linked could be called a physical interpretation.

Yes, as long as it is clear that "never" is a coordinate-dependent statement meaning "not at any finite SC coordinate time". Unfortunately, that is rarely clear.

I think that most people who start questions do understand that; but only a poll would tell us.

This is why I recommend that you read the Carroll's lecture notes. It seems to me that you don't yet understand the basic relationship between coordinate charts and manifolds. Essentially, you seem to not get the fact that a coordinate chart need not cover the entire manifold, and indeed, some simple manifolds are impossible to cover in a single chart (see p. 38). [..]

I did read and understand that, which doesn't mean that I agree with his way of thinking*; do you mean that the two parts indeed don't provide a consistent physical description?

*there turn out to be different interpretations of GR just as there are of SR: think of block universe discussions

 

[PAllen]

Of course coordinates are not absolutes, but isn't that easy solution perhaps a little too easy? For example, gravitational time dilation follows from Schwarzschild's system, and that is normal is it already appeared during the development of GR. Thus for example Moller finds dτ/dt= √(1+2g/c² - v²/c²) and such findings are what Einstein called a "physical interpretations". Similarly the Schwarzschild simulation to which I linked could be called a physical interpretation.

Actually, gravitational time dilation is a coordinate dependent convenience for calculating physical observations:

1) It is possible to define at all only for sufficiently static spacetimes such that you can introduce coordinates where the metric components are essentially unchanging with time (thus depend only on positions). This is a very special class of spacetimes. Even for these, just like any other time dilation (strictly coordinate dependent), you simply have relation between a particular class of clock (clock on static world line) and coordinate time.

2) It is never necessary to use gravitational time dilation to compute any observation. To compute clock comparison for two clocks that synchronize, separate, and come together, you just integrate proper time along their paths. This is the universal way to compute this physical observation. It works for non-static gravitational situations (e.g. the vicinity of inspiralling neutron stars, where gravitational time dilation is undefinable), and it works for coordinates for a static situation that don't manifest the static character (e.g. the extension of the local frame of free fall observer).

3) To compute redshift between an emitter and a receiver, the universal method (works for every case in SR, GR, however general) is to parallel transport the emitter 4-velocity along the null path to the receiver, and use pure SR formula for Doppler based on transported emitter velocity and null path propagation vector expressed in local frame of receiver.

So, yes, gravitation time dilation, like all forms of time dilation is a coordinate convention. The observations (2) and (3) above are independent of how different coordinate conventions differently manifest time dilation.

 

[PeterDonis]

Obviously, if it is a matter of convention whether or not it does fall through then there are conventions in which it does fall through. If there is a convention where it does fall through, then obviously it can fall through.

I agree with you on the physics, but I actually have to admit that I too find this terminology confusing. It seems to imply something that I know you didn't mean, that what is "real" is a matter of convention. IMO it would be better to say something like: the infalling object *does* fall through the horizon, but once it reaches the horizon it can't send light signals back out to the distant observer.

Then, if we wanted to talk about coordinates or simultaneity conventions, we could say that the distant observer's most natural simultaneity convention can't assign finite time coordinates to any event at or below the horizon. But even that is causing confusion; maybe it would be better to leave out coordinates altogether and insist on talking only about invariants, at least in discussions like this where the old Army axiom seems to be in full force: any statement which *can* be misinterpreted, *will* be misinterpreted.

This highlights the importance of the math over the English.

Definitely.

Again, the difficulty lies in the translation of the English word "never" into math.

Yes, and part of the problem is that many people do interpret "never" as being ontological, something that can't be a matter of convention, whereas as you put it here "never" *is* a matter of convention. The math, as you say, is unambiguous.

 

[PeterDonis]

I think that most people who start questions do understand that

I would strongly disagree; if people who start these questions understood that "never" as used in this context is coordinate-dependent, threads like this one would be over in two or three posts. You yourself give an example:

do you mean that the two parts indeed don't provide a consistent physical description?

Of course not; they do. If you understood that "never" is coordinate-dependent, you wouldn't even need to ask this question. The answer would be obvious.

 

[stevendaryl]

I understood him to mean that it will fall through according to us - he did not write that, but that is what he seemed to be arguing for several threads.

There is no "according to us". It's according to this or that coordinate system. As people have pointed out to you many times, a coordinate system only describes what happens in a patch, a region of spacetime. The fact that some event isn't included in a coordinate patch doesn't imply that the event never happens. I think you understand this well enough when it comes to Rindler coordinates or Zeno coordinates. But for some reason, you don't think it relevant when it comes to Schwarzschild coordinates. Why not?

Of course coordinates are not absolutes, but isn't that easy solution perhaps a little too easy? For example, gravitational time dilation follows from Schwarzschild's system, and that is normal is it already appeared during the development of GR. Thus for example Moller finds dτ/dt= √(1+2g/c² - v²/c²) and such findings are what Einstein called a "physical interpretations".

But t has NO direct physical meaning. It's chosen to make the metric look as simple as possible. If you use a different time coordinate, you would get a different "gravitational time dilation" formula relating proper time to coordinate time. In particular, the claim that "an infinite time is required for an infalling observer to reach the event horizon" is NOT a physically meaningful statement. It's true according to some coordinate systems, and false according to other coordinate systems.

 

[harrylin]

[..] Of course not; they do. If you understood that "never" is coordinate-dependent, you wouldn't even need to ask this question. The answer would be obvious.

As before, it seems that you misunderstood my question which doesn't relate to a time label but to what Einstein called physical interpretation, which I here called physical description, and which PAllen called "differently manifest".

 

[harrylin]

There is no "according to us". It's according to this or that coordinate system. [..]

We discussed that our common conventional time is that of the ECi frame, and that we are allowed to use it.

[..] It's true according to some coordinate systems, and false according to other coordinate systems.

Exactly, that is what we elaborated here.

 

[harrylin]

Combining the earlier mentioned opinions:

This is already a statement whose meaning is rejected by relativity. [..] 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.

. [..] it doesn't make SC uniquely represent the viewpoint of a distant observer. This implies that whether or not an object falls across the EH according to a distant observer is simply a matter of convention.

[..] Obviously, if it is a matter of convention whether or not it does fall through then there are conventions in which it does fall through. If there is a convention where it does fall through, then obviously it can fall through. So the two statements are not contradictory. [..] I was talking about whether or not it happened in a finite amount of coordinate time for the different conventions [from the point of view of the distant observer.]

[..] IMO it would be better to say something like: the infalling object *does* fall through the horizon [..]

I take it that you mean, as implied by the topic of this thread, from the point of view of the distant observer. If so, then I understood you correctly, and Dalespam misunderstood you.

So, once more: doing what you do is making a metaphysical claim that pertains to what we cannot measure - it is your physical interpretation. Others hold that GR does not make such physical claims except for the predictions of events. If an event at (x,y,z,t) has taken place according to us (or if it will take place according to our reckoning) is a matter of convention. So, while I agree with your desire to want to establish a "physical reality", the situation looks, as I said earlier today, very similar to SR in which different interpretations of physical reality are possible - and threads that are not ended voluntarily by the participants are locked when the Mentors have enough of it (of course information exchange about such ideas can be helpful; but that has been done both there and here).
It appears that the best we can do is describe the interpretations/decriptions that relate to different coordinate systems. That is constructive and helpful; philosophy debates of the block universe kind are not.

 

[PeterDonis]

I take it that you mean, as implied by the topic of this thread, from the point of view of the distant observer.

No, I meant "in reality". You can't change reality by changing point of view. So since it seems like all this talk about different points of view is only increasing the confusion, I'm no longer talking about that. I'm talking about invariant properties of reality. It is an invariant property of the reality predicted by the Schwarzschild solution of the Einstein Field Equation that objects do reach and fall through the horizon.

So, once more: doing what you do is making a metaphysical claim that pertains to what we cannot measure - it is your physical interpretation.

No, it's a physical *prediction* made by GR, and more specifically by the Schwarzschild solution of the EFE. (By which I mean not "the solution Schwarzschild published" or "what Schwarzschild believed", or "what Einstein believed about what Schwarzschild wrote", but the best modern understanding of the solution of the EFE for a spherically symmetric vacuum spacetime.) If you're going to contest that prediction, you will be claiming that the prediction doesn't match actual reality; it's not a matter of "interpretation".

How might the prediction not match reality? I can see a couple of possible ways:

(1) The EFE is wrong; or at least, it's wrong sometimes (for example, it's wrong if it predicts an event horizon). This doesn't strike me as a promising approach, and as far as I know nobody has tried to argue for it.

(2) The Schwarzschild solution isn't applicable to actual collapsed objects, because when you get close enough to forming a horizon, either the collapse stops, or the stress-energy tensor changes, in such a way that the horizon never actually forms. In other words, no real spacetime is ever vacuum down to the horizon radius; it always ends up becoming non-vacuum at some larger radius than this. This could be because:

(2a) Some classical effect intervenes: for example, there is always enough pressure present, or enough mass gets radiated away, etc., so that a horizon never actually forms; instead the collapse stops at some other stable state. This doesn't seem promising either: all the stable states of matter that we know of other than black holes have maximum mass limits, and we know of black hole candidates whose observed masses are many orders of magnitude larger than those limits.

(2b) Some quantum effect intervenes: for example, in some thread or other we have discussed possible quantum effects that could change the stress-energy tensor close enough to a horizon, so that a horizon was prevented from forming. This hasn't been ruled out, but does not appear to be promising.

Now: where in any of the above did I talk about coordinates or "interpretations"? Nowhere. I just talked about physical predictions and what sorts of physical phenomena might affect how accurate they are.

(Note: I was only talking about the horizon above, not the singularity at r = 0; that's a different can of worms. I also wasn't talking about Hawking radiation, because whether a horizon eventually evaporates away is a different question from whether it forms in the first place. If a horizon forms, then there will be some possibility of objects falling through it.)

Others claim that GR does not make such claims except for the predictions of events.

The statement that objects fall below the horizon *is* a "prediction of events".

And if an event at (x,y,z,t) has taken place according to us, is a matter of convention.

See my comments to DaleSpam on the use of the word "convention".

 

[harrylin]

No, I meant "in reality". [..]

Yes, exactly - as I specified next. I correctly understood that you were making claims about physical reality according to a distant observer. Such discussions are similar to those started by bob2c and Vandam about SR. And I clarified why I stopped participating in such discussions.

 

[Dale]

I understood him to mean that it will fall through according to us

What do you mean by "according to us"?

Of course coordinates are not absolutes, but isn't that easy solution perhaps a little too easy? For example, gravitational time dilation follows from Schwarzschild's system, and that is normal is it already appeared during the development of GR. Thus for example Moller finds dτ/dt= √(1+2g/c² - v²/c²)

No, the easy solution is not too easy, it is exactly right. What you have just demonstrated is that time dilation is a coordinate effect. Specifically dτ is coordinate independent, but dt is coordinate dependent.

I think that most people who start questions do understand that; but only a poll would tell us.

I did read and understand that, which doesn't mean that I agree with his way of thinking*;

I don't think they do, and despite your claims here I don't think that you understand either. For instance, you refer to an interpretation of GR, but that section of the lecture notes isn't even about GR. It is simply about the math of Riemannian manifolds, tensors, and coordinate charts.

Could you identify what specifically you disagree with or are having trouble with?

do you mean that the two parts indeed don't provide a consistent physical description?

Which two parts of what don't provide a consistent physical description of what?

 

[harrylin]

What do you mean by "according to us"?

We are distant observers of black holes.

[..] time dilation is a coordinate effect. Specifically dτ is coordinate independent, but dt is coordinate dependent.

Probably you meant that time dilation is coordinate dependent; a coordinate effect is no physical effect at all - in which case no prediction of a measurable effect such as gravitational time dilation of clocks could be detected.

[..] that section of the lecture notes isn't even about GR. It is simply about the math of Riemannian manifolds, tensors, and coordinate charts. Could you identify what specifically you disagree with or are having trouble with?

He starts basically the same discussion as I started in the Flowing space thread, with roughly a contrary suggestion: according to him "We therefore see the necessity of charts and atlases: many manifolds cannot be covered with a single coordinate system." However he next admits that that isn't necessarily necessary, as follows: "(Although some can, even ones with nontrivial topology." His suggestion that we need different maps that next must be linked together looks contrarian to Einstein's non-rigid measuring bodies of reference.

Which two parts of what don't provide a consistent physical description of what?

The "inner" and "outer" Schwarzschild solutions, do they - as PAllen phrased it - "manifest" such things as time dilation, speed of light, speed of particle, energy and momentum conservation etc. consistently around the Schwarzschild radius from a distant perspective?

 

[stevendaryl]

As before, it seems that you misunderstood my question which doesn't relate to a time label but to what Einstein called physical interpretation, which I here called physical description, and which PAllen called "differently manifest".

Is it possible for you to either ask a precise question, or make a precise statement? When people try to guess what you are talking about, you say that they are wrong, but you don't actually clarify what you really do mean.

So, your reason for believing that an infalling observer never crosses the horizon isn't because the Schwarzschild coordinate t goes to infinity? Then what reason do you have for believing that?

 

[stevendaryl]

Yes, exactly - as I specified next. I correctly understood that you were making claims about physical reality according to a distant observer.

Physical reality doesn't have a modifier "according to a distant observer". I think what you mean is "as described by the coordinate system of the distant observer". But that doesn't uniquely say anything, either, because the distant observer can use different coordinate systems.

 

[PeterDonis]

I correctly understood that you were making claims about physical reality according to a distant observer.

If that's what you understood, you understood incorrectly. I was making claims about physical reality, period. Reality is not "according to" any observer. It's just reality.

More precisely, as I clarified in my last post, I was making claims about a *prediction* of what physical reality is like. That prediction may turn out to be wrong, but if it is, it won't be because somebody's "interpretation" was wrong. It will be because the prediction was based on incomplete knowledge, as all predictions are. I even gave specific ways in which our knowledge might be incomplete.

Such discussions are similar to those started by bob2c and Vandam about SR.

I'm not sure I agree. In those discussions, everybody agreed on the global spacetime model that we were working with, and on all invariant quantities within that model. I don't see a similar agreement in this discussion.

 

[Dale]

I agree with you on the physics, but I actually have to admit that I too find this terminology confusing. It seems to imply something that I know you didn't mean, that what is "real" is a matter of convention. IMO it would be better to say something like: the infalling object *does* fall through the horizon, but once it reaches the horizon it can't send light signals back out to the distant observer.

Sorry about that. If even someone with your background finds the terminology confusing then I should avoid it in the future.

 

[Dale]

We are distant observers of black holes.

Yes, I am aware of that. But it doesn't clarify enough to answer the question. You need to specify which coordinate system you are referring to by "according to us", or if you are referring to invariants instead such as local observations.

according to him "We therefore see the necessity of charts and atlases: many manifolds cannot be covered with a single coordinate system." However he next admits that that isn't necessarily necessary, as follows: "(Although some can, even ones with nontrivial topology."

Yes, some manifolds cannot be covered with a single coordinate chart, he even provides an example on p 38 of a simple manifold which cannot be covered with a single chart. That in no way contradicts the fact that some manifolds can be covered in a single coordinate chart, i.e. some manifolds do require multiple charts, some manifolds do not.

It appears to me that you do not understand this point if you think that there is any discrepancy in these statements.

Also, even if a manifold can be covered by a single chart, surely you must admit that we may want to consider multiple charts, e.g. Cartesian and spherical. Do you have any objection to that?

His suggestion that we need different maps that next must be linked together looks contrarian to Einstein's non-rigid measuring bodies of reference.

Why? What seems contrarian to you in that? Again, this has nothing to do with GR at this point, it is just the mathematical framework.

Consider the manifold of a sphere. Suppose that you use non-rigid rulers to lay out a traditional lattitude and longitude grid on the sphere. If you do that, then in order for your resulting system to have the mathematical properties that Carroll identifies then you must exclude one longitude line from pole to pole. So you will require at least two such charts to cover the entire sphere. So the use of non-rigid bodies of reference does not imply that a single chart is possible.

The "inner" and "outer" Schwarzschild solutions, do they - as PAllen phrased it - "manifest" such things as time dilation, speed of light, speed of particle, energy and momentum conservation etc. consistently around the Schwarzschild radius from a distant perspective?

They don't overlap anywhere, so it is hard to see how they could either agree or disagree about anything.

 

[Nugatory]

Such discussions are similar to those started by bob2c and Vandam about SR.

I'm not sure I agree. In those discussions, everybody agreed on the global spacetime model that we were working with, and on all invariant quantities within that model. I don't see a similar agreement in this discussion.

And I am quite sure that I don't agree, for exactly PeterDonis's reason...

But that does suggest yet another way forward. Maybe we could try describing some invariant properties, without regard to ANY coordinate system, see if we agree about them.

We have two worldlines, one corresponding to Lucky who isn't falling into the black hole and one corresponding to Unlucky who is. Lucky and Unlucky both carry wristwatches which are recording their proper time along these worldlines; both Lucky and Unlucky will agree about what Lucky's watch reads at any point on Lucky's worldline, and about what Unlucky's watch reads at any point on Unlucky's worldline (one of them has to calculate it, while the other can just look at his wrist, but they'll get the same result). There is a point on Unlucky's worldline such that at and beyond that point, there is no point on the worldline from which a forward-going null geodesic can be drawn that will intersect Lucky's worldline; and for all points before that point, such a geodesic can be drawn. However, for all points on Lucky's worldline, I can draw a forward-going null geodesic that will intersect Unlucky's worldline. And finally... Unlucky's worldline terminates at the central singularity.

OK, do I have the coordinate-independent description of the physics right? (It's already been pointed out a number of times, correctly, that it's hard to be precise and accurate without the math). And if I do, then Harrylin is this the the physical situation as you understand it?

 

[Dale]

for all points on Lucky's worldline, I can draw a forward-going null geodesic that will intersect Unlucky's worldline. And finally... Unlucky's worldline terminates at the central singularity.

Some points on Lucky's worldline will have inward-going radial null geodesics that reach the singularity after Unlucky's worldline has already intersected it. But other than that minor detail everything else sounds right to me.

 

[JesseM]

There are a few reasons, but the simplest one is that if Bob calculates that Alice "never" crosses the EH, and can witness in a finite time the dissipation of the BH, then I'm having a problem accepting that Alice would ever get an opportunity to cross the EH regardless of what SC analysis shows her experience to be.

But Bob only calculates this in Schwarzschild coordinates, it's just a quirk of how that coordinate system is constructed. Even in flat spacetime it's trivial to construct coordinate systems where it takes an infinite coordinate time for someone to cross a horizon that they do cross in finite proper time, and finite coordinate time in inertial coordinates; for example, look at the Rindler horizon in Rindler coordinates. And we could construct even more trivial examples--for instance, if x and t represent coordinates in some inertial frame, then define a new coordinate system where x'=x but t'=(c * t^2)/x, in this coordinate system any event at x=0 and finite t in the original system will have an infinite t' coordinate, so anything approaching x=0 will take an infinite coordinate time to get there. Presumably you don't think this has any real physical significance, why take Schwarzschild coordinates any more seriously?

 

[bobc2]

Spot on, JesseM. And where have you been all these years since the old Michio Kaku forum days? We need Sunfist here to keep people stirred up. I wonder if Yukki is doing string theory math.

 

[Nugatory]

Some points on Lucky's worldline will have inward-going radial null geodesics that reach the singularity after Unlucky's worldline has already intersected it.

Hmmm... show me one? I'm asking, not arguing here.

[above is an edit... I initially responded "yes of course" or some such, then started thinking more about it]

 

[PeterDonis]

Hmmm... show me one? I'm asking, not arguing here.

[above is an edit... I initially responded "yes of course" or some such, then started thinking more about it]

Your first response was correct.

The easiest way to see it is to look at Lucky's and Unlucky's worldlines on a Kruskal chart. Take the point where Unlucky's worldline hits the singularity, and draw a 45 degree line down and to the right from it; this is the worldline of an ingoing light ray that hits Unlucky's worldline at the instant that worldline hits the singularity. The point where that 45 degree line intersects Lucky's worldline (which is a hyperbola in the right wedge of the diagram) is the last point on Lucky's worldline that can send an ingoing light signal to Unlucky. Any light signal Lucky sends after that will hit the singularity, not Unlucky's worldline.