On the nature of the infinite fall toward the EH

[pervect]

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

It's not really the proper time of a clock at infinity - it's still a coordinate time. I'd describe it as the coordinate time of a static observer, with the coordinate clocks normalized to run at the same rate as proper clocks at infinity.

It seems rather strange to me to ignore the readings of actual, physical clocks (proper time) in favor of some abstract coordinate time, but it seems all-too-common. My speculation is that this is based on a desire for the "absolute time" of Newtonian physics.

[add]
Static observers do have _some_ physical significance where they exist , which is outside the event horizon. This significance is derived mostly form the Killing vector field of their timelike worldlines. The Killing vector still exists at the event horizon, but it's null, so it doesn't represent any sort of "observer".

The coordinate system of static observers, where they exist, has about the same relevance to an infalling observer as the coordinate system of some "stationary" frame to somoene rapidly moving. Which in my opinion is "not very much". But I suppose opinions could vary on this point, it's not terribly critical.

The biggest difference here, and another significant underlying issue, is that static observers cease to exist at the event horizon. This makes their coordinates there problematic, as you're trying to defie a coordinate system for an observer that doesn't exist anymore. This isn't any sort of breakdown in physics - it's a breakdown of the concept of static observers.

For any actual physical observer, the horizon will always be approaching them at "c" - because any physical observer will have a timelike worldline, and the horizon is a null surface. This isn't really very compatabile with the event horizon as a "place". This is why space-time diagrams that represent the event horizon as a null surface (such as the Kruskal or penrose diagram) are a good aid to understanding the physics there, and why Schwarzschild coordinates are not.

Another sub-issue (of many) is the absolute refusal of certain posters to even consider any other coordinate systems other than Schwarzschild as having any relevance to the physics. Which gives rise to severe problems, as Schwarzschild coordinates are ill-behaved at the event horizon, for the reasons I've previously aluded to (the non-existence of static observers upon which the coordinate system is based).

This ill behavior is hardly any secret - pretty much ANY textbook is going to tell you the same thing.

 

[Dale]

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

Coordinate time always represents a simultaneity convention, which is arbitrary by definition. I.e. The way that readings on different clocks are compared is arbitrary. In the case of SC the simultaneity convention is additionally labeled to correspond with the rate of a distant clock. So the coordinate time in SC is not just proper time of that distant clock, it also necessarily involves the arbitrary simultaneity convention.

We can always do the same thing with Zeno time by judicious choice of our reference clock and our simultaneity convention. For instance, we can use a Rindler-like simultaneity convention. As long as our reference clock asymptotically approaches the worldline of the light pulse from the arrow reaching the target then that event will be at infinite coordinate time. By varying the acceleration of the reference clock we can adjust the spacing of the time coordinate between the other points on the arrows path.

 

[stevendaryl]

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

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

Thanks, I changed it.

 

[stevendaryl]

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

The relationship between proper time $\tau$ and Schwarzschild coordinate $t$ for a clock at rest in the Schwarzschild coordinates is:

$dt = d\tau/\sqrt{1-R_s/r}$

I don't immediately see any simple physical interpretation for $dt$ at finite values of $r$.

 

[harrylin]

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

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

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

It sounds as if you want to hear my opinion about how convincing that illustration may be for your arguments about the nature of Schwarzschild's physics. I won't let myself be pulled again in arguments, but will give minimal advice. t coordinates represent of course clocks (literal or virtual) and together with space coordinates they allow to calculate for example the speed of a runner or of light between different points. So, if in Zeno's story there is something to support the assumption of an effect on runner speed (similar to Schwarzschild's effect on the speed of light due to gravitation from matter), then that illustration may be helpful to explain your view.

PS. I see that A.T. gave a similar clarification:

That is an interesting comparison. One difference seems to be that "Zeno time" is the result of an arbitrary mathematical mapping, that doesn't have any physical significance. [..]

which was however obscured by what he said next (and that was probably sufficiently clarified by others).

 

[Dale]

t coordinates represent of course clocks (literal or virtual)

No, proper time represents clocks. Coordinate time represents a simultaneity convention.

 

[stevendaryl]

It sounds as if you want to hear my opinion about how convincing that illustration may be for your arguments about the nature of Schwarzschild's physics. I won't let myself be pulled again in arguments, but will give minimal advice. t coordinates represent of course clocks (literal or virtual)

No, it really doesn't. The time coordinate $t$ is related to the time $\tau$ shown on a standard clock at a constant value for $r$ by:
$t = \tau/\sqrt{1-r/R_s}$

The factor of $\sqrt{1-r/R_s}$ has no direct physical significance. $\tau$ is directly measurable. $t$ has no physical significance; it's just chosen to make the metric expression look as simple as possible.

...and together with space coordinates they allow to calculate for example the speed of a runner or of light between different points.

That's true of any coordinate system. You seem to think that there is something special about Schwarzschild coordinates, that they reflect reality in a way that other coordinates don't, but I can't get any kind of idea why you think that. Any coordinate system, as I have said, can be used equally well to describe physics within a patch. No coordinate system says anything at all about the physics outside of that patch.

So, if in Zeno's story there is something to support the assumption of an effect on runner speed (similar to Schwarzschild's effect on the speed of light due to gravitation from matter), then that illustration may be helpful to explain your view.

The Schwarzschild coordinates are not derived by computing the effect of gravity on light speed! It is derived by looking for a vacuum solution of Einstein's equations that is spherically symmetric. You are making up a physical meaning to Schwarzschild coordinates that isn't there.

Radially moving light has a certain "coordinate speed" in Schwarzschild coordinates:
$v = 1-2GM/(c^2 r)$

It has a different "coordinate speed" in Kruskal-S-whatever coordinates:
$v = 1$

You seem to think that there is a deep physical significance to the first speed, but not to the second. But you're just making things up. You're not getting that from GR. GR does not give any significance to one coordinate system over another. If you want to make up your own theory, go ahead, but from the context of GR, what you're saying makes no sense.

 

[stevendaryl]

I won't let myself be pulled again in arguments...

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?

What you're posting seems to be nonsensical. You seem to be giving a physical significance to a completely arbitrary choice. Schwarzschild coordinates are chosen for CONVENIENCE. With that choice, the metric looks the simplest. For you to go from that to the conclusion that Schwarzschild coordinates reflect reality in a way that other coordinates do not is just making stuff up. It's not part of GR. In creating GR, Einstein explicitly REJECTED the idea that some coordinates reflect reality more than other coordinates. So you're not talking about GR, you're talking about your own theory, which has an unspecified relationship with GR.

 

[Dale]

For you to go from that to the conclusion that Schwarzschild coordinates reflect reality in a way that other coordinates do not is just making stuff up. It's not part of GR. In creating GR, Einstein explicitly REJECTED the idea that some coordinates reflect reality more than other coordinates.

That, I think, is the key point of the whole thread and all of its predecessors.

 

[Dale]

I would like to expand further on the idea of coordinate time vs proper time. I have stated above that coordinate time represents a simultaneity convention. If you set coordinate time to some fixed value then you get a continuous and smooth set of events which forms some 3D hypersurface. In order to qualify as a time coordinate, this hypersurface must be spacelike everywhere, but otherwise there is no restriction to the shape of the hypersurface. This surface represents a set of all events that happened at the same time, which is, by definition, a simultaneity convention.

In contrast, proper time is only defined along a timelike worldline. If you set proper time to some fixed value, instead of getting a set of events, you get a single event. Geometrically, a fixed proper time is a point in the manifold whereas a fixed coordinate time is a hypersurface in the manifold.

Now, 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.

So, here you can say that SC represent the time of a distant clock using the Schwarzschild simultaneity convention, but you can easily make KS-like coordinates that also represent the time of the same clock using the KS simultaneity convention. So that is not a distinguishing feature of SC, i.e. 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.

 

[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 $r > 0$ and $-\infty < t < +\infty$. 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.