Is time, inside an event horizon, time-like or space-like?

In summary, the conversation is about the distinction between "space" and "space-like" dimensions inside an event horizon and the disagreement between the speakers, I and JesseM, about whether time is space-like or time-like in a realistic coordinate system. JesseM argues that time must be time-like in any realistic coordinate system, while I maintain that time is space-like inside an event horizon. We then go on to discuss a thought experiment involving a free-falling observer inside an event horizon and the behavior of two "cylinders" of free-falling particles in different coordinate systems.
  • #1
tiny-tim
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(the story so far … I maintain that, inside an event horizon, there is a useful distinction between "space" and "space-like" dimensions, and that in any realistic coordinate system, time is space-like. JesseM maintains that, in any realistic coordinate system, time must be time-like.

JesseM maintains that, "in a local coordinate system constructed out of freefalling rulers and clocks, the laws of physics must look identical to those of SR." I maintain that this is not possible … that inside an event horizon, the laws of physics are the same, but the geometry is different, and therefore the physics must look different.

now read on … :smile: )

:rolleyes: conservation of mass :rolleyes:
Hi JesseM! :smile:

a thought experiment …

In Schwarzschild coordinates, imagine a free-falling observer inside an event horizon. His path is the axis of two "cylinders" of free-falling particles, following him, but faster.

He can see them until they hit his backward light-cone ("diagonal" light does go fast enough for that). And the light by which he seems them reaches him at the same time from both "cylinders".

In your SR-physics local coordinate system, what happens to these "cylinders" when they hit the Schwarzschild light-cone?

Doesn't the Schwarzschild light-cone become a coordinate singularity which swallows them up? And isn't that singularity on the same radius as the observer?

So each "cylinder" converges to an event (possibly "at infinity") whose displacement from him is light-like … and either it is ahead of him, so he sees the particles disappearing in his own future … or it is behind him, in which case he literally has a singularity in his wake, and where is there room for the rest of space-time to fit in? :confused:

Or is it in his own present (as the Schwarzschild system requires)?

(And we can shrink the inside "cylinder", so that that event can be as "close" to the observer as we like.)

And, if the laws of physics are the same as in SR, doesn't this coordinate singularity have to be an actual singularity … which it obviously isn't? :confused:

And, if the laws of physics are the same as in SR (and if he can't see infinitely far into the future):
:smile: what happened to the mass? :smile:
 
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  • #2
and that in any realistic coordinate system, time is space-like. JesseM maintains that, in any realistic coordinate system, time must be time-like.
The Schwarzschild-time-basisvector is spacelike in any realistic local coordinate system. The local time-basisvector is different from the Schwarzschild one, and of course timelike.
 
  • #3
tiny-tim said:
(the story so far … I maintain that, inside an event horizon, there is a useful distinction between "space" and "space-like" dimensions, and that in any realistic coordinate system, time is space-like. JesseM maintains that, in any realistic coordinate system, time must be time-like.
I don't think I used the phrase "realistic coordinate system". But if an observer inside the horizon uses a network of freefalling clocks to define his time-coordinate, surely you agree this time-coordinate would be timelike? After all, the path of each clock is a timelike geodesic inside the horizon as well as outside, no?
tiny-tim said:
a thought experiment …

In Schwarzschild coordinates, imagine a free-falling observer inside an event horizon. His path is the axis of two "cylinders" of free-falling particles, following him, but faster.
I'm not sure I understand what you mean by "the axis of two cylinders of free-falling particles". How can a single path be the axis of two separate cylinders? Does one cylinder enclose the other, so their axes coincide? And are they "cylinders" in spacetime or cylinders in space?
tiny-tim said:
He can see them until they hit his backward light-cone ("diagonal" light does go fast enough for that). And the light by which he seems them reaches him at the same time from both "cylinders".
This would seem to suggest cylinders in spacetime which enclose one another, so that if we have 2 space dimensions, it's more like the guy is sitting at the center of two concentric rings in space which are moving along with him. But I don't know how to make this jibe with your statement that the particles are "following him, but faster". Maybe you could draw a spacetime diagram of what you mean, with either 1 or 2 space dimensions and 1 time dimension? For simplicity we could start out with a diagram in ordinary flat SR spacetime, then we could talk about how this would work if we extended both the observer's worldline and the worldlines of the particles forming the cylinder into a Schwarzschild or Eddington-Finkelstein diagram where they all go into a BH together (Eddington-Finkelstein would be simpler since it avoids a coordinate singularity at the horizon for infalling particles).
tiny-tim said:
In your SR-physics local coordinate system, what happens to these "cylinders" when they hit the Schwarzschild light-cone?

Doesn't the Schwarzschild light-cone become a coordinate singularity which swallows them up? And isn't that singularity on the same radius as the observer?
What do you mean by "Schwarzschild light-cone"? Do you mean a light cone in a diagram of Schwarzschild coordinates, perhaps one from an event exactly at the event horizon (as suggested by the 'coordinate singularity') comment? If we want to deal with things happening right at the event horizon, then as I said, it would probably be simpler to use Eddington-Finkelstein coordinates or some other system which doesn't blow up there.
tiny-tim said:
So each "cylinder" converges to an event (possibly "at infinity") whose displacement from him is light-like … and either it is ahead of him, so he sees the particles disappearing in his own future … or it is behind him, in which case he literally has a singularity in his wake, and where is there room for the rest of space-time to fit in? :confused:
I can't picture what you're talking about here. What event are you talking about, physically? An event on the horizon, on the wordline of the singularity, or somewhere else? Why are the cylinders "converging" to this event?
 
  • #4
Hi JesseM! :smile:
JesseM said:
I don't think I used the phrase "realistic coordinate system".

No, you didn't … what would you prefer? :smile:
But if an observer inside the horizon uses a network of freefalling clocks to define his time-coordinate, surely you agree this time-coordinate would be timelike? After all, the path of each clock is a timelike geodesic inside the horizon as well as outside, no?

In Schwarzschild coordinates, a clock moves faster than light, and so I agree its geodesic (in any coordinates, of course) is timelike.
… This would seem to suggest cylinders in spacetime which enclose one another …

Yes … as I said, I'm describing everything in Schwarzschild coordinates … in those coordinates, my (massive) particles freefall down the cylinder faster than the observer freefalls down the axis. :smile:
(Eddington-Finkelstein would be simpler since it avoids a coordinate singularity at the horizon for infalling particles).

But we're only discussing a local coordinate system, so I'm not bothered about the horizon or the singularity.
What do you mean by "Schwarzschild light-cone"?

The cones in that famous Finkelstein diagram of 1956, which I think was in Schwarzschild coordinates. :smile:
I can't picture what you're talking about here. What event are you talking about, physically? An event on the horizon, on the wordline of the singularity, or somewhere else? Why are the cylinders "converging" to this event?

I can't picture it, either.

I'm assuming, for sake of argument, that you are right in saying that an observer inside the horizon using a network of freefalling clocks (and only looking at his immediate neighbourhood) (hereafter called "your observer" :smile: ) cannot tell whether he is inside or outside the horizon.

The event I'm talking about, described in Schwarzschild coordinates, is the perfectly ordinary event of the particles crossing the light-cone of the observer.

At that event, your observer says that they disappear, doesn't he?

I was trying to describe how your observer would describe that, assuming that he cannot tell whether he is inside or outside the horizon.

I maintain that he can't … that however he tries, he ends up either with mass not being conserved, or with a coordinate singularity which is not an actual singularity.

In other words, if he assumes the physics is the same inside as outside, then he gets a contradiction.

Now, this is a perfectly simple and valid experiment, isn't it?

So … how do you say your observer describes this event (which I have defined in Schwarzschild coordinates)? :smile:
 
  • #5
JesseM said:
I don't think I used the phrase "realistic coordinate system".
tiny-tim said:
No, you didn't … what would you prefer? :smile:
Prefer for what? I never suggested that any coordinate system was better than any other. My main arguments were:

a) in a locally inertial coordinate system constructed out of freefalling rulers and clocks, the laws of physics are identical to those of SR

b) your statement that lightlike geodesics from an event would expand in all directions when projected onto a surface of constant t if the event was outside the horizon, but would only expand in a limited cone of directions if the event was inside the horizon, was just a feature of the coordinate system you were using, it has no real physical significance since we could also find coordinate systems where lightlike geodesics expand in all directions from an event inside the horizon when projected onto a surface of constant t, or even a coordinate system where lightlike geodesics would expand in a limited cone of directions outside the horizon.
JesseM said:
… This would seem to suggest cylinders in spacetime which enclose one another …
tiny-tim said:
Yes … as I said, I'm describing everything in Schwarzschild coordinates … in those coordinates, my (massive) particles freefall down the cylinder faster than the observer freefalls down the axis. :smile:
Let's think about what's going on far from the horizon, where spacetime is close to flat. If we are talking about a universe with 2 space dimensions, then at any given time do you mean the central observer would be surrounded by a ring of particles, so that the worldlines of particles in this ring forms a cylinder in spacetime? If so, what can it mean to say the particles are falling towards the black hole faster than the central observer? Surely if this were the case, then the central observer would not remain at the center of the ring, instead the center of the ring would be moving in the direction of the BH faster than the observer, so the observer would quickly pass out of the edge of the ring.
tiny-tim said:
But we're only discussing a local coordinate system, so I'm not bothered about the horizon or the singularity.
I thought that when you talked about a "Schwarzschild light cone" you might mean one on the horizon...if you were talking about an event right on the horizon then we would need to be bothered about the fact that Schwarzschild coordinates blow up there, but maybe that's not what you meant.
tiny-tim said:
The cones in that famous Finkelstein diagram of 1956, which I think was in Schwarzschild coordinates. :smile:
I am not familiar with "that famous Finkelstein diagram of 1956", unless you are just talking about a standard Eddington-Finkelstein diagram (but then you presumably wouldn't say it was in Schwarzschild coordinates, since Eddington-Finkelstein diagrams are based on Eddington-Finkelstein coordinates). Could you be a little more specific here?
tiny-tim said:
I'm assuming, for sake of argument, that you are right in saying that an observer inside the horizon using a network of freefalling clocks (and only looking at his immediate neighbourhood) (hereafter called "your observer" :smile: ) cannot tell whether he is inside or outside the horizon.

The event I'm talking about, described in Schwarzschild coordinates, is the perfectly ordinary event of the particles crossing the light-cone of the observer.

At that event, your observer says that they disappear, doesn't he?
I don't understand what you mean at all. Only instantaneous events have light-cones, an observer has a worldline made up of a continuum of events. If he is surrounded by a ring of particles, then at any given instant on his worldline there will be events on the ring that he doesn't know about because they're not in the past light cone of that instant, but he'll still learn about those events at some later instant that they are in the past light cone of. There need never be a time when the ring disappears from view, because at any given instant his past light cone will intersect with some set of events on the world-cylinder of the ring around him. But I'm still not sure if an observer surrounded by a ring (in a universe with 2D space) or a sphere (in a universe with 3D space) is really what you were talking about in the first place, you really need to state your scenario more clearly.
 
  • #6
Hi JesseM! :smile:
JesseM said:
… you really need to state your scenario more clearly.

ok … I define my "cylinder" as θ = constant, in the standard Schwarzschild coordinates (t,r,θ,φ), where θ is latitude.

(I tend to put quotes round it, since strictly it isn't a cylinder, but a cone.)
I don't understand what you mean at all. Only instantaneous events have light-cones, an observer has a worldline made up of a continuum of events.

Yes … at any instant, an observer inside a horizon has a cone (in Schwarzschild coordinates) whose rear half contains all and only the directions from which he can receive photons.

It is a different cone, of course, for different positions on his same world-line. Its half-angle becomes smaller as he falls further. :cry:

For any particular particle on the "cylinder", falling faster than the observer (in Schwarzschild coordinates), it is inside the cone up to a certain time, then it is outside it.

The event I am talking about is the event on the world-line of that particle when it changes from being inside to being outside.
… but he'll still learn about those events at some later instant that they are in the past light cone of.

No … because (in Schwarzschild coordinates) he is going faster than the photons, and that event is the last event on the world-line of that particle from which the photons can catch him.

Photons whose direction is at an angle equal to the half-angle of that cone (in other words: photons from events on his cone) reach him instantaneously (in Schwarzschild coordinates).

Photons from the same event whose direction is at a lesser angle take too long, and can be seen only by another observer following the original observer.

Having defined my event carefully … on the world-line of a particular particle, it is the last event from which your observer can receive photons … again I ask …
… how do you say your observer describes this event?​

In particular, how does he describe the "disappearance" of that particular particle from his view? :smile:
 
  • #7
tiny-tim said:
ok … I define my "cylinder" as θ = constant, in the standard Schwarzschild coordinates (t,r,θ,φ), where θ is latitude.
Outside the horizon, if we take a snapshot at any given t, that would look like an actual cone which intersects all the concentric spheres of increasing r at a fixed latitude (see the diagram here), but then if you vary t the cone is unchanging...how is anything falling in here? Are you imagining this cone's surface is made up a continuum of infalling particles?

If so, then since your claim is about the infalling particles "falling out of view" for the observer in the horizon, it seems unnecessarily complicated to talk about a continuum of particles, why not just pick a single infalling particle? I assume your claiming that if we select anyone infalling particle that the observer can see before crossing the horizon, that particle will disappear from view for him sometime after he crosses it?
tiny-tim said:
Yes … at any instant, an observer inside a horizon has a cone (in Schwarzschild coordinates) whose rear half contains all and only the directions from which he can receive photons.

It is a different cone, of course, for different positions on his same world-line. Its half-angle becomes smaller as he falls further. :cry:
If you're talking about things like "half-angles" here, presumably you're picturing something...could you please try to draw it? Your verbal explanations really aren't helping much.
tiny-tim said:
For any particular particle on the "cylinder", falling faster than the observer (in Schwarzschild coordinates), it is inside the cone up to a certain time, then it is outside it.
Are you claiming that the worldline of a particular particle can be such that it intersects the edge of the past light cone of some earlier event on the worldline of the observer, but then it doesn't intersect anywhere on the edge of the past light cone of some later event on the worldline of the observer? That seems geometrically impossible to me. Again, a diagram would be useful.
tiny-tim said:
No … because (in Schwarzschild coordinates) he is going faster than the photons, and that event is the last event on the world-line of that particle from which the photons can catch him.
But the fact that he's going "faster" than photons is just an artifact of the coordinate system, where the t in dr/dt is no longer timelike. I don't see how it would have real physical implications like photons from an object that was previously in his past light cone being unable to "catch up with him" after a certain point. And if this physical implication is correct, it should show up in any coordinate system, no? Do you think it would show up in Eddington-Finkelstein coordinates, even though in these coordinates photons on the "underside" of the future light cone of an event on the worldline of an infalling particle always move faster than the particle, while photons on the "upper side" of the future light cone move slower than the particle?

Speaking of coordinate artifacts, I think it's worth pointing out Schwarzschild coordinates look kind of ugly inside the horizon because they make the path of every object appear time-reversed, with the earliest event on their worldline inside the horizon being the collision with the singularity, and the worldline getting closer to the horizon at later values of t, with the distance from the horizon approaching zero as t approaches infinity...see the diagram on p. 848 of https://www.amazon.com/dp/0716703440/?tag=pfamazon01-20 for example. And if you have that book, perhaps you could draw a similar diagram to illustrate what you mean about photons from a particle that could previously be seen by the infalling observer being unable to "catch up" with any points on the worldline the infalling observer after a certain point. Though again, I would much prefer to see an Eddington-Finkelstein type diagram, which has nice features like the fact that photons emitted on the "underside" of the light cone always move on perfect diagonals, and the fact that it doesn't time-reverse worldlines inside the horizon like Schwarzschild coordinates do (also, I've found Eddington-Finkelstein diagrams which trace the complete worldlines of photons emitted from some event on the worldline of an infalling observer, but I can't find analogous diagrams in Schwarzschild coordinates, although the one on p. 848 does show the immediate direction of the photons after being emitted from an event).

One last thing I'd note is that even if we were talking about faster-than-light "tachyons" in ordinary flat SR spacetime using Minkowski coordinates, as far as I can tell, if some object was in the past light cone of an earlier event on the worldline of the tachyon, it would continue to be in the past light cone of all later events on the worldline of that tachyon. So even at this basic level, I don't see how the fact that something is moving faster than light in a given coordinate system would imply that any object would ever "disappear from view".
tiny-tim said:
In particular, how does he describe the "disappearance" of that particular particle from his view? :smile:
I still am inclined to think that the notion of a "disappearance" is a mistake you are making here, because you don't seem to have thought the visual diagrams through very well, and you certainly don't seem to have done any detailed mathematical analysis.
 
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  • #8
tiny-tim said:
JesseM maintains that, "in a local coordinate system constructed out of freefalling rulers and clocks, the laws of physics must look identical to those of SR." I maintain that this is not possible … that inside an event horizon, the laws of physics are the same, but the geometry is different, and therefore the physics must look different.
Let's just examine JesseM's claim about local coordinate systems inside the event horizon. I'm going to explicitly calculate such a coordinate system for you.

If we restrict ourselves to one-dimensional motion along a straight line to the centre of a black hole (so [itex]d\theta = d\phi = 0[/itex]) the Schwarzschild metric simplifies to

[tex]ds^2 = \left( 1 - \frac{2m}{r} \right ) dt^2 - \left ( 1 - \frac{2m}{r} \right )^{-1} dr^2[/tex]​

with the usual convention that units are chosen so that c = G = 1.

Consider the event (t=t0, r=r0) inside the event horizon, so that

[tex] 1 - \frac{2m}{r_0} < 0 [/tex]​

Now define new coordinates

[tex] T = \left ( \frac{2m}{r_0} - 1\right )^{-1/2} (r - r_0) [/tex]
[tex] R = \left ( \frac{2m}{r_0} - 1\right )^{1/2} (t - t_0) [/tex]​

Substitute these into the equation for the metric and you will get

[tex]ds^2 = \frac{1 - 2m/r}{2m/r_0 - 1} dR^2 - \frac{ 2m/r_0 - 1}{1 - 2m/r}dT^2 [/tex]​

When T is very small, r is approximately r0 and so ds is approximately

[tex]ds^2 \approx dT^2 - dR^2[/tex]​

But this is just the standard Minkowski metric for flat spacetime i.e. special relativity, so, very close to the event (t=t0, r=r0) i.e. (T=0, R=0), the coordinates (T,R) behave just like (and therefore are) time and space coordinates (in that order) in special relativity.

The local geometry is determined entirely by the local behaviour of the metric. So I've just shown the local geometry approximates to the geometry of special relativity.

Q.E.D.

Outside the event horizon, you would substitute

[tex] T = \left ( 1 - \frac{2m}{r_0} \right )^{1/2} (t - t_0) [/tex]
[tex] R = \left ( 1 - \frac{2m}{r_0} \right )^{-1/2} (r - r_0) [/tex]​

instead.
 
  • #9
Hi DrGreg! :smile:

Have I got this right … ?

All you've done is swap over dr and dt, and called them dT and dR, respectively, after applying a linear transformation to make c = 1?

Then the full metric becomes, locally, the standard Minkowski metric, with time and radial distance swapped over.

But … that's obvious! :smile:

I'm not disputing you can do that.

I'm disputing the claim that an observer, using local observation only, cannot tell whether he's inside or outside an event horizon.

In particular, I'm claiming that he can tell he's inside by noticing that everything goes faster than light (in the sense that a series of massive particles will overtake light), and also by noticing that faster-freefalling particles disappear from view.

Can you help on this? :smile:
 
  • #10
tiny-tim said:
Hi DrGreg! :smile:

Have I got this right … ?

All you've done is swap over dr and dt, and called them dT and dR, respectively, after applying a linear transformation to make c = 1?

Then the full metric becomes, locally, the standard Minkowski metric, with time and radial distance swapped over.

But … that's obvious! :smile:

I'm not disputing you can do that.

I'm disputing the claim that an observer, using local observation only, cannot tell whether he's inside or outside an event horizon.

In particular, I'm claiming that he can tell he's inside by noticing that everything goes faster than light (in the sense that a series of massive particles will overtake light), and also by noticing that faster-freefalling particles disappear from view.

Can you help on this? :smile:
Suppose we take a very small patch of spacetime in Schwarzschild coordinates where the worldlines of a photon and a massive particle intersect, and transform into local Minkowski coordinates on that patch. Are you claiming the massive particle will still be moving faster than the photon even in locally inertial Minkowski coordinates? If so I'm sure you're wrong just by the equivalence principle, although I don't have the expertise in GR to do the math here...maybe DrGreg or someone else can help, if indeed you would make the claim that electrons move faster than photons even in locally inertial coordinates.
 
  • #12
Hi Jennifer!

hmm … "Light cones tipping over" … I did a thread-title search on this sub-forum before starting this thread, but I didn't think to look for those words. :redface:

-> which reminds me ->​

Whenever I'm at the Restaurant At The End Of The Black Hole …

:smile: I never know how much to tip! :smile:
 
  • #13
… I-Can't-Believe-It's-Not-Water coordinates …

Hi JesseM! :smile:
JesseM said:
Are you claiming the massive particle will still be moving faster than the photon even in locally inertial Minkowski coordinates?

No! :biggrin: But it's a coordinate effect, not a physical one.

Since, in Schwarzschild coordinates, a massive particle has dx/dt > 1, it's obvious that if you swap x and t, you get dx/dt < 1! :smile:

(btw, do you accept, by the same argument, that massive particles must go faster than light in Schwarzschild coordinates?)

In other words: if an observer inside a horizon insists on using locally inertial Minkowski coordinates, then it's obvious that he will regard massive particles as slower than light.

However, that doen't alter the physical fact that the massive particles will overtake light, and in particular they will reach the singularity first. :smile:
If so I'm sure you're wrong just by the equivalence principle …

Isn't that begging the question?

I keep asking people to comment on how your observer describes the inconvenient physical fact of the disappearance from his view of a faster particle … but nobody answers!

If you start by saying "I believe that no observer using locally inertial Minkowski coordinates can tell whether he's inside a horizon", then obviously you thereby disprove the very existence of inconvenient facts! :biggrin:

Look … if I were to break through the boundary of the bowliverse, I could insist on using I-Can't-Believe-It's-Not-Water coordinates, and thereby prove that I was still in water! :smile:

I maintain that an observer crossing a horizon (traditionally described as an unremarkable event) will not suddenly decide to swap his r and t coordinates, and will notice if someone else swaps them!

In particular, he will notice that things start disappearing!

Only an observer with spaghetti for brains, or with the memory and attention-span of a goldfish, could fail to …

erm … :smile:
I … er … :redface:
:cry: d'oh! :cry:
 
  • #14
tiny-tim said:
Are you claiming the massive particle will still be moving faster than the photon even in locally inertial Minkowski coordinates?
tiny-tim said:
No! :biggrin: But it's a coordinate effect, not a physical one.
Why do you say that? Do you deny that if an observer constructs a local coordinate system out of freefalling rulers and clocks moving alongside them, using the same method that's used to construct inertial coordinate systems in SR, then this will result in the same sort of locally inertial Minkowski coordinates?
tiny-tim said:
(btw, do you accept, by the same argument, that massive particles must go faster than light in Schwarzschild coordinates?)
Of course.
tiny-tim said:
However, that doen't alter the physical fact that the massive particles will overtake light, and in particular they will reach the singularity first. :smile:
They will reach it "first" in Schwarzschild coordinate time, which is a spacelike coordinate. As I said, in a coordinate system constructed out of freefalling rulers and clocks, this might be equivalent to something like "hitting the singularity further to the left".
JesseM said:
If so I'm sure you're wrong just by the equivalence principle …
tiny-tim said:
Isn't that begging the question?
No, it's appealing to a known property of the theory of GR to answer a physical question about a GR scenario.
tiny-tim said:
I keep asking people to comment on how your observer describes the inconvenient physical fact of the disappearance from his view of a faster particle … but nobody answers!
And I keep telling you I don't believe there will be an "physical fact of the disappearance from his view", that you're confusing yourself using handwavey arguments. It would really help if you would answer some or all of the requests for clarification I made in post #7. For example, if you think the "disappearance from view" is an obvious physical fact, why do you refuse to draw a simple diagram showing how the worldline of the disappearing particle could instersect the past light cone of the infalling observer at one event on his worldline, but fail to intersect the past light cone of the same observer at a later event on his worldline?
tiny-tim said:
If you start by saying "I believe that no observer using locally inertial Minkowski coordinates can tell whether he's inside a horizon", then obviously you thereby disprove the very existence of inconvenient facts! :biggrin:
I don't believe your inconvenient "facts", which you seemed to have arrived at by vague handwavey arguments that you can't even depict in a spacetime diagram and certainly have not supported using detailed math, are facts at all. But if they are physical facts, then they should show up in any coordinate system, including locally inertial Minkowski coordinates (i.e. if the particle disappears from the past light cone of the infalling observer, it must do so when you look at the past light cone and the particle's worldline in these coordinates, no?)
tiny-tim said:
Look … if I were to break through the boundary of the bowliverse, I could insist on using I-Can't-Believe-It's-Not-Water coordinates, and thereby prove that I was still in water! :smile:
Any "physical fact" will be true in all coordinate systems.
tiny-tim said:
I maintain that an observer crossing a horizon (traditionally described as an unremarkable event) will not suddenly decide to swap his r and t coordinates, and will notice if someone else swaps them!
Who said anything about a sudden swap? If the observer uses freefalling clocks to define his time coordinate, the t coordinate will be timelike both inside and outside the horizon, nothing strange will happen at the horizon. It is in Schwarzschild coordinates where you have a "swap", but the t coordinate in Schwarzschild coordinates is not based on readings of actual physical clocks inside the horizon.
tiny-tim said:
In particular, he will notice that things start disappearing!
If you have any coherent argument for why you think this will happen, you should really be able to answer my questions in post #7. I think you're just confused.
 
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  • #15
tiny-tim said:
Hi DrGreg! :smile:

Have I got this right … ?

All you've done is swap over dr and dt, and called them dT and dR, respectively, after applying a linear transformation to make c = 1?

Then the full metric becomes, locally, the standard Minkowski metric, with time and radial distance swapped over.

But … that's obvious! :smile:

I'm not disputing you can do that.

I'm disputing the claim that an observer, using local observation only, cannot tell whether he's inside or outside an event horizon.

In particular, I'm claiming that he can tell he's inside by noticing that everything goes faster than light (in the sense that a series of massive particles will overtake light), and also by noticing that faster-freefalling particles disappear from view.

Can you help on this? :smile:
It's unfortunate that two of the Schwarzschild coordinates are labelled t and r, and often referred to as "time" and "radius" (or "distance"). That gives the false impression that these coordinates correspond to real i.e. proper time and proper length when really they don't. For a distant observer (hypothetically "at infinity") they do indeed correspond to time and radius, but as you approach the black hole they start to distort (they need to be rescaled using the second coord system I gave at the bottom of my post #8) and when you actually reach the event horizon they just break down completely. However they do work again once you are inside, but with time and space swapped as I showed. If you want to analyse what happens as you cross the event horizon, you need a different coordinate system.

I must confess I have only paddled in the shallow waters of GR, and this discussion is a learning curve for me, but a quick search of Wikipedia found the Lemaitre metric which has no discontinuity at the event horizon and also has the advantage that objects that are at rest in Lemaitre coords are free-falling. The article gives the formula for conversion from Schwarzschild coordinates.

(See also Kruskal-Szekeres coordinates and Gullstrand-Painlevé coordinates, which, at a glance, seem less useful but do contain some useful info. I haven't read any of these 3 articles in depth. yet.)

In GR the local speed of light, measured using a local frame of reference, need not be the same as the coordinate speed of light using the coordinate system of a remote observer extrapolated to the region being measured.

When you say "massive particles will overtake light", that is only from the point of view of a distant observer who interprets velocity as "dr/dt" in Schwarzschild coordinates. A local observer interprets velocity as "dR/dT" where (T,R) is a locally-inertial coord system. (Or using any other locally-inertial system related to the first one via a Lorentz transform -- choose the one in which he/she is at rest i.e. in which the coordinates of the observer satisfy dR'/ds = 0.) From the local observer's perspective, massive local particles still move slower than light. The local speed of light is always given by ds=0, in any coordinates.
 
  • #16
… a coordinate-free definition of "overtaking" …

Hi DrGreg! :smile:
DrGreg said:
When you say "massive particles will overtake light", that is only from the point of view of a distant observer who interprets velocity as "dr/dt" in Schwarzschild coordinates.

No … I can define "overtaking" in a coordinate-free way:

Imagine a series of 10 photons, freefalling inside the horizon down the same radius, labelled 1 to 10 so that 1 is nearest the singularity (either in r or in t … it makes no difference) in Schwarzschild coordinates.

And imagine a similarly labelled series of 10 massive particles freefalling down an adjacent radius.

"Adjacent" means that there is no doubt as to when one of the photons is next to one of the particles.

In Schwarzschild coordinates, |dr/dt| for a massive particle is greater than for a photon.

So, in Schwarzschild coordinates, if particle 1 starts next to photon 1, then shortly afterwards particle 2 will be next to photon 1 (assuming they're close enough together), then particle 3 …

Furthermore, in Schwarzschild coordinates, particle 1 and photon 1 hit the singularity before all the other particles and photons, respectively.

Therefore, following the world-lines of the particles and photons in any coordinate system, we can identify "particles overtaking photons" as meaning "as a particular photon gets nearer the singularity, it is successively next to freefalling particles which are further from the singularity". :smile:

Do you agree that the statement in bold is true in Schwarzschild coordinates, and therefore in any coordinate system?

(Similarly, do you agree that "diagonal" photons will overtake radial photons in any coordinate system? And that photons from particles overtaking a freefalling observer will not be able to reach the observer if they start from too great an angle … in other words, the particles will disappear from view, without any singularity, no matter how small the neighbourhood?)
 
  • #17
tiny-tim said:
No … I can define "overtaking" in a coordinate-free way:

Imagine a series of 10 photons, freefalling inside the horizon down the same radius, labelled 1 to 10 so that 1 is nearest the singularity (either in r or in t … it makes no difference) in Schwarzschild coordinates.

And imagine a similarly labelled series of 10 massive particles freefalling down an adjacent radius.

"Adjacent" means that there is no doubt as to when one of the photons is next to one of the particles.

In Schwarzschild coordinates, |dr/dt| for a massive particle is greater than for a photon.

So, in Schwarzschild coordinates, if particle 1 starts next to photon 1, then shortly afterwards particle 2 will be next to photon 1 (assuming they're close enough together), then particle 3 …

Furthermore, in Schwarzschild coordinates, particle 1 and photon 1 hit the singularity before all the other particles and photons, respectively.

Therefore, following the world-lines of the particles and photons in any coordinate system, we can identify "particles overtaking photons" as meaning "as a particular photon gets nearer the singularity, it is successively next to freefalling particles which are further from the singularity". :smile:

Do you agree that the statement in bold is true in Schwarzschild coordinates, and therefore in any coordinate system?

(Similarly, do you agree that "diagonal" photons will overtake radial photons in any coordinate system? And that photons from particles overtaking a freefalling observer will not be able to reach the observer if they start from too great an angle … in other words, the particles will disappear from view, without any singularity, no matter how small the neighbourhood?)
Look at the diagram of Eddington-Finkelstein coordinates http://io.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/Black_Holes_files/image005.jpg [Broken]. With the r=2M line as the event horizon, it shows the wordlines of photons emitted on "both sides" of various light cones--the two paths bordering the ellipses on either side. The diagram also shows a dark line representing a massive particle. Inside the event horizon, you can see that for the photons emitted on the right side of the light cones (the ones whose worldlines are curved rather than diagonal straight lines), the massive particle is moving faster than they are in these coordinates--in the diagram we see it "passing" two separate photon worldlines and hitting the singularity before they do. Is this the type of thing you're talking about?

If so, just note that the sense in which the dark particle worldline "passes" photon worldlines on the right side of the light cones inside the horizon is identical to the sense in which it passes photons worldlines on the right side of light cones outside the horizon--these are just photons moving to the right, outside the horizon, so of course if I see a series of photons moving to the right while I'm moving to the left, I'll pass a series of them in succession. Inside the horizon, it may look like these photons are moving inward just like the particle, but that's just because of the way light cones have been tilted--these are still the photons that would be moving to the right in a locally inertial coordinate system. And note that the particle does not overtake photons on the left side of the light cones, whose worldlines are always diagonal straight lines in Eddington-Finkelstein coordinates.
 
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  • #18
Hi JesseM! :smile:

ok … in your University of Winnipeg diagram, t increases upward (at large r), so my photon 1 is the diagonal line at the bottom, and my photon 8 is the diagonal line at the top.

The bold line is my particle 1 (we don't need the other particles).

The curved lines are "oufalling" photons, which we're not interested in, in this example.

photon 1 hits the singularity first, then photon 2, then … photon 5, then particle 1, then photon 6, …

So particle 1 is next to (to use a neutral expression) photon 1 first, then photon 2 …


So, according to the diagram, the particle is indeed overtaken by the photons, and therefore I am wrong and you are right. :smile:

However … all this proves is that you can prove anything with diagrams! :rofl:​

The particle line in the diagram is wrong.

It should reach an angle of 45º at the horizon … the same angle as the infalling photons, and then curve over further, overtaking the photons that had previously overtaken it. :frown:

I'll take the equations from http://en.wikipedia.org/wiki/Kepler_problem_in_general_relativity#Geodesic_equation:

For a radially falling particle (so dφ/dt = L = 0) of mass m, with c = 1, with wikpedia's 1 - rs/r = 1 - 2GM/r = δ, and with a speed "at infinity" of tanhα (so putting wikpedia's E = m.coshα), we have:

(dr/dτ)² = cosh²α - δ; (dt/dτ)² = cosh²α/δ²;

so dr/dt = ±δ√(1 - δsech²α).

In other words: "at infinity", δ = 1, and dr/dt = ±tanhα, as expected.

But very near the horizon, δ approaches 0, and dr/dt approaches ±δ … which is the same as dr/dt for photons!

And inside the horizon, δ is negative, and so of course |dr/dt| > δ, and the particle falls faster than photons.

Agreed? :smile:
 
  • #19
tiny-tim said:
ok … in your University of Winnipeg diagram, t increases upward (at large r), so my photon 1 is the diagonal line at the bottom, and my photon 8 is the diagonal line at the top.

The bold line is my particle 1 (we don't need the other particles).

The curved lines are "oufalling" photons, which we're not interested in, in this example.
The curved lines are only outgoing photons outside the horizon. The curved lines inside the horizon are falling in towards the singularity just like the diagonal lines. If at a particular moment on its worldline the observer emits photons in both left and right spatial directions as seen in locally inertial coordinates, then (assuming at this moment the observer was already inside the horizon) the left photon goes along a diagonal line until it hits the singularity, while the right photon goes on a curved line until it hits the singularity. And you can see that the infalling observer's coordinate speed is greater than the coordinate speed of the photons following curved worldlines, inside the horizon (the slope of the observer's worldline is closer to horizontal).
tiny-tim said:
o, according to the diagram, the particle is indeed overtaken by the photons, and therefore I am wrong and you are right. :smile:

However … all this proves is that you can prove anything with diagrams! :rofl:​
Not with correctly-drawn diagrams, no. Any physical conclusions you get in a correctly-drawn diagram should carry over to a correctly-drawn diagram in any other coordinate system.
tiny-tim said:
The particle line in the diagram is wrong.

It should reach an angle of 45º at the horizon … the same angle as the infalling photons, and then curve over further, overtaking the photons that had previously overtaken it. :frown:
No, you're wrong here, the diagram matches all other diagrams of Eddington-Finkelstein coordinates I have seen. For example, see the identical diagram http://www.etsu.edu/physics/plntrm/relat/eventho1.gif, from p. 829. Do you think Misner-Thorne-Wheeler made an error in their diagram too?
tiny-tim said:
I'll take the equations from http://en.wikipedia.org/wiki/Kepler_problem_in_general_relativity#Geodesic_equation:

For a radially falling particle (so dφ/dt = L = 0) of mass m, with c = 1, with wikpedia's 1 - rs/r = 1 - 2GM/r = δ, and with a speed "at infinity" of tanhα (so putting wikpedia's E = m.coshα), we have:

(dr/dτ)² = cosh²α - δ; (dt/dτ)² = cosh²α/δ²;

so dr/dt = ±δ√(1 - δsech²α).

In other words: "at infinity", δ = 1, and dr/dt = ±tanhα, as expected.

But very near the horizon, δ approaches 0, and dr/dt approaches ±δ … which is the same as dr/dt for photons!

And inside the horizon, δ is negative, and so of course |dr/dt| > δ, and the particle falls faster than photons.
I don't really follow the details of the math here, but these equations are in Schwarzschild coordinates rather than Eddington-Finkelstein coordinates, yes? And Schwarzschild coordinates are badly-behaved at the horizon--both massive particles and photons take an infinite time to reach the horizon in Schwarzschild coordinates, so naturally they must both approach the same coordinate velocity of zero as the time coordinate approaches infinity. And inside the horizon Schwarzschild coordinates are also fairly counterintuitive, since as I mentioned before they reverse the order of events, with the particles emerging from the singularity first and their proper time decreasing with increasing t, so that they get closer and closer to the horizon from the inside as t approaches infinity as well. There's a helpful diagram of the worldline of an infalling object in Schwarzschild coordinates on p. 848 of Gravitation which I've scanned and included as an attachment. You can see that even within the horizon, the object's worldline stays within the future light cone of any event on the worldline.
 

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  • #20
Hi JesseM! :smile:
JesseM said:
The curved lines are only outgoing photons outside the horizon. The curved lines inside the horizon are falling in …

I didn't call those photons outgoing, I called them "outfalling", with the quotes marks. This has the advantage of distinguishing them from infalling photons, without having to talk about "left and right spatial directions" or such-like.

I accept that it's a bit of an oxymoron … since only friends can literally fall out inside a horizon :frown:
:rolleyes: usually over map-reading! :rolleyes:
No, you're wrong here, the diagram matches all other diagrams of Eddington-Finkelstein coordinates I have seen.

ok … your first diagram is from the University of Winnipeg, and I've already commented on it … your second diagram is from the University of Syracuse NY, copied from Geroch's book, and it only shows "outfalling" photons, not infalling ones, so that doesn't help either of us (except that the slope on crossing the horizon, at U, looks like 45º to me) …

and your third diagram is from MTW's Gravitation, but it is described as "edited" … is the original diagram the same?

(And your fourth diagram, in Schwarzschild coordinates, scanned directly from p.848 of MTW, although it doesn't show a series of photons, it does show the particle going faster than infalling light, doesn't it? :smile:)
… these equations are in Schwarzschild coordinates …?

Yes. :smile:
And Schwarzschild coordinates are badly-behaved at the horizon …

Agreed … but they're fine right up to the horizon, and they show clearly that, in Schwarzschild coordinates, dr/dt approaches the local speed of light (my δ) as the particle approaches the horizon.

Do you accept that? If so, surely you accept that in the EF coordinates, photons and particles cross the horizon at the same angle … 45º? :smile:
And inside the horizon … their proper time decreasing with increasing t, so that they get closer and closer to the horizon from the inside as t approaches infinity …

Yes … a particle's "own clock" keeps going forward, but its t coordinate goes to infinity at the horizon, and then starts winding back again.

So … what?

What matters is the absolute value of dr/dt … and the equations show clearly that inside a horizon it's greater than δ. :smile:
The Schwarzschild coordinates are a perfectly valid coordinate system.

Although, in Schwarzschild coordinates, r and t both approach infinity at the horizon, dr/dt is still well-defined, and enables a valid comparison of speed both at the horizon and beyond! :smile:
 
  • #21
tiny-tim said:
I didn't call those photons outgoing, I called them "outfalling", with the quotes marks. This has the advantage of distinguishing them from infalling photons, without having to talk about "left and right spatial directions" or such-like.
OK, fair enough, although when you introduce your own nonstandard terminology it's helpful if you define it first. But do you agree that if we look at some event A inside the horizon, then the "infalling" photon worldline and the "outfalling" photon worldline that emanate from A form opposite sides of the future light cone from A? If so, then before we continue, I have a question about something you said in an earlier post:
tiny-tim said:
The particle line in the diagram is wrong.

It should reach an angle of 45º at the horizon … the same angle as the infalling photons, and then curve over further, overtaking the photons that had previously overtaken it. :frown:
Does this mean you think that inside the horizon, the worldline of a particle can actually "escape" the future light cone of an event along it, since in Eddington-Finkelstein coordinates both the "infalling" and "outfalling" worldlines from an event lie above the horizontal, but you seem to be saying the particle's worldline would be closer to horizontal than the infalling worldline? See my attempt at a diagram below.
 

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  • #22
… no continuity between infalling and outfalling photons …

Hi JesseM! :smile:
JesseM said:
But do you agree that if we look at some event A inside the horizon, then the "infalling" photon worldline and the "outfalling" photon worldline that emanate from A form opposite sides of the future light cone from A?

No … I've always found the light-cone concept extremely confusing :frown:

but it seems to me that "in a locally inertial coordinate system constructed out of freefalling rulers and clocks", the "outfalling" photon should go along the past light-cone, not the future one.

How that gets described in Schwarzschild coordinates, or in EF coordinates, I'm not sure … the two half-cones being one inside the other? :confused:

(In your Winnipeg/East Tennessee diagram, for example, it seems to me that the future light-cone must come out of the page into the θ dimension, and never get anywhere near the outfalling photon.

The circle shown in the diagram is, obviously, continuous … but what continuity can there be from an infalling to an outfalling photon? :confused:)​
Does this mean you think that inside the horizon, the worldline of a particle can actually "escape" the future light cone of an event along it, … ?
No … your diagram does accurately depict what I'm saying … but I don't accept that the cone you have drawn is a future light-cone. :smile:

I maintain that the infalling particle can only follow the same paths as infalling photons … but it goes faster.

And that there are no "horizontal" photons, and so infalling photons cannot be on the same half-cone as outfalling ones. :smile:
 
  • #23
tiny-tim said:
No … I can define "overtaking" in a coordinate-free way:
Unfortunately your definition relies on Schwarzschild coordinates, and on the assumptions that r is distance and t is time, which are invalid assumptions inside the event horizon.

tiny-tim said:
In Schwarzschild coordinates, |dr/dt| for a massive particle is greater than for a photon.
In other words, in Schwarzschild coordinates, |dt/dr| for a massive particle is less than for a photon, so inside the event horizon it moves slower than a photon.

You shouldn't really think of Schwarzschild coordinates as a single coord system. Because of the discontinuity at the event horizon, they are really two different coordinate systems, inside and out, that coincidentally have the same metric equation, although the meanings of r and t differ in the two systems.

tiny-tim said:
Furthermore, in Schwarzschild coordinates, particle 1 and photon 1 hit the singularity before all the other particles and photons, respectively.
As soon as you use the word "before", you are implicitly making use of a time coordinate. If you misinterpret a time coord as a space coord, or vice-versa, you will come to an invalid conclusion.

tiny-tim said:
Therefore, following the world-lines of the particles and photons in any coordinate system, we can identify "particles overtaking photons" as meaning "as a particular photon gets nearer the singularity, it is successively next to freefalling particles which are further from the singularity". :smile:
Sorry, that doesn't even make sense: you seem to be saying "as it gets nearer, it gets further away"!

I would suggest forgetting Schwarzschild coordinates and try the Lemaitre coords I referred to in my last post #15. I've now had a chance to read that Wikipedia page properly and, as far as I can tell, the maths is correct (you can't always trust Wikipedia but in this case it seems OK).

If you convert your problem into Lemaitre coords it may begin to make sense.
 
  • #24
I can afford … … … to board the Chattanooga choo-choo!

Hi DrGreg! :smile:
DrGreg said:
Unfortunately your definition {of "overtaking"} relies on Schwarzschild coordinates, and on the assumptions that r is distance and t is time, which are invalid assumptions inside the event horizon.

No … it's coordinate-free, even for the numbering.

(I accept that I used Schwarzschild coordinates to describe the numbering … but I didn't have to … the concept of being nearer the singularity is coordinate-free.)

For the numbering, I choose the lower numbers to be nearer the singularity. And I'll now confirm expressly what I hope was clearly implied … that both t and r are monontone decreasing (r to 0, and t from ∞), and that is what makes it irrelevant whether we use r or t. :smile:
In other words, in Schwarzschild coordinates, |dt/dr| for a massive particle is less than for a photon, so inside the event horizon it moves slower than a photon.

For coordinate-free analysis, I prefer to rely on the unambiguous concepts of a particle and a photon being "next to" each other, and of one particle (or photon) being nearer the singularity than another particle (or photon).

I said:
tiny-tim said:
In Schwarzschild coordinates, |dr/dt| for a massive particle is greater than for a photon.
because I knew that you knew that, and so it would save on explanation.

I did not claim that "greater |dr/dt|" meant "faster".

Instead, I used it to show which particles would be next to which photons, enabling me to boldly identify "particles overtaking photons" in the passage you don't like:
as a particular photon gets nearer the singularity, it is successively next to freefalling particles which are further from the singularity". :smile:
DrGreg said:
Sorry, that doesn't even make sense: you seem to be saying "as it gets nearer, it gets further away"!.

No! :smile:

Imagine I'm at track 29, Pennsylvania Station, boldly running alongside the Chattanooga choo-choo, in the same direction, and it's going faster than I am. First, I'm next to the driver, then the first car, then the second car …

So I am successively next to cars which are further from Chattanooga ! :smile:

If you now agree that the statement in bold (the original one :redface:) makes sense, do you further agree that it is correct in Schwarzschild coordinates, and therefore in any coordinate system?

Finally … of two objects, which reaches the singularity first, the one with greater |dr/dt|, or the one with greater |dt/dr|?

Well, at the singularity, r must be 0, but t can be anything.

So, using t as a parameter, if they start at the same r and t, the one that gets to r = 0 first will be the one that used up less t.

Using r as a parameter, all we can say is that the one that used up less r will get to any particular t first … but that's not the singularity!

Agreed? :smile:

(btw, any views on continuity of the light-cone between "infalling" and "outfalling" photons?)
 
  • #25
tiny-tim said:
No … I've always found the light-cone concept extremely confusing :frown:
What aspect specifically do you find confusing?
tiny-tim said:
but it seems to me that "in a locally inertial coordinate system constructed out of freefalling rulers and clocks", the "outfalling" photon should go along the past light-cone, not the future one.
What makes you say that? Of course a single photon worldline going through an event must lie on both the past and future light cones, I would say the "outfalling" photon is on the right side of the future light cone, and the left side of the past light cone.
tiny-tim said:
(In your Winnipeg/East Tennessee diagram, for example, it seems to me that the future light-cone must come out of the page into the θ dimension, and never get anywhere near the outfalling photon.

The circle shown in the diagram is, obviously, continuous … but what continuity can there be from an infalling to an outfalling photon? :confused:)​
The diagram only shows the R dimension and the t dimension, so obviously it must only show geodesics that lie within a surface of constant θ. The circle shown in the diagram represents events which can be reached by rays that emanate from the same event but don't necessarily lie in the surface of constant θ. I've attached another crude diagram to try to show what I mean.

I've also attached a second diagram to try to understand where you imagine the future light cone would be. To make things simple, I imagine we're looking at an event E right on the event horizon, in Eddington-Finkelstein coordinates--in this case, as you can see by looking at the Eddington-Finkelstein diagram I showed earlier, the "infalling" ray will lie on a 45-degree angle as always, while the "outfalling" ray will lie right along the event horizon. I say the set of events in the future light cone of E would lie in region "A" in my diagram, in the 45-degree region between the infalling ray and the outfalling ray...if you disagree, where would you put the future light cone of E? In region "B" perhaps?
 

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  • #26
… let's get back to the issue …

Hi JesseM! :smile:
JesseM said:
What makes you say that? Of course a single photon worldline going through an event must lie on both the past and future light cones,
hmm … yeah … what I said about the past light-cone is rubbish, isn't it? :frown:

What I was thinking was this …

The radially infalling photon has a "complementary" oufalling photon which is also radial … in other words, the two tangents are collinear. Similary, by the Equivalence Principle, every non-radial infalling photon must have a "complementary" oufalling photon with collinear tangent.

However, using either Schwarzschild or EF coordinates, at each point inside the horizon, the photons are restricted to a cone, and so it seems to me that there must be two "complementary" cones … and I don't understand how they fit together!

In particular, how do we get continuously from the radially infalling direction to the radially outfalling one?

Outside the horizon, there's no problem … we can just go round in a complete half-circle. But inside the horizon, the surface of the cone blocks the way. Go continuously from the axis to the surface … and then what? … bounce off it? … that's not going to look like those nice circles in your Winnipeg EF diagram, is it? :cry:

So I think I meant:
but it seems to me that "in a locally inertial coordinate system constructed out of freefalling rulers and clocks", "outfalling" photons should go along a "complementary" cone to "infalling" ones.​

Though I still have no idea where the complementary cone fits in. :cry:
I've also attached a second diagram … an event E right on the event horizon, in Eddington-Finkelstein coordinates … where would you put the future light cone of E?

hmm … in Schwarzschild coordinates, I would say the future light-cone of E fills out the whole 180º inward hemisphere … further in, at F say, the future light-cone gets narrower. I don't understand where the complementary light-cone fits in.

To specifically answer your question, I say that it's a one-space-plus-one-time-dimensional diagram, and I don't understand exactly where the future light-cone fits in, but I do say that the off-radial part of the cone goes below the 45º line, and, wherever it is, I don't understand how it can veer round through the positions shown in the Winnipeg diagram. :confused:
Thankyou for pointing out my error about the past light-cone. :smile:

But this really is a side-issue … which is why I've kept it in small print … I didn't mention outfalling photons (either by name or by implication) until my throwaway remark that I wasn't interested in them:
tiny-tim said:
The curved lines are "oufalling" photons, which we're not interested in, in this example.

Then you introduced the diagram at https://www.physicsforums.com/attachment.php?attachmentid=13650&d=1208856819, and asked me to comment on it, and I said (and still maintain) …
tiny-tim said:
… infalling photons cannot be on the same half-cone as outfalling ones

If you want to pursue this outfalling photon/light-cone business, I suggest we start another thread … :smile:

Sooo … anyway … can we please get back to the actual issue, which is whether we can define "overtaking" in a coordinate-free way, whether massive infalling particles overtake infalling photons, and how an inertial observer describes the disappearance of a faster-infalling massive particle if he can't tell he's inside an event horizon? :smile:
 
  • #27
tiny-tim said:
Hi JesseM! :smile:

hmm … yeah … what I said about the past light-cone is rubbish, isn't it? :frown:

What I was thinking was this …

The radially infalling photon has a "complementary" oufalling photon which is also radial … in other words, the two tangents are collinear.
What do you mean by "the two tangents are collinear"? The infalling photon from an event on the diagram always lies on a 45-degree angle, while the outfalling photon through the same event always lies at some other angle.
tiny-tim said:
Similary, by the Equivalence Principle, every non-radial infalling photon must have a "complementary" oufalling photon with collinear tangent.
I don't understand how you're defining "infalling" and "outfalling" photons if they don't lie on a surface of constant θ which contains the event the photons emanate from (and I assume by 'non-radial infalling photon' you mean that they don't lie on such a surface). Only on this purely radial surface does one of the two photon worldlines that intersects the surface look like a straight 45-degree line, which I thought is how you were distinguishing the "infalling" photon from the non-straight "outfalling" one.
tiny-tim said:
However, using either Schwarzschild or EF coordinates, at each point inside the horizon, the photons are restricted to a cone, and so it seems to me that there must be two "complementary" cones … and I don't understand how they fit together!

In particular, how do we get continuously from the radially infalling direction to the radially outfalling one?
Did you disagree with the first of the two diagrams in my previous post? It shows a circle of points which all lie on photon worldlines emanating from an event A, and if you take the two worldlines that lead to points on the circle that lie on the cross-section θ = 0 (the points whose 'shadows' lie right on the radial axes), one of them is the diagonal 'infalling' line and one is the curved 'outfalling' line. Can you not go in a continuous half-circle from one to the other?
tiny-tim said:
Outside the horizon, there's no problem … we can just go round in a complete half-circle. But inside the horizon, the surface of the cone blocks the way.
Blocks the way of what? Seriously, I have no idea what picture you're trying to visualize here. It would really help if you'd post your own diagrams. And if you don't have a scanner or a drawing program to do this (there are plenty of decent drawing programs which you can download for free), at least comment on my diagrams, telling me whether there's something in them you disagree with! In my first diagram, there is nothing preventing you from finding a continuous series of photon worldlines bridging the radial infalling worldline and the radial outfalling one, is there?
tiny-tim said:
Go continuously from the axis to the surface … and then what? … bounce off it? … that's not going to look like those nice circles in your Winnipeg EF diagram, is it? :cry:
Go continuously from what axis to what surface? I thought we were talking about going continuously from one light worldline to another light worldline (infalling to outfalling) via intermediate light worldlines, all of which naturally lie on the surface of the light cone. You really don't do a very good job of communicating your ideas clearly!
tiny-tim said:
To specifically answer your question, I say that it's a one-space-plus-one-time-dimensional diagram, and I don't understand exactly where the future light-cone fits in, but I do say that the off-radial part of the cone goes below the 45º line, and, wherever it is, I don't understand how it can veer round through the positions shown in the Winnipeg diagram. :confused:
But I tried to explain how it can veer around through those positions in my first diagram from the previous post. What about that diagram is unclear to you? And to be clear, when you say the off-radial part of the cone goes below the 45 degree line, are you saying that region B in my second diagram would lie within the future light cone of E, not region A?

Also, do you agree that the diagram from the MTW book Gravitation which I posted shows the same thing as the Winnipeg diagram, with the off-radial part of the cone being above the 45 degree lines? You questioned this diagram because it was described as "edited", but I looked back at the book and the image on the webpage appears identical to the one in the book, maybe they just called it "edited" because they didn't include all of the surrounding page. I can do my own scan of this diagram in the book and post it if you're skeptical. I also have a copy of the Geroch book, and he also shows the light cones as lying above the 45 degree lines, I can post his diagrams as well if you wish. If I do, will you then acknowledge that you have been visualizing light cones and particle worldlines in Eddington-Finkelstein coordinates incorrectly?
tiny-tim said:
But this really is a side-issue … which is why I've kept it in small print … I didn't mention outfalling photons (either by name or by implication) until my throwaway remark that I wasn't interested in them:
But you said your reason for not being interested was because you thought the worldline of the particle would lie below the 45 degree line, and thus it would be passing the "infalling" photons. If I can demonstrate to you that the future light cone is above the 45-degree "infalling" line and below the more vertical "outfalling" line, and that the worldline of massive particles lies inside the light cone and therefore above the "infalling" lines, this will show that the particle does not overtake the infalling lines, but only the outfalling ones. And as I said, this seems little different from the fact that a particle moving to the left in Minkowski spacetime can overtake multiple photons moving to the right, it doesn't indicate anything that would prove the the falling observer that he was inside the event horizon using purely local observations.

Also, if I can get you to agree that the Winnipeg diagram is correct, then I can add the worldline of a faster particle to the diagram, and show that it always remains within the past light cone of the slower particle--there is never a point at which it "disappears" as you seemed to imagine.
tiny-tim said:
Sooo … anyway … can we please get back to the actual issue, which is whether we can define "overtaking" in a coordinate-free way, whether massive infalling particles overtake infalling photons, and how an inertial observer describes the disappearance of a faster-infalling massive particle if he can't tell he's inside an event horizon? :smile:
This is exactly the point I'm addressing. Both the MTW Gravitation diagram and the Geroch diagrams show that the future light cones of events in EF coordinates lie above the 45-degree infalling photons from that event, and that the worldlines of massive particles going through the event are above (ie closer to vertical) than the infalling photon worldline, so the infalling photon hits the singularity "first" in these coordinates...just as depicted in the Winnipeg diagram. Again, if you don't take my word for it I can post scans, but I hope you will agree in advance that if the diagrams in these books do show this, then this is enough authority to demonstrate that your own mental visualizations are incorrect.
 
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  • #28
tiny-tim,

As you are clearly confused by light-cones, I think we need to go back to basics and go through, step-by-step, just what a light cone is. Apologies if you know all this already, but at some point down the line your understanding has gone wrong and we need to find out exactly where.

1. In 4D spacetime the forward lightcone, at a specific event E, consists of all the worldlines of all the photons emitted from E in every direction. In flat spacetime and standard inertial coordinates, these photons form an expanding sphere centred on the emission point. But it's hard to draw 4D diagrams so...

2. In flat 3D spacetime (2D space) and standard inertial coordinates, the forward lightcone is generated by an expanding circle of photons centred on the emission point. When we plot this in 3D spacetime we get an upside-down vertical-axis cone. With the convention that c=1, the sides of the cone are inclined at 45°. In non-inertial coords or in curved spacetime, the cone might be tipped over at an angle, the vertex half-angle needn't be 45°, and the cone may be distorted in shape. Most of the spacetime diagrams you have been looking at show only the tip of the cone, as the rest of the warped cone is of less interest. Note that if you choose locally-inertial coordinates at E, the tip of the cone will look like a standard upside-down 45° cone -- that's pretty much the definition of what an inertial frame is, translated into spacetime geometry.

3. In flat 2D spacetime (1D space) and standard inertial coordinates, the forward lightcone is generated by a pair of photons traveling in opposite directions from the emission point. When we plot this in 2D spacetime we get a pair of lines at 45° to the vertical. When spacetime is not flat or the coords are not inertial, the lines can be at other angles and can be curved.

In all cases, a massive particle passing through E must always lie within the sphere/circle/pair of photons (because it can't go faster than light) and so the worldline of the particle must always lie in the interior of the cone.

The worldline of any photon (on the surface of the cone), in any coordinate system, is given by ds=0. The worldline of any massive particle (inside the cone) must always satisfy ds2>0 (with the +--- signature metric we have been using in this thread), and s is the proper time measured along the worldline.

Note that in the one-space-dimension case, we must have

[tex] c_- < v < c_+ [/tex]​

where [itex] c_- [/itex] is the coordinate velocity of light in the negative direction, [itex] c_+ [/itex] is the coordinate velocity of light in the positive direction, and [itex] v [/itex] is the coordinate velocity of a massive particle. In inertial coords, [itex] c_- = -1[/itex] and [itex] c_+ = +1[/itex], but in other coords, different values are possible.

In our context of radial motion towards a black hole, [itex] c_- [/itex] is the coordinate velocity of an "infalling" photon and [itex] c_+ [/itex] is the coordinate velocity of an "outfalling" photon.

Note that the "infalling" and "outfalling" photons are simply opposite sides of a single cone. There are certainly not two different cones for each. In the 2D-space case, you have a whole (distorted) circle of photons, of which your radially infalling and radially outfalling are just two, within the circle circumference. The "continuity" comes from considering the whole "circle".

OK so far?

Now for your example of particles apparently overtaking photons. I attach a diagram to illustrate this. The problem is, in fact, due to you confusing the infalling and outfalling photons. It is perfectly true that, in the picture shown on the left, that the particles shown will get to the singularity (to the left) before the photons. The reason for this is that these photons are the outfalling photons, which are trying, but failing, to escape from the black hole. There's nothing going faster than light because the photons are going in the opposite direction to the particles. The relevant equation is

[tex] c_- < v < c_+ < 0 [/tex]​

so it is true that [itex]|v| > |c_+|[/itex], but that doesn't mean faster than light. You have to compare signed velocities, not unsigned speeds.

The picture on the right shows, in an approximate way, the bigger picture. (It is not drawn in any specific coord system -- and certainly not in Schwarzschild coords, for which JesseM has already provided the picture in post #19.)
 

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  • #29
DrGreg said:
There's nothing going faster than light because the photons are going in the opposite direction to the particles. The relevant equation is

[tex] c_- < v < c_+ < 0 [/tex]​

so it is true that [itex]|v| > |c_+|[/itex], but that doesn't mean faster than light. You have to compare signed velocities, not unsigned speeds.

Hi DrGreg! :smile:

Thanks for the assistance :smile: … I really am totally perplexed by these three-dimensional surfaces in four-dimensional space-time! :redface:

I'm working carefully through what you've written.

Would you clear up one thing which is confusing me, about the above passage?

You say "the photons are going in the opposite direction to the particles" … which presumably is in inertial coordinates … but then surely [tex]c_- < v < c_+ < 0[/tex] is wrong? … in EF, coordinates it's not clear to me what v is … and in Schwarzschild coordinates, inside the horizon, t and r both decrease (t from +∞, and r to 0), so dr/dt for any photon or particle is positive?
 
  • #30
tiny-tim said:
Hi DrGreg! :smile:

Thanks for the assistance :smile: … I really am totally perplexed by these three-dimensional surfaces in four-dimensional space-time! :redface:
It's much easier if you drop one or two spatial dimensions, so that you have either 2D surfaces in 3D spacetime (the surface of a cone) or 1D surface in 2D spacetime (the two photon worldlines emanating from an event).
tiny-tim said:
You say "the photons are going in the opposite direction to the particles" … which presumably is in inertial coordinates … but then surely [tex]c_- < v < c_+ < 0[/tex] is wrong?
I presume the < 0 part was only meant to be inside the horizon, in a coordinate system like Einstein-Finkelstein coordinates. In an inertial coordinate system, the velocity of one light beam is always -c < 0, while the velocity of the other light beam is always +c > 0.
tiny-tim said:
in EF, coordinates it's not clear to me what v is
If you're using EF coordinates with all but one space coordinate dropped (leaving only the r coordinate and the t coordinate), v is just dr/dt, which is the inverse of the slope of the worldline in the Winnipeg diagram (closer to horizontal means smaller slope but greater dr/dt, and closer to vertical means larger slope but smaller dr/dt).
tiny-tim said:
and in Schwarzschild coordinates, inside the horizon, t and r both decrease (t from +∞, and r to 0), so dr/dt for any photon or particle is positive?
It's not the coordinate time t that decreases in the diagram, but rather the proper time [tex]\tau[/tex] along the worldline of a falling particle. In 2D Schwarzschild coordinates with only the r space coordinate and the t coordinate, the falling particle seems to be moving backward in time, and one of the two photons emitted from an event on its worldline is also going back in time, so that despite the fact that the slope of the particle's worldline is closer to horizontal (so that the absolute value of dr/dt is greater for the particle than the photon), the photon hits the singularity at an earlier t coordinate than the particle (see the scanned diagram in post #19, and extrapolate the path of photons on the 'bottom' part of light cones inside the horizon to see why they'll hit the singularity earlier). So just talking about dr/dt can be especially confusing in Schwarzschild coordinates, because dr/dt doesn't tell you whether an object is moving forward or backward in time, it only tells you the slope of the worldline in these coordinates. This is another reason it would probably be easier to stick to EF coordinates!
 
  • #31
JesseM said:
tiny-tim said:
You say "the photons are going in the opposite direction to the particles" … which presumably is in inertial coordinates … but then surely [tex]c_- < v < c_+ < 0[/tex] is wrong?
I presume the < 0 part was only meant to be inside the horizon, in a coordinate system like Einstein-Finkelstein coordinates. In an inertial coordinate system, the velocity of one light beam is always -c < 0, while the velocity of the other light beam is always +c > 0.
Sorry, I got this wrong. c- and c+ must always be opposite signs, unless one of them is zero or infinite. My mental picture was switching between different coordinate systems and got confused. In fact, in Schwarzschild coords, c- = -c+.

The condition [itex] c_- < v < c_+ [/itex] applies when all three velocities are coordinate distance divided by coordinate time, where "coordinate distance" is a space-like coordinate, and "coordinate time" is a timelike or null coordinate. ("Null 'time'" allows for the possibility c+ = 0.) So in Schwarzschild coords, inside the event horizon, it applies to dt/dr but not dr/dt. It's not clear to me whether this helps at all in Eddington-Finkelstein coords, where (in the radial 1D case) there is no spacelike coordinate.

(Now I'm feeling confused!)

The above observations arise from considering that, in 2D spacetime, any metric is a quadratic form that can be factorised as

[tex]ds^2 = A(c_- dt - dx)(c_+ dt - dx)[/tex]
[tex] = A c_- c_+ dt^2 - A (c_- + c_+)dt dx + A dx^2[/tex]​
for some A. If x is spacelike, A must be negative (with a +--- metric signature - put dt=0). For a massive particle, [itex]dx = v dt[/itex], ds2 has to be positive.
 
  • #32
DrGreg said:
Sorry, I got this wrong. c- and c+ must always be opposite signs, unless one of them is zero or infinite. My mental picture was switching between different coordinate systems and got confused. In fact, in Schwarzschild coords, c- = -c+.
But they aren't opposite signs in Eddington-Finkelstein coordinates inside the horizon, are they? Both the infalling and outfalling worldlines are going from lower right to upper left (if the singularity is to the left of the event horizon).
 
  • #33
… a stitch-up … ?

Hi DrGreg! :smile:

Thanks for the clarification, and for the previous help on light-cones. :smile:

I've done a lot of thinking, and found a succinct way of describing my worries.

Imagine an observer O freefalling down a radius inside a horizon, and a particle P freefalling faster down an adjacent radius.

In Schwarzschild coordinates, O has a forward and backward light-cone which move with him. Let P meet the surface of these cones at events A and B, respectively. Then a photon from event A goes down the surface of the backward cone (infinitely fast in Schwarzschild coordinates) to meet O at C, say. At a later event D, a photon goes down the surface of the forward cone (infinitely fast in Schwarzschild coordinates) to meet P at B.

Before O reaches C, O can receive photons from P, from positions before A. But O cannot send photons to P (wherever P is).

After D, O can send photons to P, at positions after B, but cannot receive photons from P (wherever P is).

In inertial coordinates, with time and radial distance "interchanged", events A and C have the same r coordinate (because in Schwarzschild coordinates they are at the same time), and so do events B and D.

So O sees the "y" coordinate of P stay constant, but the r coordinate decrease until it is 0 (so P is on the y-axis).

Then O sees P no more … but O can wait until event D, and then send photons to P, at B and beyond; and at D, P is again on the y-axis.

And O can neither send photons to P nor receive them from P, when P is between A and B

So, if O uses inertial coordinates, O can tell he is inside a horizon because everything disappears (or appears) on the y-axis … and this is in any neighbourhood of O, however small.

Perhaps the paradox comes from correctly saying that we can interchange dt² and dx², but not noticing that that leaves an ambiguity between ±t?

The inertial coordinates are literally a stitch-up, since they open out the two Schwarzschild half-cones into hemispheres on the y-z-plane, and join them without regard either to the missing space between or to the sign of t.

:smile: Where have I gone wrong? :smile:
 
  • #34
JesseM said:
But they aren't opposite signs in Eddington-Finkelstein coordinates inside the horizon, are they? Both the infalling and outfalling worldlines are going from lower right to upper left (if the singularity is to the left of the event horizon).
In my previous post #31, I said that the lightspeeds had opposite signs when speed was (spacelike coord) divided by (timelike or null coord). Due to my confusion over what is a null coord (see this new thread), I must withdraw that and say I think it applies only if speed is spacelike divided by timelike. One of the Eddington-Finkelstein coordinates is null, so it doesn't apply in that case. (In fact one of the lightspeeds is infinite in that case -- don't forget EF diagrams are often "skewed" to show the the surfaces of constant v as cones when really they should be horizontal planes).
 
  • #35
tiny-tim

I'm having great difficulty understanding post #33. I think your idea of a light cone still differs significantly from mine. It's not clear whether each of the various letters you use refers to an event (location at a specific time, a zero-dimensional "point" in spacetime), or to a particle/photon (lots of locations spread over time, a one-dimensional worldline (curve or line) in spacetime).

Don't forget, nothing moves is spacetime, as one of the dimensions in the diagram is time. A point that moves in space corresponds to a static line in spacetime.

It would really help if you could work out a way to enclose a diagram.

E.g. if you possesses a copy of Microsoft Word, you can draw diagrams in that, then use Alt+PrintScreen to copy a snaphsot to the clipboard, paste that into Microsoft Photo Editor, crop to size, save as a PNG and then attach that (via "manage attachments") to your post. If you don't have Microsoft Word, you may have something else you can draw with.
 

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