The first probe moves toward infinity. The second probe toward the singularity.
Globally, yes. Within the LIF, it's impossible to tell, just from the way either probe is moving within the LIF, what is going to happen to it after it leaves the LIF.
Also, to the extent that "toward infinity" and "toward the singularity" can be defined within the LIF, they aren't the directions you think they are--more precisely, the latter isn't. See below.
They each move toward destinations beyond opposite sides of the LIF.
This is wrong and I've already explained why it's wrong, but I'll recap again since apparently you aren't reading my posts very carefully. The singularity is not in the negative x direction in the LIF; it's in the positive t direction, i.e., in the future. Infinity is more or less in the positive x direction, yes. But that means that both probes, as far as you can tell within the LIF, are moving in both directions: toward the singularity (in the positive t direction) and toward infinity (in the positive x direction). There's no way to tell, from within the LIF, where the probes will end up.
(You could, of course, define "toward the singularity" as "decreasing r coordinate" and "toward infinity" as "increasing r coordinate"; but as I have already shown multiple times, the respective probes satisfy those definitions while both moving in the positive x direction within the LIF; and as I've also said multiple times, the r coordinate is irrelevant to the LIF since it's global, not local.)
you wouldn't be able to show that the probes approach each other in the skydiver's frame
Wrong; I already have. But if it will help, I am working on a second spacetime diagram that shows the skydiver's LIF, to complement the one showing the black hole LIF that I already posted on PhysicsForums (see the link I posted in this thread). What that diagram will show is that the only difference between the two LIFs is how the lines of constant r coordinate appear; everything else is identical.
given the first probe moving toward higher r-coordinates and the second probe moving toward lower r-coordinates
This isn't true in the skydiver's LIF; it's only true in the black hole LIF. As I've already pointed out. And which is irrelevant to the equivalence principle anyway, since the r coordinate is global, not local. How many times are you going to repeat the same erroneous statements?
GR places additional conditions on what happens in a LIF, when it predicts that everything below a horizon falls toward the singularity
No, it doesn't. You can give a definition of what "toward the singularity" means in a given LIF (I did it above for the black hole LIF--note that it isn't what you think it is); but you can't restrict what happens within a given LIF based on which direction "toward the singularity" is. All a global definition like "toward the singularity" does is tell you the relationship between local conditions in the LIF and some global condition. That can help you to determine what the right local conditions are in an LIF to model some desired global condition; but it can't ever tell you that a given local condition that would be permissible in special relativity is not permissible in an LIF.
we must be able to set up a thought experiment that accounts for that prediction within an LIF straddling the horizon.
Yes, and I've already shown, multiple times, how the thought experiment we've been discussing does this.
For example, it's okay to have an experiment in an LIF that says "fire a particle toward Vega", even though Vega is outside the LIF, as long as the measurements take place wholly within the LIF.
Sure; and the way you would model that experiment is to determine what condition in the LIF corresponds to the global condition "fire a particle toward Vega".
Similarly, in the thought experiment under discussion, the way we model it in the LIF crossing the horizon is to determine what local condition in that LIF corresponds to "the first probe is fired at escape velocity just outside the horizon", and "the second probe is fired at a higher velocity, relative to the astronaut, than the first probe, but just inside the horizon". And then we have to check that the two probes' trajectories, in the LIF, that we come up with meet the criteria of "first probe is moving toward infinity" and "second probe is moving toward the singularity". And I've already shown, multiple times, how all these conditions are met, and how your claims that they are not met are based on an erroneous understanding of how the global conditions translate into local conditions in the LIF.
At this point, I'm tempted to conclude that you're not really trying to understand my posts, but just looking for things to disagree with based on the erroneous straw-man version of relativity that you have in your head. You keep on repeating the same errors, even though I've already shown why they're errors, and you keep on missing the things I'm telling you about how relativity actually models this scenario, that address the concerns you're raising. I really think you need to step back and take an honest look at what I've been saying.
I realize I've been confusing, for which I apologize. I change my law of physics to: Two free test particles each moving toward destinations beyond opposite sides of a LIF, as measured in a global frame containing the LIF, recede from each other as measured in that LIF. I have always accepted that the particles move in one direction as measured in the LIFs.
I've given this a lot of thought to try to see where our major disagreement is. It's hard to tell from your walls of text. I now think it's this: You disagree that any prediction for a global frame can affect experiments in the LIF. Therefore you ignore such predictions and draw spacetime diagrams as one would in SR. Of course when you do that you find no difference between two LIFs. I say you're not seeing the forest for the trees.
Forget about black holes and light rays and especially complex GR terminology. Focus on the simpler skydiver's frame. Let the analogous prediction for the global frame be this: any particle below 1 km above sea level must fall inexorably toward r = 0. Let your first probe be launched just above the 1 km mark. It doesn't need to be escaping, just always moving away from the Earth during our experiment. Let the second probe be launched just below the 1 km mark. As measured in any LIF containing both particles, they'll recede from each other. As measured in a global frame, the first probe is moving toward a destination beyond the side of the LIF that's facing away from the Earth, while the second probe is moving toward a destination beyond the opposite side of the LIF that's facing toward the Earth.
For the record I still believe my argument using megaparsec-sized frames was sound. Your counterargument didn't explicitly show that any sentence in mine was false or didn't follow from its premises. Also I still don't know from this discussion how the argument in the blog isn't sound.
Let the analogous prediction for the global frame be this: any particle below 1 km above sea level must fall inexorably toward r = 0. Let your first probe be launched just above the 1 km mark. It doesn't need to be escaping, just always moving away from the Earth during our experiment. Let the second probe be launched just below the 1 km mark. As measured in any LIF containing both particles, they'll recede from each other.
On thinking this over, I realized that even in this "skydiver" LIF, you can, in fact, set up initial conditions so that the two probes are converging, even though one probe's r coordinate is increasing and the other's is decreasing. Here's how:
At time t = minus epsilon in the LIF, the "skydiver", who is at rest in the LIF, launches the first probe. At that instant, his downward velocity, relative to an observer who is "hovering" at constant global radial coordinate r, is v1. That means that, in the skydiver LIF, an object with constant r that passes through coordinates x = 0, t = minus epsilon has velocity v1 in the positive x direction at that instant. So the skydiver launches the first probe in the positive x direction with velocity v1 + a, where a is some small constant; that means the first probe's r coordinate is increasing.
At time t = 0 in the LIF, the skydiver passes the 1 km mark.
At time t = plus epsilon in the LIF, the skydiver launches the second probe. At that instant, his downward velocity, relative to an observer who is "hovering" at constant global radial coordinate r, is v2, and v2 > v1 (because, relative to observers who are "hovering" at constant r, the skydiver is accelerating downward). So, relative to the LIF, an object with constant r that passes through coordinates x = 0, t = plus epsilon has velocity v2 in the positive x direction at that instant. So the skydiver launches the second probe in the positive x direction with velocity v2 - b, where b is some small constant; that means the second probe's r coordinate is decreasing.
Now all we have to do is choose a and b so that v2 - b > v1 + a; i.e., the velocity of the second probe, relative to the skydiver (and therefore relative to the LIF) is larger than that of the first probe. (This is always possible because v2 > v1, as above.) That means the two probes will be converging, not diverging; and yet the first probe's r coordinate is increasing while the second probe's r coordinate is decreasing.
Notice the key facts that make the above possible:
(1) The skydiver is falling downwards, with respect to the Earth, with nonzero velocity. That means an object can be moving in the positive x direction in the LIF but still be falling downwards, as long as it's falling slower, with respect to the Earth, than the skydiver.
(2) The skydiver's downward velocity, relative to observers "hovering" at constant r, increases as he falls. That is what makes v2 > v1, and thus "makes room" for the second probe's velocity, relative to the LIF, to be larger than the first probe's, so that the two converge.
You may object: but that's tidal gravity, isn't it? No, it isn't; it's just downward acceleration. The above argument holds even if the skydiver's downward acceleration, relative to observers "hovering" at constant r, does not change (which of course it can't within the LIF, since tidal gravity is by definition negligible within the LIF).
Note also that none of the above changes what I've said before; all the things I said about how the relationship between global and local is very different for the astronaut LIF as compared to the skydiver LIF are still true. But perhaps the above will help show how, even in a highly non-relativistic case (all the velocities in the above example are very small compared to the speed of light), the relationship between the global r coordinate and the local coordinates in a free-falling LIF is not quite what you might think it is.
Responding to the bulk of your post separately from the last item about the blog argument, to reduce clutter:
Two free test particles each moving toward destinations beyond opposite sides of a LIF, as measured in a global frame containing the LIF, recede from each other as measured in that LIF.
This law is fine, but it doesn't apply in the case under discussion, because the two probes are not moving "toward destinations beyond opposite sides" of the LIF in question. I've repeatedly explained why not.
You disagree that any prediction for a global frame can affect experiments in the LIF.
No; the problem is that you are incorrectly translating predictions in a global frame into predictions for experiments in the LIF. I've repeatedly explained why your translations are incorrect. One example of an incorrect translation is that you think the two probes are moving "toward destinations beyond opposite sides" of the LIF. The correct translation is that the singularity is in the positive t direction in the LIF, while infinity is in the positive x direction; these are not opposite sides of the LIF.
Forget about black holes and light rays and especially complex GR terminology.
In other words, forget about the fact that the relationship between global parameters and local parameters is different for different LIFs, which is the whole point under discussion. The relationship between global and local is different for an LIF falling through a black hole horizon than it is for a "skydiver" frame free-falling at 1 km above the Earth's surface.
Let the analogous prediction for the global frame be this: any particle below 1 km above sea level must fall inexorably toward r = 0. Let your first probe be launched just above the 1 km mark. It doesn't need to be escaping, just always moving away from the Earth during our experiment. Let the second probe be launched just below the 1 km mark. As measured in any LIF containing both particles, they'll recede from each other.
If you set up the initial conditions that way, sure. But that already makes this experiment different from the one posed in the blog post, because the initial conditions in that one were that, as measured in the LIF, the probes were approaching each other.
As measured in a global frame, the first probe is moving toward a destination beyond the side of the LIF that's facing away from the Earth, while the second probe is moving toward a destination beyond the opposite side of the LIF that's facing toward the Earth.
Sure, because this LIF is at a radial coordinate that is way, way, way larger (about a million times larger) than the Schwarzschild radius corresponding to the mass of the Earth. So the relationship between directions within the LIF and directions defined globally is very, very different than it would be in an LIF that was free-falling just at the Schwarzschild radius. As long as you continue to ignore this huge difference, even though I've explained it multiple times and given you links to a pair of spacetime diagrams that illustrate it, you will continue to make the same mistakes you've been making.
For the record I still believe my argument using megaparsec-sized frames was sound. Your counterargument didn't explicitly show that any sentence in mine was false or didn't follow from its premises.
That's because I'm not going to hold your hand when making arguments; I expect you to use your intelligence. Your argument, as I remember it, assumed that you can independently adjust the size of the LIF and the initial velocities of the probes. You can't do that, because both of those things depend on the mass of the black hole. The dependence of LIF size on the mass of the hole should be obvious. The probe initial velocities depend on the mass of the hole because that affects what the escape velocity is at the point that the first probe is launched. So there is only one adjustable parameter: the mass of the hole. And my calculation showed how the LIF size is much, much smaller than the distance it would take for the second probe to catch the first probe, regardless of the mass of the hole.
(Edit: Having looked back, I see you also claimed, as far as I can understand your argument, that if the "catch-up" happens within the skydiver LIF, even if it doesn't happen within the astronaut LIF, that somehow invalidates the equivalence principle. That's wrong, and I explained why: the smaller of the two LIF sizes determines the range of comparison. If the "catch-up" happens within the skydiver LIF but not within the astronaut LIF, it must be because the skydiver LIF is the larger of the two.)
I'm responding to this separately to reduce clutter. I'll post the rest of my response to your post after this one.
I still don't know from this discussion how the argument in the blog
isn't sound.
Sigh. Here are the basics boiled down as much as I can. I'll base my
comments on what you've said in this thread, since what you've said
here is a lot more coherent to me than the blog post itself is. I'll
give just bare statements, with no supporting argument; I've already
given the supporting arguments many, many times in this thread.
- You have claimed that, within the astronaut LIF, the
directions "toward the singularity" and "toward infinity" are opposite
directions. That's wrong; they're not.
- You have claimed that, since "toward r = 0" and "toward infinity" are
opposite directions in the skydiver LIF, they must also be opposite
directions in any LIF. (You made the claim about the astronaut LIF,
but the way you made the claim makes it clear that you would make the
same claim about any LIF whatsoever). That's wrong; see above.
- You have claimed that, if we choose the mass of the black hole
appropriately, we can make it so that, if the two probes are initially
converging, the second will catch up to the first within the astronaut
LIF. That's wrong; we can't.
- You have claimed that, if we can set up the skydiver LIF so that the second probe catches the first within that LIF, that somehow invalidates the equivalence principle, even if the second probe can't catch the first within the astronaut LIF. That's wrong; it doesn't.
- You have claimed that, if the first probe has an increasing global r
coordinate while the second has a decreasing global r coordinate, the
two probes can't be converging within the astronaut LIF. That's wrong;
they can.
- You have claimed that, since the two probes must be diverging in the
skydiver LIF if their r coordinates behave as above, they must also be
diverging in any LIF if their r coordinates behave as above. (Again,
you made the claim about the astronaut LIF, but the way you made the
claim makes it clear that you would make the same claim about any LIF
whatsoever). That's wrong; see above.
And just for comparison, I'll briefly summarize what GR actually says
about the astronaut scenario; again, just bare statements since I've
already given the supporting arguments many times in this thread:
- As the initial conditions are given, the two probes will converge
within the astronaut LIF.
- Within the astronaut LIF, the black hole's horizon is a light ray
moving in the positive x direction, which passes the astronaut halfway
between him launching the first and second probes. So clearly the
second probe will remain below the horizon, even based on observations
solely within the LIF, since it starts out behind the light ray and is
moving slower.
- Within the astronaut LIF, the horizon will (slowly) catch up to the
first probe. However, the distance it would take for the horizon to
catch up, based on its "closure rate" within the astronaut LIF, is
much, much larger than the size of the LIF. This holds regardless of
the mass of the black hole. And if the horizon can't catch the first
probe within the LIF, the second probe certainly can't either.
- Once the probes and the horizon have exited the astronaut LIF, tidal
gravity can no longer be neglected; and tidal gravity will cause the
horizon to stop catching up with the first probe and start falling
behind it. Eventually, the first probe will escape to infinity.
Globally, yes. Within the LIF, it's impossible to tell, just from the way either probe is moving within the LIF, what is going to happen to it after it leaves the LIF.
Also, to the extent that "toward infinity" and "toward the singularity" can be defined within the LIF, they aren't the directions you think they are--more precisely, the latter isn't. See below.
They each move toward destinations beyond opposite sides of the LIF.
This is wrong and I've already explained why it's wrong, but I'll recap again since apparently you aren't reading my posts very carefully. The singularity is not in the negative x direction in the LIF; it's in the positive t direction, i.e., in the future. Infinity is more or less in the positive x direction, yes. But that means that both probes, as far as you can tell within the LIF, are moving in both directions: toward the singularity (in the positive t direction) and toward infinity (in the positive x direction). There's no way to tell, from within the LIF, where the probes will end up.
(You could, of course, define "toward the singularity" as "decreasing r coordinate" and "toward infinity" as "increasing r coordinate"; but as I have already shown multiple times, the respective probes satisfy those definitions while both moving in the positive x direction within the LIF; and as I've also said multiple times, the r coordinate is irrelevant to the LIF since it's global, not local.)
you wouldn't be able to show that the probes approach each other in the skydiver's frame
Wrong; I already have. But if it will help, I am working on a second spacetime diagram that shows the skydiver's LIF, to complement the one showing the black hole LIF that I already posted on PhysicsForums (see the link I posted in this thread). What that diagram will show is that the only difference between the two LIFs is how the lines of constant r coordinate appear; everything else is identical.
given the first probe moving toward higher r-coordinates and the second probe moving toward lower r-coordinates
This isn't true in the skydiver's LIF; it's only true in the black hole LIF. As I've already pointed out. And which is irrelevant to the equivalence principle anyway, since the r coordinate is global, not local. How many times are you going to repeat the same erroneous statements?
GR places additional conditions on what happens in a LIF, when it predicts that everything below a horizon falls toward the singularity
No, it doesn't. You can give a definition of what "toward the singularity" means in a given LIF (I did it above for the black hole LIF--note that it isn't what you think it is); but you can't restrict what happens within a given LIF based on which direction "toward the singularity" is. All a global definition like "toward the singularity" does is tell you the relationship between local conditions in the LIF and some global condition. That can help you to determine what the right local conditions are in an LIF to model some desired global condition; but it can't ever tell you that a given local condition that would be permissible in special relativity is not permissible in an LIF.
we must be able to set up a thought experiment that accounts for that prediction within an LIF straddling the horizon.
Yes, and I've already shown, multiple times, how the thought experiment we've been discussing does this.
For example, it's okay to have an experiment in an LIF that says "fire a particle toward Vega", even though Vega is outside the LIF, as long as the measurements take place wholly within the LIF.
Sure; and the way you would model that experiment is to determine what condition in the LIF corresponds to the global condition "fire a particle toward Vega".
Similarly, in the thought experiment under discussion, the way we model it in the LIF crossing the horizon is to determine what local condition in that LIF corresponds to "the first probe is fired at escape velocity just outside the horizon", and "the second probe is fired at a higher velocity, relative to the astronaut, than the first probe, but just inside the horizon". And then we have to check that the two probes' trajectories, in the LIF, that we come up with meet the criteria of "first probe is moving toward infinity" and "second probe is moving toward the singularity". And I've already shown, multiple times, how all these conditions are met, and how your claims that they are not met are based on an erroneous understanding of how the global conditions translate into local conditions in the LIF.
At this point, I'm tempted to conclude that you're not really trying to understand my posts, but just looking for things to disagree with based on the erroneous straw-man version of relativity that you have in your head. You keep on repeating the same errors, even though I've already shown why they're errors, and you keep on missing the things I'm telling you about how relativity actually models this scenario, that address the concerns you're raising. I really think you need to step back and take an honest look at what I've been saying.