On the nature of the "infinite" fall toward the EH

PeterDonis

Do you realize what you've just done? You've just given a description of proper time.

Suppose that, in addition to Alice and Bob, we have Charlie, who is hovering at a constant altitude close to, but above, the horizon. Charlie's "rate of flow of time" is slower than Bob's. How do we know? Because we can compute Charlie's proper time along his worldline, and verify that it "ticks" at exactly the same rate as his instruments, physiological and cognitive processes, etc.

The *same* reasoning, and the *same* kind of computation, tells us that Alice experiences a finite amount of time to the horizon. Her instruments record a finite amount of time; her physiological and cognitive processes progress by a finite amount of time; etc. So if the reasoning applies to Charlie, it should apply to Alice as well.

pervect

I've come up with a somewhat simpler approach for presenting the solution for the EF geodesic equations.

Let r, v be the Eddington-Finklesein (EF) coordinates, which are presumed to be functions of proper time [itex]\lambda[/itex]. Then let:

[tex]\dot{r} = \frac{dr}{d\lambda} \hspace{3cm} \dot{v} = \frac{dv}{d\lambda}[/tex]

The Ingoing Eddington Finklestein metric (gemoetrized) is
http://en.wikipedia.org/w/index.php?...ldid=516198830

[tex]-(1-2m/r) dv^2 + 2\,dv\,dr [/tex]

The Christoffel symbols are:

[tex]\Gamma^{v}{}_{vv} = \frac{m}{r^2}[/tex]
[tex]\Gamma^{r}{}_{vv} = \frac{m(1-2m/r)}{r^2}[/tex]
[tex]\Gamma^{r}{}_{vr} = \Gamma^{r}{}_{rv} = -\frac{m}{r^2}[/tex]


So we can write the geodesic equations as

[tex]
\ddot{v} + \Gamma^{v}{}_{vv} \dot{v}^2 = \frac{d \dot{v}}{dr} \dot{r} + \frac{m}{r^2} \dot{v}^2 = 0
[/tex]

[tex]
\ddot{r} + \Gamma^{r}{}_{vv} \dot{v}^2 + 2 \,\Gamma^{r}{}_{vr} \dot{v} \dot{r} = \frac{d \dot{r}}{dr} \dot{r} + (1-2m/r)\frac{m}{r^2} \dot{v}^2 - 2 \frac{m}{r^2} \dot{r}\dot{v} = 0
[/tex]

Note that I've used the chain rule to write
[tex]\ddot{v} = \frac{d^2 v}{d \lambda^2} = \frac{d \dot{v}}{d \lambda} = \frac{d \dot{v}}{dr}\frac{dr}{d\lambda} = \frac{d \dot{v}}{dr} \dot{r} [/tex]

Then we can write the solution in infalling EF coordinates for m=2 as:

[tex]\dot{r} = - \sqrt{\frac{4}{r}}[/tex]
[tex]\dot{v} = \frac{\sqrt{r}}{2+\sqrt{r}}
[/tex]

And just plug them into the geodesic equations above to demonstrate that they are a solution.

Note that [itex]\dot{r}[/itex] is negative, the first post had a sign error for the equivalent expression.

The math here is only mildly obnoxious compared to the previous expressions, though I've skipped over a lot of textbook stuff like computing the EF metric (the Wiki does that), and computing the Christoffel symbols for said metric.

[add]If you want r as a function of [itex]\lambda[/itex], it remains
[tex]r = \left(9 \lambda^2\right)^{\frac{1}{3}}[/tex]

PAllen

As long as you agree not to call it GR, and not to promote it in these forums (see the rules), that is fine by me. In fact, it would be quite instructive to work out how to make such a theory precise (I just gave a hand wave description of it; I know how I would start trying to make it precise, if I cared to, but don't know what logical or mathematical conundrums might crop up).

PAllen

They do make the same predictions. Let's get to this: do you think it is somehow required in GR that if all the events (and all predictions about them) in one coordinate system are a subset of those in another coordinate system, the GR says only the coordinate system with least coverage counts? Rather than saying, woops, one coordinate system is as good as any other for what it covers, but you may have to use overlapping coordinate systems to cover the whole of existence.

PeterDonis

I see another argument looming about what "prediction" means. :sigh:

A better way to say it might be: "for all events covered by both charts, all invariants at those events come out the same when computed in both charts".

PAllen

Consider what this modification might look like, classically, and assuming we want to keep the coordinate independent nature of the equations of GR.

1) We must add a couple of new axioms the theory: Universes containing naked singularities are prohibited (as a corollary, closed universes are prohibited because event horizons cannot technically be defined for them; the required new law I give next cannot be stated for a closed universe). Eternal WH-BH are prohibited. (Much stronger than 'we think not physically plausible').

2) We supplement the EFE with a new universal boundary law: The universe is bounded (chopped in spacetime) such that the world line of any particle or fluid element always has null paths extending from it to null infinity.

Don't you find it contrived to muck up a beautiful, elegant theory with such additions?

[Edit: An example of how strange this modified theory is shown by examining the history of a late infaller for an O-S type collapse. It is a requirement of this theory that some matter disappear from existence at a certain finite local time. Freezing won't work. The reason is that a late infaller following the collapse has their world line chopped at the horizon, and this late infaller has encountered no matter on the way. There is no possible way to avoid this while keeping the EFE in any form. This means that all the orginal collapsing matter vanished, not just froze. To avoid this, we need to modify the EFE itself such that matter world lines in the collapsing body follow different spacetime trajectories than the EFE predicts, such that the dust boundary exists outside the EH when the later infaller encounters it (at the horizon). This new prediction cannot be accommodated without significant change to the EFE itself.]

rjbeery

This is true until "it isn't".

If Alice's local "rate of t" were reduced to zero then then Alice would never know it; she would simply freeze and be oblivious to it for eternity. To be clear, I'm not saying this is what happens at the EH according to GR, I'm just pointing out that the usual refutation against the distant observer proclaiming that Alice freezes is that time does not slow down locally in her frame according to her experience; this on its own is not a valid refutation.

Nugatory

That's a stronger statement than "valid coordinate systems should never make different predictions"; the latter statement allows for the possibility that one of the coordinate systems makes no prediction in some regions.

I think the weaker formulation is both more practical and more widely accepted. Consider, for example, the way that two-dimensional hyperbolic coordinates allow me to make predictions only in one quadrant of a plane, whereas Cartesian coordinates work for the entire plane. No one would seriously argue that the broader Cartesian coordinates are illegitimate because they make predictions where hyperbolic coordinates don't.

But this is basically the situation that we have when we write the Schwarzchild solution for the vacuum around a spherically symmetric non-rotationg massive body in either Schwarzchild coordinates or (for example) KS coordinates. We never get disagreeing predictions, but we do find regions of spacetime where the KS coordinates make predictions and the SC coordinates do not. Some of these predictions (both in and out of the region of overlap) may strike us as non-physical, but that's not a problem with the coordinates.

rjbeery

Actually, I don't see these as mucking anything up. On a philosophical level, I think the concept of infinity has no physicality whatsoever and the Universe should be able to be described without it.

pervect

The underlying thought process here is that there is some physically meaningful way to define a "local rate of time". Relativity doesn't necessarily say this. (I think one can make even stronger claims, but it'd start to detract from my point, so I'll refrain from now).

One can certainly say that Alice appears to freeze according to the coordinate time "t". But is this physically significant?

It might be instructive to consider Zeno's paradox. I'll use the wiki definition of the paradox.

Let's define a "zeno time" as follows. At a zeno time of 0, Achillies is 100 meters behind the tortise.

At a zeno time of 1, Achilles is 50 meters behind the tortise.

At a zeno time of 2, Achillies is 25 meters behind the tortise

At a zeno time of n, Achillies is 100/(2^n) meters behind the tortise.

Then, as n goes to infinity, Achillies is always behind the tortise.

So, in "zeno time", Achilles never does catch up with the tortise, even as "zeno time" appoaches infinity.

Are we therefore justified in claiming that Zeno was right, and that Achilles never catches the tortise? I don't think so, and I'd be more than a bit surprised if anyone really believed it. (I could imagine someone who likes to debate claiming they believed it as a debating tactic, I suppose - and to my view this would be a good time to stop debating and do something constructive).


So in my opinion, the confusion arises by taking "zeno time", which is analogous to the Schwarzschild coordinate time "t", too seriously. While it is correct to say that as t-> infinity Alice never reaches the event horizon, just as Achilles never reaches the tortise in zeno time, it still happens. It's just that that event hasn't been assigned a coordinate label.

rjbeery

That's an interesting take, Pervect. I guess Zeno's Time paradox could be dealt with by postulating a minimal quantum "unit" of time.

Additionally...
I can give physical meaning to this by simply postulating some arbitrary frame to be the preferred one. We then have a way to establish a true "local rate of time" as [tex]t_{local}\over t_{preferred}[/tex] as measured from the preferred frame.

PAllen

I wanted to re-post this edit separately, as it raises some crucial points.

stevendaryl

In general, a coordinate system is defined on a "patch": a small region of spacetime. Two different coordinate systems must make the same predictions on the overlap of the two patches. The point at which an infalling observer crosses the event horizon is not in patch described by the Schwarzschild coordinates.

rjbeery

I don't agree with the edit. Consider the graphs y = 1/x, and y = 1/(x+1). Both lines approach zero without crossing with no problem.

pervect

It's not necessarily inconsistent with relativity to postulate some "preferred frame", but when your theory *requires* it, it's getting far, far, far outside the path of conventional SR.
I notice that you aren't saying that you *do* postulate a preferred frame, are you in fact doing so? Are you saying that there isn't any other way to save your viewpoint? I'm getting a sort of debate feeling here, with this sudden lack of specificity, with all the "I could" and "I might".

PAllen

Hmm, I guess you could get that result, given that the chop produces a manifold without boundary (the EH is not in in the manifold), and you take your slices of constant time just the right way.

rjbeery

No this isn't my viewpoint; just poking holes in typical defenses of black holes. If it's true that after some time T_b, Alice cannot be saved by Bob under any circumstance as outlined in the OP, then I'm convinced that GR would allow for the formation of black holes as you and PAllen are saying. An extended discussion in this thread occurred when I brought up quantum effects, Hawking radiation, etc, and it sounds like the consensus on that is "no one knows enough to know the answer currently".