Re: the black fishing hole
- From: tessel@xxxxxx
- Date: Fri, 15 Sep 2006 19:57:54 +0000 (UTC)
On Thu, 14 Sep 2006, Gerry Quinn wrote (concerning a thought experiment in which a static observer, who uses his rocket engine to hover outside the event horizon of a Schwarzschild hole, slowly lowers a weight on a radially oriented string toward the horizon, or a similar thought experiment):
From his perspective, the dangling object will never reach the event
horizon.
Since the weight is lowered slowly, we can model it as a quasistatic observer which is simply closer to the hole (at least until the string breaks!).
Recall that we should expect that initially parallel geodesics will in general tend to either converge or diverge (depending upon their initial direction) due to the curvature of our spacetime model.
In particular, radially outgoing null geodesics -diverge- in the (exterior region of the) Schwarzschild vacuum solution. This means that light signals sent from a static observer at r=r1 will be measured to have been redshifted when they are received by a static observer at r=r2, where r2 > r1. You can easily work out the precise redshift factor and confirm that it diverges as r1 approaches r=2m (the horizon).
You can also work out the magnitude of the (radially pointing and outward) acceleration which the rocket engine of a static observer hovering at r=r1 must produce in order for this observer to remain static. Again, this diverges as r1 approaches the horizon.
However, the event horizon is only a finite distance away, and he can easily pay out more string than this.
You asserted that "the event horizon is only a finite distance away", but it's already been pointed out several times in the past few weeks why this assumption is naive. And the same point must have been made dozens of times here in the past few years!
If he does, he will find that the string goes slack,
Well, the string will break at some point before the weight reaches the horizon, because the tension in the string diverges, because of the fact about acceleration required for a bit of matter to hover outside the hole as noted above.
It is illuminating to consider what happens if he pays out string slowly, in order to maintain tension at all times, or at least periodically.
I thought that's precisely the thought experiment which we've been discussing, except that of course that in a quasistatic thought experiment the tension increases -monotonically-.
This string is very strong with a huge elastic constant, so that we can ignore its changes in length, and any variations in tension will travel up and down it at almost the speed of light. Let us say the object is dangling twenty feet above the event horizon, and he lets out six inches, and waits for the tension to be restored.
Something is wrong here. In both Newtonian gravitation and in gtr, the tension in the string should monotonically increase as the weight is slowly lowered. The computation works the same way in both theories: consider the situation at a given time and work out the total tension in the string at that time. (In the physics literature, in a discussion of this situation, one might say that the dangling weight is "quasistatic".)
The line goes slack, a signal represented by an absense of tension in the string travels down at the speed of light, the object drops six inches and is pulled up short, then an impulse of restored tension travels up again, and Zef observes that the string is once again taut, so he can pay out another six inches.
Are you talking about letting the object fall freely for a bit, then stopping it by -jerking- (and then pulling with constant magnitude) with the string? This is obviously a more complicated thought experiment, and jerking the string certainly won't help avoid snapping it. I strongly suggest you first work out the tension for the quasistatic thought experiment, comparing Newtonian gravitation and gtr.
The problem for Zef is gravitational time dilation.
Citation? (Second request.)
Are you reading a paper by someone whom you believe claims that the standard analysis in gtr textbooks is -wrong-, or what? If so, is it possible that you misunderstood?
What happens in the case where he pays out line continuously, always maintaining tension in the string, should now be obvious. He will have to pay it out slower and slower to maintain tension.
Maintain tension? If you mean, "maintain constant tension", of course he can't do that unless he pays out the string faster and faster, according to either Newtonian gravitation or gtr! (That is, he'd have to pay out the string at just the right rate to maintain constant tension, so the dangling weight would be neither quasistatic or freefalling at any time; it would be in a "controlled fall".)
I think it should be obvious that in both Newtonian gravitation and in gtr, if our hovering observer slowly lowers a weight at the end of a string (the quasistatic thought experiment), the tension will monotonically -increase- as the weight gets closer to the massive object. And you can confirm this by making the computation as I suggested.
The total length paid out will asymptotically approach the distance from Zef to the event horizon, but will never reach it.
In gtr, the string will break before the weight reaches the horizon (this should be obvious from the result noted above about the magnitude of the outward radial acceleration required for a test particle to remain static). Up to the point where that happens the length paid out will of course be finite. But if you try to compute it you should see why in curved spacetime, "length of the string" requires some assumptions in how you chose to define this notion, which is the point you forgot above!
Mostly likely you will integrate the spatial line element for the static exterior Schwarzschild chart. This is the "pedometer distance", since it gives the distance which would be reported by an ant equipped with a pedometer who crawls along the string from the observer to the dangling weight. (So this is defined in terms of a quasistatic observer.)
But if you define the "radar distance", or "optical diameter distance", etc., you should expect to obtain slightly different results!
It shouldn't be either surprising or dismaying that you should have multiple competing definitions of distance "in the large" in any curved spacetime. In any case, all reasonable definitions of "length of the string" will yield a finite length, until the string breaks, and of course they all agree in the Newtonian limit.
(Confirming all this is another good exercise.)
You -could- consider a more complicated thought experiment, but it makes good sense to work out the details of the simplest one first!
Think of it like this: in Newtonian gravitation, if you work this out for a hypothetical point mass, you intuition should say that the tension diverges as r -> 0, yes? Because the acceleration scales like 1/r^2, yes? So qualitatively, you are puzzling over the fact that in gtr, the tension diverges sooner, as r-> 2m > 0, yes? So you are puzzling over the fact that in this sense, gravitation is stronger in gtr than in Newtonian theory at a given r outside a spherically symmetric object? But that should not be so surprising!
The length of time
as measured by a static observer
for an impulse to travel between r and R (dropping all constants) is:
integral (r to R) ( dx / sqrt( 1 - M / x ) )
I think you need to back up and think about "distance" as measured by static and quasistatic observers in Schwarzschild vacuum (exterior region). So let's work the exercise I just mentioned.
Exercise: Draw a picture in the Schwarzschild static chart with R > 2m, m> 0, h >0, looking a bit like this:
R R+h
|\ |
| \ |
| \|
| /|
| / |
|/ |
(But sketch the null geodesics accurately, so that they -appear- to bend a bit in this chart-- you know their -coordinate slope- dt/dr from the line element with dtheta = dphi = 0).
Using your sketch as a guide, compute the radar distance as measured by the observer at r=R. This means: measure the elapsed proper time of a round trip lightspeed signal, as measured the observer who is computing the distance, and divide by two.
(Hint: first compute Delta t and then Delta s.).
Likewise, for compute the radar distance as measured by the observer at r = R+h:
R R+h
| /|
| / |
|/ |
|\ |
| \ |
| \|
(Again, sketch the null geodesics accurately enough to see how they appear to bend.)
Next, compute the pedometer metric for the same situation. (Argue that this given by the obvious integral obtained rom the line element with dt = dtheta = dphi = 0.)
Now for each of these three results, expand the three variable Taylor series in powers of m, h, 1/R to total degree 7. You should find
that
D_far > D_ped > D_near > h
Argue that these inequalities make sense physically.
We conclude that in the large, light travel time distance is not even symmetric. This is a global versus local issue in Lorenztian spacetimes: except over infinitesimal courses, there are multiple competing notions of distance, and these not even be symmetric. So "in the large" geometry has a non-Riemannian character in curved spaces. (Compare the discussion of radar distance in Landau & Lifschitz, Classical Theory of Fields.)
However, note that all three distances agree up to total degree two, i.e. up to first order in m/R,
D = h + m h/R
so they all reduce to h (the Newtonian result) as m -> 0 or as R -> infty.
Exercise: repeat for observers at r = r1 and r = r1, with r2 > r1 > 2m. What happens as r1 -> 2m?
Exercise: consider a similar pair of near and far observers (to the horizon) in the Rindler chart for Minkowski vacuum. Similarly define and compute pedometer distance (measured by an ant) and the light travel time distance (measured by near versus far observer). Similarly expand in a power series and draw similar conclusions. What about "the distance to the horizon"? How is this related to Darryl's computation?
Exercise: repeat, with a suitable definition of optical diameter distance.
Exercise: can you think up any more reasonable notions of "distance in th large"? What can you say about radar distance and optical diameter distance measured by Lemaitre observers? Can you find a suitable notion of pedometer distance for the Painleve chart?
Exercise: What about "the distance to the horizon" as measured in various ways by these observers? Note that since no particle can hover at the horizon, this is tricky. One approach would be to discuss "the distance" to some static observer at r = R, R > 2m, as measured in various ways (at various proper times) by some Painleve observer, and then to take the limit R -> 2m.
Integration not being my forte, I'll leave it as an exercise...
Well, if you really want to understand this stuff, you need to be willing to make your own computations to verify textbook results! You can perform the integrations for the exercise above with any standard table of integrals.
IMO, these are very good exercises, so I hope you will attempt them.
"T. Essel"
.
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