Re: black holes and information

From: Tom Roberts (tjroberts_at_lucent.com)
Date: 06/28/04 

Date: Mon, 28 Jun 2004 13:00:46 -0500

Paul Cardinale wrote:
> Tom Roberts <tjroberts@lucent.com> wrote in message news:<tKLCc.1536$nc4.285@newssvr16.news.prodigy.com>...
>>[a book falling into a Schwarzschild black hole] An intrepid explorer
>>could (in principle at least) travel in a very powerful rocket and go
>>retrieve it, intact. Assuming the book merely falls, then this explorer
>>will have to accelerate inward to catch up with the book, and then
>>quickly reverse and accelerate back out without ever crossing the
>>horizon itself.
>
> Is there a time limit (measured on a distant clock) beyond which a
> rescue is impossible?

Yes. There is a definite point on the book's worldline where it crosses
the horizon. Look on a Kruskal diagram, in which the light cone at any
point is diagonal 45-degree lines. The book is falling inward on a
roughly-vertical worldline, and that worldline will intersect the
horizon (the big X on the diagram) at the definite point mentioned
above. The rescue must occur on the book's worldline before that point.
The intrepid explorer must leave on his rescue mission from a point
within the past lightcone of the rescue point -- the radius from which
that is possible depends on the time when he leaves (in Schw. coords).
For any given radius there is a maximum value of t; but the closer to
the horizon he starts, the larger is that possible value of t.

> Doesn't an infalling observer observe a finite
> time on a distant clock when he crosses the EH?

Imagine the distant clock has a constant Schw. radius, and sends
periodic EM signals to the infalling observer. Looking again on a
Kruskal diagram, that distant observer is sliding along a hyperbola of
constant r; the infalling observer is following a roughly vertical
worldline. So there will be a last signal received by the infalling
observer before crossing the horizon. That last signal was emitted by
the distant observer at a finite value of his time coordinate. So
speaking loosely, one could say "an infalling observer observe a finite
time on a distant clock when he crosses the EH".

Tom Roberts tjroberts@lucent.com

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