Path: gmdzi!unido!mcsun!uunet!wuarchive!wuphys!ihr
From: i...@wuphys.wustl.edu (Ian H. Redmount)
Newsgroups: sci.physics
Subject: Re: Black holes and plumbs.
Summary: Lowering a string into a black hole--an old problem (long).
Message-ID: <1990Aug16.195705.24758@wuphys.wustl.edu>
Date: 16 Aug 90 19:57:05 GMT
References: <13807@megatest.UUCP>
Reply-To: i...@wuphys.UUCP (Ian H. Redmount)
Organization: Physics Dept, Washington U. in St Louis
Lines: 83
Posted: Thu Aug 16 20:57:05 1990

In article <1...@megatest.UUCP> bbo...@megatest.UUCP (Bruce Bowen) writes:
>
>Suppose we slowly...reel out a string,
>on the end of which is connected a plumb bob.  Also assume that the mass of
>the string is negligible compared to the mass of the plumb bob.  The reel
>upon which the string is suspended...is stationary
>with respect to our black hole.
>
>In the relativistic case, would the tension, measured locally along the
>string,  be constant along the length of the string?  I am inclined to
>believe not....In the limit as the plumb bob
>gets arbitrarily close to the event horizon, the tension in the string
>near the plumb bob goes to infinity, but the tension at the other end
>of the string at the reel goes to a finite value, which is a function
>of the mass of the black hole, mass of the plumb bob, and distance of
>the reel from the event horizon.
>

Apparently the good Mr. Bowen has not read Appendix B of my 1984 Caltech
Ph. D. thesis, though I cannot imagine why not :).

The dynamics---in this case, statics---of the string follows directly from
the law of conservation of stress-energy for the string.  The string's
stress energy is, in the ideal approximation, given by two quantities:
its proper linear mass-energy density U, and its proper tension T.
(As is customary, I assume units such that the speed of light c is 1.)
Stress-energy conservation implies the equation of force balance

dT/dl=(U-T)g ,

where l is proper distance up the string and g is the local proper
"gravitational acceleration" as measured by static observers.  Except
for the T on the right side, this is exactly the equation you'd expect
on Newtonian grounds:  The difference in tension across each infinitesimal
segment of string is equal to the segment's weight.  The -T contribution
to the weight density is a relativistic effect.

Relativistically, no (classical) string can have zero mass density; in
fact, the familiar classical energy conditions (e.g., positive mass
density in all reference frames) require U>T.  Since the transverse-wave
speed on the string is c*sqrt(T/U), this also follows from causality---no
signals faster than light.  The best we can do is string with T=U, and
that is mighty string indeed.  Ordinary steel piano wire has a breaking
strength T about 10^(-12)U!

For string with T<U, the above force-balance equation implies that the
tension in the string increases as you go up, from its value at the
plumb bob T(0)=Mg(0), M being the bob mass and 0 denoting that end of
the string.  The tension is highest at the top of the string, unlike
Mr. Bowen's conjecture.  It could not be otherwise, since the tension
at the top must support the weight of the string as well as the bob.
The top tension is a function of the bob mass and its position in the
black-hole spacetime.  Since g goes to infinity at the horizon, there
will be some position above the horizon at which the top tension exceeds
the breaking tension of the string---at most, U---the bob cannot be
suspended any closer that this.

For string capable of sustaining T=U, the force-balance equation implies
that the tension is constant along the string.  Such a string is effectively
weightless.  Again, it cannot support the bob any lower than the position
at which Mg(0)=U, but for arbitrarily small M this can be arbitrarily
close to the horizon.

These results are mentioned in the article ``Membrane viewpoint on black
holes:  Gravitational perturbations of the horizon," by W.-M. Suen,
R. H. Price, and I. H. Redmount, Physical Review D 37, pp. 2761-2789
(see p. 2772), 15 May 1988, and in the book ``Black Holes:  The Membrane
Paradigm," K. S. Thorne, R. H. Price, and D. A. Macdonald, eds., Yale
University Press, 1986, p. 240.  An early work on the problem is the
article by G. W. Gibbons in Nature Phys. Sci. 240, p. 77 (1972), but
Dr. Gibbons inadvertently omitted a term from one equation and got the
wrong answer.

As an aside, note that string with T=U is remarkable stuff.  (Both cosmic
and super string have this property.)  If you strung a meter of it between
posts and plucked the center a few millimeters, the waves produced would
travel at the speed of light---clearly a relativistic situation---but no
part of the string would move at more than a few times 10^(-3)c.  String
a guitar with this stuff and it (the guitar) would break.  But if it didn't,
it would play at frequencies of hundreds of megahertz!

Ian H. Redmount,
Sole Proprietor of any of the above which is not fact