Article 12060 of sci.physics:
Path: megatest!decwrl!wuarchive!wuphys!ihr
>From: ihr@wuphys.wustl.edu (Ian H. Redmount)
Newsgroups: sci.physics
Subject: Re: Black holes and Plumbs.
Summary: Tension INCREASES with height, or is constant
Keywords: Black holes, general relativity, gravity
Date: 5 Jan 91 00:13:35 GMT
Organization: Physics Dept, Washington U. in St Louis
Lines: 86

In article <14764@megatest.UUCP> megatest!bbowen@sun.UUCP () writes:
>
>Suppose we slowly (so that transients are not relevant) reel out a
>string, on the end of which is connected a plumb bob, over a black
>hole.  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 not in free fall and is stationary with respect to our
>black hole.  The whole experiment is outside the event horizon (of
>course). 
>
>In the relativistic case, would the tension, measured locally along
>the string,  be constant along the length of the string?
>[Hypothesis whereby tension decreases with height along the string
>deleted.]

I believe I was one of the respondents to the original posting of this
question; let me try to reconstruct my response.  I had to solve this
problem as part of my Ph. D. thesis, back in '83.

In summary, the solution implies that the tension either increases upward
along the string, or at best is constant, for very special string.  In no
physical case does the tension decrease upward from the plumb bob.  The
reason is the same as in the Newtonian case:  The tension at the top end
of any segment of the string must support both the weight supported at the
bottom end plus the weight of the segment.

The stress distribution on the string follows directly from the
stress-energy conservation equation for the string.  In the
quasi-static situation posited, for a string suspended radially
in the spacetime of a Schwarzschild (spherically symmetric, uncharged)
black hole, the relevant equation takes the form

	dT/ds = (u-T) g		(1)

where T is the proper tension of the string, u its proper mass-energy
density (units with c=1 are assumed), g the magnitude of the locally-measured
proper gravitational acceleration, i.e., the proper acceleration required
for stasis at the given location, and s proper length along the string,
measured upward from the plumb bob.  The only non-Newtonian feature of
this equation (except for the non-Newtonian dependence of s and g on
the radial coordinate, concealed in the above form) is the contribution
of the tension to the ``weight density'' (u-T)g .  This is a relativistic
effect---relativistically stress, e.g. tension, as well as mass gravitates.
Otherwise the equation is exactly what you'd expect:  The difference in
tension across any infinitesimal segment of string supports the segment's
weight.  The equation may be solved for T(s), given the boundary condition

	T(0) = m g(0)		(2)

where s=0 denotes the position of the plumb bob and m is the bob's mass.

Now relativity forbids the classical Newtonian ``massless string.''  The
linear mass-energy density of a string cannot be less than its tension,
else 1) in some reference frames the string density would be negative, and
2) the transverse-wave speed on the string would exceed that of light.
(Familiar materials, of course, do not come near this bound:  The mass-energy
density of steel wire, for example, exceeds its breaking tension by twelve
orders of magnitude.)  Hence the right side of Eq. (1) is nonnegative.  Thus
the tension either increases monotonically with s, upward from the plumb bob,
or is constant in the special case of a string with tension equal to its
linear mass density.

Since g diverges to infinity at the black-hole horizon, any string will
break before the plumb bob can be lowered all the way to the horizon.
A string with T = u can be lowered all the way to the horizon, provided
it is unloaded, i.e., m=0.

String with T=u is very special stuff indeed.  (Hypothetical ``cosmic
string'' and ``superstring'' both have this property.)  Since by Eq. (1)
the tension in such a string is constant, the string is effectively
weightless.  Pluck such a string and the transverse waves produced
propagate at the speed of light, even though no part of the string does
so---the string itself might only move at non-relativistic speeds.  String
a guitar with this stuff and the guitar would be crushed instantly---but
if it weren't, it would play at frequencies of hundreds of megahertz!

The problem of a string suspended over a black hole was addressed by
G. W. Gibbons [Nature Phys. Sci. 240, p. 77 (1972)], but there are
errors in some of Gibbons' equations and conclusions.  The problem is
treated in my Ph. D. thesis (Caltech, 1984), Part II, Appendix B; the
results are described in ``Black Holes:  The Membrane Paradigm,''
K. S. Thorne, R. H. Price, and D. A. Macdonald, eds. (Yale University
Press, 1986), p. 240, and by W.-M. Suen, R. H. Price, and I. H. Redmount
[Phys. Rev. D 37, p. 2761 (1988)].

Ian H. Redmount


From bbowen Sat Jan  5 21:33:53 PST 1991
Article 11962 of sci.physics:
Path: megatest!bbowen
>From: bbowen@megatest.UUCP (Bruce Bowen)
Newsgroups: sci.physics
Subject: Black holes and Plumbs.
Date: 3 Jan 91 21:02:57 GMT
Organization: Megatest Corporation, San Jose, Ca
Lines: 34


  I originally sent this a few months ago, but right after, our
newsfeed died, so I did not see any responsed, if any.  So here
is the same thought experiment again.  Is the tension in the
string constant, or does is vary?

   Here's a question for all you general relativitists.

Suppose we slowly (so that transients are not relevant) reel out a
string, on the end of which is connected a plumb bob, over a black
hole.  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 not in free fall and is stationary with respect to our
black hole.  The whole experiment is outside the event horizon (of
course). 


In the classical, non-relativistic, case, the tension in the string,
measured locally at each point on the string, is constant.

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.  My reasoning is as follows: deeper in the
gravity well, the "atoms" of the string move more sluggishly under
identical conditions due to gravitational time dialation, therefore in
order to keep the plumb bob stationary, these atoms have to "tug" more
vigorously in their reference frame.  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.  

-Bruce       megatest!bbowen@sun.UUCP