gravity's speed

[Ralph Howard]

Matthew B Hoyt writes:

> 
> 
> On Fri, 23 May 1997 09:14:38 +0200 "Paolo Cavallo"
> <ton0621 at iperbole.bologna.it> writes:
> 
> >Every fundamental interaction in Nature propagates at the speed of 
> >light - not
> >only gravity. It's required by special relativity.
> 
> Well, only sorta kinda.  During BIG gravitation events this changes. 
> Allow me
> to quote Einstein (and Max Born) from 1913 in a position I believe
> continues to be held by
> most physicists. [I assume this is a translation, my text is Fron
> _Gravitation_ by Misner,
> Thorne and Wheeler, 1970]
> 

<snipage of interesting dialogue between Einstein and Born and nice
discussion by Matt>


In the time since the Einstein-Born discussion I think that this has
become much better understood.  In special relativity it is true, more
or less by postulate, that none of the objects it discuses move faster
than than the speed of light.  But special does not include a theory
of gravity and so says nothing the speed of gravity.

General relativity is a extension of special relativity that does
cover gravity, but here things are more complicated.  There is a
condition, the "dominant energy condition" (here I quote from the _The
Large Scale Structure of Space-Time" by S. W. Hawking and
G. F. R. Ellis page 91:

  "In other words, the dominant energy condition is the weak energy
  condition with the additional requirement that the pressure should not
  exceed the energy density.  This holds for all known forms of matter
  and there is in fact good reason for believing that this should be 
  the case in all situations."

(Sorry for not give the exact condition, but it is an inequality of
tensors and thus a mess to type.)  My understanding is that if the
dominant energy condition fails that then there are regions where
matter is so dense that the speed of sound is greater than the speed
of light.  If the dominant energy does hold, then there is a rather
precise statement that the effects of gravity can not move faster than
the speed of light (the Conservation Theorem, page 94 of Hawking and
Ellis).

It is possible that since writing the above (1973) that Hawking may
have changed his mind about the dominant energy condition "should be
the case in all situations".  In his theory of evaporation of black
holes my (very limited) understanding is that part of what goes on is
that near very small black holes things are so dense that the dominant
energy condition fails and this is in part what allows the evaporation
to occur.  If I have this last part wrong wrong I will be happy to be
corrected.


Ralph

--
Ralph Howard                    Phone:  (803) 777-2913 
Department of Mathematics       Fax:    (803) 777-3783 
University of South Carolina    e-mail: howard at math.sc.edu
Columbia, SC 29208 USA          http://www.math.sc.edu/~howard/