Re: Gravitational waves in strong field limit.

LEJ Brouwer <intuitionist1@xxxxxxxxx> wrote:
> Apologies in advance if this is a "newbie" question, but in the
> presence of strong gravitational fields, will gravitational waves start
> interacting with each other, to the extent that gravitational collapse
> may take place?

The whole concept of "gravitational wave" is only meaningful if
you can write the gravitational field as a "background + wave",
i.e. if you can separate off a slowly varying part of the Riemann
tensor (the background) and call what's left (which should be small
and high-frequency relative to the background) the "wave".

But assuming we can do this, then as I understand it, the equivalence
principle says that locally, you can ignore the background, so locally,
(weak, high-frequency) gravitational waves don't interact any more
than they would in an area without the background field.


> Alternatively, or additionally, could this also be true if there many
> strong localised curvature perturbations (localised solitons?)
> introduced into the metric? If I were to agitate the surface of the
> spacetime pond by throwing in many thousands of large pebbles, could
> the resulting ripples coallesce to form a black hole? (I am assuming
> absence of massive particles throughout).

Yes. You can have a "geon" (a localized chunk of gravitational
radiation) which is so strong that it collapses to form a black hole.
I'll give a couple of references below on simulations of this process
(via numerical solution of the Einstein equations). For example,
the abstract of the Abrahams and Evens paper cited below reads:

We describe the formation of a black hole via the implosion of
an axisymmetric gravitational wave. Finite difference simulations
of the vacuum Einstein equations are used to obtain these results.
The initial data consist of nearly linear solutions to the vacuum
constraint equations that represent even-parity, ingoing wave packets
with a quadrupole angular dependence. A black hole is demonstrated
to form as a result of imploding a wave packet with a sufficiently
large value of a strength parameter, [[...]]. Black hole formation
is verified by observing [[...]] (ii) the formation of a trapped
region and marginally outer-trapped surfaces, [[...]]


I don't know of any long-term stable geons that aren't black holes.
That is, I think any given "geon" will either disperse or collapse
to form a black hole. (There are also lots of interesting Choptuik-style
critical phenomena right at the threshold between these outcomes,
but that's another story.)


Here are three references, the first two on numerical simulations of
geons collapsing to form black holes, the last a nice review talk on
Choptuik-style critical phenomena at the threshold of black hole
formation:

@article
{
Abrahams-Evans-1992-BH-geon-formation,
author = "Andrew M. Abrahams and Charles R. Evans",
title = "Trapping a Geon:
Black Hole Formation by an Imploding Gravitational Wave",
journal = "Physical Review D",
volume = 46, number = 10,
pages = "R4117--R4121",
year = 1992, month = "November 15",
snote = "3+1, maximal slicing",
}

gr-qc/9904013 (published as Phys.Rev. D61 (2000) 041501)
Gravitational Collapse of Gravitational Waves in 3D Numerical Relativity
Authors: Miguel Alcubierre, Gabrielle Allen, Bernd Bruegmann, Gerd Lanfermann,
Edward Seidel, Wai-Mo Suen, Malcolm Tobias
Abstract:
We demonstrate that evolutions of three-dimensional, strongly
non-linear gravitational waves can be followed in numerical
relativity, hence allowing many interesting studies of both
fundamental and observational consequences. We study the
evolution of time-symmetric, axisymmetric *and* non-axisymmetric
Brill waves, including waves so strong that they collapse to
form black holes under their own self-gravity. The critical
amplitude for black hole formation is determined. The
gravitational waves emitted in the black hole formation process
are compared to those emitted in the head-on collision of two
Misner black holes.

gr-qc/9803075
The (Unstable) Threshold of Black Hole Formation
Author: M.W. Choptuik
Comments: 19 pages, Latex, crckapb.sty, 8 figures included using epsf,
talk given at GR15, Pune, India, to appear in the proceedings
Abstract:
In recent years it has become apparent that intriguing phenomenology
exists at the threshold of black hole formation in a large class of
general relativistic collapse models. This phenomenology, which includes
scaling, self-similarity and universality, is largely analogous to
statistical mechanical critical behaviour, a fact which was first noted
empirically, and subsequently clarified by perturbative calculations
which borrow on ideas and techniques from dynamical systems theory and
renormalization group theory. This contribution, which closely parallels
my talk at the conference, consists of an overview of the considerable
``zoo''' of critical solutions which have been discovered thus far,
along with a brief discussion of how we currently understand the nature
of these solutions from the point of view of perturbation theory.

ciao,

--
-- "Jonathan Thornburg (remove -animal to reply)" <jthorn@xxxxxxxxxxxxxxxx>
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

.

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