State of the art in numerical simulation
Koen Delaere
Could you tell me what is the state of the art in theoretical and
numerical relativity research these days?
(sci.physics.relativity referred me to this ng)
I guess the static and spherically symmetric black hole has been
investigated by now, but what about dynamics and black holes with
charge? Is there already a library of (analytic/numeric) solutions to
Einstein's equation? Are there a lot of numerical simulations going on
about these things? Or is the numerical search impeded by slow computers
and the lack of numerical experts?
And on a side note: does there still exist software dedicated to gtr
calculations or are all you guys just plugging away in maple (maybe with
dedicated toolboxes within the common mathematical software)?
All comments are welcome,
Koen
Chris Hillman
> Could you tell me what is the state of the art in theoretical and
> numerical relativity research these days? (sci.physics.relativity
> referred me to this ng)
I'm not an expert, but for state of the art in gtr in general, up to just
-before- the positive lambda thingie :-( I would -highly- recommend that
you obtain the very recent book Black Holes and Relativistic Stars, ed. by
Robert Wald which is currently available in paperback; see the reading
list
http://www.math.washington.edu/~hillman/Relativity/reading.html
on my relativity pages for the exact citation.
> I guess the static and spherically symmetric black hole has been
> investigated by now, but what about dynamics and black holes with
> charge?
The very recent book by Frolov and Novikov, Black Hole Physics (see my
list), offers a somewhat sketchy but up to date and very readable overview
of the huge amount of work which has been done on the Schwarzschild
solution (nonrotating black hole), Reissner-Nordstrom solution (charged
nonrotating black hole), Kerr-Newman solution (rotating charged black
hole), and various generalizations, including the Vaidya null dust, the
Kinnersley photon rocket, the Aichelburg-Sexl ultrarelativistic
Schwarzschild hole, the Ernst solution (Schwarzschild hole in the Melvin
magnetic universe), etc., etc. Some of the topics covered include
perturbations and scattering of radiation, tidal deformation of extended
bodies, black hole thermodynamics, etc.
The classic book by Chandrasekhar, The Mathematical Theory of Black Holes
(see my reading list) uses the Newman-Penrose formalism extensively, so it
looks daunting, but it is a gold mine of information about geodesics and
perturbation theory. The book makes salutory reading because IMO the same
care and thoroughness is needed in investigating other important solutions
which have not yet received the attention they deserve.
Both of these books are in print in paperback :-) The Chandrasekhar
reprint is a particularly good buy (about $24 US).
These books do not discuss the recent flurry of papers which study old
solutions but replace the Minkowksi background with a de Sitter
background, which makes things a lot more interesting! There are also a
lot of papers doing the same thing, but with an anti-de Sitter background;
these are motivated by string theory rather than the recent cosmological
observations.
I am considering writing up some of my recent work in the form of a long
and completely unpublishable :-/ but systematic and well-illustrated
survey of S^2, T^2, and H^2 symmetric solutions, including electrovacs,
null dusts, and accelerating and rotating generalizations, possibly with
de Sitter backgrounds, and including matching to fluid and dust solutions.
There actually -is- a point to all the work which would be summarized in
this survey, but that's a long story :-/
> Is there already a library of (analytic/numeric) solutions to
> Einstein's equation?
Certainly. The essential sourcebook for exact solutions is the book by
Herlt et al., Exact Solutions of Einstein's Field Equations (unfortunately
out of print, very out of date, and rather hard to use). A few days spent
poring over this book will show that thousands of solutions were known by
1975, and as far as I can tell, this number has doubled by now. It is
important to understand that many very important -simple- exact solutions,
such as the Mars vacuum (1995), have been discovered -since- the
publication of Herlt et al. It is not true that all simple solutions are
known; for example, "tilted" cosmological and other solutions have not
been very well studied. I have found many interesting -simple- exact
solutions by using the tried and true method of making a metric Ansatz,
but using more imaginative possibilities than, e.g. static S^2 symmetry.
Particularly simple special cases of "known" solutions, or alternative
coordinate charts, can also yield valuable new insight.
As for computer databases, I have accumulated a libary of about a thousand
metrics (exact solutions) using GRTensor, including many apparently new
solutions as well as new coordinate charts for known solutions, and
interesting coordinate charts I have found in various recent papers, as
well as "the usual suspects" in a dozen or more coordinate charts each.
My own database hasn't been "cleaned up" for public consumption yet, but
GRTensor is a freeware package (hurrah!) which runs under maple or
Mathematica (a "toy" version runs under Java at their website). You can
get these packages here:
http://grtensor.phy.queensu.ca/
There is a much smaller library of exact solutions at the GRTensor site;
my library uses orthonormal bases exclusively (which not only simplifies
computations but greatly facilitates physical interpretation); theirs
mostly uses metrics. Another small libary of exact solutions (about two
hundred metrics) may be found at
http://www.astro.queensu.ca/~jimsk/
This on-line database is very useful because if you have found a solution
with some symmetry, chances are good you can use this utility to discover
that your solution might be locally isometric to one in this database.
One project I hope to work on if circumstances permit is to unify the best
aspects of Jim Skea's database and CLASSI with my own GRTensor database,
and to add if possible every metric ever published and then to run CLASSI
on every pair.
This would involve a huge amount of work, of course, but I think it is
doable and would be tremendously useful to people working in this field,
especially since the advent of gravitational wave astronomy is likely (as
I guess) to lead to even greater interest in theoretical models of
relativistic astrophysical and cosmological processes, and in this case it
would be very good to curtail the problem wherein a newcomer publishes a
new coordinate chart for what is (unbeknownst to him) an old solution, or
even worse, republishes a "known" coordinate chart. For example:
different people refer to the "Taub vacuum (1951)" or the "McVittie vacuum
(1929)", but in fact this solution was first published as the AIII vacuum
by Levi-Civita around 1918. To further add to the confusion, people refer
to "the Levi-Civita vacuum" which could mean any of a half dozen distinct
families of vacuum solutions. Even worse are references to "the Tolman
solution", which could refer to any of several dozen solutions :-/ So a
catalog would be useful in curtailing the monotonic growth of
terminological confusion as well.
My goal would be to create an on-line utility like the Skea database,
which would enable a user to type in a possibly new solution using
GRTensor, run CLASSI like routines (rewritten to run under maple or
Mathematica) to find generators for the isotropy and isometry groups and
various other invariant geometric characteristics such as the Ricci and
Weyl algebraic types, and then to compare, using a Karlhede type
algorithm, the user's solution with known solutions in the database which
have similar invariant properties (e.g. nonnull electrovacuum, isotropy
group SO(2), three dimensional isometry group) to see if the user's
solution is locally isometric to a known solution. The utility would then
return the known coordinate charts with comments about the physical
interpretation, and citations of the original papers.
(-Global isometry- is probably too tricky for a feasible web application,
so this would be left up to the user to determine!)
For example: many users would probably find that their coordinate chart
describes part of the Demianski nonnull electrovacuum (which has
parameters for mass, NUT charge, electrical charge, magnetic charge,
angular momentum, acceleration, and cosmological constant). This would
not neccessarily bad news, precisely since the Demianksi metric is "too
general", as far as I can see, to be of immediate use: to really
understand it, I think one must systematically study various special cases
and then painstakingly piece together geometric insight from the separate
special cases. I suspect that this would show that some "known" solutions
have been misinterpreted by at least some authors (no, I do -not- mean the
Schwarzschild and Kerr solutions, which are understood just fine!).
As for numerical solutions, see first the excellent expository article on
the Cauchy problem by Friedrich and Rendall:
http://xxx.lanl.gov/abs/gr-qc/0002074
and then see this article by Bruegmann:
http://xxx.lanl.gov/abs/gr-qc/9912009
After reading these, you'll know that the two big problems in actually
carrying out the original "ADM" approach to numerically integrating the
EFE come down to:
1. finding a suitable set of initial data
(\Sigma, h_(ab), K_(ab), rho, j_a)
where (\Sigma, h_(ab) is a Riemannian three-manifold, K_(ab) is
a symmetric tensor (will be the extrinsic curvature), rho is the
mass-energy density and j_a is the momentum density covector,
2. finding a suitable method of "gauge fixing" during the evolution, one
which avoids "kinking", extremely inhomegenous advance of "time", and
which avoids curvature singularities.
With respect to 1, the initial data for a certain black hole collision
scenario which is frequently used as a benchmark for testing new codes is
available somewhere, but I forget where. Probably the article by
Bruegmann mentions this (but I forget); if not, there are dozens of recent
preprints on numerical studies on the LANL server and I am sure some of
them say where you can find this set of initial data.
With respect to 2, there is a large (oldish) literature on gauge fixing in
the ADM formalism (see the appendix in the gtr textbook by Wald for a nice
explanation of the York formalism), but it turns out that the ADM
formalism and its close kin are for various reasons not very suitable for
simulations of phenomena involving curvature singularities (i.e., all the
interesting stuff!). So, be sure to see the article by Friedrich and
Rendall cited above for important newer formalisms which circumvent many
of the problems which, in practice, mitigate against the practical utility
of the ADM formalism.
> Are there a lot of numerical simulations going on about these things?
Gosh yes, just look at the LANL server! Try astro-phy and gr-qc.
> Or is the numerical search impeded by slow computers
>
> and the lack of numerical experts?
Au contraire! :-)
Fully three dimensional numerical simulation of realistic black hole
collisions and other astrophysical processes expected to result in strong
gravitational radiation constitute one of the Grand Challenges of the
SuperComputing Consortium or whatever it is called these days. So the
best numerical people, highly knowledgeable physicists, and the fastest
supercomputers in the world have been hard at work for quite a few years,
and on LANL you will find many preprints describing the fruit of their
efforts. They're not quite there yet in terms of the actual merger phase
during a truly realistic simulation of the collision of two black holes,
AFAIK, but they are getting very close.
> And on a side note: does there still exist software dedicated to gtr
> calculations or are all you guys just plugging away in maple (maybe
> with dedicated toolboxes within the common mathematical software)?
For symbolic computations, the real pros mostly use SHEEP, I think, but I
use GRTensor to find and investigate the properties of exact solutions
(lots of fun!)--- I check the results by hand (easy if you use Cartan's
curvature two forms) if I am preparing a preprint--- and I have noticed an
increasing number of papers which note that their computations were
checked using GRTensor (I reverse that procedure; I guess this is one
difference between mathematicians and physicists!). The most flexible
currently available GRTensor package is the latest maple version, but I
have found that the Mathematica version works better for me, even though I
have had to write somewhat clunky "patches" to define useful tensor
calculus operations, because I find that the Mathematica simplfication
routines work better. But others have had the opposite experience, and
maple's core is public domain (unlike Mathematica's core, which is rather
buggy and is proprietary), so it is best to try several packages, see
which works best for the applications you have in mind, and if possible
check results by hand or at least use two different software packages and
make sure that they give the same results.
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/personal.html
Eric Forgy
http://jean-luc.ncsa.uiuc.edu/
You can find all you ever wanted to know about the state-of-the-art in
numerical relativity there.
Cheers,
Eric
Robert C. Helling
>Dear All,
>
>Could you tell me what is the state of the art in theoretical and
>numerical relativity research these days?
>
>investigated by now, but what about dynamics and black holes with
>Einstein's equation? Are there a lot of numerical simulations going on
>about these things? Or is the numerical search impeded by slow computers
>
Check out for example the web pages of the numerical GR group at our
institute: http://jean-luc.aei-potsdam.mpg.de
Robert
--
.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Albert Einstein Institute Potsdam
Max Planck Institute For Gravitational Physics
and
2nd Institute for Theoretical Physics
DESY / University of Hamburg
Email hel...@x4u2.desy.de Fon +49 40 8998 4706
<href=http://www.aei-potsdam.mpg.de/~helling>
John Baez
Koen Delaere <koen.d...@esat.kuleuven.ac.be> wrote:
>Could you tell me what is the state of the art in theoretical and
>numerical relativity research these days?
No, but I have friends who can - try this:
http://xxx.lanl.gov/abs/gr-qc/9912009
Bernd Bruegmann
Numerical Relativity in 3+1 Dimensions
Abstract: Numerical relativity is finally approaching a state where the
evolution of rather general (3+1)-dimensional data sets can be computed
in order to solve the Einstein equations. After a general introduction,
three topics of current interest are briefly reviewed: binary black hole
mergers, the evolution of strong gravitational waves, and shift conditions
for neutron star binaries.
>I guess the static and spherically symmetric black hole has been
>investigated by now, but what about dynamics and black holes with
>charge?
See above. Colliding black holes and neutron stars are a hot topic
since LIGO is designed to detect the gravitational radiation produced
by these suckers.
Here's a little information on how it's going, taken from "week135":
..................................................................
In the Classical and Quantum Physics of Strong Gravitational Fields
conference at U. C. Santa Barbara on June 22-26 1999, Richard
Price gave a nice talk on computer simulation of black hole
collisions. Quantitatively understanding the gravitational
radiation emitted in black hole and neutron star collisions is a
big business these days - it's one of the NSF's "grand challenge"
problems. The reason is that folks are spending a lot of money
building gravitational wave detectors like LIGO:
7) LIGO project home page, http://www.ligo.caltech.edu/
8) Other gravitational wave detection projects,
http://www.ligo.caltech.edu/LIGO_web/other_gw/gw_projects.html
and they need to know exactly what to look for. Now, head-on
collisions are the easiest to understand, since one can simplify
the calculation using axial symmetry. Unfortunately, it's not
very likely that two black holes are going to crash into each
other head-on. One really wants to understand what happens when
two black holes spiral into each other. There are two extreme
cases: the case of black holes of equal mass, and the case of
a very light black hole of mass falling into a heavy one.
The latter case is 95% understood, since we can think of the
light black hole as a "test particle" - ignoring its effect
on the heavy one. The light one slowly spirals into the
heavy one until it reaches the innermost stable orbit, and then
falls in. We can use the theory of a relativistic test particle
falling into a black hole to understand the early stages of this
process, and use black hole perturbation theory to study the
"ringdown" of the resulting single black hole in the late stages
of the process. (By "ringdown" I mean the process whereby an
oscillating black hole settles down while emitting gravitational
radiation.) Even the intermediate stages are manageable, because
the radiation of the small black hole doesn't have much effect on
the big one.
By contrast, the case of two black holes of equal mass is less
well understood. We can treat the early stages, where relativistic
effects are small, using a post-Newtonian approximation, and
again we can treat the late stages using black hole perturbation
theory. But things get complicated in the intermediate stage,
because the radiation of each hole greatly effects the other,
and there is no real concept of "innermost stable orbit" in this
case. To make matters worse, the intermediate stage of the process
is exactly the one we really want to understand, because this is
probably when most of the gravitational waves are emitted!
People have spent a lot of work trying to understand black hole
collisions through number-crunching computer calculation, but
it's not easy: when you get down to brass tacks, general relativity
consists of some truly scary nonlinear partial differential equations.
Current work is bedeviled by numerical instability and also the
problem of simulating enough of a region of spacetime to understand
the gravitational radiation being emitted. Fans of mathematical
physics will also realize that gauge-fixing is a major problem.
There is a lot of interest in simplifying the calculations through
"black hole excision": anything going on inside the event horizon
can't affect what happens outside, so if one can get the computer
to *find* the horizon, one can forget about simulating what's going
on inside! But nobody is very good at doing this yet... even using
the simpler concept of "apparent horizon", which can be defined
locally. So there is some serious work left to be done!
.................................................................
>Is there already a library of (analytic/numeric) solutions to
>Einstein's equation?
There are books of analytic solutions, and there's a lot of software
for numerical calculations... but I'm not an expert on this kind of
stuff.
>Are there a lot of numerical simulations going on about these things?
Yes indeed!
>And on a side note: does there still exist software dedicated to gtr
>calculations or are all you guys just plugging away in maple (maybe with
>dedicated toolboxes within the common mathematical software)?
I don't use any of this software since my calculations only involve
pencil and paper, but here is some of the free stuff that's out there:
.......................................................................
Sheep is a Lisp-based computer algebra system for tensor manipulation
especially in gravity and differential geometry. With its extensions
Classi and Stensor it has had more than 50 man years of development.
The source code for Sheep and Classi version 2.059 and a number of
useful auxiliary files is now available by
anonymous FTP from galois.maths.qmw.ac.uk (138.37.80.15) or
euclid.maths.qmw.ac.uk (138.37.80.16) in the directory
homeftp/pub/sheep. Information about obtaining Stensor, and binary
images, is also given there.
The full capabilities of Sheep, Classi and Stensor would take too long
to detail here. An account can be found in the forthcoming review:
J.E.F.\ Skea and M.A.H.\ MacCallum, ``Sheep: a computer algebra system
for general relativity'', to appear in Proceedings of the First
Brazilian School on Computer Algebra, vol. 2, ed.\ W. Roque and
M. J. Reboucas, to appear, Oxford University Press (1993).
but in brief...
Sheep provides the underlying mechanisms for fast and efficient tensor
handling, and the commands to define new tensors, as well as basic
algebraic routines for simplification, differentiation and so on. Its
lack of features such as polynomial factorization or division is
compensated by rather good substitution handling mechanisms. If more
general purpose algebraic facilities are required they can be provided
by the combined Reduce/Sheep images rsheep etc, which give Sheep users
access to Reduce's facilities.
Classi is named because it contains an extensive implementation of the
methods for invariant and classification of metrics based on the
theory developed by Christoffel, Cartan and Karlhede, but it also
provides more user-friendly ways of defining new tensors, spinors and
programs for their manipulation, and a significant number of other
useful programs e.g. for handling electromagnetic fields, perfect and
imperfect fluids and so on. Calculations can be done in coordinate or
tetrad form or in Newman-Penrose form, and variants exist for
non-Riemannian geometries or variant theories of gravity.
The remarks made below about GRG by Zhytnikov more or less apply
completely to Classi also.
Stensor provides an indicial tensor manipulator, handling all types of
indexed objects with symbolic (as distinct from numerical) indices. It
can deal with such problems as octonion calculations, Kaluza-Klein
splitting, gamma matrices, and the generation of component-wise
calculations from symbolic input. There are many more features
described in the documentation.
GRG 3.1
Computer Algebra System for
Differential Geometry,
Gravitation and
Field Theory
The computer algebra program GRG is intended for
calculation in differential geometry, field theory
and gravitation. It manipulate with various field
quantities and equations in any concrete coordinate map.
The main purpose of GRG is to automate as much as possible
various standard field-theoretical calculations.
* Main Features
GRG works in spinorial and tetrad formalism with arbitrary
tetrad (Lorentzian, null, semi-null etc) and uses modern
language of vectors and differential forms. All calculations
can be performed in Riemann or Riemann-Cartan spaces with
torsion. GRG can work with real and complex coordinates,
functions and expressions. All calculations can be curried
out in either holonomic or arbitrary unholonomic basis.
Fast saving/restoring of results of calculations is available.
* Built-in Formulas
GRG know more than 130 built-in geometrical and field
objects and equations and more than 130 formulas for
their calculation:
- Connection and curvature calculation in various representations.
- Curvature and torsion irreducible decomposition.
- Petrov-Penrose classification of Weyl spinor.
- Field equations for arbitrary gravitational Lagrangian.
- Electromagnetic field.
- Scalar field with nonminimal coupling.
- Yang-Mills field.
- Dirac field.
- Killing and Killing-Yano equations.
Large number of known objects and formulas allows to
solve many standard problems automatically without any
special programming. User-defined new objects and formulas
are available too.
* Covariant Operations
GRG knows covariant properties of built-in and
user-defined mathematical objects. It can perform
coordinates transformations, tetrad and spinorial
rotations. GRG calculates covariant differentials and
Lie derivatives of any geometrical and field quantities.
* Input and Output Language
GRG has simple and clear input languge and readable output.
The input language is as close as possible to the ordinary
English and traditional mathematical notation. Example of output:
2 2 1 2 2 2 2 2 2
d s = ( - A) d t + (---) d r + (r ) d th + (SIN(th) *r ) d ph
A
* Technical Information
GRG is distributed free of charge.
GRG 3.1 works with PSL-based (Portable Standard Lisp)
REDUCE 3.3 or 3.4 on various computers.
Current version has been tested on IBM 386/486 (MS-DOS)
personal computers, IBM RISC System/6000, VAX (VMS),
Sun SPARC and Sun 386i.
* Address for Correspondence
V.V. Zhytnikov
Department of Physics,
National Central University,
Chung-li, Taiwan 320
Phone: 886-3-422-7151+5332 or +5337
Fax: 886-3-425-1175
E-mail: vv...@phy.ncu.edu.tw
Jonathan Thornburg
Koen Delaere <koen.d...@esat.kuleuven.ac.be> asked
KD? Could you tell me what is the state of the art in theoretical and
KD? numerical relativity research these days?
In article
<Pine.OSF.4.21.00030...@goedel2.math.washington.edu>,
Chris Hillman <hil...@math.washington.edu> replied
CH: for numerical solutions, see first the excellent expository article on
CH: the Cauchy problem by Friedrich and Rendall:
CH:
CH: http://xxx.lanl.gov/abs/gr-qc/0002074
CH:
CH: and then see this article by Bruegmann:
CH:
CH: http://xxx.lanl.gov/abs/gr-qc/9912009
I also highly recommend Bruegmann's article as a general review of the
state-of-the-art in numerical relativity.
CH: After reading these, you'll know that the two big problems in actually
CH: carrying out the original "ADM" approach to numerically integrating the
CH: EFE come down to:
More accurately, "two of the big problems" -- there are several other
"big problems" as well, and many of us don't think that these two are
in fact the biggest problems.
CH: 1. finding a suitable set of initial data
CH:
CH: (\Sigma, h_(ab), K_(ab), rho, j_a)
CH:
CH: where (\Sigma, h_(ab) is a Riemannian three-manifold, K_(ab) is
CH: a symmetric tensor (will be the extrinsic curvature), rho is the
CH: mass-energy density and j_a is the momentum density covector,
In my opinion, this problem is fairly well under control: several
reasonably-good solutions are known. (Of course, research is continuing
on better solutions.) See Bruegmann's review for details.
CH: 2. finding a suitable method of "gauge fixing" during the evolution, one
CH: which avoids "kinking", extremely inhomegenous advance of "time", and
CH: which avoids curvature singularities.
Yes, this is a significant problem. A number of methods have been
tried, some of which seem to work fairly well, but many people (myself
included) are (still) actively researching better methods.
[[...]]
CH: With respect to 2, there is a large (oldish) literature on gauge fixing in
CH: the ADM formalism (see the appendix in the gtr textbook by Wald for a nice
CH: explanation of the York formalism), but it turns out that the ADM
CH: formalism and its close kin are for various reasons not very suitable for
CH: simulations of phenomena involving curvature singularities (i.e., all the
CH: interesting stuff!). So, be sure to see the article by Friedrich and
CH: Rendall cited above for important newer formalisms which circumvent many
CH: of the problems which, in practice, mitigate against the practical utility
CH: of the ADM formalism.
Many numerical relativists (myself included) would strongly disagree
with this (negative) assessment of the ADM approach. In my opinion,
the ADM approach has a great deal of "practical utility" for studying
(say) dynamic black hole (BH) spacetimes (which of course contain
singularities within the black holes). Some of the literature on
coordinate choice ("gauge fixing") is "oldish"... but some if it is
rather recent, too.
Many numerical relativists (myself included) have used the ADM approach
successfully to study (among others) dynamic BH spacetimes. Notably,
most of the work on simulating the decay/coalescence of BH binaries has
used the ADM approach, and there's a lot of current work continuing to
use it.
I think the "important newer formalisms" Friedrich and Rednall describe
are interesting ideas worthy of further investigation, but at this point
it remains unproven how well they will work in practice. It's certainly
not in any way clear that they will prove preferable to the ADM approach;
I think it likely the two approaches will coexist (and coexist with still
other approaches, i.e. the various 2+2 null-based approaches) in the
field for a long time, i.e. different researchers will prefer different
techniques.
Koen Delaere <koen.d...@esat.kuleuven.ac.be> also asked
KD? Are there a lot of numerical simulations going on
KD? about these things? Or is the numerical search impeded by slow computers
KD? and the lack of numerical experts?
Both statements are true: Yes, there are a lot of numerical simulations
being done. Yes, there is a lot of current research on better ways to
do these simulations. But yes, this is a small field (counting grad
students, there are at most only a few hundred researchers worldwide),
so progress could surely be faster if there were more people working,
including "numerical experts".
As to slow computers: I've been doing numerical relativity for a bit
over 15 years now, and I don't remember ever hearing a numerical
relativist complain that her/his computers were too fast or had too
much memory or disk space. Today's supercomputers are easily adequate
for (say) the simulation of the sparaling decal/coalescence of BH binaries
(a.k.a. "the 2BH problem"), but not all researchers (especially grad
students) have easy access to these, and today's desktop systems aren't
really adequate, in any of cpu speed, memory size, or disk space.
To give a (very) rough idea of the scale of problems: To test a new
code I'm working on, last week I did a 3-spatial-dimensions x time
ADM numerical evolution of Kerr-spacetime initial data for a period
of t=20m. Using a rather low-resolution grid covering only 1/8 of the
neighbourhood of the BH outside the horizon, this took about 145 MB
of memory, and ran for a bit over 2 cpu days on a Pentium 333.
Of course much faster computer systems exist, and are rather common,
but it's often more convenient to use a local "slow system" than a
distant "fast system". (Long-haul internet access is often painfully
slow, in particularly trying to visualize monster datasets over a
long-haul network connection is often impractically slow.)
But having said all this, my opinion is that today, slow computers
aren't the biggest limiting factor in (say) the 2BH problem. That is,
even given much bigger/faster computers, today we still don't know how
to fully solve the 2BH problem. Things are getting better, though,
and I expect the first published results (asymptotic h_+ and h_x
waveforms) for the radiated gravitational radiation in an equal-mass
2BH binary coalescence, in about 3+/-2 years from now.
Koen Delaere <koen.d...@esat.kuleuven.ac.be> also asked
KD? And on a side note: does there still exist software dedicated to gtr
KD? calculations or are all you guys just plugging away in maple (maybe with
KD? dedicated toolboxes within the common mathematical software)?
All of these are common. Many people use "tensor packages" for
Macsyma/Maple/Mathematica. (My impression is that at least in the
numerical relativity community, M/M/M are much more common nowdays
than specialized-for-relativity systems like Sheep.) Many people
also take the result of symbolic manipulations and have have the
symbolic algebra system (M/M/M) generate Fortran/C/C++/etc code
for the number-crunching.
--
-- Jonathan Thornburg <jth...@galileo.thp.univie.ac.at>
http://www.thp.univie.ac.at/~jthorn/home.html
Universitaet Wien (Vienna, Austria) / Institut fuer Theoretische Physik
"Washing one's hands of the conflict between the powerful and the powerless
means to side with the powerful, not to be neutral." - Freire / OXFAM
John Baez
>the Aichelburg-Sexl ultrarelativistic Schwarzschild hole [...]
What's that? I mainly know Aichelburg as the extremely courteous
and friendly Viennese host of the Schrodinger Institute's program
on loop quantum gravity a few summers ago. I didn't know he had
an ultrarelativistic black hole!
Please pretend I am a high school student and explain without any
equations... I just want a rough idea of the physics behind this
solution.
>The essential sourcebook for exact solutions is the book by
>Herlt et al., Exact Solutions of Einstein's Field Equations (unfortunately
>out of print, very out of date, and rather hard to use). A few days spent
>poring over this book will show that thousands of solutions were known by
>1975, and as far as I can tell, this number has doubled by now.
I believe that GRG and/or the Journal of Classical and Quantum Gravity
now have editorial policies saying that they will not publish exact
solutions of general relativity, even if they are new, unless the author
explains what is *interesting* about the solution. So I guess there's
no shortage of such solutions! I like your idea of a computerized
database.
Chris Hillman
On Thu, 9 Mar 2000, Jonathan Thornburg wrote:
> CH: After reading these, you'll know that the two big problems in
> actually CH: carrying out the original "ADM" approach to numerically
> integrating the CH: EFE come down to:
>
> More accurately, "two of the big problems" -- there are several other
> "big problems" as well, and many of us don't think that these two are
> in fact the biggest problems.
OK.
> CH: 1. finding a suitable set of initial data
> CH:
> CH: (\Sigma, h_(ab), K_(ab), rho, j_a)
> CH:
> CH: where (\Sigma, h_(ab) is a Riemannian three-manifold, K_(ab) is
> CH: a symmetric tensor (will be the extrinsic curvature), rho is the
> CH: mass-energy density and j_a is the momentum density covector,
>
> In my opinion, this problem is fairly well under control: several
> reasonably-good solutions are known. (Of course, research is continuing
> on better solutions.) See Bruegmann's review for details.
I didn't express myself very clearly--- what I was trying to say is that
coming up with initial data is one of the things you have to accomplish to
get the ADM aproach off the ground. What I didn't say clearly enough is
that this problem has been well understood and considered "tractable" for
a long time now, as readers of the appendix in the textbook by Wald will
appreciate. I probably should have said that the paper by Friedrich and
Rendall doesn't -discuss- the problem of coming up with initial data at
all, precisely because it is so well understood!
> Many numerical relativists (myself included) would strongly disagree
> with this (negative) assessment of the ADM approach. In my opinion,
> the ADM approach has a great deal of "practical utility" for studying
> (say) dynamic black hole (BH) spacetimes (which of course contain
> singularities within the black holes). Some of the literature on
> coordinate choice ("gauge fixing") is "oldish"... but some if it is
> rather recent, too.
I stand corrected :-/
> Many numerical relativists (myself included) have used the ADM
> approach successfully to study (among others) dynamic BH spacetimes.
> Notably, most of the work on simulating the decay/coalescence of BH
> binaries has used the ADM approach, and there's a lot of current work
> continuing to use it.
I didn't express myself sufficiently clearly, and I apologize.
In a long post to this newsgroup or another I did try to distinguish
between the late stage orbital decay, merger, and "ringdown" phases of the
coalesence of two black holes (or a black hole and a neutron star),
stressing that the decay and ringdown phases are well understood from
stable numerical simulations and from perturbation theory (right?).
However, IIRC, recent papers on the theory of interferometric
gravitational wave detectors by Schutz and others do say that the merger
phase is still incompletely understood. And IIRC, at least one recent
paper on numerical simulations (can't remember which) does say that the
ADM approach (using mean curvature gauge fixing) runs into trouble in
simulating gravitational collapse or binary mergers because the
hyperslices "curl up". Is this incorrect? Am I wrong in thinking that
the ADM approach is ill-suited to the merger phase in numerical
simulations of black hole coalescence?
> I think the "important newer formalisms" Friedrich and Rednall
> describe are interesting ideas worthy of further investigation, but at
> this point it remains unproven how well they will work in practice.
Friedrich and Rendall don't present them as "battle-tested" formalisms.
I should have made that clear.
> It's certainly not in any way clear that they will prove preferable to
> the ADM approach; I think it likely the two approaches will coexist
> (and coexist with still other approaches, i.e. the various 2+2
> null-based approaches) in the field for a long time, i.e. different
> researchers will prefer different techniques.
Well, this is good news for me, then, since so far the ADM approach is the
only one I -think- I have a good handle on, conceptually, although I've
been studying the alternatives described by Friedrich and Rendall bit by
bit. In a recent post to sci.physics,
http://www.math.washington.edu/~hillman/PUB/efe
I attempted to complement John Baez's explanation of the Einstein field
equation using the Raychaudhuri equation by sketching the ADM formulation,
based in part on the discussion of Friedrich and Rendall, and trying (once
more unto the breach!) to give some intution for why the local versus
global distinction is so important in gtr (cf. the bit about smooth versus
analytic manifolds). If you have a chance to look at that and catch any
errors or misrepresentations, I'd appreciate correction!
> Both statements are true: Yes, there are a lot of numerical
> simulations being done. Yes, there is a lot of current research on
> better ways to do these simulations. But yes, this is a small field
> (counting grad students, there are at most only a few hundred
> researchers worldwide), so progress could surely be faster if there
> were more people working, including "numerical experts".
[...]
I stand corrected on many, many points! Thanks for your very interesting
and informative post! I've felt for a long time that it would be good to
have an entry in the sci.physics.relativity FAQ on numerical relativity---
maybe your post could be adapted for that purpose, if you're willing.
Chris Hillman
Aichelburg-Sexl ultrarelavistic boost of a Schwarzschild hole, which is
strictly speaking not a new coordinate chart for the Schwarzschild vacuum,
but a coordinate chart for a "magnified piece" of it. Robert Helling
provided a good high school level explanation:
> The idea is that that you take the Schwarzschild solution and then
> boost it (meaning define new coordinates x',t' as in SRT) the trick is
> somehow to 'renormalize' the mass in this boost such that you end up
> with something when you have boosted to the speed of light.
before I could get back to John :-/
The "renormalization" involves boosting the hole by velocity v, setting
mu = m/sqrt(1-v^2)
where m is the mass of the hole, and then keeping mu constant as you let v
-> 1. So, you increase v and decrease m in a controlled way. The result
is an axially symmetric -impulsive- pp wave:
ds^2 = -2 mu delta(U) log(X^2 + Y^2) dU^2 - 2 dU dV + dX^2 + dY^2
where delta is a Dirac delta "generalized function" (linear functional).
Pirani recognized as early as 1959 that this could be used to study the
"ultrarelativistic collision" of two black holes, by having two such
impulsive PP waves collide "head on" (if the waves are not propagating in
the same direction, a "head on" collision can always be arranged by a
change of coordinates), and he further recognized that the renormalization
changes the algebraic type of the Weyl tensor from type D (two double
principal null directions) to type N (one quadruple principle null
direction).
[snip great stuff about PP waves]
Wow, I am going to try to work that into the introduction to my
forthcoming introduction to plane waves (special case of PP waves).
PP waves are a large class of solutions found by Brinkmann (1923) and have
a whole bunch of amazing properties. In Brinkmann's coordinate chart the
line element is
ds^2 = -H(U,X,Y) dU^2 - 2 dU dV + dX^2 + d^2
They model radiation (gravitational radiation, incoherent massless
radiation, or mixtures) far from some compact source, and are the natural
generalization to gtr of the familiar plane waves introduced in most first
year college physics courses which cover Maxwell's theory of
electromagnetism. Here, H is assumed to be C^2, and if it is harmonic in
X,Y (as in the Aichelburg-Sexl example), then the Einstein tensor
vanishes, so that this is a pure gravitational wave (disturbance in Weyl
curvature propagating at the speed of light). Of course, this implies
that H blows up for large X,Y, so the Brinkmann chart is not entirely well
behaved. The geometrical-physical reason for this behavior is explained
in my forthcoming eprint (which will be too long and expository to be
publishable). By the way, if H is harmonic in X,Y, then H_(XX) controls
on linear polarization mode and H_(XY) the other, and in this case the
Riemann tensor is "conserved" in the sense that
R^(abcd)_(;d) = 0
Indeed, as Penrose noticed, all the 14 CM scalar invariants vanish
identically, and Jordan et al. showed that -all- scalar invariants which
can be built out of covariant derivatives (of any order) of the Riemann
tensor vanish identically for individual gravitational PP waves. This
makes it a real challenge to show that some plane waves are not
geodesically complete after all. The (correct) theorem saying that they
are assumes that H is defined on R^3, but if one drops this assumption, I
can give simple explicit examples of "thunderbolts" (Hawking's term) or
"waves of death" (my term), which are gravitational planes waves which
propagate through what initially appears to be Minkowski spacetime,
destroying spacetime as they go.
The earliest explicit example of an exact solution consisting of the
head-on collision of two PP waves was the famous Penrose-Khan solution
(see the gtr textbook by D'Inverno). This provided a shock (no pun
intended!) because the nonlinear interaction post-collision leads to the
formation of a curvature singularity, and the global structure of this
solution turns out to be very surprising.
Since we presumably experience the collision of (very weak!) gravitational
waves in our vicinity fairly frequently, but have not observed curvature
singularities, it was initially hoped that perturbations of the extreme
symmetry of the colliding waves would destroy the singularities. This
does sometimes happen, but apparently perturbation is more likely to
-strengthen- the singularity. This is important because it tends to
suggest that Penrose's cosmic censorship conjecture might not be true.
Certainly naked singularities are more common than "hidden" ones in exact
solutions; Penrose and others hoped that these would all be unrealistic,
but there is a disturbing amount of evidence that this might not be true,
at least if one takes gtr at face value.
Let me point out that there is one other possible "classical physics"
explanation of why we do not observe naked curvature singularities in our
vicinity--- these may indeed form, but so far in the future that we have
not yet been around long enough to observe any of them. I haven't had the
courage to plug numbers in the Penrose-Khan solution to see if that is
realistic :-/
Maybe experts here can provide alternative explanations--- I have tried to
provide in the introduction to my preprint a brief survey of what is known
about PP waves on the basis of reading or at least skimming all the papers
I could grab off the web, and my paper has about 75 references, including
the book by Griffiths on colliding impulsive PP waves.
Let me just mention a few more amazing facts:
1. PP waves traveling in the -same- direction superimpose linearly, but
if they travel in -opposite- directions, they interact nonlinearly.
2. Colliding impulsive gravitational PP waves are equivalent via an
internal symmetry of the Einstein field equation to interesting vacuum
solutions, e.g Chandrasekhar and Ferrari showed:
a) Penrose-Khan colliding impulsive PP waves <--> Schwarzschild vacuum
b) Nutku-Halil colliding impulsive PP waves <--> Kerr vacuum
3. Ferrari et al. showed that the solition solution generating technique
of Belinksy and Sakharov, which starts with a "seed solution", yields
a) Minkowski vacuum --> Schwarzschild vacuum
b) Kasner vacuum --> Penrose-Khan colliding impulsive PP waves
4. These tricks depend upon the presence of two commuting Killing vector
fields (as in stationary axisymmetric vacuums and gravitational plane
waves), and Schmidt has shown that the EFE restricted to spacetimes with
two commuting Killing fields can be reformulated as a metric-dilaton
theory in two dimensions with two scalar fields. (I think this explains
why such solutions have the property that their geodesic equations can
often be solved in closed form, for example.)
5. PP waves do not occur in dimensional reductions of gtr, but higher
dimensional PP waves are dual to fundamental strings and turn up in the
Newtonian (!) n-body problem.
By the way, Charles Torre (who I hope is reading this!) has characterized
the EFE in terms of plane-symmetric solutions, but I am not sure how this
fits in with my collection of remarkable facts concerning PP waves or the
special case of plane waves, since not every plane symmetric vacuum is a
plane wave (c.f. the Levi-Civita type AIII or McVittie or Taub vacuum,
whose Riemann [Weyl] tensor has algebraic type D).
In my paper, I introduce some interesting ONB's, and then specialize to
the study of Baldwin and Jeffery plane waves
ds^2 = (X^2-Y^2) f(U) + 2 XY g(U) + (X^2 + Y^2) h(U)
(which have a five or six dimensional isometry group acting on the
hyperslices U = U_0; I list independent Killing vector fields explicitly).
An interesting consequence of these additional symmetries is that one can
change to a new coordinate chart, the Rosen chart, which for a linearly
polarized gravitational wave is
ds^2 = -dt^2 + dz^2 + P(t-z)^2 dx^2 + Q(t-z)^2 dy^2
and then the optical scalars of the null congruence k = e_t + e_z vanish,
but those of k = e_t - e_z do not, so that "we cannot see the plane wave
entering the station (by looking through the advancing wavefronts at
distant objects), but we can see it leave (by looking through the
departing wavefronts at distant objects)". This is another way of
understanding the nonlinearity of these waves.
In the rest of the paper, I study in detail some instructive examples,
such as uniform EM waves (well known under various names in the
literature), explicit pulse waves causing linear divergence of initially
stationary test particles after passage, or leaving them displaced but
again stationary, and waves with exponentially growing amplitude, plus an
explicit wave of death propagating through Minkowski vacuum, plus waves
with cylindrical wavefronts on H^3 and Hopf torus wavefronts on S^3 (I
have already discussed these in detail in previous posts to this group).
As always, I would welcome any misstatements in what I've said from the
experts here (and I'd like to hear about any additional remarkable facts
about PP waves which I didn't mention above--- I know about a great number
of papers studying chaos in the geodesics, PP waves arising from the
collapse of domain walls, PP waves arising as traveling waves on cosmic
strings, interactions of PP waves with Yang-Mills fields and exotic
matter, and classifications of PP waves by their symmetries (Killing
vector fields), and other such stuff, but might have overlooked some
"remarkable facts" found by these authors.).
Charles Torre
> By the way, Charles Torre (who I hope is reading this!) has characterized
> the EFE in terms of plane-symmetric solutions, but I am not sure how this
> fits in with my collection of remarkable facts concerning PP waves or the
> special case of plane waves, since not every plane symmetric vacuum is a
> plane wave (c.f. the Levi-Civita type AIII or McVittie or Taub vacuum,
> whose Riemann [Weyl] tensor has algebraic type D).
I guess it depends on what you mean by "plane symmetry". I
get the feeling that your use of "plane symmetry" refers to
a two dimensional Abelian isometry group. The work of mine
you are referring to characterizes plane waves (not the
more general pp waves) in terms of their symmetry group,
which you might call the "plane wave symmetry group". This
group is 5 dimensional and has null hypersurfaces for
orbits. According to Petrov ("Einstein spaces") there is
only one (up to diffeo) 5-d transformation group that
admits an invariant Lorentz metric and whose orbits are
null hypersurfaces. If you write down the most general
metric admitting this group as an isometry group and you
demand it be Ricci flat, you get the (vacuum) plane wave
metrics. This is essentially the route taken by Bondi,
Pirani and Robinson in their work on plane waves in the
late 1950's.
Charles Torre
Chris Hillman
> Chris Hillman <hil...@math.washington.edu> writes:
> > By the way, Charles Torre (who I hope is reading this!) has characterized
> > the EFE in terms of plane-symmetric solutions, but I am not sure how this
> > fits in with my collection of remarkable facts concerning PP waves
[snip]
> I guess it depends on what you mean by "plane symmetry". I
> get the feeling that your use of "plane symmetry" refers to
> a two dimensional Abelian isometry group.
I meant a group generated by two independent -spacelike- Killing vector
fields :-/
> The work of mine you are referring to characterizes plane waves (not
> the more general pp waves) in terms of their symmetry group, which you
> might call the "plane wave symmetry group". This group is 5
> dimensional and has null hypersurfaces for orbits.
Wow! This is very good news! Sorry I didn't understand the paper better,
but this is -exactly- what I -thought- you were talking about when you
mentioned the paper, but when I read it I thought I'd misunderstood :-(
Silly me.
The "striking facts" I listed in my previous post all concerned pp waves,
but after the second section, the rest of the paper is about -plane-
waves, so I'll rewrite the introduction to stress your result.
> According to Petrov ("Einstein spaces") there is only one (up to
> diffeo) 5-d transformation group that admits an invariant Lorentz
> metric and whose orbits are null hypersurfaces. If you write down the
> most general metric admitting this group as an isometry group and you
> demand it be Ricci flat, you get the (vacuum) plane wave metrics. This
> is essentially the route taken by Bondi, Pirani and Robinson in their
> work on plane waves in the late 1950's.
Hmm... the isometry group admits an invariant Lorentz metric?
I'll try to grep this too so I can add a summary to the introduction,
contrasting the list of interesting things one can say about pp waves in
general and the interesting things one can say about plane waves in
general. That will be nice because the next section gives some details
concerning pp waves (e.g. reduction of geodesic equations) and the one
after that starts in on plane waves (e.g. discussing Rosen and Bondi
coordinate charts). The remaining sections study explicit plane waves
which exhibit interesting effects on test particles.
I never wanted to try to write a "review" of the enormous literature of pp
waves (I am obviously unqualified to do that!), but I did want to try to
point novice readers to some of the interesting stuff which is out there,
like the Aichelburg-Sexl "ultrarelativistic boosted Schwarzschild hole" we
were discussing. My hope is that the paper will be comprehensible to
graduate students, blah, blah, blah.
Charles Torre
> On 15 Mar 2000, Charles Torre wrote:
>> I guess it depends on what you mean by "plane symmetry". I
>> get the feeling that your use of "plane symmetry" refers to
>> a two dimensional Abelian isometry group.
> I meant a group generated by two independent -spacelike- Killing vector
> fields :-/
Fine.
>> According to Petrov ("Einstein spaces") there is only one (up to
>> diffeo) 5-d transformation group that admits an invariant Lorentz
>> metric and whose orbits are null hypersurfaces. If you write down the
>> most general metric admitting this group as an isometry group and you
>> demand it be Ricci flat, you get the (vacuum) plane wave metrics. This
>> is essentially the route taken by Bondi, Pirani and Robinson in their
>> work on plane waves in the late 1950's.
> Hmm... the isometry group admits an invariant Lorentz metric?
I am not sure why this makes you scratch your head, but let me
try to clarify anyway (and make more precise my terse summary
above). I know that a lot of the following is obvious to you,
but it doesn't hurt to spell things out. I am afraid it is
a long and boring story...
The game Petrov plays is as follows. Given a manifold M you can
consider the group of diffeomorphisms f: M -> M. Given a
Lorentz-signature metric g, a diffeomorphism is a symmetry of
the metric (an isometry) if f*g = g, where f* is pull-back. And,
of course, the set of diffeos preserving a given metric forms a
subgroup of the group of diffeos. It is easy to see that most
subgroups of diffeos are such that there is NO Lorentz metric
for which they are symmetries. (For example, a necessary
condition for a diffeo subgroup to preserve a Lorentz-signature
metric is that its linear isotropy group at a point be a
subgroup of the Lorentz group. It is easy to create group
actions - at least local ones - that do not have this property.)
One can then ask: which subgroups of diffeos can be isometry
groups of some (class of) Lorentz metric(s)? Petrov answered
this question for the local, connected subgroups of the diffeo
group. That is, he found all Lie algebras of vector fields that
are Killing vector fields for some Lorentz signature metric, and
he exhibited the form of said metric in some specialized
coordinates. More succinctly, Petrov classified all spacetimes
with symmetry (no field equations imposed, though).
Actually, he used a lot of other people's work too.
He found that there are something like 100 symmetry classes.
(Here at USU we are working very hard to further develop, extend,
and apply Petrov's classification.)
Let me remind you how many textbooks like to derive, say, the
Schwarzschild solution (stationary, spherically symmetric solution)
to the EFE. One picks a four dimensional
transformation group (a subgroup of diffeos) with orbits R x
S^2, corresponding to time translations and rotations. Up to
diffeomorphism (or coordinate choice, if you like), this group
action is essentially unique. One looks for all
Lorentz-signature metrics that are invariant under this group
action and such that the spheres are spacelike and the R is a
timelike curve. One finds that the metrics depend on 4 functions
of one variable, one plugs the result into the Einstein tensor,
etc.
Ok, similarly, if you look at all those Lie algebras of vector
fields that Petrov found and ask which are (1) 5 dimensional,
(2) have orbits which are null with respect to all the metrics
for which they are Killing vector fields, you find a that these
Lie algebras of vector fields are parametrized (up to coordinate
choice) by two arbitrary functions of one variable. The abstract
Lie algebra is uniquely determined by (1) and (2), but the
vector fields depend upon those two functions. So, unlike the
stationary, spherically symmetric case, the group action here is
not unique. Anyway, when you write down the metrics for which
these vector fields are Killing vector fields, you get a family
of metrics that depend upon a number of arbitrary functions, as
well as those two functions that went into the definition of the
vector fields in the first place. Imposing the Ricci flat
condition on this family of metrics yields plane wave metrics.
Conversely, every plane wave metric admits one of the 5-d Lie
group actions defined by (1) and (2) as (a subgroup of) its
isometry group for some choice of those two functions.
Physically, the two functions that appeared in the Killing
vector fields turn out to control the amplitude and polarization
of the wave.
Charles Torre
Chris Hillman
> Chris Hillman <hil...@math.washington.edu> writes:
>
> > Hmm... the isometry group admits an invariant Lorentz metric?
>
> I am not sure why this makes you scratch your head,
I thought you meant that the isometry group itself could be given an
invariant semi-Riemannian metric (there is a canonical way of doing this
for "isometry groups" arising from quadratic forms on R^d, so my
misinterpretation of the phrase "admits an invariant Lorentz metric" was
not entirely unnatural).
> I am afraid it is a long and boring story...
Actually, I thought this was very interesting! I'll certainly add
something about this to the introduction of my eprint.
> Imposing the Ricci flat condition on this family of metrics yields
> plane wave metrics.
What if you just ask for a null dust, i.e. that in an appropriate ONB the
Einstein tensor have the form
[ 1 1 0 0 ]
G^(ab) = [ 1 1 0 0 ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
i.e. if you permit massless radiation sharing the same wavefronts as any
gravitational radiation?
By the way, a minor clarification for the benefit of others--- people who
have read more than two or three of my own posts :-/ or who have glanced
through a modern gtr textbook have probably noticed references to "the
Petrov type of a spacetime" (more properly, of the Weyl tensor at an event
in a spacetime) or the "Petrov classification of the algebraic type of the
Weyl tensor" (at some event in a spacetime). That's not the same thing as
the Petrov classification Charles is talking about in this post.
Unless I was reading -much- too fast :-/