finding all Killing vectors
Urs Schreiber
given manifold systematically? Or at least to find out how many there are?
There is the Killing equation, but it's usually hard to finds all its
solutions. Also, the commutator of two Killing vectors may give a new one.
What else?
Ralf Muschall
> Are there any algorithms/methods to find *all* the Killing vectors of a
> given manifold systematically? Or at least to find out how many there are?
IIRC this is not as hard as it seems, since the Killing equations
are overdetermined (i.e. they usually have no solutions at all, but
if the have solutions, they can be found).
Look for a person named Thomas Wolf (formerly in Jena/Germany - I have
no idea what/where he is doing now). He wrote a REDUCE package
("crackstar") to solve overdetermined PDEs (for another purpose), but
adaption should be possible.
The process is similar to Buchberger's algorithm for Groebner bases
(which are useful to find solution for DEs of order zero :-)), just
different (i.e. more complicated and heuristically optimized).
IIRC even manual search for KVs was only boring and lengthy, but not
specially interesting or hard.
Ralf
Urs Schreiber
It looks like I am really only concerned with metrics of constant
scalar curvature for now. This should simplify things. I found a paper
(gr-qc/9403036) which gives an explicit formula for all Killing
vectors of such symmetric spaces - but in terms of the geodetic
distance! For me this quantity is right now just as inaccessible as
the complete set of Killing vectors itself, so this did not help much.
Or is there an easy way to find the geodetic distance? (Easy here
means that I can make Mathematica do it or at least help me do it.)
Also, I have now begun to search the web and did - of course! - find
some postings by Chris Hillman. In a 1999 posting Chris Hillman
mentions "Jim Skea's CLASSI database" and his "GRTensorM database of
approximatly 1300 known metrics expressed in terms of orthonormal
frames". I am under the impression that these databases do contain
information on Killing vectors, but I could not connect to the one by
Jim Skea (a problem that is also already mentioned in 1999) and I did
not find the one based on GRTensorM on the web (is it accessible?).
Also, I am not sure if it would be any help, since most of the metrics
whose Killing vectors I would like to know are of higher than four
dimensions. On the other hand, they are generically rather simple:
diagonal, all except one or two entries constant, the non-constant
entries always of the form Exp[a1 x1 + a2 x2 +...] with ai constant.
Apparently all cases I am concerned with also have constant negative
scalar curvature. Maybe there is a general algorithm for such special
cases? (Actually, on first sight I thought that this should be easy
enough to solve by looking at Killing's equation long enough, but by
that method I am only able to find n=dim[metric] Killing vectors,
usually.)
I have also searched the web for computer algebra packages that might
do the trick, but from reading the descriptions it did not appear that
solving the Killing equation is a standard feature of these. If I am
wrong here, please let me know. From my experience with Mathematica
the general problem is that the DEs involved are PDEs. If I could find
a slick ansatz that reduces the problem to that of a system of ODEs
Mathematica will probably be of great help. Can anyone provide me with
some general advice on how to tackle systems of PDEs with computer
algebra systems?
Thanks very much for help,
Urs Schreiber
Norbert Dragon
An analysis of the Killing equation for the vectorfield xi shows that
the second covariant can be expressed in terms of xi and the Riemann
tensor
D_l D_m xi_n = R_nml^k xi_k
Differentiating one obtains a linear system for xi and its first
antisymmetrised derivatives of the form
(D-R-components) * xi + R-components * (D xi) = 0
(eq. D.16 in my script "Geometrie der Relativitaetstheorie")
In principle one can determine the rank of this system.
It has rank 0 in maximally symmetric spaces.
Further differentiation yields further restrictions.
Thereby one obtains an upper bound on the dimension of the
vector space spanned by the Killing vector fields.
--
Norbert Dragon
dra...@itp.uni-hannover.de
http://www.itp.uni-hannover.de/~dragon
Chris Hillman
On Mon, 13 Aug 2001, Urs Schreiber wrote:
> "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
>
> > Are there any algorithms/methods to find *all* the Killing vectors of a
> > given manifold systematically? Or at least to find out how many there are?
Just a general comment--- I am pretty sure that Cartan developed a
systematic approach, as part of his solution of a general class of
equivalence problems in differential geometry, but I am not yet familiar
with this yet. But see the books by Peter J. Olver cited in another very
recent post of mine (due to vagaries of moderation, it might appear
shortly -after- this one, but you'll know it when you see it).
Ralf Muschall sketched a procedure which sounds like the kind of approach
which would arise from Cartan's work, which in a sense explains -all- the
trickery one learns in a traditional course on ODEs, like integrating
factors, in terms the symmetries of the equation to be solved. As I
mentioned in the other post, the marriage of computational algebraic
invariant theory (Groebner bases play a crucial role here) with the deep
work of Cartan seem to be producing a practical method of -systematic-
attack on many problems which previously could only be attacked by various
special tricks.
[snip]
> Or is there an easy way to find the geodetic distance? (Easy here
> means that I can make Mathematica do it or at least help me do it.)
At first I assumed "geodetic distance" is just a synonym for "geodesic
distance" but a VERY quick look at the paper failed to confirm this.
> Also, I have now begun to search the web and did - of course! - find
> some postings by Chris Hillman. In a 1999 posting Chris Hillman
> mentions "Jim Skea's CLASSI database" and his "GRTensorM database of
> approximatly 1300 known metrics expressed in terms of orthonormal
> frames".
I currently have 696 entries in my GRTensorM for Maple database, and
approximately 1960 entries in my older GRTensorM for Mathematica database.
However, both are "dirty", especially the older one, and none are on-line.
Kayll Lake and Mustapha Ishak at Queens University have an experimental
on-line database currently at
http://130.15.26.66/servlet/GRDB2.GRDBServlet
So far it's fairly rudimentary in comparison to Jim Skea's database.
> I am under the impression that these databases do contain information
> on Killing vectors, but I could not connect to the one by Jim Skea (a
> problem that is also already mentioned in 1999)
I have been unable to contact Jim Skea since about 1999--- but fortunately
there is a mirror of his database at
http://www.astro.queensu.ca/~jimsk/
Skea's database consists of runs of his CLASSI on a large number of
published metric tensors (defined on some coordinate chart), and IIRC, in
the course of each run, CLASSI explicitly computes the isometry and
isotropy groups, by systematically determining the Killing vectors. I
don't really understand how CLASSI works, but maybe someone at Queens
University does. IIRC, if you look at the entries in the database, you
can see the Killing vectors (infinitesimal generators of the local
isometry group) written out.
> I have also searched the web for computer algebra packages
[snip]
> Can anyone provide me with some general advice on how to tackle
> systems of PDEs with computer algebra systems?
Maple already incorporates some of Lie's methods in its ODE solver, but
apparently not yet in its PDE solver (which appears to attempt only a
separation of variables attack). As I said in my other post, I would like
to know if anyone has developed a Maple package implementing the
techniques described in Olver's books and other recent books on the
Lie-Cartan theory of symmetries of differential equations. Because these
methods are -systematic-, they are ideally suited for implementation in a
computer algebra system.
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
NOTE: Since I post under my real name, as an anti-spam measure, I have
installed a mail filter which deletes incoming messages not from the
"*.edu" or "*.gov" domains or overseas academic domains.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Charles Torre
This approach is worked out more or less explicitly by Eisenhart in his
book on tranformation groups. So you can look there for the formulas
and theorems.
It is possible to automate the procedure described above. We have
some Maple code written here at Utah State that will take a metric
and (with a little luck) use this linear system to compute the
Lie algebra of the Killing vector fields. It may not be quite ready for
public consumption, but send me email if you are interested. Note, though,
that I did not say that our code will compute the actual Killing vector
fields, just their Lie algebra. So it will tell you how many there are and
what their brackets are. To get formulas for the vector fields themselves,
it seems that you have to solve the DEs.
-charlie
Urs.Sc...@uni-essen.de
<Pine.OSF.4.33.010813...@goedel1.math.washington.e
>
> On Mon, 13 Aug 2001, Urs Schreiber wrote:
> > Or is there an easy way to find the geodetic distance? (Easy
here
> > means that I can make Mathematica do it or at least help me do
it.)
>
> At first I assumed "geodetic distance" is just a synonym for
"geodesic
> distance" but a VERY quick look at the paper failed to confirm
this.
First of all: Thank you very much for the wealth of information and
references that you provided in this thread and another one.
Concerning "geodetic" versus "geodesic": I hope it all referes to
the same concept (because otherwise I am missing something :-) The
paper I was referring to mentions a
"[...] world function (or geodetic interval) [...]"
in the third paragraph on page 11 and then says
"One can show that the geodetic interval characterizes the geometry
of the manifold in full, i.e. it gives an alternative description of
a curved space [...]"
Gordon D. Pusch
> Concerning "geodetic" versus "geodesic": I hope it all referes to
> the same concept (because otherwise I am missing something :-) The
> paper I was referring to mentions a
>
> "[...] world function (or geodetic interval) [...]"
>
> in the third paragraph on page 11 and then says
>
> "One can show that the geodetic interval characterizes the geometry
> of the manifold in full, i.e. it gives an alternative description of
> a curved space [...]"
One of the earliest extensive discussions of the ``world function'' approach
to curved spacetime is in J.L. Synge's ``Relativity: The General Theory.''
IIRC, deWitt also used the ``world function'' extensively in a book on
quantum field theory in curved spacetime, but I'm afraid I've misplaced
the title... :-(
-- Gordon D. Pusch
perl -e '$_ = "gdpusch\@NO.xnet.SPAM.com\n"; s/NO\.//; s/SPAM\.//; print;'
Robert Low
>du> Chris Hillman <hil...@math.washington.edu> writes:
>> At first I assumed "geodetic distance" is just a synonym for
>"geodesic
>> distance" but a VERY quick look at the paper failed to confirm
>this.
>the same concept (because otherwise I am missing something :-) The
>paper I was referring to mentions a
There's a (not very widely observed) convention of distinguishing
between a 'geodesic' (noun) and a 'geodetic' (adjective)
quantity. I used it in my thesis, and my examiners (neither of
whom could safely be accused of vast ignorance) simply
inferred that I couldn't make up my mind how to spell 'geodesic'.
Ah well.
Urs.Sc...@uni-essen.de
<Pine.OSF.4.33.010813...@goedel1.math.washington.e
> > Or is there an easy way to find the geodetic distance? (Easy
here
> > means that I can make Mathematica do it or at least help me do
it.)
>
> At first I assumed "geodetic distance" is just a synonym for
"geodesic
> distance" but a VERY quick look at the paper failed to confirm
this.
First of all: Thank you very much for the wealth of information and
references that you provided in this thread and another one.
Concerning "geodetic" versus "geodesic": I hope it all referes to
the same concept (because otherwise I am missing something :-) The
paper I was referring to mentions a
"[...] world function (or geodetic interval) [...]"
in the third paragraph on page 11 and then says
"One can show that the geodetic interval characterizes the geometry
of the manifold in full, i.e. it gives an alternative description of
a curved space [...]"
Gordon D. Pusch
The ``world function'' is a biscalar (i.e., a ``two-point'' scalar)
`function' whose value is the integral of the line-element along
a geodesic (or more generally, an extremal path) connecting the two
points. (Calling it a ``function'' is not strictly speaking correct,
since it can obviously be multivalued if more than one geodesic can
connect the two points.)
Urs Schreiber
news:m2ae0yh...@pusch.xnet.com...
> It occurs to me I neglected to define what the ``world function''
*is*.
>
> The ``world function'' is a biscalar (i.e., a ``two-point'' scalar)
> `function' whose value is the integral of the line-element along
> a geodesic (or more generally, an extremal path) connecting the two
> points. (Calling it a ``function'' is not strictly speaking
correct,
> since it can obviously be multivalued if more than one geodesic can
> connect the two points.)
So it *is* the geode?ic distance, after all!?
P.S. My apologies to the moderators that my last posting probably
arrived a dozen of times. The machine I was posting from kept telling
me that it could not and did not send it.