Dear Dr. Meier,

May I request your opinion on the initial value problem.

Your astro-ph/0312052 v1 [Ref. 1] has been quoted as ref [3] in a recent paper by Cayetano Di Bartolo, Rodolfo Gambini, and Jorge Pullin [Ref. 2].

Perhaps the problem cannot be resolved numerically in principle, that is, not even by a truly four-dimensional grid method [Ref. 1].

Consider this. In present-day models of spacetime, we inevitably start with some *finite* time intervals to describe the dynamics of some system, which we inevitably consider as a sub-system from 'the only truly isolated system' (=the whole universe).

Hence the initial value problem of the sub-system under consideration is inevitably linked to the initial value problem of 'the whole universe'. To resolve the latter, we need new ideas about spacetime: the sliding cut-off needed to fix any *finite* time interval cannot be found in any sub-system, the *current* universe included,

http://God-does-not-play-dice.net/Schwarz.html

http://God-does-not-play-dice.net/Linder.html#Cauchy

I wonder if this is the *conceptual basis* of the initial value problem. If this is the case, I would guess that it cannot be solved by any computer, since fixing the initial conditions for any finite sub-system will lead to the task of finding the initial conditions of a larger system, etc., ad infinitum.

Please be assured that your feedback and the opinion of your colleagues will be kept strictly private.

Regards,

Dimi Chakalov
--
http://God-does-not-play-dice.net
 

References

[Ref. 1] David L. Meier, Constrained Transport Algorithms for Numerical Relativity. I. Development of a Finite Difference Scheme, astro-ph/0312052 v1, 2 December 2003

pp. 23-24: "But a serious problem still remains. There is no means of enforcing the Hamiltonian constraint at t = ^1t.
...

"The elegant, and proper, method of solving this problem is to solve all ten of the Einstein field equations simultaneously on the initial hypersurfaces. (...) The reader will, of course, recognize that this is more than solving an initial data problem; in actuality the proposed scheme solves the initial data problem plus the first file evolutionary time step simultaneously. This is done to ensure that the initial data on the first three hypersurfaces are solutions of the discrete, staggered evolutionary field equations. No constraint violation will be introduced implicitly other than what is naturally present in the evolutionary method already."
...

p. 26: "This would be a truly four-dimensional grid method and involve solving the entire spacetime structure in one giant iterative procedure. Present-day computers still struggle with three-dimensional explicit schemes, so a four-dimensional implicit one is clearly beyond current technology."
 
 

[Ref. 2] Cayetano Di Bartolo, Rodolfo Gambini, and Jorge Pullin, Consistent and mimetic discretizations in general relativity, gr-qc/0404052 v3, 1 June 2004
 

Note: I hope to hear from David L. Meier and his colleagues, since the issue is of fundamental nature. See Vladimir Mashkevich, "The completeness of the Einstein equation and the Cauchy problem", quoted here, and the basic philosophy of GR, from David Hilbert here. More from Hermann Weyl here.

It's getting very interesting, isn't it? Just try to squeeze it on a poster!
 

D. Chakalov
June 2, 2004