Discretizations of Classical and Quantum Gravity: A New Paradigm

[wolram]

http://arxiv.org/abs/gr-qc/0505052

Authors: Rodolfo Gambini, Jorge Pullin
Comments: 8 pages, one figure, fifth prize of the Gravity Research Foundation 2005 essay competition
Report-no: LSU-REL-051105

We argue that recent developments in discretizations of classical and quantum gravity imply a new paradigm for doing research in these areas. The paradigm consists in discretizing the theory in such a way that the resulting discrete theory has no constraints. This solves many of the hard conceptual problems of quantum gravity. It also appears as a useful tool in some numerical simulations of interest in classical relativity. We outline some of the salient aspects and results of this new framework.

 

[marcus]

http://arxiv.org/abs/gr-qc/0505052

Authors: Rodolfo Gambini, Jorge Pullin
...
We argue that recent developments in discretizations of classical and quantum gravity imply ...

good find. they call their approach "consistent discretizations". Ashtekar lists it as one of the most promising 3 or 4 attempts to handle QG dynamics. as you say they solve the problem of the QG Hamiltonian constraint by getting rid of it! Put simply, they make time discrete and make time-evolution advance in discrete steps (but in a way they say is consistent with the classical picture).
Gambini and Pullin are in the minority, but they do get recognized and they do attract some young researchers.
For example, "Edgar1813" who sometimes posts at PF is a post-doc who has worked with them and is interested in the consistent discr. approach.
About recognition, Jorge Pullin is one of the invited speakers at this year's Loop conference
https://www.physicsforums.com/showthread.php?t=74889

 

[Stingray]

Wow. Very interesting. I'll have to read more about it.

 

[wolram]

At the bottom page 3, it states, These extra degrees of freedom characterize
the freedom to choose the lagrange multipliers (the lapse and shift). since the
lapse is determined dynamically, this implies that the "time steps" taken by the
evolution change over time.

Im not sure i understand this, but it seems "open ended",or is there a constraint
to the evolution?

 

[marcus]

At the bottom page 3, it states, These extra degrees of freedom characterize
the freedom to choose the lagrange multipliers (the lapse and shift). since the
lapse is determined dynamically, this implies that the "time steps" taken by the
evolution change over time.
...

the size of the time step varies but apparently stays small, so that their method is good for numerical models (computer simulations of evolving geometry) they say, as well as for theory.

I don't understand what keeps the size of the time step small, why should it stay "in bounds"? to better understand this one has to go back and read earlier papers, I guess, because this paper is a brief summary of their work that skips over much detail

 

[wolram]

I guess it is related to the vagueness of clocks, if one can not have
a perfect clock the time intervals have a bounded variance.

 

[Stingray]

I don't understand what keeps the size of the time step small, why should it stay "in bounds"? to better understand this one has to go back and read earlier papers, I guess, because this paper is a brief summary of their work that skips over much detail

Although I haven't read this in any detail yet, the lapse is a dimensionless number. It's more of a normalized time step than an actual one. The real time steps are basically lapse*dt for an arbitrary dt (just make it small enough). In any case, I doubt that there's any particular mechanism to keep the lapse from growing too large. Even defining what exactly is meant by that varies considerably from problem to problem (and even between different regions in the same problem).

In the past, some people in numerical relativity have tried to postulate evolution equations for the lapse and shift. These often resulted in lapses going either to zero or infinity in finite time - either of which would crash the program. It would be interesting to see if these problems go away in Gambini and Pullin's framework.