Differential Geometry/Topology in GR: Research Needed?

[andytoh]

Does anyone have a good idea of how big the holes are in our current knowledge of the differential geometry and topology that would make general relativity a much better understood area of research? Or are we already fully equipped in that regard and only need to seek further physical ramifications of general relativity? I am considering going in that direction, but only if there is really a need for it.

 

[robphy]

I think there is still work to do in differential geometry/topology aspects of GR.

Some folks still work on aspects of causal structure and singularity theorems.
Some folks would like to find ways of generating solutions of the field equations... and this could include exploiting symmetries and classifying spacetimes according to available structures [e.g. symmetries imposed, choice of matter fields]. There are probably aspects of differential geometry/topology in the "space of solutions".
Some folks are looking to reformulate GR in terms of new variables and new structures which might prove simpler for formulating initial-value problems, for analysis, for computation, for "quantization", or for further generalization (including quantum field theory in curved spacetime and approaches to quantum gravity and unified theories).

Certainly more differential geometric/topological tools (including development of pedagogy) could be helpful in trying to help others extract the physics from it.

Some possibly interesting reading...
"Results and Open Problems in Mathematical General Relativity" - Yvonne Choquet-Bruhat
http://www.springerlink.com/content/964186644455l058/
Singularity Theorems in General Relativity: Achievements and Open Questions - José M.M. Senovilla
http://arxiv.org/abs/physics/0605007
83 years of general relativity and cosmology: progress and problems - George F R Ellis
http://www.iop.org/EJ/abstract/0264-9381/16/12A/303
...
http://www.google.com/search?hl=en&q=open+(problems+OR+questions)+"general+relativity"

 

[Entropia]

There is definitely a need for further research in differential geometry and topology in the context of general relativity. While our current understanding of these mathematical concepts has allowed for the development of the theory of general relativity, there are still many unanswered questions and potential areas for improvement.

One area that could benefit from further research is the understanding of the singularities that arise in general relativity, such as the singularity at the center of a black hole. These singularities are currently described by mathematical models, but a deeper understanding of the underlying geometry and topology could potentially lead to a more complete and accurate description.

Additionally, there is ongoing research into the application of differential geometry and topology in the study of gravitational waves and their propagation through spacetime. This is a relatively new area of research and there is still much to be discovered and understood.

Furthermore, the connection between general relativity and other areas of physics, such as quantum mechanics, is an active area of research. Understanding the underlying geometry and topology of spacetime could potentially shed light on the unification of these two theories.

In summary, while we have made significant progress in using differential geometry and topology to understand general relativity, there is still much to be explored and discovered. As such, there is a definite need for further research in this area. If you are considering pursuing this direction, there are certainly many exciting and important questions waiting to be answered.