Einstein theory is just a special case of the Cartan geometry, where torsion vanishes. The good thing about it is that we have only the metric as the fundamental field (gravitational one). In general the Cartan theory might contain torsion as independent degrees of freedom. This lead to a larger theory with all features of Einstein theory are retained.
My question is whether we can have another ansatz of the Cartan structure equation leading to a theory where torsion can be calculated from the metric. I have found one. But it is very difficult to keep the global Lorentz symmetry SO(3,1) on such an ansatz. Is such ansatz meaningful?
I have no answer, but I think it is a good question to consider. I have a question. What happens if we can? You said "This leads to a larger theory with all features of Einstein theory are retained". I mean is there any problem with Einstein theory to generalize or what would be the benefits of such generalization.
Up to now there are many works to consider torsion as a new and independent variables. They found particle like solutions without singularity, the cosmology models derived from it might be able to explain dark matter data without assuming the existence of new matter, torsion indicated the source of spin,.... Since torsion is an independent variable, its introduction cannot change Einstein theory, but it can add new feature, but it is less predictive.
My question is that if torsion can be calculated from the metric. In that case we do not have the torsion free condition like in Einstein theory, but we have some different condition. The good thing about it is that the theory is as predictive as the Einstein theory. I wonder if I can achieve that at the cost of the Lorentz symmetry, then what happens.
If I remember correctly, Einstein himself did use connections with torsion in his attempt to find a unified field theory. However, he discarded this theory using connections with torsion later.
Sincerely,
B.-Y. Chen
The two articles by Einstein using a non-symmetric tensor are the following:
1. A. Einstein, A generalization of the relativistic theory of gravitation, Ann. Math. 46 (1945), 578-584.
2. A. Einstein and E. C. Straus, A generalization of the relativistic theory of gravitation, II, Ann. Math. 47 (1946), 731-741.
I found an article published in Living Review in Relativity
(http://relativity.livingreviews.org/Articles/lrr-2014-5/articlese8.html)
In this article, it mentioned that Einstein pointed to a technical weakness of his theory using a non-symmetric tensor.
I still want keep the symmetric metric tensor. Additionally, both connection and torsion can be calculated from the vierbein (square root of metric). Therefore the curvature cann be expressed in terms of vierbein as well.
We have imposed a constraint on the connection (instead of imposing the torsion free). Einstein had imposed the zero connection condition in a so called tele-parallel gravity. He also proposed to replace GR with this theory since it looks like a Maxwell theory. In this theory, there is no curvature. Our condition make keep non-zero curvature and make the torsion look like a nonabelian gauge field strength. The good news is that we found a soliton, since the attraction from the curvature balances the repulsion from the torsion (both are calculated from vierbein)
At the moment, I do not understand the physical meaning of this solution yet.
professor friedrich hehl, cologne university , did a lot of research about gravity theory with torsion. have a look at spires.
- Badges
- Science topic
- Similar topics
- Control Systems Engineering
- Filtering