Two Questions on General Relativity & String Theory

[IRobot]

Hi, I have two questions dealing with general relativity:

-assume that we want to use a manifold with torsion, an we also want to add electromagnetism, Faraday's tensor ($$F^{\mu\nu}=\nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu}$$) is no longer gauge-invariant, I have been looking for a way of constructing a cinetic term for the vector field which is gauge invariant without any results, maybe electromagnetism forbids the manifold from having torsion

-in the prologue of Kiritsis book on String Theory, he writes: "Three of the four fundamental forces known are described by gauge theories. The fourth, gravity, is also based on local gauge invariance, albeit of a different type, and so far stands apart." I know that this gauge transformation is $$g^{\alpha\beta} \rightarrow g^{\alpha\beta} + \partial^{\alpha}\epsilon^{\beta} + \partial^{\beta}\epsilon^{\alpha}$$ the infinitesimal form of a diffeomorphism transformation, but I don't see the conceptual difference with local U(1), SU(2), or SU(3) gauge transformation.Thanks.

 

[bcrowell]

-in the prologue of Kiritsis book on String Theory, he writes: "Three of the four fundamental forces known are described by gauge theories. The fourth, gravity, is also based on local gauge invariance, albeit of a different type, and so far stands apart." I know that this gauge transformation is $$g^{\alpha\beta} \rightarrow g^{\alpha\beta} + \partial^{\alpha}\epsilon^{\beta} + \partial^{\beta}\epsilon^{\alpha}$$ the infinitesimal form of a diffeomorphism transformation, but I don't see the conceptual difference with local U(1), SU(2), or SU(3) gauge transformation.

One difference is that although global gauge symmetry in E&M gives charge conservation via Noether's theorem, you don't get any similar conserved quantity in GR. (GR doesn't have a conserved mass-energy that can be defined in all spacetimes.)

 

[Mentz114]

-in the prologue of Kiritsis book on String Theory, he writes: "Three of the four fundamental forces known are described by gauge theories. The fourth, gravity, is also based on local gauge invariance, albeit of a different type, and so far stands apart." I know that this gauge transformation is $$g^{\alpha\beta} \rightarrow g^{\alpha\beta} + \partial^{\alpha}\epsilon^{\beta} + \partial^{\beta}\epsilon^{\alpha}$$ the infinitesimal form of a diffeomorphism transformation, but I don't see the conceptual difference with local U(1), SU(2), or SU(3) gauge transformation.

There is more than one way of getting gravity from local gauge invariance. This paper discusses some of them.

"On the Gauge Aspects of Gravity" by Frank Gronwald and Friedrich W. Hehl, arXiv:gr-qc/9602013.

The simplest is probably gauging translations, which gives a theory with local conservation of energy and momentum ( which seems right because changes in energy/momentum are the generators of translations).

[edit]on reflection, I don't think this helps with your question, but I'll leave the post in case any readers want to find out more about gauge gravity.