Large diffeomorphisms in general relativity

[crackjack]

I do not know if there is an explicit formula for a "winding number" of general diffeomorphism.

If you are looking for topological indices that characterize gravitational instantons ( ~ Asymptotically Locally Euclidean solutions), the following might help:
http://empg.maths.ed.ac.uk/Activities/GT/EGH.pdf

 

[atyy]

Carlip comments explicitly on this in http://arxiv.org/abs/gr-qc/0409039 , section 2.6.

Also interesting are his comments in http://arxiv.org/abs/gr-qc/0501033 about gauge/real symmetries depending on boundary.

 

[atyy]

More interesting comments on diffeomorphisms when a boundary is present: "But the presence of a boundary alters the gauge invariance of general relativity: the infinitesimal transformations must now be restricted to those generated by vector fields with no component normal to the boundary, that is, true diffeomorphisms that preserve the boundary of M. As a consequence, some degrees of freedom that would naively be viewed as "pure gauge'' become dynamical, introducing new degrees of freedom associated with the boundary. http://math.ucr.edu/home/baez/week41.html"

A similar thing happens with other sorts of gauge structures: "The failure of gauge invariance under large gauge tranformations is also reflected in the properties of Chern-Simons theory on a surface with boundary, where the Chern-Simons action is gauge invariant only up to a surface term. http://arxiv.org/abs/0707.1889"