GR with Riemannian ( ) metric?

[petergreat]

GR with Riemannian (++++) metric?

Neither the Einstein field equations nor the Einstein-Hilbert action requires pseudo-remannian metrc. Is there anything interesting about GR with Riemannian metric? Are there still black holes?

Also, is it possible to find a solution to GR in which the metric is Riemannian in half of the manifold and pseudo-Riemannian in the other half? Joined up like how a black hole and a white hole are joined in Kruskal metric?

 

[atyy]

http://arxiv.org/abs/gr-qc/9605066
From Euclidean to Lorentzian General Relativity: The Real Way
J. Fernando Barbero G.

http://arxiv.org/abs/hep-th/0303174
Can black holes have Euclidean cores?
T. Hirayama, B. Holdom

http://arxiv.org/abs/0711.1656
The Arrow Of Time In The Landscape
Brett McInnes
(See section 4.1)

 

[petergreat]

Thanks! It'll take me some time to look at the papers.

 

[bcrowell]

If a metric is going to switch over from one signature to another, it's going to be degenerate on the boundary. Einstein's formulation of GR has problems when the metric is degenerate. You can't raise and lower indices, etc. The Ashtekar formulation of GR can handle degeneracies. There are some interesting remarks on this in Rovelli, "Ashtekar formulation of general relativity and loop-space non-perturbative quantum gravity: a report." Apparently there is some non-uniqueness in the extension of Einstein's GR to formulations that allow degeneracies.

 

[bcrowell]

I've written a discussion of this: http://www.lightandmatter.com/html_books/genrel/ch06/ch06.html#Section6.4