Can Lorentzian Signature Metrics Always Be Continued into Euclidean Signatures?

[sheaf]

Back in the 1980s, a lot of work (Hawking et al) was done on deriving semiclassical and quantum gravity results on spacetimes with Euclidean signature. The connection with the Lorentzian signature versions was then made by analytic continuation. Does anyone know - are there any generic conditions which can be stated which ensure that a Lorentzian signature metric can be analytically continued into the complex domain and there posess a Euclidean signature section ? Or alternatively, what are the obstructions to analytic continuation into the Euclidean domain ?

 

[Careful]

Back in the 1980s, a lot of work (Hawking et al) was done on deriving semiclassical and quantum gravity results on spacetimes with Euclidean signature. The connection with the Lorentzian signature versions was then made by analytic continuation. Does anyone know - are there any generic conditions which can be stated which ensure that a Lorentzian signature metric can be analytically continued into the complex domain and there posess a Euclidean signature section ? Or alternatively, what are the obstructions to analytic continuation into the Euclidean domain ?

There could be many obstructions I guess, such as the lack of an everywhere well defined timelike vectorfied which is generically the case for Morse-like ''Lorentzian'' spacetimes in topology changing configurations (such as the trousers); another obstruction could be the development of poles due to analytic continuation such as happens for the metric

- 1/(a^2 + t^2) dt^2 + dx^2

if you rotate t --> i t , then you get poles at t = +/- a.

 

[sheaf]

Thanks. I guess it's a case by case basis. I was trying to get a feel for how typical the results obtained by Hawking and co (e.g. those for the Euclidean Schwarzschild solution) were.

 

[Careful]

Thanks. I guess it's a case by case basis. I was trying to get a feel for how typical the results obtained by Hawking and co (e.g. those for the Euclidean Schwarzschild solution) were.

Yes, it is a case by case basis. Analytic continuation works rather well for QFT's on a fixed background spacetime and I think (not sure though) Hawking radiation on Schwarzschild has been computed with and without it (I remember the original calculation was Lorentzian). However, in quantum gravity, the technique is disputed; my personal take on it is that it is not correct, but I have no proof of this.

Careful