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In their famous paper Action integrals and partition functions in quantum gravity, Gibbons and Hawking argue that in order to avoid the singularity of a Schwarzschild black hole you can complexify spacetime and then integrate over a Euclidean region when evaluating the action functional. He further claims that it does not matter which region you integrate over, as long as it is homologous to the Euclidean region.

Here is an excerpt from the paper (below eq. (2.18)):

We have evaluated the action on a section in the complexified spacetime on which the induced metric is real and positive-definite. However, because $R$, $F_{ab}$ and $K$ are holomorphic functions on the complexified spacetime except at the singularities, the action integral is really a contour integral and will have the same value on any section of the complexified spacetime which is homologous to the Euclidean section even though the induced metric on this section may be complex.

How does the action integral become a contour integral and why is it independent of the region?

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    $\begingroup$ The latter is not a question pertaining to the action integral - contour integrals are always invariant under homologous deformation of the contour. For the former - could you write down the explicit integral Hawking (presumably) uses somewhere in there? $\endgroup$ – ACuriousMind Nov 20 '14 at 20:04
  • $\begingroup$ If you only consider the time integral part, and time can be treated as a complex variable, the action integral can be viewed as a contour integral in the complex plan of (complex) time. $\endgroup$ – Drake Marquis Nov 20 '14 at 20:07
  • $\begingroup$ He's just using the residue theorem: en.wikipedia.org/wiki/Residue_theorem $\endgroup$ – Jold Nov 20 '14 at 22:15
  • $\begingroup$ @ACuriousMind could you point me to a reference about the homologous deformation? Hawking integrates over the Einstein--Hilbert action with GHY boundary term (en.wikipedia.org/wiki/…). He does this for example for the Schwarzschild metric (en.wikipedia.org/wiki/Schwarzschild_metric) along real values of $\theta$, $\phi$, $r<r_0$ and imaginary time $it$ which is periodic. If you don't have access to the paper, Bañados also made it available here: srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/… $\endgroup$ – Friedrich Nov 20 '14 at 22:15
  • $\begingroup$ @jld but the Wikipedia article only talks about the complex plane, not a four dimensional complex manifold. Could you point me to any generalizations? $\endgroup$ – Friedrich Nov 20 '14 at 22:18
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One can treat the spacetime coordinates as complex and therefore turn spacetime integrals into contour integrals in the complex plane. The value of the integral of a function that is holomorphic except at certain (singular) points is now determined by its singularity structure: the Cauchy's residue theorem tells us that the integral is given solely in terms of the residue of the function evaluated at these singular points. The residue corresponds to the expansion coefficient of order $-1$ in the Laurent series of the function. The key point is that the precise integration contour does not play a role as long as it contains the singularity.

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  • $\begingroup$ Your first sentence is exactly what Hawking writes. My question is how do you do that? I don't even know what is meant by a contour if we are talking about four dimensional integrals. Cauchy's residue theorem, as far as I understand, only talks about the complex plane. How do you generalise to four dimensional complex manifolds? $\endgroup$ – Friedrich Nov 20 '14 at 22:26

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