Scalar product of fields in Schwarzschild space-time

Question

The scalar product of fields in curved space-time is defined by (Birrel, Davies)
$$left(phi_{1}, phi_{2}right)equiv-mathrm{i} int_{Sigma} phi_{1}(x) overset{leftrightarrow} {partial_mu}phi_{2}^{*}(x)left[-g_{Sigma}(x)right]^{frac{1}{2}} mathrm{~d} Sigma^{mu}$$
where $mathrm{d} Sigma^{mu}=n^{mu} mathrm{d} Sigma$, with $n^mu$ a future-directed unit vector orthogonal to the spacelike hypersurface $Sigma$ and $mathrm{d}Sigma$ is the volume element in $Sigma$. If $Sigma$ is taken to be a Cachy surface in the globally hyperbolic spacetime, then the scalar product is independent of $Sigma$.
I don't know how to apply this formula to a particular example. For instance, consider a Schwarzschild spacetime. How will this formula look like when I choose $Sigma$ to be the past light-like infinity (of course, one can choose from many coordinate systems)?

Valter Moretti · Accepted Answer

First of all that is not a  scalar product but it is a symplectic form.
Yes you can use the lightlike infinity provided
(a) no information escapes through the timelike infinity,
(b) you have rewritten that integral in the language of differential forms ,
(c)  the field is massless (otherwise it vanishes too fast before reaching the past light infinity).
I used lots of times  that mathematical technology in the past.
See for instance this couple of   papers https://arxiv.org/abs/gr-qc/0512049 and
https://arxiv.org/abs/gr-qc/0610143 I wrote in the past (around Eq.(47) in the former there is an explicit discussion about the expression of the symplectic form in terms of differential forms)   and the rigorous construction of the Unruh state in the Kruskal-Schwarzshild manifold  I obtained in collaboration with two colleagues https://arxiv.org/abs/0907.1034. Also this brief    monography  https://doi.org/10.1007/978-3-319-64343-4 may be useful.
In these papers you can find the explicit version,  in terms of forms,  of that symplectic form in a manner that can be used for light like 3-surfaces.

Scalar product of fields in Schwarzschild space-time

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