I have found this article recently published in Classical and Quantum Gravity giving the exact solution of a scalar field in the Kerr-Newman metric. These authors also derived Hawking radiation for this black hole here. So, the question is do similar works exist for the Schwarzschild metric or, in this case, only approximate technique are used? I imagine that, after so much research on the Hawking radiation, there should be a large body of literature about this. Is it so?

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    $\begingroup$ Ehm... can't you recover the Schwarzschild solution from the charge and angular momentum equal to zero case? $\endgroup$ – Danu Nov 20 '14 at 18:39
  • $\begingroup$ Yeah, the schwarzschild metric is a special case of the kerr-newman metric. It's quite an achievement exactly solving the Klein-Gordon equation in a Kerr background. $\endgroup$ – Jerry Schirmer Nov 20 '14 at 18:40
  • $\begingroup$ @JerrySchirmer confluent Heun functions... never heard of them! $\endgroup$ – Danu Nov 20 '14 at 18:41
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    $\begingroup$ @Danu -- not confluent, DOUBLE confluent! $\endgroup$ – Jerry Schirmer Nov 20 '14 at 18:42
  • $\begingroup$ @Danu Yes, you are right. Thanks. Indeed, the authors use the asymptotic behaviuor of Heun functions to get back Hawking radiation. But just look this tcpa.uni-sofia.bg/heun/bibliography.html $\endgroup$ – Jon Nov 20 '14 at 19:12

You can find an explanation of scalar fields and associated quantum effects in the Schwarzschild background in chapter four of these lecture notes. The article also contains references which might be of use to you.

  • $\begingroup$ Thanks for pointing this out. I upvoted your answer because that lectures are very good but do not appear to fit the bill properly. The radial part of the wave equation is explicitly given without being solved. No explicit exact solution is really presented. $\endgroup$ – Jon Nov 21 '14 at 10:45

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