Short-distance contribution to the spectrum of Hawking radiation

The Hawking effect can be rederived in terms of two-point functions and in such a way that it makes it possible to estimate, within the conventional semiclassical theory, the contribution of ultrashort distances at $I^{+}$ to the Planckian spectrum. The analysis shows that, for Schwarzschild astrophysical black holes, the Hawking radiation (for both bosons and fermions) is very robust up to very high frequencies (typically two orders above Hawking’s temperature). Below this scale, the contribution of ultrashort distances to the spectrum is negligible. We argue, using a simple model with modified two-point functions, that the above result seems to have a general validity and that it is related to the observer independence of the short-distance behavior of the corresponding two-point function. The above suggests that only at high emission frequencies could an underlying quantum theory of gravity potentially predict significant deviations from Hawking’s semiclassical result.

Figure 1

Penrose diagram of a collapsing body producing a Schwarzschild black hole.

Figure 2

Plot comparing the Planckian spectrum $N(w,\kappa )=(e^{2\pi w{\kappa }^{-1}}-1{)}^{-1}$ (solid line) with the contributions $I^B$ (dashed line) and $I_{iϵ}^B$ (dotted line) coming from distances $|z|\lesssim \alpha$ according to the normal-ordering prescription and the $iϵ$ prescription, respectively. We have taken $\kappa =0.1$ and $\alpha =1$, 10, 30, and 100 (in Planck units), respectively.

Figure 3

Plot comparing the Dirac-Fermi distribution (solid line) $N(w,\kappa )=(e^{2\pi w{\kappa }^{-1}}+1{)}^{-1}$ with the contribution $I^F$ coming from distances $|z|<\alpha$ according to the normal-ordering prescription (dashed line). For completeness we have also plotted the result obtained with the $iϵ$ prescription (dotted line). We have taken $\kappa =0.1$ and $\alpha =1$, 40, ${10}^3$, and ${10}^4$ (in Planck units), respectively.