On the origin of the particles in black hole evaporation

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A potentially related point is the connection between acceleration and temperature due to quantum effects unruh . Of course, we can do the math and derive this connection – but this does not mean that we fully understand the underlying reason for it

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Note that the geometric mean of group and phase velocity for a massive particle with ω 2 = c 2 ​ k 2 + m 2 ​ c 4 \omega^{2}=c^{2}k^{2}+m^{2}c^{4} yields c c , while the arithmetic or harmonic mean, for example, diverges at k → 0 k\to 0

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Of course, one should also ask the question of whether the assumed 1 − 2 ​ M / r 1-2M/r behavior of the metric is still valid is this extremal regime

Cited in the paper.

In order to motivate the third example, one could argue that it should be impossible to resolve spatial details smaller than the Planck length ℓ P = ℏ ​ G N / c 3 ≈ 1.6 × 10 − 35 \ell_{\rm P}=\sqrt{\hbar G_{\rm N}/c^{3}}\approx 1.6\times 10^{-35} m, because Heisenberg’s uncertainty principle would then require concentrating a large (trans-Plankian) energy and thus mass in a small volume – which should form a black hole. In analogy to special relativity, where the acceleration of massive particles towards the speed of light is prohibited by a diverging energy E = m ​ c 2 / 1 − v 2 / c 2 E=mc^{2}/\sqrt{1-v^{2}/c^{2}} , one could speculate that quantum gravity impedes a spatial resolution better than the Planck length ℓ P \ell_{\rm P} in a similar manner ω = c ​ k / 1 − ℓ P 2 ​ k 2 \omega=ck/\sqrt{1-\ell_{\rm P}^{2}k^{2}} . Even though the two relations ω 2 = k 2 + k 4 \omega^{2}=k^{2}+k^{4} and ω = k / 1 − k 2 \omega=k/\sqrt{1-k^{2}} can probably not be distinguished via observations in the low-energy sector (e.g., energy-dependent time-delay measurements of signals from distant galaxies), they induce drastic differences for black holes

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