Do Evaporating 4D Black Holes Form Photospheres and/or Chromospheres?

Microscopic black holes are of great interest in astrophysics and cosmology. 1 A 4D black hole with a Hawking temperature 2 T bh = 1.06 GeV/ M bh /10 13 g should emit all available particle species which appear non-composite compared with the wavelength of the radiated energy. Once T bh Λ QCD ≃ 200 -300 MeV, quarks and gluons should be directly emitted 3 and then decay into stable species. It has been claimed that interactions between emitted particles significantly modify the astrophysically observable spectra from such black holes. Most scenarios are based on the Heckler 4 model in which two-body bremsstrahlung and pair-production interactions form a QED photosphere at T bh 45 GeV and a QCD chromosphere at T bh Λ QCD . Here we summarize our recent detailed analysis 5,6 of interaction models and discuss various points which prevent the development of photospheres and chromospheres.

The Heckler model takes the two-body bremsstrahlung (and corresponding pairproduction) cross-section 7 to be σ brem ≈ 8α 3 /m e 2 ln (2E/m e ) where E is the energy of the initial electrons (e ± ) in the center-of-momentum (CM) frame and m e is the electron mass. In the CM frame, the average momentum exchanged is ∼ m e (implying particles must be within ∼ 1/m e of each other to interact), the average angle between the final (‘on-shell’) electron and outgoing photon is φ av ∼ m e /2E, and the average energies of the final electron and photon is ∼ E/2. Even if a photosphere develops, most particles continue to move radially away from the black hole. Since the average energy 8 of the Hawking emitted electrons is E peak ≃ 4T bh and φ av ∼ m e /2E, a particle emitted by a T bh ∼ 1 GeV black hole would have to undergo N ∼ 10 6 scatterings to deviate 0.1 radians from the radial direction. The form of σ brem is accurate but, as discussed below, its application must be modified in the black hole context. The following corrections to the Heckler model must be applied:

(i) The black hole rest frame can be shown 5 to be the CM frame for most pairs of emitted particles but two particles moving in a similar direction, whose CM frame is highly Lorentz-boosted relative to the black hole, will not interact near the black hole. Thus each particle is surrounded by an ’exclusion cone’ of particles with which it can not interact significantly. Once the particle is a distance d from the black hole, the transverse distance to the nearest one with which it can interact is x T ∼ d. However, from above we require x T 1/m e . Thus particles must be within ∼ 1/m e of the black hole to interact. This point was omitted in the Heckler model.

(ii) Because the black hole flux radiates outward from a central point, the mean free path formula λ ≈ (nσ brem v rel )

-1 employed in the Heckler model is not appropriate and must be replaced by a radial description. Additionally, particles created at the black hole do not originate at an infinite distance from their interaction region, contrary to assumptions in the σ brem derivation. Thus the effective interaction cross-section for particles emitted by the black hole must be less than σ brem .

(iii) Any two particles must be in causal contact to interact. For two particles within ∼ 1/m e of the black hole, it can be shown 5 that this requires the time between their Hawking emission, ∆t e , to be less than the causal interval ∆t c ∼ 1/(γm e ) where γ ∼ E peak /m e . However, for the Hawking flux ∆t e ∼ 200/E peak » ∆t c . Thus a negligible fraction of emitted particles are causally connected within ∼ 1/m e of the black hole. This point was omitted in the Heckler model.

(iv) The Heckler model assumes that the distance required for the formation of a final on-shell electron and its bremsstrahlung photon is d f orm ∼ 1/m e in the CM frame. Applying the Heisenberg Uncertainty Principle to outgoing particles separated by φ av ∼ m e /2E, the correct distance is d f orm ∼ E/m e 2 in the CM frame. Thus any multiple interactions experienced by an electron within ∼ 1/m e of the black hole are off-shell interactions and so are strongly suppressed by the LPM effect. 9 This was omitted in the Heckler model.

We conclude that the causal contact requirement and the suppresion of multiple scatterings prevent the development of a Heckler QED photosphere. Analoguous arguments also prevent a QCD photosphere developing when T bh » Λ QCD .

3. QCD Chromosphere when T bh ∼ Λ QCD ?

When T bh ∼ Λ QCD (or more strictly, E peak ≃ Λ QCD ), the Hawking flux is damped by the grey-body factors which apply near rest mass thresholds. 3,8 This increases ∆t e and strengthens the causality constraint. Additionally, the limited available energy per emitted particle (E ∼ Λ QCD ) and particle physics conservation laws severely constrain the number of final particles that can be produced per emitted particle. In particular, gluon bremsstrahlung is insignificant when T bh ∼ Λ QCD because the lowest colourless state is a pion and m π ∼ 140 MeV. Theref

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