Does Compton–Schwarzschild duality in higher dimensions exclude TeV quantum gravity?

Abstract

In three spatial dimensions, the Compton wavelength (R C ∝ M −1 ) and Schwarzschild radius (R S ∝ M ) are dual under the transformation M → M 2 P /M , where M P is the Planck mass. This suggests that there could be a fundamental link -termed the Black Hole Uncertainty Principle or Compton-Schwarzschild correspondence -between elementary particles with M < M P and black holes in the M > M P regime. In the presence of n extra dimensions, compactified on some scale R E exceeding the Planck length R P , one expects R S ∝ M 1/(1+n) for R P < R < R E , which breaks this duality. However, it may be restored in some circumstances because the effective Compton wavelength of a particle depends on the form of the (3 + n)-dimensional wavefunction. If this is spherically symmetric, then one still has R C ∝ M −1 , as in the 3-dimensional case. The effective Planck length is then increased and the Planck mass reduced, allowing the possibility of TeV quantum gravity and black hole production at the LHC. However, if the wave function of a particle is asymmetric and has a scale R E in the extra dimensions, then R C ∝ M −1/(1+n) , so that the duality between R C and R S is preserved. In this case, the effective Planck length is increased even more but the Planck mass is unchanged, so that TeV quantum gravity is precluded and black holes cannot be generated in collider experiments. Nevertheless, the extra dimensions could still have consequences for the detectability of black hole evaporations and the enhancement of pair-production at accelerators on scales below R E . Though phenomenologically general for higher-dimensional theories, our results are shown to be consistent with string theory via the minimum positional uncertainty derived from D-particle scattering amplitudes.

Self-complete and GUP-modified charged and spinning black holes

“…However, the solution becomes complex when M falls below √ α M Pl , corresponding to a minimum mass, and it then connects to the positive branch of Eq. (15). This asymptotes to 2ηM/α, which is presumably unphysical since it exceeds the Planck temperature.…”

“…(2) and (3) suggests another view, in which there is some deep connection between the Uncertainty Principle (which underlies the Compton expression) on small scales and black holes on large scales, so that there is a smooth transition between the two expressions. This is termed the Black Hole Uncertainty Principle (BHUP) or Compton-Schwarzschild (CS) correspondence [13][14][15] and is manifested in a unified expression for the Compton wavelength on sub-Planckian mass scales (M < M Pl ) and the Schwarzschild radius on super-Planckian (sometimes termed trans-Planckian) mass scales (M > M Pl ). So the issue is whether there is some way of merging the expressions for r C and r S .…”

“…(2) because the free parameter is associated with the first term rather than the second. However, this seems plausible because the Compton scale arises in various physical contexts, each corresponding to a different value of β [15]. It would be unnatural to associate the free parameter with the second term because the expression for the Schwarzschild radius is exact.…”

Generalised uncertainty relations from superpositions of geometries

“…Though tentative, an identification of the form (109) in the super-Planck mass regime would provide a concrete realisation of the BHUP correspondence, but not one based on modified de Broglie relations applied to fixed-background states [81][82][83][84], or on the inclusion of gravitational torsion [87][88][89], as in previous approaches.…”

Horizons and the Wave Function of Planckian Quantum black holes

“…Accordingly, one expects a similar behavior for a QBH. The effects of gravity lead to the Generalized Uncertainty Principle (GUP) whose purpose is to put quantum particles and QBHs on the same footing [2,3,4,5,6,7,8,9,10,11,12,13]. The GUP can be written in a most general form as [14] ∆x∆p…”