The Generalized Uncertainty Principle and Higher Dimensions: Linking Black Holes and Elementary Particles

Abstract

Black holes play an important role in linking microphysics with macrophysics, with those of the Planck mass ($M_P\sim 10^{-5}$g) featuring in any theory of quantum gravity. In particular, the Compton-Schwarzschild correspondence posits a smooth transition between the Compton wavelength ($R_C\propto 1/M$) below the Planck mass and the Schwarzschild radius ($R_{\mathrm{S}}\propto M$) above it. The duality between $R_{\mathrm{C}}$ and $R_{\mathrm{S}}$ implies a form of the Generalized Uncertainty Principle (GUP) and suggests that elementary particles may be sub-Planckian black holes. The simplest possibility is that the ADM mass has the form $M+\beta M_P^2/M$ for some constant $\beta$ and this model can be extended to charged and rotating black holes, clearly relevant to elementary particles. Another possibility is that sub-Planckian black holes may arise in loop quantum gravity and this explicitly links black holes and elementary particles. Higher dimensions may modify both proposals. If there are $n$ extra dimensions, all with the same compactification scale, one expects $R_{\mathrm{S}}\propto M^{1/(1+n)}$ below this scale but $R_{\mathrm{C}}$ depends on the form of the higher-dimensional wave-function. If it is spherically symmetric, then $R_{\mathrm{C}}\propto M^{-1}$, so duality is broken and the Planck mass is reduced, allowing the possibility of TeV quantum gravity. If the wave-function is pancaked in the extra dimensions, $R_{\mathrm{C}}\propto M^{-1/(1+n)}$ and so duality is preserved but the Planck mass is unchanged.

17 pages, 5 figures

References in corpus (4)

• Physics of Trans-Planckian Gravity
• Self-dual Black Holes in LQG: Theory and Phenomenology
• Generalized Uncertainty Principle and Self-dual Black Holes
• Search for new phenomena in different-flavour high-mass dilepton final states in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector

Cited by in corpus (1)

• Dimensionally-dependent uncertainty relations, or why we (probably) won't see micro-black holes at the LHC, even if large extra dimensions exist