Bernard Carr

The Generalized Uncertainty Principle (GUP) and Extended Uncertainty Principle (EUP) are modifications to the Heisenberg Uncertainly Principle (HUP), expected to apply as the energy approaches the Planck scale. Here we consider a possible combination of these modifications (GEUP) and analyse the implications in various regions of the ($Δx$, $Δp$) plane. We also consider an alternative combination (EGUP) which exhibits duality between $Δp$ and $Δx$, showing that this has some unusual features. The parameters which describe these models are usually assumed to be positive but we extend our analysis to include negative values. All these proposals entail a link between black holes and the various types of Uncertainty Principle. In particular, the GEUP predicts a new kind of strong-gravity black hole and this implies an interesting link between black holes and elementary particles.

One of the major problems in quantum gravity research is the lack of signals at the reach of present or near-future experimental facilities. In this paper, we show that this is not the case. Contrary to previous claims, the quantum decay of de Sitter space into black hole spacetimes can be significant even after inflation and can be observed on galactic scales. Using the instanton formalism within the no-boundary proposal for a class of short-scale, quantum-gravity-improved black holes, we show that de Sitter space decay would result in the production of $10^{60}$ stable Planck-size black hole remnants within the current Hubble horizon, which is the number required to explain dark matter.

We review the experimental constraints on the parameter $α$ associated with the Generalized Uncertainty Principle (GUP) and the parameter $β$ associated with the Generalized Event Horizon (GEH). The Compton-Schwarzschild correspondence implies a relationship between $α$ and $β$, with both parameters being of order 1. This presents a problem for our previous `$M+1/M$' model since the extra gravitational force at sub-Planckian masses contravenes observations. Various resolutions of this problem are discussed.

We review numerous arguments for primordial black holes (PBHs) based on observational evidence from a variety of lensing, dynamical, accretion and gravitational-wave effects. This represents a shift from the usual emphasis on PBH constraints and provides what we term a positivist perspective. Microlensing observations of stars and quasars suggest that PBHs of around $1\,M_{\odot}$ could provide much of the dark matter in galactic halos, this being allowed by the Large Magellanic Cloud microlensing observations if the PBHs have an extended mass function. More generally, providing the mass and dark matter fraction of the PBHs is large enough, the associated Poisson fluctuations could generate the first bound objects at a much earlier epoch than in the standard cosmological scenario. This simultaneously explains the recent detection of high-redshift dwarf galaxies, puzzling correlations of the source-subtracted infrared and X-ray cosmic backgrounds, the size and the mass-to-light ratios of ultra-faint-dwarf galaxies, the dynamical heating of the Galactic disk, and the binary coalescences observed by LIGO/Virgo/KAGRA in a mass range not usually associated with stellar remnants. Even if PBHs provide only a small fraction of the dark matter, they could explain various other observational conundra, and sufficiently large ones could seed the supermassive black holes in galactic nuclei or even early galaxies themselves. We argue that PBHs would naturally have formed around the electroweak, quantum chromodynamics and electron-positron annihilation epochs, when the sound-speed inevitably dips. This leads to an extended PBH mass function with a number of distinct bumps, the most prominent one being at around $1\,M_{\odot}$, and this would allow PBHs to explain many of the observations in a unified way.

Black holes play an important role in linking microphysics with macrophysics, with those of the Planck mass ($M_P \sim10^{-5}$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 ($R_{\rm S} \propto M$) above it. The duality between $R_{\rm C}$ and $R_{\rm S}$ 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 $M + βM_P^2/M$ for some constant $β$ 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 $R_{\rm S} \propto M^{1/(1+n)}$ below this scale but $R_{\rm C}$ depends on the form of the higher-dimensional wave-function. If it is spherically symmetric, then $R_{\rm C} \propto M^{-1}$, 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, $R_{\rm C} \propto M^{-1/(1+n)}$ and so duality is preserved but the Planck mass is unchanged.

Black holes play an important role in linking microphysics with macrophysics, with those of the Planck mass (M_P ∼ 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 ∝ 1/M) below the Planck mass and the Schwarzschild radius (R_S ∝ M) above it. The duality between R_C and R_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+βMP2/M for some constant β 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_S ∝ M^1/(1+n) below this scale but R_C depends on the form of the higher-dimensional wave-function. If it is spherically symmetric, then R_C ∝ 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_C ∝ M^−1/(1+n) and so duality is preserved but the Planck mass is unchanged.

We completely classify the Friedmann-Lema\^{i}tre-Robertson-Walker solutions with spatial curvature $K=0,\pm 1$ for perfect fluids with linear equation of state $p=w\rho $, where $\rho$ and $p$ are the energy density and pressure, without assuming any energy conditions. We extend our previous work to include all geodesics and parallelly propagated curvature singularities, showing that no non-null geodesic emanates from or terminates at the null portion of conformal infinity and that the initial singularity for $K=0,-1$ and $-5/3<w<-1$ is a null non-scalar polynomial curvature singularity. We thus obtain the Penrose diagrams for all possible cases and identify $w=-5/3$ as a critical value for both the future big-rip singularity and the past null conformal boundary.