What have electrons, black holes and Clifford Algebra ever done for us?

  Several independent semiclassical arguments converge on $$\boxed{\Delta A=8\pi l_{\mathrm{Pl}}^2}, \qquad l_{\mathrm{Pl}}^2=\frac{G\hbar}{c^3}$$ I got interested in this reading the 2018 paper  Bekenstein, I, and the quantum of black-hole surface area . Even though I disagree with Hod, its a great paper.  Honourable mentions also go to:   The Physical Interpretation of the Spectrum of Black Hole Quasinormal Modes , 2008  The Off-Shell Black Hole , 1994     On Black Hole Spectroscopy via Adiabatic Invariance , 2012 An Analytical Computation of Asymptotic Schwarzschild Quasinormal Frequencies , 2003  Asymptotic Black Hole Quasinormal Frequencies , 2003  1. Horizon Action Near any nonextremal horizon, imaginary time turns the normal geometry into a plane: $$ds_E^2\simeq d\rho^2+\rho^2d\Theta^2+\cdots.$$ Smoothness requires one horizon cycle: $$\oint d\Theta=2\pi.$$ The gravitational action identifies the momentum conjugate to $\The...

      We previously  discussed Verlinde's connection of the MOND acceleration scale to the entropy of de Sitter space. A different route appears in superfluid dark matter (a 2026 review is here ), where baryons interact with a phonon field whose nonlinear dynamics generate a MOND-like force. The superfluid theory introduces a characteristic scale $\Lambda_{\rm SF}$, which must be of order meV to reproduce the MOND scale. This is also the order of the vacuum-energy scale, $\rho_{\rm DE}^{1/4}\sim{\rm meV}$.  So, we postulate $\Lambda_{\rm SF}^4=\rho_{\rm DE}$.    To be clear, this idea is related to, does not actually require  a unified dark sector , e.g. 2019 Unified Superfluid Dark Sector , 2026 A solid unification of the dark sector , and 2026 Unified dark sector approach to cosmological tensions among many others. What we are doing here is identify the MOND phonon scale exactly with vacuum energy, plus postulate a dimensionless infrared...

  A Minimal Entropic Re-Derivation of MOND-like Gravity Motivation Verlinde  (2016) proposed that de Sitter space contains a volume-scaling entropy associated with dark energy. Matter removes part of this entropy. Beyond a critical scale, the remaining volume entropy dominates and produces an apparent dark gravitational component.  Peach  (2019) reformulated the idea using an entropic-screen construction.  The argument below combines these ideas and fixes the force normalisation using Verlinde's spherical response coefficient. Competing Entropies For a sphere of radius $r$, the de Sitter entropy is $$S_{\rm DE}(r) = \frac{r}{L}\frac{A(r)c^3}{4G\hbar}, \qquad A(r)=4\pi r^2,$$ while a central mass $M$ removes $$S_M(r)=\frac{2\pi Mc}{\hbar}r.$$ Thus $S_{\rm DE}\propto r^3$ and $S_M\propto r$. Define $a_\Lambda=c^2/L$. Their ratio is $$\frac{S_{\rm DE}}{S_M} = \frac{a_\Lambda r^2}{2GM} = \left(\frac{r}{r_c}\right)^2,$$ where $$\boxed{r_c=\sqrt{\frac{2GM}{...

  Consider a spherical Schwarzschild horizon of radius $R$. Its energy, temperature, and entropy are $$E=\frac{c^{4}R}{2G},\qquad k_BT_H=\frac{\hbar c}{4\pi R},\qquad S=\frac{k_BA}{4L_P^2},$$ where $A=4\pi R^2$ and $L_P^2=\hbar G/c^3$. 1. Quantized Horizon The horizon's thermal period defines an angular frequency $$\omega_H=\frac{2\pi k_BT_H}{\hbar}=\frac{c}{2R}.$$ Treat the Euclidean horizon cycle as a periodic quantum motion and define its action by $dI=dE/\omega_H$. Since $dE=c^4dR/2G$, we obtain $dI=c^3RdR/G$, and therefore $$I=\frac{c^3R^2}{2G} =\frac{\hbar R^2}{2L_P^2} =\frac{\hbar A}{8\pi L_P^2}.$$ Equivalently, using $dE=T_HdS$, $$I=\frac{\hbar S}{2\pi k_B}.$$ Quantizing this periodic motion gives $$I_n=\left(n+\frac12\right)\hbar, \qquad n=0,1,2,\ldots$$ where the half-unit is the ground-state contribution associated with the regular centre of the Euclidean horizon geometry. Hence $$A_n=4\pi(2n+1)L_P^2, \qquad R_n=\sqrt{2n+1},L_P.$$ Thus the first nonzero hor...

  A positive cosmological constant sets the future cosmic event-horizon radius of an expanding FLRW universe: $$l_\Lambda=\sqrt{\frac{3}{\Lambda}}.$$ Associate two lengths with a positive mass $m$: $$\bar\lambda_C=\frac{\hbar}{mc} \qquad r_g=\frac{2Gm}{c^2}$$ where $\bar\lambda_C$ is the reduced Compton wavelength and $r_g$ is the gravitational radius. Requiring both lengths not to exceed the cosmic horizon, $$\bar\lambda_C\le l_\Lambda \qquad r_g\le l_\Lambda,$$ gives $$\boxed{ m_{\min}= m_s=\frac{\hbar}{cl_\Lambda} =\frac{\hbar}{c}\sqrt{\frac{\Lambda}{3}} }$$ $m_s$ is the minimum mass first pointed out by Wesson , 2003. This was also adopted by  Bohmer and Harko , 2005 to derive a  minimum density for static bodies  using the cosmological Buchdahl inequality. and $$\boxed{ m_{\max}=\frac{c^2l_\Lambda}{2G} =\frac{c^2}{2G}\sqrt{\frac{3}{\Lambda}} \equiv m_{\rm CEH}. }$$ Wesson also conjectured a maximum mass, but was off by a factor of 2, as well ...