Schwarzschild solution from a Raychaudhuri-Huygens relation

Abstract

Null-congruences encode the causal structure of spacetime, describing a phase space of radiation channels, whose expansion scalar $\theta$ satisfies the Raychaudhuri equation. In spherical symmetry, the expansion scalar of wave fronts of area $A=4\pi r^2$ satisfies $\theta =A^{\prime }/A=2/r$. About a mass $M$, this carries an encoding area $A_E=\lambda \phi {\ell }_p^2=\lambda R_{g}r\le A$ in a UV-IR consistent coupling ${\ell }_p^2=G\hslash /c^3$ to its Compton phase $\phi =(mc/\hslash )r\propto 1/\hslash$. Here, $\lambda =8\pi$ is fixed by the Newtonian limit at large distances and the saturation limit $A_E=A$ at horizon about $M$ based on the non-local extension of the equivalence principle in Rindler spacetime, without using the Einstein equations. In spherical symmetry, this recovers the Schwarzschild metric from a geometric embedding of $p=A_E/A$, which is directly observable in gravitational lensing. The result shows a non-thermal origin of the scaling dimension of two of gravitational phase space, previously derived based on the second law of thermodynamics by Bekenstein (1973), with possible implications for cosmological spacetime.

arXiv comment3 pages, 1 figure, based, in part, on invited talk to Corfu2025, Summer Institute 2025 'School and Workshops on Elementary Particle Physics and Gravity', 233, doi 10.22323/1.509.0233