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 $θ$ satisfies the Raychaudhuri equation. In spherical symmetry, the expansion scalar of wave fronts of area $A=4πr^2$ satisfies $θ= A^\prime/A = 2/r$. About a mass $M$, this carries an encoding area $A_E = λφ\ell_p^2 = λR_g r \le A$ in a UV-IR consistent coupling $\ell_p^2= G\hbar/c^3$ to its Compton phase $φ= (mc/\hbar)r \propto 1/\hbar$. Here, $λ= 8π$ 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.