Testing the $\mathrm{ER}=\mathrm{EPR}$ conjecture with entangled photons

Abstract

We regularize the Aichelburg-Sexl shock-wave metric for massless particles by smearing the point-like source over a string-inspired length scale $l_0$, obtaining a singularity-free gravitational potential. A coordinate transformation reveals that the transverse geometry is a zero-throat Einstein-Rosen wormhole, providing an explicit geometric realization of the ER=EPR conjecture for entangled photons. Crucially, we show that the gravitational self-energy depends on the photon's longitudinal extent $L$ (its wavelength) and, for a transversely separated photon pair, is suppressed by a factor $1/L$, giving $E^{\mathrm{GSE}}\sim 4G(\hslash \ddot{I}‰{)}^2/(c^{4}L)\ln (d^2/l_0^2)$. For the coincident back-to-back pair created in $e^{+}{e}^{-}\to 2{\hat{I}}^3$, the wormhole carries no additional binding energy; the logarithmic interaction energy emerges only after the entangled photons separate to a distance $d$, stretching the ER bridge. We further provide an entanglement-entropy interpretation: by computing the entanglement entropy of null intervals in the shock-wave geometry and introducing an effective entanglement temperature $k_B{T}_{\mathrm{ent}}\sim \hslash c/(2\ddot{I}€L)$, we recover the same scaling and normalization of the gravitational self-energy. For optical photons the corresponding collapse time exceeds $10^{30}$ years, making isolated photons immune to gravity-induced wave-function collapse. These findings establish a rigorous playground for testing ER=EPR and reveal a deep suppression of quantum-gravity effects for ultra-relativistic quanta.

10 pages, 1 figure