A regularized photon-pair geometry motivated by the ER=EPR conjecture

Abstract:We regularize the Aichelburg-Sexl shock-wave geometry for massless particles by smearing the point-like source over a string-inspired length scale $l_0$. A radial extension of the transverse geometry admits a zero-throat Einstein-Rosen interpretation. For a photon wave packet of longitudinal extent $L$, the regularized gravitational self-energy of a transversely separated pair is $E_{\rm GSE}\sim \frac{4G(\hbar\omega)^2}{c^4L} \ln\!\left(\frac{d^2}{l_0^2}\right)$, where $d$ is the transverse separation. The factor $1/L$ strongly suppresses the interaction, giving gravitational stability times greater than $10^{30}$ years for optical photons. We also examine an effective two-dimensional entanglement-entropy description of the shock-wave geometry. The entropy construction reproduces the same dependence on the scales $d$, $L$, and $l_0$, and can be calibrated to reproduce the normalization of the direct self-energy calculation. However, we find that the gravitationally induced excess entropy, and hence the self-energy inferred from this prescription, is the same for Bell-entangled and unentangled photon pairs having identical energy-momentum distributions. This agrees with the direct geometrical calculation, since the classical two-shock metric depends on the stress-energy tensor but not on the polarization entanglement. The present construction should therefore be understood as a semiclassical ER-like geometry motivated by ER=EPR, rather than as a complete realization of the conjecture. A geometry whose connectivity tracks the amount of quantum entanglement would require a quantum-gravitational description sensitive to state-dependent correlations beyond the one-point stress-energy tensor.

10 pages, 1 figure. V2: 10 pages; change in title, discussion added. Matches published version

arXiv:2606.02943 [gr-qc]

(or arXiv:2606.02943v2 [gr-qc] for this version)

https://doi.org/10.48550/arXiv.2606.02943 Focus to learn more arXiv-issued DOI via DataCite

Phys. Lett. B 881 (2026) 140886

https://doi.org/10.1016/j.physletb.2026.140886 Focus to learn more DOI(s) linking to related resources

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