K. Jusufi, F. S. N. Lobo, D. Singleton

Abstract
We regularize the Aichelburg--Sexl shock-wave geometry for massless particles by smearing the point-like source over a string-inspired length scale l₀. 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 EGSE~4G(ℏω)2c4Lln(d2l02), where d is the transverse separation. The factor 1/L strongly suppresses the interaction, giving gravitational stability times greater than 10³⁰ 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 l₀, 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.

Keywords
ER=EPR conjecture / Entangled photons

Physics Letters B
Volume 881, Number 140886
2026 October

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