Testing the ${\rm ER=EPR}$ 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 $$E^{\rm GSE}\sim 4G(\hbarω)^2/(c⁴ L)\ln(d<sup>2/l_0<sup>2)$$. For the coincident back-to-back pair created in $e^{+} e<sup>{-}\to2γ$, 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 $k_B T_{\rm ent}\sim\hbar c/(2πL)$, we recover the same scaling and normalization of the gravitational self-energy. For optical photons the corresponding collapse time exceeds $10<sup>{30}$ 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.

• The paper constructs a singularity-free Aichelburg–Sexl geometry whose transverse section forms a zero-throat Einstein–Rosen wormhole connecting the asymptotic regions of an entangled photon pair.
• The paper finds that finite photon wave-packet length suppresses gravitational self-energy by 1/L, yielding collapse times above 10³⁰ years for optical photons and about 10¹⁴ years at the 511 keV scale.
• The paper derives the same self-energy scaling from gravitational calculations and calibrated entanglement entropy, while emphasizing that the two-shock geometry and effective central charge remain linearized and model-dependent assumptions.

Overview

This paper constructs a regularized Aichelburg–Sexl (AS) shock-wave geometry for massless particles and uses it to develop a quantitative, photon-based realization of the ER=EPR conjecture (2606.02943). The central technical move is to smear the point-like null source over a string-inspired zero-point length $l_0 \sim \sqrt{\alpha'}$, which removes the logarithmic curvature singularity of the standard AS metric. The authors then show that the transverse section of the regularized metric is a zero-throat Einstein–Rosen wormhole, identify the gravitational self-energy (GSE) functional for extended photon wave packets, compute Diósi–Penrose collapse times, and provide an entanglement-entropy derivation that reproduces the same self-energy scaling.

Regularized shock-wave geometry

The unregularized AS metric carries a transverse potential $\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$, singular at $\rho = 0$. The paper replaces this with

$$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$

sourced by the normalized density $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$, whose Laplacian exactly solves the two-dimensional Poisson equation governing $pp$-waves. The potential is finite at the origin and recovers the logarithmic AS behavior at large $\rho$. The regularization is motivated by T-duality arguments implying a fundamental minimal length, and is consistent with prior work on nonlocal black-hole and wormhole geometries.

A coordinate transformation $r^2 = \rho^2 + l_0^2$ maps the transverse plane onto

$$d\Sigma^2 = \frac{dr^2}{1-l_0^2/r^2} + (r^2-l_0^2)\,d\theta^2,$$

a two-dimensional wormhole with throat at $|r| = l_0$ and vanishing throat radius — a "zero-throat" wormhole — regular in the original $\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$0 coordinates. This identification supplies the geometric side of the ER=EPR correspondence: the two asymptotic regions $\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$1 are interpreted as the geometric duals of the two entangled photons.

An important caveat stated explicitly: the back-to-back two-shock metric superposing shocks on $\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$2 and $\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$3 is not an exact solution of the full Einstein equations; it is justified only in the linearized regime where the two gravitational fields do not interact.

Gravitational self-energy and collapse times

The key dimensional insight is that a photon wave packet of longitudinal extent $\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$4 distributes its energy as a linear density $\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$5 along the null direction, so only a fraction of the energy contributes per $\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$6-slice. The resulting GSE for a transversely separated pair ($\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$7) is

$\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$8

The $\Phi(\rho) = -(8GE/c^4)\ln(\rho/\rho_0)$9 suppression is the paper's main quantitative claim: naive estimates omitting the packet length drastically overstate the effect. Taking $\rho = 0$0 for a quasi-monochromatic optical photon gives a GSE of order $\rho = 0$1 J or smaller, and a Diósi–Penrose collapse time

$\rho = 0$2

For optical photons ($\rho = 0$3 eV, $\rho = 0$4 m, $\rho = 0$5 m) this exceeds $\rho = 0$6 years; even at the 511 keV annihilation scale it remains around $\rho = 0$7 years. The implication is strong: isolated photons are effectively immune to gravity-induced wave-function collapse, and tabletop tests of Penrose-type collapse must rely on massive systems. Only in the TeV–PeV regime would timescales become short, but the authors concede their approximations may fail there.

ER=EPR from positronium annihilation

The para-positronium channel $\rho = 0$8 produces a maximally polarization-entangled Bell pair emitted back-to-back with $\rho = 0$9. In the dual gravitational picture, the superposed two-shock metric describes an ER bridge connecting the two photons' asymptotic regions. At creation ($$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$0) the bridge carries no additional binding energy; the logarithmic interaction energy emerges only after separation stretches the bridge. The same expression for $$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$1 governs both the collapse analysis and the stretched-wormhole binding energy, so all numerical conclusions transfer directly.

A consistency check: setting $$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$2 and $$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$3 yields a collapse time of order the Planck time, matching the independent result that exotic matter supporting these regularized wormholes vanishes at the same scale. The energy density falls off as $$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$4 and satisfies the null energy condition everywhere, avoiding macroscopic energy-condition violations.

The conclusion distinguishes two notions of "collapse": spontaneous Diósi–Penrose collapse of the undisturbed bridge is cosmologically slow, whereas measurement-induced disconnection requires only the minuscule energy $$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$5 J supplied by the apparatus, after which the stress-energy supporting the throat disappears and classical pinch-off follows. The extreme smallness of the binding energy is thus presented as necessary for consistency with ordinary quantum measurement rather than contradictory.

Entanglement-entropy interpretation

Suppressing the transverse sector, the shock induces a coordinate shift $$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$6, so a null interval straddling the shock acquires effective conformal length $$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$7. Applying the Calabrese–Cardy formula with UV cutoff $$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$8 gives an entanglement entropy involving precisely the scales $$\Phi(\rho) = -\frac{4GE}{c^4}\ln\!\left(1+\frac{\rho^2}{l_0^2}\right),$$9, $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$0, and $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$1 that enter the GSE. For the back-to-back pair, the excess entropy is

$\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$2

Introducing an effective entanglement temperature $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$3 and forming $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$4 reproduces the direct gravitational result in the weak-shock regime, provided the effective central charge is calibrated as $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$5, i.e., $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$6 for $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$7. The authors are careful to state that this is a dimensionless calibrated measure of the null degrees of freedom in the reduced description, not the fundamental central charge of the four-dimensional photon field — a substantive assumption underlying the claimed equivalence of the two derivations.

Limitations and open questions

Several limitations are conceded within the text. The two-shock superposition is valid only at linearized order; nonlinear photon dynamics and genuine wormhole formation from photon collisions remain untreated. The identification $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$8 is a crude approximation for quasi-monochromatic packets, and the treatment may be unreliable at TeV–PeV energies where collapse times become short. The entanglement-temperature construction is calibrated rather than derived, and the effective central charge is fitted to match normalization. Open questions posed by the paper include dynamical wormhole formation in photon collisions, numerical study of the nonlinear equations, and whether the tiny per-unit-length interaction accumulates observably over astrophysical distances.

Conclusion

The paper provides a singularity-free AS geometry whose transverse section is a zero-throat ER wormhole, yielding an explicit massless-sector realization of ER=EPR anchored to the experimentally clean process $\rho_E(\rho) = E l_0^2/[\pi(\rho^2+l_0^2)^2]$9. Its principal quantitative results — the $pp$0-suppressed self-energy, collapse times exceeding $pp$1 years for optical photons, and the entropy-based reproduction of the self-energy functional — jointly imply that quantum-gravitational effects on ultra-relativistic quanta are deeply suppressed, and that any experimental probe of ER=EPR via photons must exploit measurement-induced disconnection rather than spontaneous collapse.