Shockwaves and Time Delays in Einstein-Maxwell Effective Field Theory

TL;DR

The paper tackles how four-derivative operators in Einstein–Maxwell EFT modify charged black hole shockwaves and the time delay experienced by a probing photon. It develops the EFT framework, computes corrections to the Reissner–Nordström solution, and derives the EFT-corrected shockwave geometry by ultra-relativistic boosting, including backreaction effects. A key result is that both the EFT-induced modification of the shockwave and the backreaction of the probe on the geometry are essential to obtain a physically meaningful, field-redefinition–invariant time delay. The findings illuminate causality constraints in gravity–gauge EFTs and provide a framework for connecting EFT coefficients to bounds motivated by the Weak Gravity Conjecture, with future work aimed at completing the EFT correction program and cross-checking with amplitude methods.

Abstract

We derive the shockwave metric in four-dimensional Einstein-Maxwell effective field theory (EFT) by performing an ultra-relativistic boost of the charged black hole solution accompanied by a rescaling of its mass and charge, including leading order EFT corrections. In contrast to the neutral (Schwarzschild) case, where higher derivative operators leave the shockwave geometry unchanged, we show that electrically charged shockwaves receive nontrivial EFT corrections. We then compute the time delay experienced by a probe photon traversing the resulting charged shockwave. We find that two EFT contributions, the correction to the shockwave geometry and the backreaction induced by the probe photon, are both essential for obtaining a physical time delay that is invariant under field redefinitions of the metric.

Figures (2)

• Figure 1: \ref{['fig: time-delay-cartesian']} Spacetime diagram, in Cartesian coordinates, describing the effect of the shockwave in the $(y,z)$ plane on a probed particle moving backward in the $x$-direction. Two initially synchronised clocks separated by different impact parameters ($y$-direction) become out of sync after crossing the shockwave plane. The clock with smaller impact parameter will experience a larger absolute time delay, i.e. $|\Delta v_1 | > |\Delta v_2|$. \ref{['fig: time-delay-lightcone']} Spacetime diagram, in light-cone coordinates, describing the effect of the shockwave localised at $u = 0$ on a probed particle moving forward in the $u$-direction.
• Figure 2: Illustration of three types of EFT contribution to the time delay. Thick single and double wavy lines denote the background electromagnetic and gravitational fields of the shockwave, $F_{\mu\nu}^{\rm B}$ and $g_{\mu\nu}^{\rm B}$, respectively. While the EM background remains unmodified by EFT operators, we distinguish the unperturbed shockwave metric $(g_{\mu\nu}^{\rm B})^{(0)}$ from its EFT corrections $(g_{\mu\nu}^{\rm B})^{(c_i)}$. Thin single and double wavy lines represent the probe photon $f_{\mu\nu}$ and metric perturbation $h_{\mu\nu}$. The solid rectangle indicates interactions from EFT operators. The solid rectangle represents the interaction from EFT operators. Diagrams $(a,b)$ correspond to the first kind of contribution. Diagram $(c)$ corresponds to the second contribution. Diagrams $(d,e)$ correspond to the third contribution.