The Refractive Index of Curved Spacetime II: QED, Penrose Limits and Black Holes

TL;DR

This work develops a geometric framework for quantum corrections to light propagation in curved spacetime, showing that the full frequency-dependent refractive index is determined by geodesic deviation encoded in the Penrose plane-wave limit and the Van Vleck–Morette matrix. By extending scalar to spinor QED, the authors derive the vacuum polarization in general plane-wave backgrounds and demonstrate that causality is preserved even when the low-frequency phase velocity exceeds c. They apply the formalism to black holes, cosmological spacetimes, and gravitational waves, revealing rich polarization-dependent effects, gravitational birefringence, and novel analytic structures that modify conventional dispersion relations. A notable finding is that the optical theorem can appear violated in curved spacetime, prompting further investigation into the interpretation of Im n(ω) and the interplay between unitarity and curvature. Overall, the paper provides a comprehensive phenomenology of quantum gravitational optics with broad implications for causality, unitarity, and the behavior of light near strong gravitational fields.

Abstract

This work considers the way that quantum loop effects modify the propagation of light in curved space. The calculation of the refractive index for scalar QED is reviewed and then extended for the first time to QED with spinor particles in the loop. It is shown how, in both cases, the low frequency phase velocity can be greater than c, as found originally by Drummond and Hathrell, but causality is respected in the sense that retarded Green functions vanish outside the lightcone. A "phenomenology" of the refractive index is then presented for black holes, FRW universes and gravitational waves. In some cases, some of the polarization states propagate with a refractive index having a negative imaginary part indicating a potential breakdown of the optical theorem in curved space and possible instabilities.

Figures (8)

• Figure 1: The two Feynman diagrams that contribute to the vacuum polarization to order $\alpha$.
• Figure 2: The refractive index $n(\omega)-1$ of a conformally flat symmetric plane wave, in units of $\alpha\sigma^2/(2\pi m^2)$, plotted as a function of $\log\omega\sigma/m^2$ for scalar QED (left) and spinor QED (right).
• Figure 3: The refractive index $n(\omega)-1$ of a Ricci flat symmetric plane wave, in units of $\alpha\sigma^2/(2\pi m^2)$, plotted as a function of $\log\omega\sigma^2/(2\pi m^2)$: continuous (real part, polarization $i=1$); big dashes (imaginary part, $i=1$); small dashes (real part, polarization $i=2$) and dots (imaginary part, $i=2$), for scalar QED (left) and spinor QED (right).
• Figure 4: Analytic structure of the $t$ integrand of ${\cal F}^\text{scalar}(u;z)$ in the near-singularity region of Schwarzschild spacetime. The branch point lies at the position of the singularity and the other points are double poles.
• Figure 5: $n(\omega)-1$ for the near singularity region of the Schwarzschild black hole plotted as a function of $\log\omega$: continuous (real part, polarization $i=1$); big dashes (imaginary part, $i=1$); small dashes (real part, polarization $i=2$) and dots (imaginary part, $i=2$), for scalar QED (left) and QED (right).