Hall scrambling on black hole horizons

We explore the effect of the electrodynamics $\theta$ angle on the macroscopic properties of black hole horizons. Using only classical Einstein-Maxwell-Chern-Simons theory in ($3+1$) dimensions, in the form of the membrane paradigm, we show that in the presence of the $\theta$ term, a black hole horizon behaves as a Hall conductor, for an observer hovering outside. We study how localized perturbations created on the stretched horizon scramble on the horizon by dropping a charged particle. We show that the $\theta$ angle affects the way perturbations scramble on the horizon, in particular, it introduces vortices without changing the scrambling time. This Hall scrambling of information is also expected to occur on cosmological horizons.

Figure 1

Rindler coordinates plotted on a Minkowski diagram. The dashed lines correspond to the Rindler horizons. The thin blue line corresponds to the worldline for a free falling charge.

Figure 2

Maximal analytic extension of Schwarzschild solution in Kruskal-Szekeres coordinates. Region I is the Schwarzschild patch and the dashed lines correspond to the horizons. The worldline of a free falling charge from $V=0$ is shown in brown.

Figure 3

A schematic diagram of Hall scrambling on a black hole horizon. Lines represent surface electric current on the black hole horizon after a positive charge is dropped at point $P$, as observed by an outside observer for (a) $\theta =0$, (b) $\theta >0$, (c) $\theta <0$.