Modular Fluctuations from Shockwave Geometries

Modular fluctuations have previously been shown to obey an area law $\langle \Delta K^2 \rangle = \langle K \rangle = {A}/{4 G_N}$. Furthermore, modular fluctuations generate fluctuations in the spacetime geometry of empty causal diamonds. Here we demonstrate the physical origin of these fluctuations, showing that the modular area law, in $d-$dimensionsal Minkowski space, can be reproduced from shockwaves arising from vacuum fluctuations. The size of the vacuum fluctuations is fixed by commutation relations in light-ray operators, of the same form postulated by 't Hooft in the context of black hole horizons.