Geometry of the Infalling Causal Patch

TL;DR

This paper investigates the causal structure of black holes to determine if an infalling observer can access all the information needed to resolve the firewall paradox, focusing on the visibility of interior modes within a single causal patch.

Contribution

It provides a geometric analysis showing that no single observer can see both early Hawking radiation and interior modes in static black holes across dimensions.

Findings

No observer can see both the early radiation and interior modes in static black holes.

The causal patch geometry limits the information accessible to infalling observers.

Schwarzschild black hole analysis illustrates the closest approach to measuring relevant modes.

Abstract

The firewall paradox states that an observer falling into an old black hole must see a violation of unitarity, locality, or the equivalence principle. Motivated by this remarkable conflict, we analyze the causal structure of black hole spacetimes in order to determine whether all the necessary ingredients for the paradox fit within a single observer's causal patch. We particularly focus on the question of whether the interior partner modes of the outgoing Hawking quanta can, in principle, be measured by an infalling observer. Since the relevant modes are spread over the entire sphere, we answer a simple geometrical question: can any observer see an entire sphere behind the horizon? We find that for all static black holes in 3+1 and higher dimensions, with any value of the cosmological constant, no single observer can see both the early Hawking radiation and the interior modes. We present a detailed description of the causal patch geometry of the Schwarzschild black hole in 3+1 dimensions, where an infalling observer comes closest to being able to measure the relevant modes.