[Paper Review] Scrambling Time and Causal Structure of the Photon Sphere of a Schwarzschild Black Hole

Peter W. Shor|arXiv (Cornell University)|Jul 11, 2018

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This paper investigates the causal structure of the photon sphere in a Schwarzschild black hole to assess the feasibility of fast quantum information scrambling, concluding that a scrambling time of order $M\log M$ violates causality unless non-standard physics occurs outside the stretched horizon. The analysis shows that information transfer across the black hole's photon sphere requires at least $O(M^2)$ time under general relativity and quantum field theory assumptions, challenging the widely cited fast scrambling conjecture.

ABSTRACT

Recently, physicists have started applying quantum information theory to black holes. This led to the conjecture that black holes are the fastest scramblers of information, and that they scramble it in time order M log M, where M is the mass of the black hole in natural units. As stated above, the conjecture is not completely defined, as there are several possible definitions of scrambling times. It appears that not all papers that refer to this conjecture interpret it the same way. We consider a definition of scrambling time stronger than the one given in the paper that first proposed this conjecture [Sekino and Susskind, JHEP 0810:065 (2008)], and show that this stronger version of the conjecture appears to be incompatible with a number of other widely-believed and reasonable-sounding properties of black holes. We argue that for the scrambling time of a black hole to be this fast, either relativity is violated or non-standard physics must be occurring outside the stretched event horizon of a black hole. More specifically, either information is being transferred faster than relativity would permit, the information is not carried by the Hawking radiation and thus must be carried by unknown physics, or the Hawking radiation carries much more information than standard thermodynamics would permit.

Motivation & Objective

• To assess whether the fast scrambling conjecture—suggesting black holes scramble information in $M\log M$ time—can be consistent with general relativity and quantum field theory.
• To analyze the causal structure of the photon sphere from the perspective of an outside observer who never crosses the event horizon.
• To test whether information can be transmitted across the black hole quickly enough to achieve fast scrambling, under standard assumptions about Hawking radiation and spacetime causality.
• To challenge the viability of the fast scrambling conjecture by demonstrating that $O(M^2)$ time is required for maximal entanglement across the black hole's equatorial plane.

Proposed method

• Divides the photon sphere into discrete cells to model information transmission across the black hole's equatorial plane.
• Applies computer-science-style information-theoretic arguments to bound the minimum time required to transmit entanglement-inducing information between opposite sides of the black hole.
• Assumes that Hawking radiation carries the information involved in scrambling, and that physics outside the stretched horizon is governed by quantum field theory and general relativity.
• Uses the causal structure of spacetime as dictated by general relativity to constrain possible information transfer speeds.
• Considers the implications of Planck-scale physics near the stretched horizon, including the formation and evaporation of Planck-sized black holes.
• Evaluates speculative mechanisms like wormholes connecting opposite sides of the black hole, showing they lead to causality violations or fail to support fast scrambling.

Experimental results

Research questions

• RQ1 Can the $M\log M$ scrambling time conjecture be compatible with the causal structure of spacetime as described by general relativity?
• RQ2 Is it possible for information to be transmitted across the photon sphere in less than $O(M^2)$ time under standard assumptions about Hawking radiation and quantum field theory?
• RQ3 What are the consequences of assuming that Hawking radiation carries the information needed for fast scrambling, and how does this affect causality?
• RQ4 Do speculative structures like wormholes in the photon sphere provide a viable mechanism for fast scrambling without violating causality?
• RQ5 What happens to the causal structure when one attempts to extend the cell-based model below the Planck scale near the stretched horizon?

Key findings

• The minimum time required to maximally entangle opposite sides of the black hole via the photon sphere is $O(M^2)$, contradicting the $M\log M$ scrambling time conjecture.
• A faster-than-$M\log M$ scrambling time would require information to be transmitted faster than light, violating relativistic causality.
• If Hawking radiation carries the scrambling information, it must carry significantly more information than allowed by standard black hole thermodynamics.
• The existence of stable, traversable wormholes in the photon sphere cannot support fast scrambling without leading to causality violations or requiring unphysical assumptions.
• Planck-scale black holes near the stretched horizon are unlikely to enable fast scrambling, as they form and evaporate rapidly and cannot sustain long-lived information transfer.
• The analysis rules out firewall-like mechanisms as solutions to the fast scrambling problem, since they do not resolve the underlying causality conflict from the outside observer's perspective.