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The Typical-State Paradox: Diagnosing Horizons with Complexity
Leonard Susskind
TL;DR
The paper addresses the tension between opaque horizons among typical black-hole states and transparent horizons in black holes formed naturally. It resolves the tension by distinguishing thermal from complexity equilibrium and uses increasing state complexity as a transparency diagnostic; opaque horizons can become transparent after a thermal-photon perturbation.
Problem
The paper asks how typical black-hole states can have opaque horizons while all black holes formed by natural processes have transparent horizons.
Source: §1 · p. 3Method
It distinguishes thermal equilibrium from complexity equilibrium and relates horizon transparency to whether quantum-state complexity is increasing.
Source: §2 · p. 5, §5 · p. 12Results
Natural-process black holes remain far from maximal complexity for an exponential time, while states with small but growing complexity occupy a doubly exponentially small fraction of state space.
Source: §4 · pp. 7–8Takeaways & Limitations
A single thermal photon can render an opaque horizon transparent within a scrambling time.
Source: §1 · p. 2The long-time thermofield-double puzzle remains unresolved: the paper offers possible resolutions but leaves the issue for future work.
Source: §6 · p. 15
Abstract
from arXiv · showhide
The concept of transparent and opaque horizons is defined. One example of opaqueness is the presence of a firewall. Two apparently contradictory statements are reconciled: The overwhelming number of black hole states have opaque horizons; and: All black holes formed by natural processes have transparent horizons. A diagnostic is proposed for transparency, namely that the computational complexity of the state be increasing with time. It is shown that opaque horizons are extremely unstable and that the slightest perturbation will make them transparent within a scrambling time.
1 Introduction
The paper defines transparent horizons as freely crossable by infalling observers and asks what determines horizon transparency. It proposes that natural black holes have transparent horizons, while opaque horizons are fragile.
1 Introduction
- A horizon is transparent when an infalling observer can freely cross it; opacity is unstable and a single thermal photon can make it transparent within a scrambling time.A firewall is one proposed source of opacity.Source: §1 · pp. 2–3
- White holes are as common as black holes in the Hilbert space but cannot be entered across their horizons, unlike the depicted black hole.The figure contrasts their bridge volumes: increasing in the black hole and decreasing in the white hole.Source: §1 · p. 2, Figure 1 · p. 3
- The paper asks which quantum-state criteria determine horizon transparency, whether one criterion suffices, and whether natural black holes satisfy it and for how long.It also asks how robust opaque and transparent horizons are.Source: §1 · p. 3
- Natural black holes are proposed to satisfy a single transparency criterion, while opaque horizons are fragile and transparent horizons robust.In anti de Sitter space, natural black holes are proposed to remain transparent for at least an exponentially long time but no longer than a doubly exponential time.Source: §1 · p. 3
2 The Typical State Paradox
The paradox juxtaposes typical black-hole states with opaque horizons and natural black holes with transparent horizons. Its resolution separates thermal equilibrium from complexity equilibrium: natural black holes remain low-complexity, non-typical states long after thermalization.
2 The Typical State Paradox
- The paradox assumes typical black-hole states are opaque, while natural black holes have transparent horizons; the paper adopts the typical-state opacity scenario as a worst-case assumption.The opacity claim is attributed to Marolf and Polchinski [4].Source: §2 · p. 4
- A collapsing shell illustrates why a smooth horizon can persist: its interior is flat before arrival, and no known semiclassical mechanism creates a firewall from a smooth horizon.The shell may have arbitrarily small energy density as it crosses the Schwarzschild radius.Source: §2 · p. 4
- The paper locates the error in identifying thermal equilibrium with typicality: quantum systems have a gulf between maximum entropy and maximum complexity.Creating a typical maximum-complexity state takes an exponentially long time in entropy.Source: §2 · pp. 4–5
- The proposed resolution distinguishes thermal equilibrium, reached quickly, from complexity equilibrium, which takes at least an exponential time in entropy to reach.States with sub-exponential complexity are extraordinarily rare and not typical.Source: §2 · p. 5
- Natural black holes remain far from maximal complexity for an exponential time and are non-typical thermal-equilibrium states.The argument applies to formation processes on timescales much shorter than exponential.Source: §2 · p. 5
3 Criteria for Transparency
The paper proposes three related sufficient criteria for horizon transparency, beginning with expansion of the black hole interior.
3 Criteria for Transparency
- The first proposed sufficient criterion is an expanding black-hole interior geometry.The paper says the three criteria are not independent and are believed to be the same.Source: §3 · p. 5
- The expansion criterion is expressed geometrically through the future interior, depicted as an expanding bridge-to-nowhere.The geometry is foliated by maximal spacelike slices.Source: §3 · p. 5
- The section presents expansion as one of three sufficient criteria, while leaving the other two criteria to later discussion.The criteria are described as related rather than independent.Source: §3 · p. 5
Expansion Criterion
The expansion criterion links transparent horizons to increasing interior volume and increasing quantum complexity. The paper also gives a past-singularity criterion and assumes complexity grows as rapidly as possible for as long as possible.
Expansion Criterion
- The Page criterion is motivated by time reversal: contracting interiors imply past singularities, while expanding geometries have future singularities.Objects with past singularities are described as white holes, despite the paper's use of the common black-hole terminology.Source: Expansion Criterion · pp. 5–6
- Black holes formed from non-singular Cauchy data, with no past singularity, have transparent horizons.The paper presents this as the Page criterion.Source: Expansion Criterion · p. 6
- A horizon remains transparent while the quantum complexity of its state is increasing.The paper names this the Increasing Complexity Criterion.Source: Expansion Criterion · p. 6
- In the complexity/volume duality, state complexity corresponds to the volume of maximal slices through the interior bridge, which increases in black holes and decreases in white holes.The paper associates decreasing white-hole complexity with opaque horizons.Source: Expansion Criterion · p. 6
- Complexity can grow for an exponential time, so transparency can persist far beyond scrambling or Page times; bounded complexity eventually limits eternal classical expansion.The paper assumes maximal-rate linear growth until complexity levels off.Source: Expansion Criterion · p. 7
4 How Rare is Transparency?
In chaotic systems, evolution through the black-hole state space explores a torus, while complexity grows from simple states toward a maximum. States with small but growing complexity occupy a doubly exponentially small fraction, reconciling typical opacity with transparency in naturally formed black holes over sub-exponential times.
4 How Rare is Transparency?
- Chaotic evolution traces a torus in CP(eS), with incommensurate energy eigenvalues making the motion ergodic.The state space consists of normalized pure states that are macroscopically indistinguishable and behave externally like black holes.Source: §4 · p. 7
- Ergodicity in Hilbert space makes time averages equal averages over the state's torus.Source: §4 · p. 8
- Starting from a simple state, complexity must increase and is expected to continue growing until it is exponential in S.The passage says early growth can be linear; stalling before exponential complexity would imply an unexpected complexity-class collapse.Source: §4 · p. 8
- Small but growing-complexity states occupy a doubly exponentially small fraction of CP(eS), while the rest may have opaque horizons.Over the doubly exponential recurrence time, states return quasi-periodically to low complexity and can grow again for a singly exponential time.Source: §4 · p. 8
- Typical horizons can be opaque even though naturally formed horizons remain transparent for sub-exponential times.In flat space, real black holes evaporate within polynomial time.Source: §4 · p. 8
5 Evidence for Criterion
This section tests increasing complexity as protection against firewalls using eternal-hole and shock examples; the shock calculation links falling complexity with opacity and its reversal with transparency.
5 Evidence for Criterion
- In the eternal two-sided example, increasing t_R raises ERB volume and complexity, diagnosing the right horizon as transparent; the same argument applies on the left.The example uses maximal slices anchored at the two boundaries.Source: §5 · pp. 9, 10, 13
- Equation 5.2 is geodesic length rather than actual maximal-slice volume, but it has the same features and serves as a stand-in.The section uses this formula in its discussion of the volume as a function of boundary times.Source: §5 · p. 11
- For a left-injected shock initially within a Planck distance, complexity decreases while the horizon is opaque, then both complexity and transparency rise sharply at |t_w|−t_*.The shock separates from the horizon at that time, matching exactly when complexity stops decreasing.Source: §5 · pp. 11–12, Figure 5 · p. 13
- With multiple shocks from either side, ERB volume decreases until the last shock exits, further supporting the conjectured transparency criterion.For shocks entering from the left, the first shock to enter is the last to leave the green region.Source: §5 · pp. 13–14
- Localized shocks are left for future work; their horizons are expected to become transparent non-uniformly over time.The proposed pattern is a wave of transparency spreading out over a period of time.Source: §5 · p. 14
6 Very Long Times
Natural black holes begin at low complexity and remain in a rising-complexity black-hole era for an exponentially long time, unlike typical states that may occupy opaque grey- or white-hole phases. The later complexity history includes a long, uncertain grey-hole period and an unresolved puzzle about the exact thermofield-double state.
6 Very Long Times
- Natural black holes start at low complexity and remain black holes while complexity increases; after it reaches its maximum, they exhibit white-hole behavior.Figure 6 depicts the low-complexity start, exponentially long rising era, complexity equilibrium, and eventual quasi-periodic decline.Source: §6 · p. 14, Figure 6 · p. 14
- After a doubly exponential quantum recurrence time, complexity becomes small and decreases during a white-hole period, when horizons are vulnerable to instabilities.Between the rising black-hole era and this decline, the system can spend exp exp(S) in complexity equilibrium; the paper assumes grey-hole horizons may be opaque, while noting this could be overly pessimistic.Source: §6 · p. 15
- The exact thermofield-double state poses a puzzle: if grey-hole horizons are opaque, transparent horizons would be negligible at any time.The proposed resolutions are that grey holes are often transparent or that the thermofield-double state has special protection; the paper leaves the puzzle unresolved.Source: §6 · p. 15
7 The Fragility of Opacity
Opacity is costly to create but fragile: a thermal photon can reverse decreasing complexity and restore transparency within a scrambling time. The paper also describes this recovery geometrically and argues that a photon can clear a grey hole for a similar exponential period.
7 The Fragility of Opacity
- Creating opacity is computationally expensive, requiring a complex precursor or waiting until the black hole passes through exponentially many mutually orthogonal states.Once opacity forms, a thermal photon can render the horizon transparent in a scrambling time.Source: §7 · pp. 15–16
- A single low-energy disturbance, such as a thermal photon, quickly reverses decreasing complexity; the required spreading time is the scrambling time.The paper relates this spreading to the switchback effect [2][9] and concludes that opacity gives way to transparency in about a scrambling time.Source: §7 · p. 16
- A photon colliding with the shock shifts it relative to the horizon; the shock exits the Planckian region in a scrambling time, leaving the horizon transparent.Figure 7 depicts the shock wave, photon, and Planck-length region.Source: Figure 7 · p. 16, §7 · p. 17
- An observer unsure whether a horizon is transparent can drop in a thermal photon and wait a couple of scrambling times to clear a firewall.After the shock exits the Planckian region, the paper says it falls into the singularity in another scrambling time.Source: §7 · p. 17
- In a black hole at complexity equilibrium, one thermal photon restarts complexity growth and is conjectured to make the horizon transparent for a similar exponential period.The photon increases entropy by about one bit, doubling the exponential of the entropy and taking the black hole out of complexity equilibrium.Source: §7 · p. 17
- A horizon on an increasing-complexity trajectory is extremely robust, whereas an opaque horizon can be destroyed by a slight perturbation.A firewall may endanger an infalling observer, but its thin Planckian shell is fragile.Source: §7 · p. 17
8 Remark About Linearity
The paper questions whether being a black hole can be a conventional observable: the spans of black-hole, white-hole, and grey-hole states each fill the Hilbert space. It leaves the resulting nonlinearity paradox for future work.
8 Remark About Linearity
- The spans of rising-complexity black-hole states, complexity-equilibrium states, and white-hole states each cover the full Hilbert space, so black-hole status cannot be a conventional observable.The paper says the equilibrium states are vastly over-complete.Source: §8 · p. 17
- Complexity is not linear: the superposition of two low-complexity states will generally have high complexity.Source: §8 · pp. 17–18
- The paper defers the nonlinearity paradox raised by this account to a separate paper.Source: §8 · p. 18