PaperlayerView exact revision on arXiv

Source-linked AI summary

Black Holes Hint Towards De Sitter-Matrix Theory

Leonard Susskind

arXiv:2109.01322v1hep-thgr-qc

TL;DR

The paper asks how de Sitter black holes can probe the holographic degrees of freedom of de Sitter space, for which no clear RT-like framework has been formulated. It studies black-hole configurations and fluctuation contributions, including matrix descriptions of the relevant entropy relations. The Nariai geometry contributes a universal non-perturbative term and leads the inside-out process, while the proposed holographic description is not a union of independent horizon blocks.

  • Problem

    Although the two de Sitter static patches are entangled in a thermofield-double state, no clear framework analogous to the Ryu–Takayanagi formula has been formulated for de Sitter space.

    Source: §1 · p. 4

  • Method

    The paper probes de Sitter holography through black-hole configurations and fluctuation probabilities, using matrix degrees of freedom to reproduce a Schwarzschild formula.

    Source: §2 · p. 8, §5.2 · p. 22

  • Results

    The Nariai geometry contributes a universal non-perturbative term to fluctuation probabilities and gives the leading contribution to the inside-out process.

    Source: §4 · p. 18

  • Takeaways & Limitations

    The paper argues that the hologram is associated with the dominant path-integral saddle’s horizon, even when no single horizon component in an individual wave-function branch is large enough to describe it.

    Source: §9 · p. 36

    The horizon union does not yield independent block Hilbert spaces, because off-diagonal matrix components remain and constraints cannot be infinitely tight.

    Source: §9 · p. 35

Abstract

from arXiv · show

De Sitter black holes and other non-perturbative configurations can be used to probe the holographic degrees of freedom of de Sitter space. For small black holes evidence was first given in seminal work of Banks, Fiol, and Morrise; and followups by Banks and Fischler; showing that dS is described by a form of matrix theory. For large black holes the evidence given here is new: Gravitational calculations and matrix theory calculations of the rates of exponentially rare fluctuations match one another in surprising detail. The occurrence of the Nariai geometry and the "inside-out" transition are especially interesting examples which I explain.

1 Entanglement in de Sitter Space

The paper sketches an RT-like prescription for entanglement between opposing de Sitter static patches, whose holographic degrees of freedom are associated with stretched horizons. Separating those horizons prevents the proposed minimal surface from collapsing to zero, while the time-dependent maxmin construction reduces to the anchoring points in the depicted case.

1 Entanglement in de Sitter Space

  • The proposed de Sitter holographic description is assumed to place static-patch degrees of freedom nominally on the stretched horizon.The discussion is limited to four dimensions; generalizing the mechanisms and calculations is described as non-trivial.Source: §1 · p. 3
  • Separating the stretched horizons allows the proposed minimal surface to lie between them, avoiding the zero-area collapse of the initial homology-only prescription.The initial homologous curve can shrink to a point, which would imply vanishing pode–antipode entanglement.Source: §1 · pp. 5–6
  • The proposed pode–antipode entropy is 1/4G times the minimal area of a surface homologous to one stretched horizon and lying between both.Source: §1 · p. 6
  • For time-dependent geometries, the prescription maximizes over connecting spacelike slices the minimum area of a two-sphere cutting each slice.In the depicted case, the minimum sphere lies at the anchoring points, making that maximization redundant.Source: §1 · pp. 7–8

2 From Small Black Holes to Nariai

The paper uses four-dimensional de Sitter black-hole geometries to probe holographic degrees of freedom and argues that the underlying description must be a form of matrix quantum mechanics. Its parameterization distinguishes configurations in which the two horizons exchange roles.

2 From Small Black Holes to Nariai

  • The paper argues that de Sitter holographic degrees of freedom must be described by a form of matrix quantum mechanics.The black-hole properties provide the evidence for this conclusion.Source: §2 · pp. 8–9
  • The Schwarzschild–de Sitter horizon condition has two positive roots, r− and r+, identifying the black-hole and cosmic horizons, respectively.The metric has a naked singularity outside the stated allowable mass range.Source: §2 · pp. 9–10
  • The black-hole and cosmic horizons exchange roles as x passes through zero, where their radii become equal.The paper treats the x < 0 and x > 0 ranges as physically different configurations.Source: §2 · p. 10

3 Entropy Deficit

The section relates de Sitter entropy deficits to the probabilities of black-hole fluctuations. It distinguishes small black holes, whose deficits scale with the square root of de Sitter entropy, from large black holes, whose deficit follows a detailed function of the horizon parameter x.

3 Entropy Deficit

  • The equilibrium de Sitter entropy is maximal at fixed average energy, and a fluctuation to lower entropy has probability determined by its entropy deficit.Source: §3 · pp. 11–12

3.1 Small Black Holes

  • For fixed-mass small black holes, the entropy deficit scales as S^1/2, and the Planck-mass example has probability equal to the Boltzmann weight e^−βM.The paper presents the small-hole relation as a hint toward matrix degrees of freedom, with a further relation for large holes offered as stronger support.Source: §3.1 · pp. 13–14

3.2 Large Black Holes

  • For large black holes, the entropy deficit varies with x in a detailed relation that the paper identifies as characteristic of theories with matrix degrees of freedom.The deficit is symmetric under x → −x and smooth at x = 0, where the black-hole and cosmic horizons cross.Source: §3.2 · p. 16

4 Probabilities and the Entropy Deficit

The entropy deficit determines black-hole fluctuation probabilities, while integrating over possible nucleation configurations separates a microphysics-sensitive contribution from a universal Nariai saddle. For the inside-out process, the Nariai geometry supplies the leading contribution.

4 Probabilities and the Entropy Deficit

  • The entropy deficit determines the probabilities of Boltzmann fluctuations, including the probability for a black hole of mass M to appear at the pode.The passage introduces the probability formula but does not provide its expression.Source: §4 · pp. 16–17
  • The total probability for nucleation anywhere in the static patch is integrated over X from 0 to about 1, where the black hole is small or near the horizon at the boundary.The boundary details are stated to be unimportant if the components of X are of order 1.Source: §4 · pp. 17–18
  • The first term appears to come from very small black holes near the horizon that fall back in, but a microscopic-mass cutoff could change that term.The passage does not specify the cutoff-dependent replacement completely.Source: §4 · p. 18
  • The Nariai saddle is non-perturbative in G and universal, independent of microphysics, and gives the leading contribution to the inside-out process.The first term in the total-probability result appears perturbative and microphysics-sensitive.Source: §4 · p. 18

5 dS-Matrix Theory

The paper argues that de Sitter holographic degrees of freedom are matrices and uses constrained matrix states to reproduce black-hole entropy deficits. The match extends across small and large black holes, with a numerical discrepancy at the Nariai point and an N-dependent cost in higher dimensions.

5 dS-Matrix Theory

  • The paper states that de Sitter holographic degrees of freedom must be matrices [2][3][4][5], and that the Hamiltonian should include a term whose role is to enforce certain constraints.Source: §5 · p. 19

5.1 Small Blocks

  • The small-block matrix calculation reproduces the small-black-hole entropy deficit within a factor of 2.The model treats the separated black hole and cosmic horizon as matrix blocks, with off-diagonal degrees of freedom constrained to their ground states and carrying no entropy [2][3][4][5].Source: §5.1 · pp. 20–21

5.2 Remark on Higher Dimensions

  • The d-dimensional Schwarzschild entropy-deficit formula can be reproduced with matrix degrees of freedom, at the cost of making entropy per degree of freedom depend on N.Further discussion of the higher-dimensional generalization is left for future work.Source: §5.2 · p. 22

5.3 Large Blocks

  • For large blocks, the matrix and gravitational relations coincide in form, but at the Nariai point the matrix entropy deficit is S0/2 rather than gravity's S0/3.The relation is not derived from general statistical or thermodynamic principles; it follows from matrix degrees of freedom and the chosen subsystem decomposition, and the coefficient may be adjustable.Source: §5.3 · pp. 23–24

6 Dynamics of the Constraints

The paper proposes enforcing matrix constraints through energy-dependent dynamics rather than implementing de Sitter symmetries directly. The off-diagonal modes can carry no entropy when their oscillator frequency is high, while relaxing the constraints changes the numerical coefficients in the entropy-deficit relations.

6 Dynamics of the Constraints

  • The proposed mechanism uses energy considerations to enforce constraints, since directly implementing the de Sitter symmetries is a difficult problem the paper does not attempt to solve.The authors describe their aim as illustrating a dynamical mechanism, not providing a detailed model.Source: §6 · p. 24
  • The added matrix R is intended to encode radial position through its eigenvalues, and the assumed Lagrangian term couples it to the other bosonic and fermionic matrices.The object is modeled as well separated from the cosmic horizon by taking R approximately block diagonal.Source: §6 · pp. 24–25
  • When ω is much larger than the other energy scales, the oscillators are forced into their ground states, so off-diagonal degrees of freedom carry no entropy; the prior analysis applies unmodified.Source: §6 · p. 26
  • When c is approximately 2π, constraints are not tight and off-diagonal modes carry some entropy, but relaxing them changes only the coefficients in the entropy-deficit relations.The coefficient in (5.35) might be changed from 2 to the gravitational value 1; bringing the factor in (5.42) exactly to 1/3 may require subtle, possibly fine-tuned properties of H.Source: §6 · p. 26

7 The Inside-Out Process

The inside-out transition exchanges the two de Sitter horizons through the Nariai state, and its rate is set by that state’s exponentially small probability. In the matrix description, the black-hole and cosmic-horizon blocks grow and shrink until they are exchanged.

7 The Inside-Out Process

  • In gravity, a fluctuation transfers energy between the equal-temperature Nariai horizons, driving one to shrink toward a small black hole and the other to grow toward the de Sitter horizon.The reverse direction is also possible.Source: §7 · p. 27
  • The horizon exchange takes order S0R, the Page time, and increases entropy from 2/3S0 to S0 rather than violating the second law.The exchange can be viewed as a massive Boltzmann fluctuation from de Sitter through Nariai and back to de Sitter.Source: §7 · pp. 27–28
  • The transition must pass through the Nariai state at x = 0, whose probability e^(-S0/3) gives the leading contribution and makes the process non-perturbative.The Nariai saddle controls the rate, with no contribution from small black holes.Source: §7 · pp. 28, 30
  • The transition lasts order the Page time, δt, while the average interval ∆t between events is much longer; its rate is controlled by the Hawking–Moss instanton.The rate is set by the probability of reaching the top of the barrier at Nariai, not by a Euclidean solution interpolating between the endpoints.Source: §7 · p. 29
  • If terminal-vacuum decay takes its longest possible time, e^S0, many inside-out events can occur before the de Sitter vacuum decays.The prefactor in the rate estimate is unreliable, but the exponential is e^(-S0/3).Source: §7 · p. 30
  • The small black-hole block grows as the cosmic-horizon block shrinks; after they become equal at Nariai, they continue until the blocks are exchanged.The two equal matrix blocks represent the Nariai geometry.Source: §7 · p. 31

8 Instantons and Giant Instantons

This section compares gravity and matrix-theory fluctuation probabilities with instanton amplitudes, finding matching scalings in several cases. It relates black-hole formation to single-eigenvalue tunneling and compares giant transitions with non-perturbative processes in large-N theories.

8 Instantons and Giant Instantons

  • The section compares the probabilities of black-hole formation and inside-out transitions with instanton amplitudes in matrix and gauge theories.The comparison is intended to identify parallels between these non-perturbative processes.Source: §8 · p. 31, §8.1 · p. 31

8.1 Some Probabilities

  • The matrix and gravity probabilities for forming a minimal black hole are essentially the same, while their Nariai-state probabilities scale in the same way.The comparison uses S = σN2.Source: §8.1 · pp. 31–32

8.2 Instantons

  • The large-N scaling of single-eigenvalue tunneling matches the smallest-black-hole fluctuation, suggesting Planck-mass black-hole formation is instanton-mediated in dS-matrix theory.The matrix model reduces to N fermions representing the eigenvalues, each of which can tunnel between the wells.Source: §8.2 · p. 33

8.3 Giant Instantons

  • The giant-instanton transition scales as exp(−S0), like the inside-out transition, while giant transitions in Yang–Mills theory scale as exp(−N2).An ordinary SU(N) instanton rate scales as exp(−N), whereas the giant process involves all N/2 commuting SU(2) subgroups.Source: §8.3 · p. 34

9 Remarks about the Holographic Principle in dS

Large fluctuations can produce higher-topology geometries and multiple horizons, raising the question of where the holographic degrees of freedom reside. The paper proposes a single holographic system sized to the largest connected cosmic horizon, even when no individual horizon in a wave-function branch is large enough to describe the whole.

9 Remarks about the Holographic Principle in dS

  • Large Boltzmann fluctuations can produce higher topologies such as Nariai and break the horizon into multiple components, leaving the location of holographic degrees of freedom unclear.The alternatives raised are the outermost horizon, the union of horizons, or a more abstract, nonlocalized hologram.Source: §9 · p. 35
  • Near Nariai, the largest horizon has entropy slightly above S0/3, insufficient to describe both horizons with combined entropy 2S/3, and horizon sizes can swap over time.Source: §9 · p. 35
  • A union-of-horizons description would map to a union of matrix blocks, but the Hilbert space does not factor into block Hilbert spaces because off-diagonal components connect them.Those components may be unexcited when constraints are tight, but matching numerical coefficients requires constraints that are not infinitely tight.Source: §9 · p. 35
  • The dS-matrix hologram is one system with degrees of freedom sized to the largest area of the cosmic horizon when it forms a connected whole, sufficient for any state but not much larger.An individual wave-function branch may have no single horizon component large enough to describe the whole.Source: §9 · pp. 35–36

A Nariai Geometry

As the black-hole mass increases, the two horizon roots meet at the Nariai limit, whose geometry is dS2 × S2 and whose Euclidean continuation is S2 × S2. The Nariai geometry contributes a saddle to the path integral, while transitions between geometries in the full quantum theory prevent either semiclassical symmetry from being exact.

A Nariai Geometry

  • At the Nariai limit, the roots r− and r+ meet, and the geometry is dS2 × S2 with Euclidean continuation S2 × S2.The near-Nariai function f(r) is expanded to quadratic order to obtain the limiting metric.Source: A Nariai Geometry · pp. 36–38
  • Each Nariai horizon has entropy S0/3, and the two horizons together have total entropy SN.Source: A Nariai Geometry · p. 36
  • The entropy deficit is the de Sitter entropy minus the state's entropy, and the Nariai deficit is evaluated using this definition.Source: A Nariai Geometry · pp. 36–37
  • The Nariai S2 × S2 geometry contributes a saddle-point term to the Euclidean path integral alongside the original de Sitter S4 contribution.The Nariai contribution is obtained from the classical action of the S2 × S2 geometry.Source: A Nariai Geometry · p. 38
  • The dS symmetry is larger than the Nariai symmetry but does not contain it as a subgroup; in the full quantum theory, transitions mean neither symmetry can be exact [6].In the infinite-entropy semiclassical limit, transition probability between the geometries is zero.Source: A Nariai Geometry · pp. 38–39

B The Equilibrium Shell

A Schwarzschild–de Sitter black hole has an unstable equilibrium shell where df/dr = 0, between the black-hole and cosmic horizons. In the Nariai limit, the shell lies midway between the horizons, which remain a finite distance apart.

B The Equilibrium Shell

C The Temperature of the Nariai Geometry

Because de Sitter space lacks spatial infinity, the Nariai temperature is defined at the equilibrium shell. There the horizons share a temperature and are in unstable thermal equilibrium.

C The Temperature of the Nariai Geometry

Loading 2109.01322v1…