Very long time scales and black hole thermal equilibrium

The product of the proposed resolution's time scale and amplitude respects the lower bound.

“although their product does conspire so as to respect the bound.”

The Gaussian-noise damping factor removes all possible recurrences as time goes to infinity.

“a damping factor that eliminates all possible recurrences as t → ∞.”

• AssumptionThe authors state this as an assumption: The operator C is non-degenerate in the energy basis.

The time scale supplied by the proposed resolution is much shorter than expected.

“The time scale it provides is much shorter than could be expected”

In the semiclassical bulk approximation, the correlation functions in the presence of eternal black holes vanish.

“The correlators indeed vanish in a semiclassical bulk approximation.”

• Also stated in: § JHEP11(2003)047 · plain

  “semiclassically, on the bulk side, the correlation function does decay to zero in the presence of a black hole and thus violates the bound.”

The amplitude supplied by the proposed resolution is much smaller than expected.

“the amplitude much smaller than could be expected”

For two-point functions on different thermo-field doubles, the time average L_LR is approximately exp(-K(β)), with K(β) = 2β(F(β) - F(β/2)) - S(β).

“L LR ∼ exp(-K(β)) , where K(β) = 2β (F (β) -F (β/2)) -S(β) ,”

Poincaré recurrences can be absent in non-unitary systems even when the entropy is finite.

“we show that Poincaré recurrences can indeed be absent for non-unitary systems.i.e. a system with noncoherent quantum evolution [10] will show no recurrences despite having finite entropy.”

• cites [10] J.L.F. Barbón, 1999, Nucl. Phys. sourceNot checked yet

  Not checked in the free run.

A direct analysis of correlation functions in a black hole background yields L = 0 in perturbation theory.

“However, a direct analysis of correlation functions in the background of a black hole yields L = 0 in perturbation theory.”

In Lorentzian signature, the wave functions on the exponential potential are Bessel functions.

“In lorentzian signature, the wave functions on the exponential potential are given by Bessel functions”

For r near r0, the metric can be approximated by Rindler space with a pure exponential potential in (3.11).

“For r ≈ r 0 we can approximate the metric by Rindler space, with a pure exponential potential in (3.11).”

The time averages L_αβ vanish identically for any black-hole background.

“At any rate, the time averages vanish identicaly for any black-hole background.”

The Euclidean Rindler space near the horizon becomes a copy of flat R^d.

“the euclidean Rindler space in the vicinity of the horizon r ≈ r 0 becomes a copy of flat R d space:”

The zero-time Green's functions G_α(0) are of order 1 in the semiclassical limit.

“The zero-time Green's functions G α (0) are of O(1) in the semiclassical limit.”

The total Euclidean correlator is given by an average over the different backgrounds.

“The total euclidean correlator is then given by an average over the different backgrounds.”

The total measure splits into perturbative measures around each background, with suppression by the classical Euclidean action of each background.

“In this situation the total measure dµ(φ, . ..) splits into the different perturbative measures dµ α around each background, dµ → α e -I cl (Xα) dµ α ,”

On X, G_E can be written as a sum over n and ω of e^{2πin(τ−τ')/β} φ_ω(r,Ω) φ*_ω(r',Ω') divided by ω2 + 4π2n2/β2.

“Therefore, on X we can write G E (τ, τ ) = 1 β n ω e 2πin(τ -τ )/β φ ω (r, Ω) φ * ω (r , Ω ) ω 2 + 4π 2 n 2 /β 2 .”

The Green's function G_E satisfies the same differential equation as the Hartle-Hawking Green's function when restricted to X.

“satisfies the same differential equation as the Hartle-Hawking Green's function (3.17), when restricted to X .”

The final estimate is L_inst ≈ e^{−2ΔI} C_{Y/X}^2 L_Y.

“In this way we arrive at the final estimate: L inst ≈ e -2 ∆I C Y /X 2 • L Y ,”

Near a non-degenerate horizon, the effective potential has the universal scaling V_eff ∝ exp(4πr*/β) as r* → -∞.

“Near a non-degenerate horizon, g(r 0 ) = 0, the effective potential shows the universal scaling V eff (r * ) ∝ e 4πr * /β , as r * → -∞”

Perturbative corrections in the black hole background will not discretize the spectrum unless perturbation theory itself breaks down.

“Perturbative corrections in the black hole background are constructed in terms of the free Green's functions, and will not discretize the spectrum (unless perturbation theory itself breaks down.)”

In the same approximation, the perturbative partition function Tr exp(-βH) is divergent.

“In the same approximation, the perturbative partition function Tr exp(-βH) that appears in the normalization of (3.2) is divergent.”

Putting the system in a large box with a reflecting wall at finite radius does not discretize the spectrum because of the exponential tail.

“In particular, enclosing the system in a large box by placing a reflecting wall at large and finite r fails to discretize the spectrum, due to the exponential tail (3.11).”

Anti-de Sitter spaces behave effectively as finite-volume cavities.

“we recover the known fact that AdS spaces behave effectively as finite-volume cavities.”

For a bounded system with a discrete energy spectrum and unitary evolution, thermal dissipation cannot become complete at very long times.

“the thermal dissipation cannot be complete at very long times.”

The vanishing of L is tied to the continuous spectrum of the perturbative Hamiltonian H = i∂/∂t.

“The vanishing of the recurrence index, L = 0, in the previous discussion is tied to the continuous spectrum of the perturbative hamiltonian H = i ∂/∂t.”

The Heisenberg time is the smallest possible Poincaré time.

“the Heisenberg time is the smallest possible Poincaré time.”

A perfectly unitary evolution can still fail to show Poincaré recurrences when the spectrum is continuous.

“one may have a perfectly unitary evolution, and still miss the Poincaré recurrences because the spectrum is continuous.”

The numerical value of the unitarity-based bound comes from the holographic dual field theory.

“The value of the bound is obtained from the holographic dual field theory.”

The absence of Poincaré recurrences in this black-hole case may be an artefact of the semiclassical approximation.

“Thus the absence of Poincaré recurrences in this case may be an artefact of the semiclassical approximation.”

The continuous spectrum is a property of the bare horizon as described by General Relativity.

“This is a property of the “bare” horizon, as described by General Relativity, and could be modified by nonperturbative dynamical effects.”

Such non-unitary systems can generically be modelled as ordinary quantum mechanics coupled to Gaussian random noise.

“It was shown in [14] that these systems can be generically modelled by ordinary quantum mechanics coupled to a gaussian random noise.”

• cites [14] M. SrednickiNot checked yet

  Not checked in the free run.

In the free approximation, the frequency spectrum in the presence of horizons is continuous.

“The frequency spectrum in the presence of horizons is thus continuous in the free approximation.”

The vanishing of the correlation functions is caused by the continuous energy spectrum that exists when event horizons are present.

“We trace the origin of their vanishing to the continuum energy spectrum in the presence of event horizons.”

The infinite redshift of an event horizon allows a quantum field to accumulate infinitely many modes there at finite energy cost.

“the infinite redshift of an event horizon implies that a quantum field can accumulate an infinite number of modes there, with a finite cost of energy (c.f.[15]).”

• cites [15] W. ThirringNot checked yet

  Not checked in the free run.

The expected recurrence index for black holes scales nonperturbatively as exp(-1/G_N).

“Thus, the expected recurrence index for black holes has a nonperturbative scaling with Newton's constant: L ∼ exp(-1/G N ).”

• AssumptionThe authors state this as an assumption: The asymptotic boundary conditions make the canonical ensemble stable, meaning the specific heat is positive.

The time average L is bounded strictly above zero when the thermodynamic entropy is finite.

“Thus, L is bounded strictly above zero provided the thermodynamic entropy is finite.”

• AssumptionThe authors state this as an assumption: The quantity βΓ is finite in the thermodynamic or semiclassical limits.
• AssumptionThe authors state this as an assumption: The operators A and B have matrix elements around E(β) forming a band of width Γ/2 E(β), and the integrands can be approximated by their mean value in that band.

The spectrum of positive real frequencies is continuous independently of the potential's behaviour away from the horizon.

“We see that the spectrum of positive real frequencies will be continuous, quite independently of the properties of the potential for r r 0 .”

Giving the thermal gas background a role restores agreement with a time-averaged unitarity bound.

“We find that assigning a role to the thermal gas background, as suggested in the above work, does restore the compliance with a time-averaged unitarity bound.”

• Also stated in: § JHEP11(2003)047 · plain

  “While it does lead to a result respecting a lower bound on a time averaged normalized correlation function”

• cites [2] S.S. Gubser, 1998, Phys. Lett. sourceNot checked yet

  Not checked in the free run.

The large-time decay of the retarded correlators can be understood through complex frequencies, analogous to thermal quasinormal modes.

“Since tt > 0 in our Green's functions, they are actually retarded correlators, and the large-time decay can be understood in terms of complex frequencies, a thermal analog of the so-called “quasinormal modes” [9].”

• cites [9] E. Witten, 1998, Adv. Theor. Math. Phys. sourceNot checked yet

  Not checked in the free run.

The Poincaré patch of vacuum AdS has a continuous spectrum because r = 0 is a degenerate horizon.

“However, the Poincaré patch of vacuum AdS, with g(r) = r 2 /R 2 has continuous spectrum, as r = 0 is a degenerate horizon with g(0) = g (0) = 0.”

The recurrence or Poincaré time scales as a double exponential in the entropy.

“Since N ∼ exp(2S), the recurrence or Poincaré time scales as a double exponential in the entropy.”

• AssumptionThe authors state this as an assumption: The N different frequencies ω_ij = E_i - E_j are rationally independent.

Hawking-type non-unitary time evolutions violate quantum coherence while preserving hermiticity, positivity and normalization of the density matrix.

“Time evolutions of this sort violate quantum coherence but preserve hermiticity, positivity and normalization of the density matrix.”

• cites [10] J.L.F. Barbón, 1999, Nucl. Phys. sourceNot checked yet

  Not checked in the free run.

Within the perturbative formalism, L = 0 in the presence of static horizons.

“we will find L = 0 in the presence of static horizons, at least within the perturbative formalism sketched here.”

For black holes, the bound should hold with β equal to the inverse Hawking temperature and S equal to the Bekenstein-Hawking entropy.

“In that case we expect the bound (2.22) to hold with β the inverse Hawking temperature and S(β) = A H /4G N the Bekenstein-Hawking entropy.”

• AssumptionThe authors state this as an assumption: The asymptotic boundary conditions make the canonical ensemble stable, meaning the specific heat is positive.

Poincaré recurrences can serve as a criterion for unitarity if the energy spectrum is genuinely discrete.

“This result implies that Poincaré recurrences may be used as a criterion for unitarity, provided the energy spectrum is truly discrete.”

The denominator of (4.4) is dominated by the manifold with the largest partition function at each temperature.

“Hence, the denominator of (4.4) is dominated by the manifold with largest partition function at each temperature.”

The background Y that dominates the numerator of (4.4) does not contain horizons.

“Hence, the background that dominates the numerator of (4.4), denoted Y , does not contain horizons.”

The instanton approximation to L agrees exactly with bound (2.22) when X = Y, i.e. when the thermodynamical free energy is dominated by a thermal gas with no black holes.

“Hence, the instanton approximation to L agrees exactly with the bound (2.22) when X = Y , i.e. when the thermodynamical free energy is dominated by a thermal gas with no black holes.”

Such modifications of the boundary conditions are not consistent with the systematics of the semiclassical expansion.

“Of course, such modifications of the boundary conditions are not consistent with the systematics of the semiclassical expansion.”

Equations (3.31) and (3.32) should be viewed only as order-of-magnitude estimates of the true black hole's damping frequencies.

“It should be noted however that (3.31), (3.32) must be viewed only as order-of-magnitude estimates of the true black hole's damping frequencies.”

The absence of recurrences is traced to the topological properties of the Euclidean manifold X.

“We have traced the absence of recurrences to the peculiar topological properties of the euclidean manifold X.”

The final result is the Green's function of the Laplacian on R^d, i.e. the Hartle-Hawking Euclidean Green's function.

“The final result is the Green's function of the laplacian on R d , i.e. the Hartle-Hawking euclidean Green's function.”

This estimate yields L ∼ exp(−A_H/4G_N) for a system whose entropy is dominated by a black hole phase.

“This estimate yields L ∼ exp(-A H /4G N ) for a system whose entropy is dominated by a black hole phase.”

Because the Rindler approximation treats the horizon as flat R^d, the results are not accurate on scales of the Schwarzschild radius, which can be comparable to |Im ω|^{-1}.

“Since the Rindler approximation regards the horizon as flat R d , results are not accurate on scales of the Schwarzschild radius, which can be of the same order of magnitude as | Im ω | -1 .”

Because (3.17) and (3.25) satisfy the same differential equation, G_E is expected to give exactly the Hartle-Hawking Green's function on X when extended by continuity from X.

“Since (3.17) and (3.25) satisfy the same differential equation, we expect that G E gives exactly the Hartle-Hawking Green's function on X, when extended by continuity from X .”

Nonperturbative effects of order e^{−1/G_N} must effectively discretize the energy spectrum to capture the recurrences.

“Nonperturbative effects of O(e -1/G N ) arising in the semiclassical expansion must effectively discretize the energy spectrum in order to capture the phenomenon of the recurrences.”

The topology makes the analytic continuation to real time blind to the spacing of black hole energy levels.

“At the same time, it renders the analytic continuation to real time completely blind to the spacing of black hole energy levels.”

If the divergent entropy is equated with the Bekenstein-Hawking entropy, the phenomenological parameter ε is adjusted a posteriori.

“If this is equated with the Bekenstein-Hawking entropy, we have an a posteriori adjustment of the phenomenological parameter, ε.”

For systems with discrete spectrum, the recurrence index is estimated as L ∼ exp(−S).

“The recurrence index was estimated as L ∼ exp(-S) for systems with discrete spectrum.”

The recurrence index L behaves quite differently from the partition function Z(β).

“In this respect, the recurrence index L seems to behave quite differently from the partition function Z(β).”

Applying the general discussion of section 2 gives L_Y ∼ exp(−S(Y)).

“Applying the general discussion of section 2 we obtain L Y ∼ exp(-S(Y )).”

The details of unitary time evolution can only be retrieved by new nonperturbative effects.

“implies that the details of unitary time evolution can only be retrieved by new nonperturbative effects.”

The special topology guarantees the emergence of a classical contribution to the entropy.

“This special topology guarantees the emergence of a classical contribution to the entropy [17].”

• cites [17] G. 't Hooft, 1985, Nucl. Phys. sourceNot checked yet

  Not checked in the free run.

Bulk correlation functions built from local bulk operators can be interpreted as correlators of spatially nonlocal operators in the CFT.

“Correlation functions such as (3.16), in terms of local operators in the bulk, can be interpreted as correlators of spatially nonlocal operators in the CFT.”

By explicit calculation, a real-time thermal correlator with continuous spectrum at the horizon is the analytic continuation of a Euclidean correlator that is smooth near r = r0.

“Thus, we have shown by explicit calculation that a real-time thermal correlator with continuous spectrum at the horizon is the analytic continuation of an euclidean correlator that is perfectly smooth in the vicinity of r = r 0 .”

This conclusion holds for any static horizon, including de Sitter in the static patch.

“This conclusion holds for any static horizon, including de Sitter in the static patch”

The phenomenon is traced to a subtlety in the analytic continuation from the Euclidean path integral to the Lorentzian correlator, originating from the peculiar topology of the Euclidean black hole manifold X.

“This phenomenon can be traced back to a subtlety in the analytic continuation from (3.16) to (3.2), which in turn originates on the peculiar topology of the euclidean black hole manifold X.”

Restoration of the recurrences depends on dynamical effects that change the topology of X.

“Hence, the restoration of the recurrences depends on dynamical effects that change the topology of X.”