Disturbing Implications of a Cosmological Constant

May combine two claims.

For early inflationary eras the entropy is probably about 10^10, while the equilibrium entropy is about 10^120, giving a recurrence time of order exp(10^120).

“The fluctuations we have in mind correspond to early inflationary eras during which the entropy is probably of order 10 10 , while the equilibrium entropy is of order 10 120 .Thus T r ∼ exp 10 120 .”

May combine two claims.

For the ring solution with α > 0, the ring has positive energy density and requires tension to remain static.

“This corresponds to a positive energy density in the ring (for α > 0), with some tension to hold it static.”

May combine two claims. The quote only partly states this claim.

The back reaction on the geometry is small but makes the Penrose diagram slightly taller, pushing the horizon point toward the antipodal point; the stretched horizon temporarily moves away from the observer and then returns near the equator.

“The back reaction on the geometry is small, but it results in making the Penrose diagram slightly taller.”

May combine two claims. The quote only partly states this claim.

For a potential that is a constant plus a small step-like function, FRW solutions roll from negative to positive φ over the step at a localized time, with a small decrease in the cosmological constant and small back reaction.

“There are solutions which in FRW coordinates roll from negative to positive φ, going over the step at some fairly localized time.”

May combine two claims.

For a spherically symmetric matter distribution localized along a thin ring, the solution outside the ring has a deficit angle, and the horizon area is 2πα ≤ 2π.

“Note that the area of the horizon is 2πα ≤ 2π.”

May combine two claims.

Inside the ring, the solution is pure de Sitter space with no deficit angle.

“Inside the ring, the solution is pure de Sitter space with no deficit angle.”

The quote only partly states this claim.

If the transition occurs during expansion, the domain wall originates at the observer and falls toward the horizon, again increasing horizon entropy.

“In this case the domain wall originates at the observer and falls toward the horizon.”

May combine two claims. The quote only partly states this claim.

In O-coordinates, if the transition occurs during FRW contraction, a domain wall or wave of decreased cosmological constant originates at the horizon and propagates inward, releasing energy and increasing horizon entropy.

“In the former case, a domain wall or wave of decreased cosmological constant originates at the horizon (ρ = ∞) and propagates inward toward the observer.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The ring solution is unstable under perturbations that are not spherically symmetric.

“This solution is unstable to non-spherically symmetric perturbations, but this is not important for our purposes.”

May combine two claims.

In the extreme case of a very large change in the cosmological constant, the horizon intersection moves far toward the north pole and may reach it, so the causal patch may temporarily become a closed sphere with no horizon.

“In some cases it may even reach the pole, so that the causal patch may temporarily become a closed sphere, with the horizon disappearing for a time.”

Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing), S4 (based). May combine two claims. The quote only partly states this claim.

The boundary term at the origin is special to 2+1 dimensions and vanishes in higher dimensions.

“This”

May combine two claims.

The entropy of inflationary starting configurations is very low; in a typical high-scale theory the initial de Sitter-like entropy can be as low as 10^10, while the final de Sitter entropy is about 10^120.

“The entropy of such configurations is very low.In a typical high-scale inflationary theory, the entropy of the initial de Sitter-like space can be as low as 10 10 .The entropy of the final de Sitter space (assuming the observed value for the dark energy) is of order 10 120 .”

May combine two claims. The quote only partly states this claim.

The global AdS black hole geometry resembles closed FRW cosmologies, beginning in a big bang singularity and ending in a final crunch, with an observer at the spatial center experiencing the crunch in finite time.

“It starts with a big bang singularity and ends with a final crunch.”

May combine two claims.

De Sitter space lacks an ultraviolet boundary analogous to that of AdS, so one cannot modify boundary conditions to intervene from outside and create a particular initial condition.

“The reason is that the causal patch does not have the analog of an ultraviolet boundary.Because AdS has such a boundary it is possible to modify the boundary conditions without changing the fact that the bulk theory still satisfies the bulk field equations.Without such a boundary there is no way to do the same for de Sitter space.”

• AssumptionThe authors state this as an assumption: The ideas of holography and complementarity can be applied to de Sitter space.

Classified as probably not a substantial claim.

On average the universe is in a steady state of thermal equilibrium.

“On the average it is in a steady state of thermal equilibrium.”

• cites [23] sourceNot checked yet

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• AssumptionThe authors state this as an assumption: The usual connections between quantum statistical mechanics and thermodynamics hold.

May combine two claims. The quote only partly states this claim.

Poincaré recurrences in the AdS black hole system cause the black hole to reassemble into an energy shell and repeat at intervals of order exp S; these recurrences cannot be described by classical general relativity.

“One important fact is that these recurrences cannot be described by classical general relativity.”

May combine two claims.

High-scale inflation implies an initial holographic entropy around 10^10, which is extremely small compared with the present visible entropy.

“This implies an initial holographic entropy of about 10 10 , which is extremely small by comparison with today's visible entropy.”

• AssumptionThe authors state this as an assumption: The universe began with an inflationary phase and will end in an accelerated de Sitter phase.

May combine two claims. The quote only partly states this claim.

The classical global AdS black hole geometry can be interpreted as a cosmology; the boundary observer's causal patch describes the black hole exterior and is represented by a thermal density matrix in the holographic theory.

“It is evident from the diagram that the classical geometry can be interpreted as a cosmology.”

May combine two claims.

The quantum description of a causal patch should obey ordinary quantum mechanics and unitary evolution, treating it as a closed isolated box bounded by the observer's horizon.

“In other words, the theory describes a closed isolated box bounded by the observer's horizon, and makes reference to no other region.”

• cites [2] sourceNot checked yet

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• AssumptionThe authors state this as an assumption: The holographic and horizon complementarity principles are valid assumptions for describing horizons.

May combine two claims.

The time symmetry of the global geometry is unnatural and implies that the universe starts in a very special, extremely fine-tuned state.

“This is very unnatural and presumably means that the universe starts in very special, extremely fine tuned state.”

Classified as probably not a substantial claim. May combine two claims.

The de Sitter causal patch is described thermodynamically; its entropy is the logarithm of the number of effective quantum states of the thermal ensemble, equal to Trρ log ρ.

“The entropy usually ascribed to de Sitter space is that of the causal patch.”

May combine two claims.

Intermittent fluctuations temporarily disturb equilibrium, and interesting events and objects form during the return to equilibrium.

“However, intermittent fluctuations occur which temporarily disturb the equilibrium.It is during the return to equilibrium that interesting events and objects form.”

May combine two claims.

If the entropy is finite, then the energy levels are discrete and only finitely many levels lie below any given energy.

“From the fact that the entropy is finite, it follows that the energy levels are discrete and that the number of levels below any given energy is finite.”

• AssumptionThe authors state this as an assumption: The physics inside an observer's causal patch is exactly described by a dual quantum system with a Hamiltonian and a space of states, and the state is a thermal density matrix exp(−βH) at the appropriate temperature.

May combine two claims.

Because the causal patch boundary is a horizon, the box has finite temperature T = 1/(2πR) and entropy S = 4πR^2/(4G).

“Because the boundary of the box is a horizon, the box has a finite temperature and entropy:”

May combine two claims. The quote only partly states this claim.

The deficit angle can be identified with mass, and the mass is proportional to the horizon entropy used up in forming the object; more precisely, the number of degrees of freedom borrowed from the stretched horizon equals the product of the mass and the de Sitter radius.

“More precisely, the number of degrees of freedom borrowed from the stretched horizon is the product of the mass and the de Sitter space radius.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

The low-entropy initial condition is the fundamental reason the universe has an arrow of time, and without it the second law would be meaningless.

“The low entropy starting point is the ultimate reason that the universe has an arrow of time, without which the second law would not make sense.”

May combine two claims.

A boundary observer's observations are completely described by a unitary holographic quantum mechanics with no information loss, and the black hole is the thermal state of the holographic dual.

“Observations by the observer are known to be completely described by a unitary holographic quantum mechanics with no loss of information.”

Probably not a substantial claim: may not pass S1 (asserted). May combine two claims.

The conventional picture is that the universe ends in a de Sitter phase where matter is infinitely diluted and comoving points lose causal contact.

“The conventional view is that the universe will end in a de Sitter phase with all matter being infinitely diluted by exponential expansion.”

• AssumptionThe authors state this as an assumption: The universe began with an inflationary phase and will end in an accelerated de Sitter phase.

May combine two claims.

On timescales of order exp S, a finite system undergoes Poincaré recurrences that can restore rare low-entropy configurations and allow cosmology to begin again.

“Then on time scales of order T r ∼ exp S (1.1) the system will undergo Poincaré recurrencesfoot_2 [8,9].”

• cites [8] W. FischlerNot checked yet

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• cites [9] L. Dyson, 2002, J. High Energy Phys. sourceNot checked yet

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May combine two claims.

Given the stated assumptions and a final cosmological constant, the universe is eternal but finite, with the observable universe's entropy bounded.

“These assumptions-together with the existence of a final cosmological constant-imply that the universe is eternal but finite.”

• AssumptionThe authors state this as an assumption: The universe began with an inflationary phase and will end in an accelerated de Sitter phase.
• AssumptionThe authors state this as an assumption: A final cosmological constant exists.
• AssumptionThe authors state this as an assumption: The usual connections between quantum statistical mechanics and thermodynamics hold.

May combine two claims.

In a recurrent cosmology, the second law and the arrow of time are not fundamental laws but anthropic environmental conditions needed for life; life and interesting events occur only during brief out-of-equilibrium periods.

“In other words, the second law and the existence of a time arrow are anthropic.They are part of the environmental conditions needed for life.”

The quote only partly states this claim.

Starting from a high-entropy dead configuration, a fluctuation will eventually make the inflaton climb to the top of its potential and begin a cycle of inflation, reheating, conventional cosmology, and heat death.

“Starting in a high entropy,”

• AssumptionThe authors state this as an assumption: The physics inside an observer's causal patch is exactly described by a dual quantum system with a Hamiltonian and a space of states, and the state is a thermal density matrix exp(−βH) at the appropriate temperature.
• AssumptionThe authors state this as an assumption: The maximum entropy the system can ever have is the entropy of a de Sitter space whose size is set by the cosmological constant.

Probably not a substantial claim: may not pass S2 (checkable).

The remedy is to restrict attention to a single causal patch, following the black hole analogue.

“The remedy suggested by the black hole analogue is obvious; restrict all attention to a single causal patch [5,8,9].”

• cites [5] sourceNot checked yet

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• cites [8] W. FischlerNot checked yet

  Not checked in the free run.

• cites [9] L. Dyson, 2002, J. High Energy Phys. sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The transplanckian problem in Hawking radiation is an artifact of using quantum field theory on both sides of the horizon; black hole complementarity instead describes the black hole with strongly interacting Planck- or string-scale degrees of freedom outside the horizon.

“We now know that this is an artifact of trying to describe the complex interacting degrees of freedom of the horizon by quantum field theory defined on both the interior and exterior of the black hole.”

• cites [1] L. SusskindNot checked yet

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• cites [2] sourceNot checked yet

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• AssumptionThe authors state this as an assumption: The holographic and horizon complementarity principles are valid assumptions for describing horizons.

Probably not a substantial claim: may not pass S4 (based).

Applying ordinary unitary quantum mechanics to a system with horizons requires restricting attention to the region on the observer's side of the horizon.

“For reasons that have been explained in many places, if we want to apply the ordinary principles of unitary quantum mechanics to a system with horizons, we need to restrict attention to the region on the observer's side of the horizon.”

May combine two claims.

The rapid time dependence near the early singularity is far from adiabatic, so large entropy production should occur.

“Under these kind of circumstances there should be large amounts of entropy production.”

May combine two claims. The quote only partly states this claim.

For geometries that begin and end in de Sitter phases, the early and late stretched horizons match eternal de Sitter, so the causal patch horizon entropy in the far future and past is bounded in the ordinary way.

“This implies that the horizon entropy within the causal patch in the far future and past is bounded in the ordinary way.”

• AssumptionThe authors state this as an assumption: The universe began with an inflationary phase and will end in an accelerated de Sitter phase.

The quote only partly states this claim.

A more reasonable cosmology can begin from a far-from-equilibrium configuration, such as empty AdS excited by an infalling shell of energy from the boundary.

“A more reasonable cosmology can result if we started with a far from equilibrium configuration.”

Classified as probably not a substantial claim.

Horizon Complementarity describes the universe, for any observer, as a closed and finite system.

“Horizon Complementarity then says the universe is described by any observer as a closed and finite system.”

• AssumptionThe authors state this as an assumption: Horizon Complementarity is taken to mean that physics in the causal patch can be described as a closed or isolated system using conventional quantum mechanics.

May combine two claims.

The AdS/Schwarzschild black hole system is unrealistic as a cosmology but is extremely well understood through AdS/CFT duality.

“This system, although unrealistic as a cosmology, is extremely well understood through the AdS/CFT duality.”

• cites [19] S. Hawking, 1983, Comm. Math. Phys. sourceNot checked yet

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May combine two claims.

Quantum gravity is a special case of quantum mechanics; every nonperturbative well-defined string theory formulation has a Hamiltonian and a state space, including matrix theory and AdS/CFT.

“Perhaps the most important conceptual lesson that we have learned from string theory is that quantum gravity is a special case of quantum mechanics.”

May combine two claims. The quote only partly states this claim.

In such worlds the helium abundance would be incomprehensible from usual arguments, and statistically miraculous events would be needed to assemble and preserve fragile nuclei.

“In all of these worlds statistically miraculous (but not impossible) events would be necessary to assemble and preserve the fragile nuclei that would ordinarily be destroyed by the higher temperatures.”

May combine two claims.

Although each such history is extremely unlikely, their greater number means they would vastly dominate the livable universes produced by Poincaré recurrences.

“However, although each of the corresponding histories is extremely unlikely, there are so many more of them than those that evolve without”

May combine two claims.

A perturbation on the past boundary cannot influence the causal patch because the signal would have to pass through the past horizon at infinite past, and a finite-entropy thermal cavity re-thermalizes after finite time.

“For a perturbation on the past boundary to influence the causal patch, the signal must pass through the past horizon of the observer.However the past horizon is just the infinite past t = -∞.”

• Also stated in: § JHEP10(2002)011 · boosted

  “Obviously, any such perturbation of a finite entropy thermal cavity will re-thermalize after a finite time.In fact, at any finite time, an infinite number of Poincaré recurrences have already occurred.”

• AssumptionThe authors state this as an assumption: The physics inside an observer's causal patch is exactly described by a dual quantum system with a Hamiltonian and a space of states, and the state is a thermal density matrix exp(−βH) at the appropriate temperature.
• AssumptionThe authors state this as an assumption: The maximum entropy the system can ever have is the entropy of a de Sitter space whose size is set by the cosmological constant.

May combine two claims.

L may be extremely small for large finite systems but does not vanish because the entropy is finite; it vanishes for infinite systems.

“Although L may be incredibly small for sufficiently large systems, it will not vanish since the entropy of a finite system is finite.It will vanish, however, for systems that are infinite.”

May combine two claims.

If an unknown agent restarted the universe in a low-entropy inflationary state, only the first occurrence would be consistent with usual expectations; later recurrences would be extremely unlikely to be consistent, so the anthropically acceptable part of spacetime would be dominated by peculiar conditions.

“Only the first occurrence would evolve in a way that would be consistent with usual expectations.Subsequently the recurrences would be extremely unlikely to be consistent, in this sense.Thus the anthropically acceptable part of spacetime would be dominated by peculiar, incomprehensible conditions.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The quantum Poincaré recurrence theorem states that a system with discrete energy eigenvalues will return arbitrarily close to its initial state in a finite time.

“The quantum Poincaré Recurrence theorem is similar and can be stated as follows: given a system in which all energy eigenvalues are discrete, a state will return arbitrarily close to its initial value in a finite amount of time.”

• cites [25]-[27]Not checked yet

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May combine two claims.

In the limit of a continuous energy spectrum, the quantum recurrence theorem breaks down, corresponding to a classically unbounded system and breakdown of the classical theorem.

“Additionally, in the limit of a continuous energy spectrum, the quantum theorem breaks down.This breakdown corresponds to a classically unbounded system and hence the breakdown of the classical theorem as well.”