The Firewall Paradox

Emanuele Ottino1, ⇤ and Maria Walch1, †

1University of Heidelberg

The implications of the Hawking radiation led to the formulation of some paradoxes via a seriesof thought experiments. While the information loss can be solved with di↵erent strategies, it isdiscussed whether the idea of complementarity can completely overcome the problem, also for thefirewall paradox. If the firewall paradox then holds, at least one between the well establishedprinciples of the modern physics must be dropped.

Contents

I. Introduction 2

II. The Information Paradox And Its Solutions 2A. No-Hair-Theorem 2B. The Information Paradox 2C. Suggested Solutions To The Information Paradox 3

1. The Information Comes Out With The Hawking Radiation 32. Stable Remnants 33. The Information Comes Out At The End 34. The Information Is Encoded In Quantum Hair 45. The Information Escapes To A Baby Universe 46. Black Hole Complementarity 4

III. Black Hole Complementarity 4A. Unitarity or Information Conservation 4B. Equivalence Principle 5C. Xerox Principle 5D. Can an Observer outside the horizon of a black hole recover information about the interior? 6E. Xeroxing Paradox 8F. Resolution Of The Xeroxing Paradox 9G. Black Hole Complementarity 10

Principles Of Black Hole Complementarity 11

IV. Underlying QM 12A. Maximal entanglement 12B. Monogamy of entanglement 13C. Entropy of a subsystem 13

V. The Unruh State 13

VI. Postulates 14A. Unitarity 14B. E↵ective QFT 14C. Equivalence principle 14

VII. AMPS thought experiments 15A. Double entanglement 15B. Mining argument 16

References 16

⇤Electronic address: ottino@stud.uni-heidelberg.de

†Electronic address: M.walch@stud.uni-heidelberg.de

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I. INTRODUCTION

Gravity is always attractive. This follows from the principle of equivalence, which states that gravity couplesto the energy-momentum tensor of matter, and the quantum-mechanical requirement that energy should bepositive. In any reasonable theory of gravitation, singularities appear. Singularities are places where the classicalconcepts of space and time break down as do all known laws of physics being formulated on a classical space-time background. This breakdown constitutes a limitation of our ability to predict the future called ”ignoranceprinciple”. It arises because in general relativity the interaction region can be bounded not only by an initialsurface on which data are given and a final surface on which measurements are made but also a ”hidden surface”about which the observer has only limited information such as mass, angular momentum, and charge. Thehidden surface emits with equal probability all configurations of particles that are compatible with the observer’sknowledge.

II. THE INFORMATION PARADOX AND ITS SOLUTIONS

A few decades ago, Hawking showed that black holes radiate because of the inevitable production of particlesand antiparticles from the vacuum in pairs. One particle always falls into the hole while the other possiblyescapes to infinity. Furthermore, he showed that this radiation is exactly thermal, meaning that no subtle orsecret correlations exist between the emitted particles. However, as conservation of energy has to be conformed,the radiation will carry energy away from the black hole and therefore shrinks its mass. As the mass shrinks,the surface gravity increases and therefore the temperature. (As the surface gravity of the black hole is equal tothe temperature of the black body spectrum of the Hawking radiation.) This is a self-catalyst process in whichthe entire mass evaporates away in a finite time.

For astrophysical black holes, this procedure is slow, their lifetime being of the order

⌧BH =

✓M

mpl

◆tpl ⇠

✓M

Msun

◆3

1071s,

with mpl ⇠ 10�5 g being the Planck mass and tpl ⇠ 10�43 s being the Planck time. This implies that theestimated life time of a solar mass black hole exceeds the age of the universe by 53 orders of magnitude. (withthe Hubble time being: H�1

0 ⇠ 1018 s. However, the life time of mini black holes could be such, that primordialblack holes could reach their end already today.

The detailed form of the radiation, however, should not depend on the detailed structure of the body thatcollapsed to from the black hole. The state of the radiation is determined only by the geometry of the black holeoutside the horizon. The black hole as no hair that records any detailed information about the collapsing body.

A. No-Hair-Theorem

The no-hair or uniqueness theorem applying to the stationary solutions of the Einstein-Maxwell equationsasserts that all electrovac black hole space-times are characterized by their mass, angular momentum andelectric charge. This bears a striking resemblance to the fact that a statistical system in thermal equilibriumis described by a small set of state variables, whereas considerably more information would be required tounderstand its dynamical behaviour. However, since Bartnik and McKinnon discovered the first self-gravitatingYang-Mills soliton in 1988 [1], a variety of new black hole configurations which violate the generalized no-hairconjecture have been found. These include non-Abelian black holes, and black holes with Skyrme, Higgs ordilaton fields.

The first example of black hole solutions with hair was the Bekenstein solution, describing a conformallycoupled scalar field in an extreme Reissner-Nordstrom spacetime.

B. The Information Paradox

Even if we assume that Hawking’s semiclassical approach is not exact, the emitted radiation can be onlyweakly correlated with the state of the collapsing body. The key constraint is posed by causality: Once the

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collapsing body is behind the horizon, it is incapable of influencing the radiation.

The radiation outside the black hole is in a mixed (thermal) state. There are correlations between degrees offreedom (quantum fields) that are accessible outside the horizon, and the inaccessible degrees of freedom behindthe horizon. These correlations are the reason that the radiation detected by observers outside the horizon is ina mixed state.

If the black hole continues to evaporate until it disappears completely, the radiation literally is the wholesystem. This would imply that an initially pure quantum state evolved to a mixed state by collapsing to a blackhole and subsequently evaporating completely. This means that even if we would know the initial quantumstate precisely, we could not predict with certainty what the final quantum state will be, only probabilities canbe assigned to various alternatives.

This states the information loss paradox. If we try to analyze the evolution of a black hole using the usualprinciples of relativity and quantum theory, we are led to a contradiction, for these principles forbid the evolutionof a pure state to a mixed state.

C. Suggested Solutions To The Information Paradox

1. The Information Comes Out With The Hawking Radiation

If we literally burn information (e.g. by burning a book) we cannot retrieve the information from the radiationbecause it has too complicated correlations with the internal state of the fire. However, if we wait long enoughuntil the fire has settled down to its unique quantum ground state and stopped radiating, then only correlationsamong the emitted quanta can possibly carry information. The information could be encoded in correlationsamong quanta emitted at di↵erent times. If these could be measured, it should be possible to regain theinformation.

One idea to solve the information paradox uses this ansatz. However, if one assumes that black holes haveno hair, it becomes di�cult to explain how the black hole records the information about the quanta that it hasalready emitted. In fact, Susskind concluded that if the information propagates out encoded in the Hawkingradiation, then there must be a mechanism to strip away the information about the collapsing body as the bodyfalls trough the apparent horizon. Until now, such a mechanism has not been reasonably found.

2. Stable Remnants

Another suggestion is that the Hawking radiation does contain at most only little information about thecollapsing body. Then an intuitive solution would be that information is retained inside the black hole - itwould be carried in a stable remnant at Planck scale after the black hole has radiated away. However, thelogical conclusion from that would be that the remnant has to be capable of carrying the information. Theinformation of a black hole with mass M should be of the order of 2/M2

Planck (holographic principle). As there isno constraint on M , there should be one on the number of di↵erent species of stable remnants as well. However,these remnants should all have masses comparable to the Planck mass MPlanck. This again makes it hard toreconcile the infinite degeneracy of stable remnants with causality and unitarity.

3. The Information Comes Out At The End

One idea to circumvent problems of solution 1 is to assume that the radiation remains truly thermal until theblack hole radiates down to the Planck size. Then the information should reappear - encoded in correlationsbetween the thermal quanta emitted earlier and the late quanta. This emission process should depend on themass of the black hole, as this determines the amount of information storable in it. The Bekenstein-Hawkingentropy finds the entropy and thus the number of states to be S2. The upper bound of the emission time hencecan be found to be: t4remnant. The black hole itself evaporates down to Planck size in time M3, while the time forthe remnant to disappear has to be at least M4 ( One quantum has energy M�2 and wavelength M2. The timefor the emission of one quantum is M2 and there are M2 quanta, hence it takes total time M4.) However, if thereis an infinite number of species with mass of order the Planck mass and with lifetime greater than googolplex,

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the same problems as with stable remnants appear. Other solutions suggest therefore a slight variation of thisansatz with metastable remnants.

4. The Information Is Encoded In Quantum Hair

There could be additional possibilities for hair that are missed in the analysis of black hole solutions of theclassical field equations. However, until now such possibilities of additional hair arise only in theories with aspecial matter content.

5. The Information Escapes To A Baby Universe

Hawking, Dyson and Zeldovich suggested that quantum gravity e↵ects prevent the collapsing body fromproducing a true singularity inside the black hole. Instead, the collapse produces a closed baby universe, whichcarries away the collapsing matter and with it detailed information about its quantum state. This of coursestates is own problems, as the universe should by construction be causally connected from us.

6. Black Hole Complementarity

In its simplest form, black hole complementarity just says that no observer ever witnesses a violation of a lawof nature.

Thus, for an external observer it says: a black hole is a complex system whose entropy is a measure of itscapacity to store information. For a freely falling observer, black hole complementarity tells us that the equiva-lence principle is respected. This means that as long as the black hole is much larger than the infalling system,the horizon is seen as flat featureless space-time. No high temperatures or other anomalies are encountered. Noobvious contradiction is posed for the external observer, since the infalling observer cannot send reports frombehind the horizon. But a potential contradiction can occur for the infalling observer.

III. BLACK HOLE COMPLEMENTARITY

Three fundamental laws of Physics have been challenged by the information loss paradox: 1.) Unitarity(equivalently: information conservation), 2.) the equivalence principle and 3.) the quantum xerox principle.

Now lets first recap them shortly.

A. Unitarity or Information Conservation

In quantum theory, the norm of a state |si at time t = 0 should be the same at a later time ⌧ . This meansthat:

hs, t = 0|s, t = 0i = hs, ⌧ |s, ⌧i.

From this follows that the Hamiltonian should be Hermitian, H†=H, as:

|s, ti = eiHt|s, 0i.

It also means then that the S-matrix should be unitary, since S=eiHt. Explicitly:

S†S = 1.

This formulates Information conservation, as it implies that the S-matrix has an inverse. Thus for a state| fini governed from an initial state | ini via:

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| fini = S| iniwe would expect to be able to go back to the initial state via:

| ini = S�1| fini

B. Equivalence Principle

On scales ⌧ rBH for observers entering after the scrambling time (i.e. the minimum time for ”localized”information to become inaccessible without measuring fraction O(1) of the whole system), the horizon should belocally indistinguishable from Minkowski space. More precisely, this means that the horizon is not a specialplace. In its simplified form, the equivalence principle tells us that a gravitational field is locally equivalent toan accelerated frame. More exactly, it says that a freely falling observer will not experience the e↵ects of gravityexcept for the tidal forces, or curvature components. The magnitude of curvature at the black hole horizon tendsto zero as the mass of the black hole increases. The curvature typically satisfies:

R ⇠ 1

(MG)2

with M being the mass of the black hole.Therefore a freely falling system of size smaller than (MG) will not be distorted or otherwise disrupted by the

presence of the horizon.

C. Xerox Principle

The xerox principle says that a particular kind of apparatus cannot be built, i.e. the xerox machine. It is amachine into which any system can be inserted that will produce an exact copy of it - hence it produces theoriginal and a duplicate. To see that this is not possible, imagine inserting a spin into such a machine. If thespin is in the up state with respect to the z-axis, it is duplicated:

| "i ! | "i| "i

Figure 1: xerox-machine

Now suppose that the spin is inserted with its polarization along the x-axis, i.e. 1/p2(|"i+|#i).

The general principles of quantum mechanics require this state to evolve linearly, i.e.:

1p2(| "i+ | #i) ! 1p

2(| "i| "i+ | #i| #i)

On the other hand, a true quantum Xerox machine is required to duplicate the spin:

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1p2(| "i+ | #i) ! 1p

2(| "i+ | #i) 1p

2(| "i+ | #i) = 1

2| "i| "i+ 1

2| "i| #i+ 1

2| #i| "i+ 1

2| #i| #i

This is obviously not the same equation as would have been required by the linearity of QM. Therefore thelinearity of QM forbids the existence of a quantum Xerox machine. If we could construct Xeroxed quantumstates, we would be able to violate the Heisenberg uncertainty principle by a set of measurement on those states.

D. Can an Observer outside the horizon of a black hole recover information about the interior?

Now back to the initial question: ”Can an observer outside the horizon of a black hole recover informationabout the interior?” Consider therefore a black hole that formed by the gravitational collapse of a star inasymptotically flat space (fig.2).

Figure 2: Black hole formed by gravitational collapse. The horizontally shaded region is the causal diamond of an observer(red worldline) who remains forever outside the black hole. The vertically shaded region is the causal diamond of anobserver (blue worldline) who falls into the black hole. There are infinitely many observers, whose diamonds have di↵erentendpoints on the conformal boundary (the singularity inside the black hole)

Fig. 2 shows the spacetime region that can be probed by an infalling observer, and the region accessible to anobserver who remains outside the horizon. At late times, the two observers are out of causal contact, but in theremote past their causal diamonds have considerable overlap.

Let A be an observable behind the horizon as shown in fig.3A is a low energy operator that can be described by conventional physics. The question above can be re-

formulated in whether there is an operator outside the horizon on future light-like infinity, that has the sameinformation as A. Call it Aout. Aout shall be an operator in the Hilbert space of the outgoing Hawking radiationthat can be measured by the outside observer, and has the same probability distribution as the original operatorA when measured by the in-falling observer.

First, one has to show that there is an operator in the remote past Ain, that has the same probabilitydistribution as A. In the causal diamond of the infalling observer, all of the evolution leading to A is low energy

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Figure 3: An observable A inside the black hole can be approximately defined and evolved using local field theory. Inthis way, it can be propagated back out of the black hole to an operator Ain defined on the asymptotic past boundary.From there, it is related by an exact S-matrix to an operator Aout that can be measured in the Hawking radiation (wigglydashed line) by a late time observer outside.

physics. Consider an arbitrary foliation of the infalling causal diamond into Cauchy surfaces, and let each slicebe labeled by a coordinate t. Let us choose t = 0 for the slice containing A. Let U(t) be the Schrodinger picturetime-evolution operator, and let | (t)i be the state on the Cauchy surface t. One can write | (0)i in terms ofthe state at a time �T in the remote past:

| (0)i = U(T )| (�T )iThe expectation value of A can be written in terms of this early-time state as:

h (�T )|U†(T )AU(T )| (�T )iThus the operator

Ain = U †(T )AU(T )

has the same expectation value as A. More generally, the entire probability distributions for A and Ain arethe same. In the limit T ! 1, Ain becomes an operator in the Hilbert space of the incoming scattering states.

Since the two diamonds overlap in the remote past, Ain may also be thought of as an operator in the spaceof states of the outside observer. Now let us run the operator forward in time, while working in the causaldiamond of the outside observer. The connection between incoming and outgoing scattering states is throughthe S-matrix. Thus:

Aout = SAinS†

or:

Aout = limT!1

SU†(T )AU(T )S†

The operator Aout, when measured by an observer at asymptotically late time, thus has the same statisticalproperties as A if measured behind the horizon at time zero. The S-matrix should have a completely precisedefinition but is hard to compute in practice. Information that falls onto the horizon is radiated back out in acompletely scrambled form.

This is the content of black hole complementarity: observables behind the horizon are not independent vari-ables. They are related to observables in front of the horizon by unitary transformation. The transformationmatrix is limT!1

⇥U(T )S†⇤. It is however not operationally possible to check whether a measurement of A and

Aout agree. It is enough that every observable behind the horizon has a complementary image among the dof ofthe Hawking radiation that preserves expectation values and probabilities.

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Now one could ask for the most general form of operators A inside the black hole, that can be written as anoperator Aout. Naively, one could guess that any operator with support inside the black hole can be representedin that way, since any operator can be evolved back to the asymptotic past using local field theory. But thismethod is not completely exact, so there must be situations where it breaks down completely. For example, onewould be free to consider operators with support both in the Hawking radiation and in the collapsing star andevolve them back; this would lead us to conclude that either information was xeroxed or lost to the outside.

E. Xeroxing Paradox

Now consider a black hole, shown in fig. 4 along with an infalling system A. System A is assumed to containsome information. According to observations done by A, it passes through the horizon without incident. Next,consider an observer B who hovers above the black hole monitoring the Hawking radiation. According toassumption, the photons recorded by observer B encode the information carried in by system A. After collectingsome information about A (from the Hawking radiation), observer B then jumps in into the black hole. We donot actually need observer B to decode the information. All we really need is a mirror outside the black holehorizon to reflect the Hawking radiation back into the black hole.

Figure 4: Information exchange from external to infalling observer.

This means that there is an instant of time at which the collapsing materia inside the black hole as well as theHawking radiation that carries away its quantum state, are in regions of negligible curvature, where semiclassicalgravity should be valid. This would mean that the quantum state of the star has been copied: it exists bothinside and outside the black hole. This violates the linearity of quantum mechanics.

The paradox is resolved by sacrifying the omniscient observer. The question to pose is of course if observer Bcan ever discover the duplicate information at point C. If he would do so, he would have discovered a quantumXeroxing of information. However, complementarity says that he will never do so. Either he stays outside, seeingonly the Hawking radiation or the observer falls in and sees only the star. There is no observer whose causalpast can include both copies of the quantum state.

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Figure 5: A black hole forms by the collapse of an object in the pure state | i. After the black hole has evaporated,the Hawking cloud is in the quantum state | i by unitarity. At the same time inside the black hole the object is stillcollapsing and remains in the quantum state | i by the equivalence principle. Thus the black hole acts as a quantumxeroxing machine | i to | i � | i. This violates the linearity of QM

F. Resolution Of The Xeroxing Paradox

Figure 6: Alice falls in with the star and sends a bit of information to Bob. Bob waits until he can recover the bit fromthe Hawking radiation and then jumps in to receive Alice’s message. However, he has to wait at least O(R3

Bh) until thefirst bit of information comes out with the Hawking radiation. Alice would need energy O(expR2) to send a signal thatreaches Bob before he hits the singularity. So in practice, Bob will never see both copies of the information

No Information will be emitted until about one half of the entropy of the black hole has evaporated. Thistakes a time of order:

t⇤ ⇡ M3G2

Where does this come from?

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In terms of the Schwarzschild time, the flux of quanta is of the order of 1/MG. Furthermore each quantacarries an energy at infinity of order the Schwarzschild temperature of 1/8⇡MG. The resulting rate of energyloss then is of the order of 1/M2G2. Energy conservation implies that the black hole should lose mass at justthis rate:

dM

dt= � C

M2G2

with C being a constant of order of unity. Integrating this gives the time of evaporation to be:

tevap ⇠ M3G2

The observer B hovers above the horizon at a distance at least of the order of the Planck length lP . B’sposition then corresponds to Rindler coordinates satisfying:

!⇤ � t⇤4MG ⇡ M2G

⇢⇤ � lP

In terms of the light cone coordinates xpm=⇢epm! we have:

x+⇤ x

�⇤ > l2P

x⇤ & lP exp!⇤

The singularity is given by:

x+x� = (MG)2

Thus, observer B will hit the singularity at a point with:

x� . (MG)e�!⇤

The implication is that if A is to send a signal that B can receive, it must all occur at:

x� < (MG)e�!⇤

This means that A has a time of order �⇡(MG)e�!⇤to send the message. In quantum mechanics, sending a

single bit requires at least one quantum. Since the quantum must be emitted between x�=0 and x�=MGe�!⇤,

its energy must satisfy:

E >1

MGe!

⇤

Therefore we see that this energy is exponential in the square of the black hole mass. In other words, forobserver A to be able to signal observer B before observer B hits the singularity, the energy carried by observerA must be many orders of magnitude larger than the black hole mass. It is obvious that A cannot fit into thehorizon, an the experiment cannot be done.

G. Black Hole Complementarity

Black Hole Complementarity tries to reconcile the conflict between the equivalence principle and unitarity. Itstates that no contradictions can be verified by any observer in spacetime. One should note that ”observer” inthis case is a locally bounded expression meaning that there is no omniscient superobserver.

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Figure 7: Bob already controls half of the Hawking radiation, then Alice throws in a bit. Since Bob’s radiation ismaximally entangled with the black hole, Alice’s bit can be recovered as soon as the black hole has completely thermalizedit. Speculative arguments suggest that the scrambling time can be as fast as O(RBH logRBH). In this case, Bob fails tosee both copies only barely.

Principles Of Black Hole Complementarity

• The Hawking radiation is in a pure state determined by a unitary S-matrix;

• E↵ective Field Theory is valid outside the stretched horizon;

• A black hole has a Hilbert space of dimension exp(A/4);

• An infalling observer encounters nothing special on crossing the horizon.

Complementarity implies that there must be a theory for every causal diamond, but not necessarily forspacetime regions that are too large to be contained in any causal diamond. If we attempt to describe suchregions, we may encounter contradictions, but such contradictions cannot be verified in any experiment.

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IV. UNDERLYING QM

From now on, the discussion will lay on the attempt by Almheiri, Marolf, Polchinski and Sully (AMPS) [2]to build up a sharp paradox that goes beyond and cannot be solved inside BH complementarity. In order toapproach their conclusions it is useful to remind some of the fundamental concepts of Quantum mechanics as asolid base to reach the contradiction between the postulates they set.

A. Maximal entanglement

Let us consider a bipartite pure state | iAB on a product tensor Hilbert space H = HA ⌦HB and its densitymatrix as:

⇢AB = | ih |.We can now define for the subsystem A:

• The number of degrees of freedom

NA = log|HA|;

• Its desity matrix obtained as partial trace of ⇢AB over B

⇢A = TrB⇢AB =X

b

hb|⇢AB |bi;

• The Von Neumann entropy

SA = �trA⇢Aln⇢A.

The state | iAB is defined entangled when its writing as direct product is not possible:

| ABi = | iA ⌦ | iB .

In particular for a pure state, it is maximally entangled if SA = NA (or equivalently ⇢A = e�NA · , whichimplies NA NB).

These correlations between states are the carrier of the information of a quantum system. As the Hawkingradiation is supposed to be conserving the information initially stored in the pure state that collapsed into theblack hole, it must preserve its maximal entanglement with the interior of the black hole. The most simplemaximally entangled states one can define are the four states of the so called Bell basis:

|�±iAB =1p2(|00iAB ± |11iAB)

| ±iAB =1p2(|01iAB ± |10iAB) .

Therefore these states are conveniently identified with the unity of quantum information, or qubit.

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B. Monogamy of entanglement

One can argue whether a quantum subsystem could share its maximal entanglement with more than oneother subsystem, considering for instance a Hibert space built as: H = HA ⌦HB ⌦HC with B simultaneouslyentangled with both A and C.

This is proven to be impossible, and it is justified by the strong subadditivity of quantum entropy (SSA):

SAB + SBC � SB + SABC .

It follows from subadditivity of entropy SA + SB � SAB : entropies of the two subsystem are increasinglycancelled by the degree of mutual entanglement. SSA is violated when we assume the tripartite state to bemaximally entangled twice as SAB and SBC are zero.

C. Entropy of a subsystem

According to a paper by D. Page [3], if we go back to the (random) pure state AB with NA NB , the smallersubsystem will be nearly maximally entangled with:

SA ⇡ NA � 1

2eNA�NB .

Furthermore, one can define the quantum information of a system as:

I = NA � SA = NA � SB 1

2.

This is useful to state that to recover any information all about the purity of a state ⇢AB it is necessary toknow more than half of it.

Figure 8: Following Page’s statemet: chosen a random smaller subsystem, it is not possible to extract information of theglobal state. For an evaporating black hole, the state of radiation grows in entropy until half of the entropy inside theblack hole is radiated away. From the halfway point information is extracted and radiation rebuilds the pure state.

V. THE UNRUH STATE

Let us consider a minkowskian region of spacetime ideally separated by a surface: its vacuum state can beexpressed as the product of modes living on left and right Rindler wedges. This is the Unruh state:

|0iM =Y

j

1X

nj=0

e�⇡!jnj |njiR ⌦ |njiL.

Note that the surface can be the horizon of the black hole, on condition that the observer is small enoughcompared to rs and thus the surrounding spacetime can be considered minkowskian.

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Figure 9: Penrose diagram of Minkowski space divided in left and right wedge

Unruh state is highly entangled, in particular for wavepackets both very close on the two sides of the surface.Tracing over the left Rindler wedge gives a thermal state: ⇢R = trL|0iMM h0| = e�2⇡H , with Unruh temperatureT = 1

2⇡ . This thermal state is a good description of Hawking radiation quanta as measured by an observer ata fixed distance from the horizon, thus constantly accelerating and experiencing a thermal bath in its Rindlerframe.

VI. POSTULATES

In their literature representative work, AMPS suggest two thought experiments to rise the hypothesis of afirewall at the stretched horizon. Starting from the formulation of black hole complementarity by L. Susskindand L. Thorlacius [Ref.] a further postulate is added to preserve the equivalence principle. The subsequentconflict of the initial postulates with the latter made them indicate the sacrifice of the equivalence principle forthe region of the black hole horizon as the less destructive choice.

A. Unitarity

The life of the black hole from the collapsing of some pure state into a gravitational singularity to its completeevaporation through Hawking radiation must be globally described inside unitarity. This means the existence ofa unitary S matrix that describes the evolution of the pure state. Preserving unitarity implies the purity of theinitial state is conserved into the state of the entire radiation.

B. E↵ective QFT

In the region outside the stretched horizon holds a field theory that allows the definition of lowering operatorsand the evolution backwards in time for the Hawking outgoing modes. Giving up this postulate equals to theadmission that some exotic new dynamics must appear in a region ⇠ rs outside the black hole.

C. Equivalence principle

As the event horizon should be for equivalence principle a place where nothing special happens, and theinfalling observer does not encounter a firewall (no drama), it must verify entanglement between the outgoingmodes and those trapped behind the horizon.

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VII. AMPS THOUGHT EXPERIMENTS

A. Double entanglement

Let us return again to the black hole created by collapse of a pure state, old enough to have radiated out halfof its initial entropy. If we decide to separate the full state of radiation into that was already emitted (earlyradiation) and that is still to be emitted (late radiation), we get:

| i =X

i

| iiE ⌦ | iiL.

It is fundamental to notice that this state has to be maximally entangled, as the complete radiation must bepure to conserve trace of black hole information. Thus the late radiation by the time it is emitted graduallypurifies it. If we consider an outgoing mode of the of the late radiation B, the field theory allows us to set alowering operator b related to this mode and project onto eigenspaces of b†b. This operator must be observer-independent. Since one can construct operators for the early radiation that act on | i as projection operatorson late radiation, measures on early radiation can give information about the outgoing modes. We are allowedalso to relate the outgoing mode to one of earlier times and propagate it backwards from infinity to find out itis greatly blueshifted. For the free falling observer though, it is correct to assign its own lowering operator a. Ifwe consider:

b =

Z 1

0d!(B(!)a! + C(!)a†!),

recalling this the reason for the Hawking radiation to appear, we see the full state of radiation cannot be bothvacuum for a (a| i = 0) and and an eigenstate of b†b. The application of the first two postulated led to theconclusion that the infalling observer sees high-energy modes.

Figure 10: Penrose diagram for the infalling (red) and outside (blue) observers’ causal diamonds. Their overlap is markedin purple. The infaller encounters the outgoing mode B and must see entanglement with its partner C trapped behindthe horizon. The observer at infinity collects the early radiation A and witness its entanglement with the outgoing mode.

We can restate the problem this way: if the distant observer must verify the entanglement of the outgoingmode with the early radiation and the infalling one sees entanglement between partner modes on the two sidesof the horizon, monogamy of entanglement is violated. If the infaller does not see entanglement at the horizon,this means spacetime is not continuous and it must encounter a barrier.

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The violation monogamy implies violation of SSA:

SAB + SBC � SB + SABC ,

with A the early Hawking modes, B the outgoing mode with its partner inside C. In a black hole past itshalfway point (old black hole), entropy is decreasing: SAB > SA, the absence of barrier at the horizon meansSBC = 0 and SABC = SA. Thus: SA � SB + SA. Following Page’s argument, for and old black hole entropydecrease is maximal, SAB = SA�SB , which goes to SA � 2SB +SA, and a even more stronger violation of SSA.

While for complementarity it is accepted that no observer seeing two copies of the same state is enough, in thiscase it seems one observer can witness double entanglement. The infaller could see the outgoing B mode beforecrossing the horizon and include its Hilbert space. Then, still decide to return to infinity and add the Hilbertspace for early radiation. Since unitarity requires B to be maximally entangled to purify the full radiation state,and free fall without a firewall asks for maximal entanglement with the inner mode C, we came to a contradiction.

B. Mining argument

AMPS suggest a second thought experiment that strengthen their argument. At infinity Hawking radiation isdominated by low angular momentum modes, that is consequence of the fact that high angular momentum modesare trapped behind a potential barrier. But one could imagine to decide to probe the region immediately overthe stretched horizon, trap high-l modes and extract them out. This is ideally made with a box (equivalently: adetector) hovering at a safe distance, catching these modes and then retired to infinity. As in principle there isno upper bound for those high energetic modes to appear, the conclusion seems to be an infaller will encountera divergent stress tensor at the stretched horizon and thus a firewall.

[1] R. Bartnik, J. McKinnon - Particle - Like Solutions of the Einstein Yang-Mills Equations, Phys.Rev.Lett. 61 (1988)141-144.

[2] A. Almheiri, D. Marolf, J. Polchinski, J. Sully - Black Holes: Complementarity or Firewalls?, (2013) [arXiv:1207.3123[hep-th]].

[3] D. Page - Average entropy of a Subsistem, (1993) [arXiv:gr-qc/9305007 [gr-qc]][4] S. Hawking - Information Loss in Black Holes, (2005) [arXiv:hep-th/0507171v2 [hep-th]].[5] J. Preskill - Do Black Holes Destroy Information?, (1992) [arXiv:hep-th/9209058v1 [hep-th]].[6] J. Preskill - Quantum information and black holes, Lectures for QIP 2014, Barcelona (2014).[7] R. Bousso - Black Holes: Complementarity vs. Firewalls, Lectures for Strings 2012, Munich (2012).