Black hole information via adm reduction
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Black hole information via ADM reduction. Inyong Park Philander Smith College. KIAS, Seoul Dec 2013. Contents 2/27. 1. Black hole information (BHI) Review of BHI problem , black hole complementarity (BHC), Firewall 2. Assessment

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Black hole information via ADM reduction

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Black hole information via ADM reduction

Inyong Park

Philander Smith College

KIAS, Seoul Dec 2013

Contents 2/27

  • 1. Black hole information (BHI)

    Review of BHI problem , black hole complementarity (BHC), Firewall

  • 2. Assessment

    Limitations of semiclassical description,

    quantum gravity effect:

    bleaching and/or blackening (potential presence of these mechanisms will lead to a very different solution of BHI problem)

  • 3. Reduction of BTZ to hypersurfaces of foliation

    appearance of Liouville type theories

  • 4. Conclusion


  • Hawking discovered that a BH radiates (Hawking radiation) once quantum effects are taken into account

  • Hawking radiation is thermal (this is modified in BHC)

  • Evolution of pure state of initial matter into thermal state of radiation ; non-unitary evolution, but how? (BHI problem)

  • Different causes with the same effects. Hawking originally suggested modification of quantum principles. (Later he conceded.)

  • There are (at least) three stances one can take:

  • Agree with Hawking’s original opinion: modification of quantum principles

  • Try to come up with an approach that accommodates

    BH evaporation and QM: BHC

  • Keep QM but end up abandoning something else

    (Firewall contradicts with equivalence principle)


  • Schwarzschild coordinate (stationary observer)

    vsKruskal coordinate (free-falling observer)

  • Scalar field theory in these two coordinates, creation and annihilation operators.

  • Two different Fockvacuua :Schwarzschild vacuum |S> vsKruskal vacuum |K>

  • They are inequivalent

    In particular, the Kruskal vacuum appears as a radiating BH to a Schwarzschild observer (stationary observer)

    (One can see this by looking at expectation value of the stress-energy tensor; it displays blackbody radiation at a certain temperature)

    (This is a long-known fact, known much before complementarity.)

  • BH radiates and eventually evaporates with information

    information loss ?

According to Firewall argument

  • Even for a free-falling observer, |K> cannot be a vacuum state if the Hawking radiation is taken to be pure as one assumes in BHC.


black hole complementary

  • L. Susskind, L. Thorlacius and J. Uglum; C. R. Stephens, G. 't Hooft and B. F. Whiting

  • Unitarity is one of the foundational principles of quantum physics

  • Assume: somehow the BH evaporation process is unitary

    (What is given up is locality)

    a set of 4 postulates

  • 1. Unitary evolution: in particular, there exists a unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking-like radiation. (Purity of Hawking radiation)

  • 2. Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations. (Semiclassical description of geometry)

  • 3. To a distant observer, a black hole appears to be a quantum system with discrete energy levels. The dimension of the subspace of states describing a black hole of mass M is the exponential of the Bekenstein entropy.

  • 4. A freely falling observer experiences nothing out of the ordinary when crossing the horizon.

    (No drama (no trouble), equivalence principle)


  • Firewall

  • Ahmed Almheiri, Donald Marolf, Joseph Polchinski, James Sully

  • Infalling observer experiences violent horizon due to outgoing high energy radiation,

  • Firewall: an infalling observer will burn up at the horizon

  • Violation of Equivalent Principle:

  • Infalling observer must see smooth event horizon according to Equivalence Principle

  • Many and ongoing debates after their paper



In a few words: Firewall is Violent horizon due to high energy radiation, inconsistent with equivalence principle

(high energy radiation)


Inconsistent with Equivalence Principle according to which

Smooth horizon



  • 3 out of 4 BHC postulates cannot all be true:

  • Purity of Hawking radiation, Emission of information from horizon (semiclassical description), No drama cannot all be true

  • At least for sufficiently old black holes (after the Page time: the black hole has emitted half of its initial Bekenstein-Hawking entropy), there must be a firewall (a violent horizon)

  • The relevance of Page time is technical; we will not be concerned

  • (earlier related observation by S. L. Braunstein: energetic curtain)


  • Divide the BH system into 3 parts:

    Mode : early (before Page time) Hawking radiation

    Mode : late (after Page time) Hawking radiation

    Mode : the interior “partners” of the late Hawking radiation

    Mode expansion

    Modes in Schwarzschild coordinates:

    Modes in Kruskal coordinates:

    (These are schematic notations)


Purity of Hawking radiation implies

Consider an outgoing Hawking mode in the late radiation.

Stationary observer measures on the early radiation. This projects

the state of late radiation into an eigenstate of

Next consider an infalling observer and the associated set of infalling modes . The afore-mentioned eigenstate of cannot be a vacuum of the number operator of a mode . This is because the Schwarzschild vacuum and Kruskal vacuum are inequivalent.

Therefore the infalling observer sees radiation and this radiation has high frequency due to blueshift (to see blueshift, propagates backwards in time)


  • From slightly different angle:

  • The purity of the Hawking radiation implies

  • ; the late radiation is fully entangled with the early radiation

  • No drama for the infalling observer implies late radiation is fully entangled with the BH, i.e., the modes behind the event horizon

  • Because entanglement can not be shared (otherwise information cloning), the late radiation can not be entangled with the black hole.

  • the latter property (,i.e., entanglement between late radiation and BH) is believed to be required for the BH quantum state to locally look like Minkowski vacuum near the horizon, one concludes that the infalling observer cannot pass through the horizon smoothly, i.e., without experiencing a drama.

  • The dramatic event is high energy (due to blue-shift) radiation

2 Assessment 14/27

Let us pause and assess the situation

  • Unrenormalizability of 4D gravity

    semiclassical description

  • Limitations of semiclassical description

    Semiclassical description; geometry is fixed, i.e., nonfluctuating, usually free (,i.e., quadratic) QFT description, gravity theories are interacting theories, gravitational Bremsstrahlung might be relevant in various circumstances but semiclassical description is not adequate for quantum field theory interactions

2 Assessment

  • In semiclassical picture, information enters BH intact and BH remains black throughout

  • In full quantum gravity description: Information bleaching and blackening through meta-black states

  • Presence of these phenomena would have direct implications for BHI itself but may also shed light on Firewall

  • Study of BHI in a different approach (quantum gravity of hypesurface of foliation)

2 Assessment

  • Quantization issue can be “bypassed” by reduction to lower dimensions, reduction to 3D

2 Assessment

  • Interestingly: informtion obliteration observed by Susskind, Thorlacius and Uglum (hep-th/9306069)

    “~” means related by S-matrix


Must be independent of the initial state !

“obliteration of information”

This is in the semiclassical picture; in the full quantum picture

it should be information bleaching

3 Reduction to hypersurface of foliation

  • Variation of Kaluza-Klein reduction but not the same

3 Reduction to hypersurface of foliation 19/27

  • AdS/CFT is bulk/bd correspondence with bd at r=infinity

  • Bulk/bd correspondence seems to be quite general phenomenon by now

  • Could bulk/bd surface correspondence be generalized to “bulk/hypersurface correspondence”?

  • The answer seems affirmative. The former may be derived from the latter

  • Should examine physics on the hypersurface

  • The concept of foliation is important

  • Will apply the reduction scheme to BTZ BH spacetime

  • Employ ADM decomposition: frequently used initial value

    problem and numerical relativity,

    Hamiltonian formulation, time evolution

3 Reduction of BTZ to hypersurfaces of foliation


hypersurfaces of foliation (ex: onion)

r = r0

Bulk as a collection of

hypersurfaces (foliation),

motivated by derivation of AdS/CFT

Is this approach useful?

ADM reduction applied to IIB setup

M. Sato and A. Tsuchiya (hep-th/0211074)

E. Hatefi, A. J.Nurmagambetov and IYP (1210.3825)

Consider 10D IIB supergravity and compactify IIB supergravity on S^5

  • 5D AdS gravity

  • Solve its field equation after going through ADM splitting and Hamilton-Jacobi formulation

  • The field equation takes a form of partial differential eq of the principal function

  • The solution of the PDE can be written as a 4D action of gauge field in the curved background

  • Seems to be a direct verification of “abelianAdS/CFT”

3 Reduction of BTZ to hypersurfaces

of foliation



3 Reduction of BTZ to hypersurfaces of foliation

  • Consider

  • BTZ black hole solution (non-rotating)

  • ADM formulation : 1 (r direction) + 2 (i directions)

  • The original action can be re-expressed as

  • n_r: lapse function, N_i: shift function

3 Reduction of BTZ to hypersurfaces of foliation

  • Set

    But be careful: this does not mean is a scalar.

    Eventually we go to 2D; there it should be ok; reduce according to


    Naively, the system seems to reduced to

  • Not so fast, what about the virtual boundary effects?

  • Does the system satisfy the “reduced BH solution”?

Virtual boundary



Region I


Region II

3 Reduction of BTZ to hypersurfaces of foliation

  • With the bd effects

  • Gauge-fix the metric (which induces the Virosora constraint)

  • The final form of the action is

4 Conclusion

  • Have reviewed BHI problem and BHC

  • Have introduced Firewall argument

  • It may well be the inadequacy of semiclassical description that is causing problems: geometry is fixed, i.e., non-fluctuating, usually free (,i.e., quadratic) QFT description

  • Studying information bleaching mechanism bleaching should be interesting

  • ADM reduction of BTZ spacetime => Interacting 2D QFT, Liouville type theory

  • One can use Liouville theory to study various aspects of BTZ BH

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