a perspective on black hole horizons from the quantum...

A perspective on Black Hole Horizonsfrom the

Quantum Charged Particle

Jose Luis Jaramillo

Laboratoire de Physique des Oceans (LPO)

Universite de Bretagne Occidentale, [email protected]

XXIII International Fall Workshop on Geometry and PhysicsAftermath Week, IEMath Granada

Granada, 8 September 2014

Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 1 / 39

mailto:[email protected]

Scheme

1 Stability of Marginally Outer Trapped Surfaces (MOTS)

2 Motivations and Problem formulationBlack Hole Horizon Geometric Inequalities.A Young-Laplace Law for Black Holes.

3 MOTSs and the Quantum Charged ParticleMOTSs and Spinors.Spectrum analyticity in the “fine structure constant”.

4 Conclusions and Perspectives


Stability of Marginally Outer Trapped Surfaces (MOTS)

Outline







Establishment’s picture of the gravitational collapse

Heuristic chain of Theorems and Conjectures:

1 Singularity Theorems (Theorem) [Penrose 65, Hawking

67, Hawking & Penrose 70, Hawking & Ellis 73]:Sufficient “energy” in a compact region, then lightrays converge: notion of Trapped Surface.

Trapped surfaces ⇒ spacetime singularity(spacetime geodesically incomplete)

2 (Weak) Cosmic Censorship (Conjecture) [Penrose

69]:The singularity should not be visible from a distantobserver. Preservation of predictability.Black Hole region as a region of no-escape.Event Horizon as the Black Hole region boundary.

3 Black hole spacetime ’stability’ (Conjecture):General Relativity gravitational dynamics drivespacetime to stationarity.

4 BH uniqueness (“Theorem”) [Chrusciel et al. 12]:The final state of the evolution is a Kerr black hole.



A physical motivation: (future) outer trapping horizons

Let S be an orientable closed spacelike(codimension 2) surface with induced metric qab:

Normal plane spanned null vectors à and ka

Normalization: àka = −1Boost-rescaling freedom:

`′a = fà, k′a = f−1ka , with f > 0

Define the expansions:θ(`) ≡ qab∇a`b = 1√

qL`√q

θ(k) ≡ qab∇akb = 1√qLk√q

Marginally Outer Trapped Surface (MOTS):θ(`) = 0



A physical motivation: (future) outer trapping horizons

Trapping Horizon [Hayward 94]:

A Trapping Horizon is (the closure of)a hypersurface H foliated by closedmarginal surfaces:H =

⋃t∈R St, with θ(`) =

∣∣St

0.

Sign of θ(k): controls if singularity occurs in the future or in the past.

Sign of δkθ(`): controls the (local) outer- or inner character of H.

Trapping Horizons are called:

i) Future: if θ(k) < 0. Past: if θ(k) > 0.

ii) Outer: if δkθ(`) < 0 . Inner: if δkθ

(`) > 0.



MOTS Stability: Stability operator

Stably outermost MOTS in a normal direction va [Andersson, Mars & Simon 05, 08]

Definition. Given a closed orientable marginally outer trapped surface S and avector va orthogonal to it, we will refer to S as stably outermost with respect tothe direction va iff there exists a function ψ > 0 on S such that the variation ofθ(`) with respect to ψva fulfills the condition

δψvθ(`) ≥ 0

MOTS Stability operator

The MOTS stability operator along a normal direction va to S is given by:

Lvψ ≡ δψvθ(`)

In particular, for va = −ka, we write LS ≡ L−k:

LSψ =

[∆ + 2Ω(`)

a Da −(

Ω(`)a Ω(`)a −DaΩ(`)

a −1

2RS +Gabk

a`b)]

ψ



MOTS Stability: Stability operator

Stably outermost MOTS in a normal direction va [Andersson, Mars & Simon 05, 08]

Definition. Given a closed orientable marginally outer trapped surface S and avector va orthogonal to it, we will refer to S as stably outermost with respect tothe direction va iff there exists a function ψ > 0 on S such that the variation ofθ(`) with respect to ψva fulfills the condition

δψvθ(`) ≥ 0

MOTS Stability operator

The MOTS stability operator along a normal direction va to S is given by:

Lvψ ≡ δψvθ(`)

Imposing Gab + Λgab = 8πTab, let us define also the operator L∗S associated tomatter:

L∗Sψ =

[−∆ + 2Ω(`)

a Da −(

Ω(`)a Ω(`)a −DaΩ(`)

a −1

2RS + 8π Tabk

a`b)]

ψ



MOTS Stability: Spectral characterization

Principal eigenvalue of Lv

Let us consider the eigenvalue problem:Lvφ = δφvθ

(`) = λ φDefinition. The eigenvalue λo with the smallest real part, is called the principaleigenvalue.

Spectral characterization of MOTS stability [Andersson, Mars & Simon 05, 08]

Lemma. The principal eigenvalue λo of Lv is real. Moreover, the correspondingprincipal eigenfunction φo is either everywhere positive or everywhere negative.

Lemma. Let S be a MOTS and let λo be the principal eigenvalue of thecorresponding operator LS = L−k. Then S is stably outermost iff:

λo ≥ 0

Remark: notion of generic Non-Expanding-Horizon [Ashtekar, Beetle & Lewandowski 02]

A non-extremal NEH (see later) is generic if LS = L−k has trivial kernel.



Rayleigh-Ritz-like characterization of λo

Theorem [Andersson, Mars & Simon 08]

The principal eigenvalue λo can be written as

λo = infψ>0

∫S

[|Dψ|2 +

(1

22R−Gabka`b

)ψ2 − |Dωψ + z|2ψ2

]dA

where Ω(`)a = za +Daf (with Daz

a = 0, for any closed Riemannian S),∫S ψ

2dA = 1 and ωψ satisfies, for a given ψ > 0

−∆ωψ −2

ψDaψD

aωψ =2

ψzaDaψ


Motivations and Problem formulation

Outline






Motivations and Problem formulation Black Hole Horizon Geometric Inequalities.

Outline







MOTS Stability and Horizon Area Inequalities

Stationary Black Holes cannot rotate arbitrarily fast

Their angular momentum J is bounded by their mass M :

J ≤M2

Is there a (quasi-local) dynamical version of this bound?

Requirements on the closed surfaces S (“sections” of the Black Hole horizon)

Need of:

i) Geometric characterization of S in a Black Hole spacetime.

ii) Stability condition.



MOTS Stability and Horizon Area Inequalities

Integral characterization of MOTS stability

Lemma. Given an axisymmetric closed marginally trapped surface S (with axialKilling ηa on S) satisfying the stably outermost condition for an axisymmetricXa = γ`a − ψka, then for all axisymmetric α it holds∫

S

[DaαD

aα+1

2α2 2R

]dS ≥∫

S

[α2Ω(η)

a Ω(η)a + αβσ(`)ab σ

(`)ab +Gabα`a(αkb + β`b)

]dS ,

where β = αγ/ψ.

Remarks:

The inequality can be obtained from the Rayleigh-Ritz-like characterizationof the principal eigenvalue λo.

Spacetime expression in which the positivity of the rhs is guaranteed byenergy conditions: form of a “energy-flux inequality”.



Horizon Geometric Inequalities

Area-angular momentum inequality for outermost stably MOTS

Theorem [JLJ, Reiris & Dain 11]. Given an axisymmetric closed marginally trappedsurface S satisfying the (axisymmetry-compatible) spacetime stably outermostcondition, in a spacetime with non-negative cosmological constant and fulfillingthe dominant energy condition, it holds the inequality

A ≥ 8π|J |

where A and J = 18π

∫S Ω

(`)a ηadS are the area and (Komar) angular momentum

of S. If equality holds, then i) the geometry of S is that extreme Kerr throatsphere, and ii) if Xa is spacelike then S is a section of a non-expanding horizon.

Last step in a series of works along two lines of research: [Ansorg & Pfister 08, Ansorg,

Cederbaum & Hennig 08, 10, 11] & [Dain 10, Acena, Dain & Gabach-Clement 11; Dain & Reiris 11]

Clarification of the relation (variational problem): [Chrusciel et al. 11; Mars 12; Gabach-Clement

& JLJ 12].

Non-existence of equilibrium aligned rotating BHs: [Neugebauer & Hennig 09, 11, 12].



Horizon Geometric Inequalities

Area-angular momentum-Charge inequality for outermost stably MOTS

Theorem [Gabach-Clement, JLJ & Reiris 12]. Given an axisymmetric closed marginallytrapped surface S satisfying the (axisymmetry-compatible) spacetime stablyoutermost condition, in a spacetime with non-negative cosmological constant andfulfilling the dominant energy condition, it holds the inequality

(A/(4π))2 ≥ (2J)2 + (Q2

E +Q2M)2

where A is the area of S and:J = J

K+ J

EM= 1

8π

∫S Ω

(`)a ηadS + 1

4π

∫S

(Aaηa)Fab`

akbdS

QE = 14π

∫S Fab`

akbdS , QM = 14π

∫S∗Fab`

akbdS.If equality holds, then i) the geometry of S is that extreme Kerr-Newman throatsphere, and ii) if Xa is spacelike then S is a section of a non-expanding horizon.

Last step in a series of works along two lines of research: [Ansorg & Pfister 08, Ansorg,

Cederbaum & Hennig 08, 10, 11] & [Dain 10, Acena, Dain & Gabach-Clement 11; Dain & Reiris 11]

Clarification of the relation (variational problem): [Chrusciel et al. 11; Mars 12; Gabach-Clement

& JLJ 12].

Non-existence of equilibrium aligned rotating BHs: [Neugebauer & Hennig 09, 11, 12].



Some remarks on the Area Geometric Inequalities

Cosmological constant Λ shift of the princioal eigenvalue λo

Need of solving Variational Problem when Symmetry is present.

Area-Charge inequality [Dain, JLJ & Reiris 11]: A ≥ 4π(Q2

E +Q2M

).

No need of symmetry requirements. No need of variational principle.

Extension to include the Cosmological constant Λ [Simon 11]:

λ∗oA2 − 4π(1− g)A+ (4π)2

∑i

Q2i ≤ 0

with λ∗o = λo + Λ , where LSφo = λoφo , L∗Sφo = λ∗oφo .

Principal eigenvalue λo acts as a Cosmological Constant Λ


Motivations and Problem formulation A Young-Laplace Law for Black Holes.

Outline







Young-Laplace law for equilibrium soap bubbles

Young-Laplace Law

In equilibrium, each point at the interface between two fluids satisfies

∆p = pinn − pout = γ

(1

R1+

1

R2

),

where:

pinn and pout are the pressures of the “inner” and “outer” fluids.

γ is the surface tension at the interface: δE = γ δA.

R1 and R2 are the principal radii of curvature: H ≡ 1R1

+ 1R2

is the meancurvature.



Principal eigenvalue for stationary axisymmetric horizons

Principal eigenvalue λo for equilibrium axisymmetric BH horizons

Theorem [Reiris 13]. Given an axisymmetric Isolated Horizon H in arbitrarydimensions (*) with null generator `a and non-affinity coefficient κ(`):

There exists an (axisymmetric) foliation of H =⋃t SYL

t with constantingoing expansion θ(k).

The principal eigenvalue λo evaluated in these sections is

λo = −κ(`)θ(k)

The principal eigenfunction φo is given by

φo = e2χ

with Ω(`)a |SYL = Daχ+ za, where Daza = 0.

Remark: In an IH, the principal eigenvalue λo does not depend on the section[Mars 12].





Theorem [Reiris 13]. Given an axisymmetric Isolated Horizon H with null generator`a and non-affinity coefficient κ(`):




λo = κ(`)(−θ(k))


φo = e2χ







Theorem [Reiris 13]. Given an axisymmetric Isolated Horizon H with null generator`a and non-affinity coefficient κ(`):




λo/(8π) = κ(`)/(8π) (−θ(k))


φo = e2χ





Comparison to the Young-Laplace law I

First factor in the right-hand-side: κ(`) as a surface tension

i) Thermodynamical perspective (energy density):

Horizon fluid analogy [Smarr 73] based on BH 1st Law (δM = κ(`)

8π δA+ ΩδJ),

and Smarr formula for the BH mass (M = 2κ(`)

8π A+ 2ΩJ), leads to:

γBH = κ(`)/(8π)

ii) Mechanical perspective (2D-”pressure”):

Horizon evolution equations for θ(`) and Ω(`)a ...

δ`θ(`) − κ(`)θ(`) = −1

2θ(`)2

− σ(`)ab σ

(`)ab − 8πTab`a`b ,

δ`Ω(`)a + θ(`) Ω(`)

a = 2Da

(κ(`) +

θ(`)

2

)− 2Dcσ

(`)c

a + 8πTcd `cqda



Comparison to the Young-Laplace law I

First factor in the right-hand-side: κ(`) as a surface tension

i) Thermodynamical perspective (energy density):

Horizon fluid analogy [Smarr 73] based on BH 1st Law (δM = κ(`)

8π δA+ ΩδJ),

and Smarr formula for the BH mass (M = 2κ(`)

8π A+ 2ΩJ), leads to:

γBH = κ(`)/(8π)

ii) Mechanical perspective (2D-”pressure”):... as energy/momentum eqs in the “membrane paradigm” [Damour 78, 79; Znajek

77, 28; Price & Thorne 86...], under ε ≡ −θ(`)/8π and πa ≡ −Ω(`)a /(8π)

δ`ε+ θ(`)ε = −(κ(`)

8π

)θ(`) − 1

16π(θ(`))2 + σ

(`)cd

(σ(`)cd

8π

)+ Tab`

a`b

δ`πa + θ(`)πa = −2Da

(κ(`)

8π

)+ 2Dc

(σ

(`)ca

8π

)− 2Da

(θ(`)

16π

)− qcaTcd`d



Comparison to the Young-Laplace law II

Second factor in the right-hand-side: : −θ(k) is a mean curvature H

Considering S embedded in an appropriately boosted 3-slice Σ so that`a = na + sa and ka = (na − sa)/2. Then

θ(`) = qab (∇anb +∇asb) = P +Hθ(k) = 1

2qab (∇anb −∇asb) = 1

2 (P −H)

=⇒ H = −θ(k) +

1

2θ(`)

For MOTS θ(`) = 0, so that: −θ(k) = H






θ(`) = qab (∇anb +∇asb) = P +Hθ(k) = 1


2 (P −H)

=⇒ H = −θ(k) +

1

2θ(`)


Interpretation proposal: λ0/(8π) as a pressure difference

Matching of λo/(8π) = κ(`)/(8π) (−θ(k)) with the form of a Young-Laplacelaw achieved, if λo/(8π) is formally identified with a pressure difference:

λ0/(8π) ≡ ∆p = pinn − pout






θ(`) = qab (∇anb +∇asb) = P +Hθ(k) = 1


2 (P −H)

=⇒ H = −θ(k) +

1

2θ(`)


Interpretation proposal: λ0/(8π) as a pressure difference

Matching of λo/(8π) = κ(`)/(8π) (−θ(k)) with the form of a Young-Laplacelaw achieved, if λo/(8π) is formally identified with a pressure difference:

λ0/(8π) ≡ ∆p = pinn − pout

Does this make sense?



Principal eigenvalue λo and pressure

The principal eigenvalue λo as a pressure

i) λo has the same nature as the Cosmological Constant: Λ “shifts” theeigenvalues

LSφ = λψ , L∗Sφ = λ∗φ =⇒ λ∗ = λ+ Λ

ii) The Cosmological Constant Λ IS a pressure: Pcosm = −Λ/(8π)



Principal eigenvalue λo and pressure

The principal eigenvalue λo as a pressure

i) λo has the same nature as the Cosmological Constant: Λ “shifts” theeigenvalues

LSφ = λψ , L∗Sφ = λ∗φ =⇒ λ∗ = λ+ Λ

ii) The Cosmological Constant Λ IS a pressure: Pcosm = −Λ/(8π)

Stability operator as a “Pressure Operator”

Consider the evolution vector ha on the horizon H =⋃t∈R St, written as

ha = `a − Cka. Then the trapping horizon condition, δhθ(`) = 0 writes

δ`−Ckθ(`) = 0 ⇐⇒ δ−Ckθ

(`) = −δ`θ(`)

LSC = σ(`)abσ

(`)ab + 8πTab`a`b

The rhs fixes the physical dimensions the stability operator:[LS/(8π)] = Energy · Time−1 ·Area−1 = Pressure



MOTS-stability from a BH Young-Laplace law perspective

BH Young-Laplace “law” [JLJ 13]

For stationary axisymmetric IHs, there exists a foliation in which theidentifications

κ(`)/(8π) → γBH

−θ(k) → H = (1/R1 + 1/R2)

λo/(8π) → ∆p = pinn − pout

permit to recast the principal eigenvalue in the form of a Young-Laplace law:

λo/(8π) = κ(`)/(8π) (−θ(k)) ⇐⇒ ∆p = pinn − pout = γBHH

In this view, MOTS-stability (λo ≥ 0) is interpreted as the result of anincrease in the pressure of the trapped region.



Problem Proposal: Full spectral analysis of LS

The whole Stability Operator LS argued to represent a “PressureOperator”, [LS/(8π)] = Presure.

Beyond the principal eigenvalue λo, interest in the full spectrum of LS .

Complex λn’s, might play a role in the analysis of instabilities (characteristicfrequencies and timescales): LSφn = λnφn = [Re(λn) + iIm(λn)]φn

In particular, LS not self-adjoint for not vanishing 2Ω(`)a Da term, i.e. with

rotation (rotational instabilities, superradiance (?)...):

J =1

8π

∫S

Ω(`)a ηadS

Proposal: “Can one hear the stability of a Black Hole horizon?” [JLJ 13]

Systematic full spectrum analysis of LS as a probe into BH horizon(in)stability (kind of “inverse spectral problem” [cf. problem by Kac 66]).

Semiclassical tools for qualitative aspects of the LS spectrum? [e.g. Berry,

Nonnenmacher...].


MOTSs and the Quantum Charged Particle

Outline







From MOTS-stability to the quantum charged particle

Based on [JLJ 14, in preparation].

MOTS-stability operator

The operator LS is not self-adjoint:

LS = −∆ + 2Ω(`)aDa −(

Ω(`)a Ω(`)a −DaΩ(`)

a −1

2RS +Gabk

a`b)

Structural similarity with the quantum charged particle

Ω(`)a →

iq

~cAa , RS →

4mq

~2φ , Gabk

a`b → −2m

~2V

the MOTS-stability operator becomes ~2

2mLS → H where

H = − ~2

2m∆ +

i~qmc

AaDa +i~q2mc

DaAa +

q2

2mc2AaA

a + qφ+ V



From MOTS-stability to the quantum charged particle

Based on [JLJ 14, in preparation].

MOTS-stability operator

The operator LS is not self-adjoint:[−(D − Ω(`)

)2

+1

2RS −Gabka`b

]ψ = λψ

Structural similarity with the quantum charged particle

H =1

2m

(−i~D − q

cA)2

+ qφ+ V

Hamiltonian operator of a non-relativistic (spin-0) quantum particle of massm and charge q moving on S under a magnetic and electric fields with vectorand scalar potentials given by Aa and φ, and an additional external potential V .



Gauge freedom in the MOTS-stability problem I

Boost/null rescaling freedom

Under the normalization `aka = −1, we have the freedom`′a = f`a , k′a = f−1ka

with f > 0 to preserve time orientation.

MOTS condition preserved

The expansion rescales

θ(`′) = fθ(`)

so that θ(`) = 0 is invariant.

Haicek or rotation 1-form Ω(`)a transformation

The form Ω(`)a = −kcqda∇d`c, transforms as a connection

Ω(`′)a = Ω(`)

a +Da(lnf)



Gauge freedom in the MOTS-stability problem II

Interpretation of the Haicek form

Geometrically is indeed a connection in the normal plane:

D⊥a (α`b + βkb) = (Daα+ Ω(`)a α)`b + (Daβ − Ω(`)

a β)kb

Physically is a kind of angular momentum density: J [φ] = 18π

∫S Ω

(`)a φadA

Spectral problem invariance: analogy to Schrodinger equation U(1)-invariance

Under the gauge transformations `′a = f`a, k′a = f−1ka,Ω(`′)a = Ω

(`)a +Da(lnf)

the MOTS-stability operator transforms as:

(LS)′ψ = fLS(f−1ψ)

Consider the eigenfunction transformation: ψ′ = fψ .

Then the spectral problem (stationary Schrodinger equation) is invariant:

LSψ = λψ → (LS)′ψ′ = λψ′



MOTS-stability and quantum charged particle similarities

Spectral problem: LS ↔ H

LSψ = λψ (MOTS) , Hψ = Eψ (stationary quantum charged particle)

Abelian gauge symmetry

Aa → Aa −Daσ , ψ → eiqσ/(c~)ψ , (quantum charged particle)

Ω(`)a → Ω

(`)a −Daσ , ψ → fψ = e−σψ , (MOTS-spectral problem)

Phase U(1) (charged particle) and rescaling R+ (MOTS) gauge symmetries.

Operators obtained by “minimal coupling” of the gauge potentials

i~∂t → i~∂t − qφ , −i~Da → −i~Da − qcAa

Da → Da − Ω(`)a

MOTS stability and quantum stability

λo ≥ 0 and E bounded below



MOTS and negative fine structure constant α

Stable MOTS as quantum particles with negative α = e2/(~c). Terminology

Setting ~ = m = c = 1 (φ ≡ −φ/e) and the fine-structure constant: α = e2

LS2

= −1

2∆ + Ω(`)aDa +

1

2DaΩ(`)

a −1

2Ω(`)a Ω(`)a +

1

4RS −

1

2Gabk

a`b

H = −1

2∆ + i

√αAaDa +

i√α

2DaA

a +α

2AaA

a − αφ+ V

∆: kinematical term,

Ω(`)a : magnetic potential vector.

RS/4: electric potential (actually RS/4 ∼ Re(Ψ2) + ...).

Ω(`)a Ω(`)a: diamagnetic term.

2Ω(`)aDa: paramagnetic term.

Gabka`b/2: external mechanical potential.

DaΩ(`)a : gauge-fixing term.


MOTSs and the Quantum Charged Particle MOTSs and Spinors.

Outline







First-order operator and SpinorsInterest in a first-order version of stability

MOTS-stability as a 1st-order condition to be used as an inner boundarycondition (Witten’s proof of positivity of mass M ≥ 0, Penrose conjecture...).

Reduction of the spectral problem to that of a 1st-order operator.

...

Spinor characterization of MOTS-stability.

Ideally: Klein-Gordon equation as square of the Dirac equation

[(i~)2 +m2c2

]Ψ = 0 (−p2

0 + ~p 2 = −m2c2)(i~γiDi +mc

)Ψ = 0

with Dirac-gamma matrices γµ

γµ,γν = 2ηµν1 , ψ =

ψ1

ψ2

ψ3

ψ4



Pauli’s approach to minimal coupling

Pauli’s way to spinors

Note: ∆ = DaDa = (σiDi)2, with

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)

Minimal coupling: two possibilites

Starting from the non-charged particle: (−i~σiDi)2 →

(σi(−i~Di − q

aAi))2

leads to Pauli’s equation

i~∂tΨ =

[1

2m(−i~Da −

q

cAa)2− ~q

2mcσiBi + qφ+ V

]Ψ

for the non-relativistic spin- 12 quantum charged particle with giromagnetic factor

g = 2 (elementary particle) .



MOTS-stability and Pauli operator

Lichnerowicz-Weitzenbock... formula

(i /DA)2 = (iγa(Da −Aa))2

= −(Da −A)2 +1

4RS +

1

4[γa,γb]FAab

with FAab = DaAb −DaAb +AaAb −AbAa.

MOTS case: Aa → Ω`a

(i /DΩ)2 = (iγa(Da − Ωa))2

= −(D − Ω)2 +1

4RS +

1

4[γa,γb]FΩ

ab

MOTS-stability operator:

LS = (i /DΩ)2 +1

4RS −

1

4[γa,γb]FΩ

ab +Gabka`b

with 14RS electric potential, − 1

4 [γa,γb]FΩab standard correction to the

giromagnetic factor (non-elementary particle), Gabka`b external potential.



MOTS-stability and Pauli operator

Dimention d = 2 case

LS = (i /DΩ)2 +1

4RS −

1

4[σa,σb]FΩ

ab +Gabka`b

Then 14 [σa,σb]FΩ

ab = 2iσ3 14εabFΩ

ab.

Penrose-Rindler complex scalar K

K =1

4RS + i

1

4εabFΩ

ab ∼ Re(Ψ2) + iIm(Ψ2) + ...

where Ψ2 is one of the (complex) Weyl scalars: components of the Weyl tensor,namely the traceless part of the Riemann curvature tensor. In fact, Re(Ψ2) isreferred as the “Coulombian part” and Im(Ψ2) as the “rotation part”.

Note: “Isolated Horizon” Multipoles as spherical harmonics of Re(K)→Mn andIm(K)→ Jn.



MOTS-stability operator as a “non-relativistic limit”

From Dirac to Pauli: “Non-relativistic limit” approach

Pauli equation can be recovered from Dirac equation in the limit c→∞.

Following the “non-relativistic” strategy (d = 2)

Let ε be a dimensionless number and L any length dimension. Define

/Dε,LΩ = −ε−1σa(Da − Ωa) + σ3

(ε−2

L

)+ L

(Re[K]− 2 Im[K]−Gabka`b

)

i) Obtain the eigenvalues: /Dε,LΩ = λε,LΨ

ii) Choose the set of eigenvalues whose eigenfunction does not vanish in thelimit ε→ 0, then the eigenvalues λ to LSψ = λψ can be obtained as

λ = limε→0

λε,L

And they are independent of L. The ε→ 0 plays the role of the c→∞ limit.


MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.

Outline







A suggestive simple example: ”Landau levels”

“Landau levels” for MOTS

Consider S2 with Ω(`)a = εa

bDbω +DaσChoose the simplest case ω = a cos θ:

qab = dθ2 + sin2 θdϕ2 , RS = 2r2, Ω

(`)a = (0, a sin2 θ) , Ω

(`)a Ω(`)a = a2

r2sin2 θ

Then LSψ = λψ exactly solved by:

λ = (λ`m + 1− a2) + i2am , ψ = Slm(a, cos θ)eimϕ

where Slm(a, cos θ) are the “prolate” spheroidal functions, going to the standardspherical harmonics for a→ 0: λ`m → `(`+ 1) and Slm(a, cos θ)→ P`m(cos θ).




“Landau levels” for MOTS

Consider S2 with Ω(`)a = εa

bDbω +DaσChoose the simplest case ω = a cos θ:

qab = dθ2 + sin2 θdϕ2 , RS = 2r2, Ω

(`)a = (0, a sin2 θ) , Ω

(`)a Ω(`)a = a2

r2sin2 θ

Then LSψ = λψ exactly solved by:

λ = (λ`m + 1− a2) + i2am , ψ = Slm(a, cos θ)eimϕ

where Slm(a, cos θ) are the “prolate” spheroidal functions, going to the standardspherical harmonics for a→ 0: λ`m → `(`+ 1) and Slm(a, cos θ)→ P`m(cos θ).

Notice:

This spectral problem can be recast as LaSψ = λψ, with

LaS =

[−∆ + 2aΩ(`)aDa + aDaΩ(`)

a − a2Ω(`)a Ω(`)a +

1

2RS −Gabka`b

]and Ω

(`)a = (0, sin2 θ).




MOTS: λ = (λ`m + 1− a2) + i2am , ψ = Slm(a, cos θ)eimϕ (prolate)





Relevant remark: complex rotation a→ ia

The operator LiaS is now self-adjoint and the problem:

LiaS ψ =

[−∆ + 2iaΩ(`)aDa + iaDaΩ(`)

a + a2Ω(`)a Ω(`)a +

1

2RS −Gabka`b

]ψ = λψ

namely a stationary quantum-charged particle (QCP), has as solutions:

QCP: λ = (λ`m + 1 + a2)− 2am , ψ = Slm(ia, cos θ)eimϕ (oblate)





Relevant remark: complex rotation a→ ia

The operator LiaS is now self-adjoint and the problem:

LiaS ψ =

[−∆ + 2iaΩ(`)aDa + iaDaΩ(`)

a + a2Ω(`)a Ω(`)a +

1

2RS −Gabka`b

]ψ = λψ

namely a stationary quantum-charged particle (QCP), has as solutions:

QCP: λ = (λ`m + 1 + a2)− 2am , ψ = Slm(ia, cos θ)eimϕ (oblate)

Moral: self-adjoint “trick”

(Landau) MOTS-spectrum problem solved by considering the self-adjoint problemLiaS ψ = λψ and making a→ −ia in eigenvalues and eigenfunctions.



MOTS-stability operator and the fine-structure constant α

Analyticity Conjecture

Given an orientable closed surface S and the one-parameter family of operators

LS [√α] = −

(D − i

√αA)2 − αφ+ V

= −∆ + 2i√αAaDa + i

√αDaAa + αAaA

a − αφ+ V

in the (squared-root) of the fine-structure constant α ≡ e2

~c :the MOTS-spectrum (α = −1) can be recovered as an “analytic continuation” ofthe spectrum in the quantum charged particle spectrum (α = 1) self-adjointproblem.

Hopes in a difficult problem in (perturbation) theory of linear operators [Kato 80]

No boundary conditions (assume topological conditions, if needed).

Functions qab, Aa, φ and V as well-behaved as necessary.

LS [√α] is a self-adjoint holomorphic family of type (A) [Kato 80]:

consequences on analytic continuation of eigenvalues and eingenfunctions...?



From Black Holes to charged particle Quantum Mechanics

If the Analyticity Conjecture proves valid...

The MOTS-stability spectrum problem is “essentially” reduced to that of theself-adjoint problem of the stationary non-relativistic quantum charged particle.

“Inverse” application: ground state energy of the quantum charge particle

A gauge-invariant characterization of the ground state Eo is given by

Eo = infψ>0

∫S

(|Dψ|2 +

(eφ+ V+|Dωψ + z|2

)ψ2)dA

where Aa = za +Daf (with Daza = 0, for any closed Riemannian S),∫

S ψ2dA = 1 and ωψ satisfies, for a given ψ > 0

−∆ωψ −2

ψDaψD

aωψ =2

ψzaDaψ

Note: Gauge invariant and the paramagnetic term is recast as a diamagnetic one.



An avenue to semi-classical tools...

Classical Hamiltonian to the LS [√α] problem

According with the “quantization rule”, pi → −iDi, (valid in the selfadjointα > 0 case) consider the classical Hamiltonian:

Hcl[√α] = (p−

√αΩ(`))2 + 1

2RS −Gabka`b

Semiclassical approach to the LS spectral problem

Considering semiclassical tools analysis (e.g. WKB... [e.g. Berry...]) based onthe classical trajectories of the Hamiltonian Hcl[

√α] in the phase space.

On the resulting estimations for eigenvalues and eigenfunctions, make√α→ −i.

Remark: analogue to the study of the spectrum of the Laplacian operator,∆S , from the properties of geodesics on S.

Tools employed in Quantum Chaos? [e.g. Berry 80’s, Nonnenmacher 10, many others...].

Spectral zeta function ζLS (s) =∑λ

1λs [Berry..., Aldana]. Semiclassical

approximations... (Anecdote with [Berry 86]...)


Conclusions and Perspectives

Outline








Conclusions

Stable MOTS as Quantum Particles with “negative fine-structureconstant”: formal analogy between the MOTS-stability operator of BlackHole apparent horizons and the Hamiltonian of a non-relativistic quantumcharged particle.

Self-adjoint “shortcut” to the spectral MOTS-problem: solution of thequantum charged particle problem and analytic extension to negative valuesof the fine-structure constant. Transfer of tools from quantum theory.

Semiclassical tools and MOTS-stability: different potential applications,in particular an avenue to the (very important and very complicate) Kerr case.

MOTS and Spinors: an avenue to the reformulation of MOTS-stability interms of spinors. Towards a 1st-order formulation. “Non-relativistic” limit.

Others: Gauge-invariant expression for particle ground state, MOTS“Aharonov-Bohm effect”, signature of quasi-normal modes/superradiance,“BH horizon degrees of freedom” from 2-nd quantization of QCP...?




Perspectives I: An object in the “vertex” of different lines

Seed for a Research Program:

Analyticity of the spectrum operator: principal eigenvalue λo of Kerr fromQuantum Particle ground state Eo.

Spectrum statistics and Spectral zeta function: random matrices(Extremal Kerr... and Riemann conjecture for the zeros of the Riemann zetafunction?).

Semiclassical tools, Dynamical Systems: “high-eigenvalue” asymptoticsand link to quantum billiards.

Spinors and Geometric Inequalities: inner boundary conditions for Penroseconjecture, A ≤ 16πM2. Link to Sen-Witten connection. Quasi-localgravitational mass. Superradiance...

ANR project NOSEVOL: ”Nonselfadjoint operators, semiclassical analysisand evolution equations”.GDR DYNQUA: “Quantum Dynamics”.




Perspectives II: An object in the “vertex” of different lines

Higher (BH) dimensions. Richer topologies and fields:Hodge-decomposition Ω(`) = dα+ δβ + γ, with γ harmonic.

Variational formulations: Action from Wess-Zumino term in aChern-Simons action, Ginzburg-Landau functionals (link to Seiberg-Wittentheory in the dim(S) = 4 self-dual case)...

(Quantum and) Semiclassical Gravity: model for inner “fluid” pressure,insight into BH entropy from statistics of the spectrum, Young-Laplace as a“classical-limit test” for quantum inner pressures...

Oceanography: generalized “potential vorticity” q in quasi-geostrophicmotions. Physical mechanism for effect of “fast motions” on “slow motions”:

q = ∆ψ −(

1rRossby

)2

ψ + η , ∂tq + v ·Dq = 0

Statistical physics of 2-dimensional flows: Turbulence and large coherentstructures (vortices...).







(Quantum and) Semiclassical Gravity: model for inner “fluid” pressure,insight into BH entropy from statistics of the spectrum, Young-Laplace asclassical limit test for quantum inner pressures..


−∆ψ + RS2

2ψ = η − q ⇔ LSψ = |σ(`)|2 + Tab`

a`b

Statistical physics of 2-dimensional flows: Turbulence and large coherentstructures.

...







(Quantum and) Semiclassical Gravity: model for inner “fluid” pressure,insight into BH entropy from statistics of the spectrum, Young-Laplace asclassical limit test for quantum inner pressures..


−∆Ψ + RS2

2ψ = η − q ⇔ LSψ = |σ(`)|2 + Tab`

a`b

Statistical physics of 2-dimensional flows: Turbulence and large coherentstructures.


a perspective on black hole horizons from the quantum...

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