a perspective on black hole horizons from the quantum...
TRANSCRIPT
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
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 2 / 39
Stability of Marginally Outer Trapped Surfaces (MOTS)
Outline
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 3 / 39
Stability of Marginally Outer Trapped Surfaces (MOTS)
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 4 / 39
Stability of Marginally Outer Trapped Surfaces (MOTS)
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 4 / 39
Stability of Marginally Outer Trapped Surfaces (MOTS)
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 `a and ka
Normalization: `aka = −1Boost-rescaling freedom:
`′a = f`a, 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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 5 / 39
Stability of Marginally Outer Trapped Surfaces (MOTS)
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 5 / 39
Stability of Marginally Outer Trapped Surfaces (MOTS)
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 6 / 39
Stability of Marginally Outer Trapped Surfaces (MOTS)
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 6 / 39
Stability of Marginally Outer Trapped Surfaces (MOTS)
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 7 / 39
Stability of Marginally Outer Trapped Surfaces (MOTS)
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 8 / 39
Motivations and Problem formulation
Outline
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 9 / 39
Motivations and Problem formulation Black Hole Horizon Geometric Inequalities.
Outline
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 10 / 39
Motivations and Problem formulation Black Hole Horizon Geometric Inequalities.
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 11 / 39
Motivations and Problem formulation Black Hole Horizon Geometric Inequalities.
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”.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 11 / 39
Motivations and Problem formulation Black Hole Horizon Geometric Inequalities.
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].
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 12 / 39
Motivations and Problem formulation Black Hole Horizon Geometric Inequalities.
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].
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 12 / 39
Motivations and Problem formulation Black Hole Horizon Geometric Inequalities.
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 13 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
Outline
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 14 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 15 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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].
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 16 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
Principal eigenvalue for stationary axisymmetric horizons
Principal eigenvalue λo for equilibrium axisymmetric BH horizons
Theorem [Reiris 13]. Given an axisymmetric Isolated Horizon H 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].
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 16 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
Principal eigenvalue for stationary axisymmetric horizons
Principal eigenvalue λo for equilibrium axisymmetric BH horizons
Theorem [Reiris 13]. Given an axisymmetric Isolated Horizon H 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/(8π) = κ(`)/(8π) (−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].
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 16 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 17 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 17 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 18 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 18 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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
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?
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 18 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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)
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 19 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 19 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 20 / 39
Motivations and Problem formulation A Young-Laplace Law for Black Holes.
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...].
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 21 / 39
MOTSs and the Quantum Charged Particle
Outline
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 22 / 39
MOTSs and the Quantum Charged Particle
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 23 / 39
MOTSs and the Quantum Charged Particle
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 .
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 23 / 39
MOTSs and the Quantum Charged Particle
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)
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 24 / 39
MOTSs and the Quantum Charged Particle
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′
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 25 / 39
MOTSs and the Quantum Charged Particle
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 26 / 39
MOTSs and the Quantum Charged Particle
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 27 / 39
MOTSs and the Quantum Charged Particle MOTSs and Spinors.
Outline
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 28 / 39
MOTSs and the Quantum Charged Particle MOTSs and Spinors.
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 29 / 39
MOTSs and the Quantum Charged Particle MOTSs and Spinors.
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) .
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 30 / 39
MOTSs and the Quantum Charged Particle MOTSs and Spinors.
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 31 / 39
MOTSs and the Quantum Charged Particle MOTSs and Spinors.
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 31 / 39
MOTSs and the Quantum Charged Particle MOTSs and Spinors.
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 32 / 39
MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.
Outline
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 33 / 39
MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.
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).
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 34 / 39
MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.
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).
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).
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 34 / 39
MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.
A suggestive simple example: ”Landau levels”
MOTS: λ = (λ`m + 1− a2) + i2am , ψ = Slm(a, cos θ)eimϕ (prolate)
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 34 / 39
MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.
A suggestive simple example: ”Landau levels”
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)
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 34 / 39
MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.
A suggestive simple example: ”Landau levels”
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)
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 34 / 39
MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.
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...?
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 35 / 39
MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.
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.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 36 / 39
MOTSs and the Quantum Charged Particle Spectrum analyticity in the “fine structure constant”.
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]...)
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 37 / 39
Conclusions and Perspectives
Outline
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
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 38 / 39
Conclusions and Perspectives
Conclusions and Perspectives
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...?
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 39 / 39
Conclusions and Perspectives
Conclusions and Perspectives
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”.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 39 / 39
Conclusions and Perspectives
Conclusions and Perspectives
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...).
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 39 / 39
Conclusions and Perspectives
Conclusions and Perspectives
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 asclassical limit test for quantum inner pressures..
Oceanography: generalized “potential vorticity” q in quasi-geostrophicmotions. Physical mechanism for effect of “fast motions” on “slow motions”:
−∆ψ + RS2
2ψ = η − q ⇔ LSψ = |σ(`)|2 + Tab`
a`b
Statistical physics of 2-dimensional flows: Turbulence and large coherentstructures.
...
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 39 / 39
Conclusions and Perspectives
Conclusions and Perspectives
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 asclassical limit test for quantum inner pressures..
Oceanography: generalized “potential vorticity” q in quasi-geostrophicmotions. Physical mechanism for effect of “fast motions” on “slow motions”:
−∆Ψ + RS2
2ψ = η − q ⇔ LSψ = |σ(`)|2 + Tab`
a`b
Statistical physics of 2-dimensional flows: Turbulence and large coherentstructures.
Jose Luis Jaramillo (LPO) Black hole Horizons and the quantum charged particle Granada, 8 September 2014 39 / 39