Gedanken experiment to destroy a BTZ black hole Baoyi Chen ( TAPIR, - - PowerPoint PPT Presentation

Based on arXiv:1902.00949 with Feng-Li Lin and Bo Ning APS April Meeting, April 14 2019

Gedanken experiment to destroy a BTZ black hole

Baoyi Chen (TAPIR, Caltech)

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Weak cosmic censorship conjecture (WCCC)

“Nature abhors a naked singularity” (Roger Penrose 1969) A general proof is notoriously difficult Gedanken experiments in the Kerr-Newman spacetime A way to probe its validity— overspin or overcharge a Kerr-Newman black hole by throwing particles into it Black hole Extremal black hole Naked singularity M2 < (J/M)2 + Q2 M2 ≥ (J/M)2 + Q2 M2 = (J/M)2 + Q2

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Weak cosmic censorship conjecture (WCCC)

An extremal Kerr-Newman black hole cannot be overcharged or overspinned (Wald 1974) A naked singularity may be created by carefully throwing particles into a near- extremal black hole (Hubeny 1999) A near-extremal Kerr-Newman black hole cannot be overcharged or overspinned (Sorce & Wald 2017) In (3+1)-dimension, provided the null energy condition for the falling matter,

self-force? finite-size effect? gives a conclusive answer to whether WCCC is violated no self-force calculations

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What about gravity in n-dimensions? AdS black holes?

No curvature singularity but a conical singularity Described by its mass and angular momentum Asymptotically dual CFT description Solutions to a general category of gravity theories in (2+1)-D A (2+1)-D AdS black hole —Banados-Teitelboim-Zanelli (BTZ) black hole AdS3

Einstein gravity, Chiral gravity (with/without torsion), etc

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Gedanken experiment to destroy an extremal BTZ black hole

r = ∞ Σ0 Σ1 H r = 0 δφ = 0

Linear variational identity

Σ = H ∩ Σ1

δM − ΩHδJ − THδS = − Z

Σ

δCξ

“first law” “null energy condition”

We use this identity to constrain the sign of

M(λ), J(λ) M(0), J(0)

= 2λ p −Λeff|J| ⇣ δM − p −ΛeffδJ ⌘ + O(λ2) f(λ) = M(λ)2 + ΛeffJ(λ)2

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Gedanken experiment to destroy an extremal BTZ black hole

In torsional chiral gravity, whether WCCC holds depends on a relation between the spin angular momentum and its coupling to torsion In chiral gravity, provided the null energy condition, extremal BTZ cannot be destroyed, thus WCCC is preserved In Einstein gravity, provided the null energy condition, extremal BTZ cannot be destroyed, thus WCCC is preserved

What about throwing matter into a near-extremal BTZ black hole?

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Hubeny-type violations of WCCC

• J

M BTZ BH Conical Singularity δS = 0 J M BTZ BH Conical Singularity δS = 0

• No Hubeny-type violation in chiral gravity

Einstein gravity has Hubeny-type violations need second order variations! no need to check second order!

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Gedenken experiment to destroy a near-extremal BTZ black hole

B r = 0 r = ∞ Σ0 Σ1

δ2φ = 0 δφ = 0

H

H

Second order variational identity

“first law” “null energy condition”

We use this identity to constrain the sign of

Σ = H ∩ Σ1

δ2M − ΩHδ2J = EΣ − Z

Σ

iξ(δE ∧ δφ) − Z

Σ

δ2Cξ

Canonical energy

f(λ) = M(λ)2 + ΛeffJ(λ)2

WCCC is preserved!

=(M 2 + ΛJ2) + λg1(δM, δJ) + λ2g2(δM 2, δJ2, δ2M, δ2J) + O(λ3) = ✓ Mε + λΛJδJ M ◆2 ≥ 0

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The takeaway

In (2+1)-dimension, provided the null energy condition and torsionless limit, WCCC is preserved for a BTZ black hole with a conical singularity WCCC may be violated in presence of torsion Our gedanken experiment around BTZ is holographically mapped to the cooling

• f the boundary CFT

Our results indicate the third law of thermodynamics holds for the boundary CFT Generalizations to higher dimensional AdS black holes can be done in the future

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Additional slides

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Wald’s Lagrangian approach

The Noether current associated with a Lagrangian and a Killing vector field is

ξ

jξ = Θ(φ, Lξφ) − iξL jξ = dQξ + Cξ

L

According to the Noether theorem, the Noether current can also be written as

surface term in variation “Noether charge” “Constraint of the theory”

Variation of both equations gives the fundamental linear variational identity

• ∂Σ

δQξ − iξΘ(φ, Lξφ) =

• Σ

Ω(φ, δφ, Lξφ) −

• Σ

δCξ −

• Σ

iξ(E ∧ δφ)

“Simplectic current” equation of motion

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Mielke-Baekler model

L = L(ea, ωa) = LEC + LΛ + LCS + LT + Lm LEC = 1 πea ∧ Ra LCS = − θL (ωa ∧ dωa + 1 3 ϵabcωa ∧ ωb ∧ ωc ) LΛ = − Λ 6π ϵabcea ∧ eb ∧ ec LT = θT 2π ea ∧ Ta

Motivated by writing the gravity theory as a Poincaré gauge theory Einstein-Cartan term Cosmological term with Λ < 0 Chern-Simons term Torsion term coupling constant

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Three limits of MB model

(i) Einstein gravity (ii) Chiral gravity (iii) Torsional chiral gravity

θL → 0 θT → 0 θL → − 1/(2π −Λ) 𝒰(θL, θT) = −θT + 2π2ΛθL 2 + 4θTθL → 0 coefficient of the torsion tensor θL → − 1/(2π −Λ) 𝒰(θL, θT) = π −Λ/2 ≠ 0 (1) (2) (1) (2) “chiral limit” “torsionless limit”