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

Gedanken experiment to destroy a BTZ black hole Baoyi Chen ( TAPIR, Caltech ) Based on arXiv:1902.00949 with Feng-Li Lin and Bo Ning APS April Meeting, April 14 2019

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

gives a conclusive answer to whether WCCC is violated self-force? finite-size effect? no self-force calculations Weak cosmic censorship conjecture (WCCC) In (3+1)-dimension, provided the null energy condition for the falling matter, 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) � 3

Einstein gravity, Chiral gravity (with/without torsion), etc What about gravity in n-dimensions? AdS black holes? A (2+1)-D AdS black hole —Banados-Teitelboim-Zanelli (BTZ) black hole No curvature singularity but a conical singularity Described by its mass and angular momentum Asymptotically dual CFT description AdS 3 Solutions to a general category of gravity theories in (2+1)-D � 4

“first law” “null energy condition” Gedanken experiment to destroy an extremal BTZ black hole Linear variational identity Z δ M − Ω H δ J − T H δ S = − δ C ξ Σ r = 0 M ( λ ) , J ( λ ) Σ 1 H M (0) , J (0) We use this identity to constrain the sign of Σ 0 r = ∞ δφ = 0 f ( λ ) = M ( λ ) 2 + Λ e ff J ( λ ) 2 ⇣ ⌘ p p + O ( λ 2 ) = 2 λ − Λ e ff | J | δ M − − Λ e ff δ J Σ = H ∩ Σ 1 � 5

What about throwing matter into a near-extremal BTZ black hole? 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 � 6

no need to check second order! need second order variations! Hubeny-type violations of WCCC M M � � BTZ BH BTZ BH δ S = 0 � � � � � � δ S = 0 � Conical Singularity � Conical Singularity J � � � � � J � � � � � No Hubeny-type violation in chiral gravity Einstein gravity has Hubeny-type violations � 7

WCCC is preserved! Canonical energy “null energy condition” “first law” Gedenken experiment to destroy a near-extremal BTZ black hole Second order variational identity r = 0 Z Z δ 2 M − Ω H δ 2 J = E Σ − δ 2 C ξ i ξ ( δ E ∧ δφ ) − Σ Σ H Σ 1 H Σ = H ∩ Σ 1 δφ = 0 r = ∞ δ 2 φ = 0 Σ 0 We use this identity to constrain the sign of B f ( λ ) = M ( λ ) 2 + Λ e ff J ( λ ) 2 =( M 2 + Λ J 2 ) + λ g 1 ( δ M, δ J ) + λ 2 g 2 ( δ M 2 , δ J 2 , δ 2 M, δ 2 J ) + O ( λ 3 ) ◆ 2 ✓ M ε + λ Λ J δ J = ≥ 0 M � 8

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 of 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 � 9

Additional slides � 10

equation of motion surface term in variation “Simplectic current” “Noether charge” “Constraint of the theory” Wald’s Lagrangian approach The Noether current associated with a Lagrangian and a Killing vector field is ξ L j ξ = Θ ( φ , L ξ φ ) − i ξ L According to the Noether theorem, the Noether current can also be written as j ξ = dQ ξ + C ξ Variation of both equations gives the fundamental linear variational identity � � � � δ Q ξ − i ξ Θ ( φ , L ξ φ ) = Ω ( φ , δφ , L ξ φ ) − i ξ ( E ∧ δφ ) δ C ξ − ∂ Σ Σ Σ Σ � 11

coupling constant Mielke-Baekler model Motivated by writing the gravity theory as a L = L ( e a , ω a ) = L EC + L Λ + L CS + L T + L m Poincaré gauge theory L EC = 1 π e a ∧ R a Einstein-Cartan term L Λ = − Λ 6 π ϵ abc e a ∧ e b ∧ e c Cosmological term with Λ < 0 L CS = − θ L ( ω a ∧ d ω a + 1 ) 3 ϵ abc ω a ∧ ω b ∧ ω c Chern-Simons term L T = θ T 2 π e a ∧ T a Torsion term � 12

“torsionless limit” “chiral limit” coefficient of the torsion tensor Three limits of MB model (i) Einstein gravity θ L → 0 θ T → 0 𝒰 ( θ L , θ T ) = − θ T + 2 π 2 Λ θ L (ii) Chiral gravity (1) (2) → 0 θ L → − 1/(2 π −Λ ) 2 + 4 θ T θ L (iii) Torsional chiral gravity (1) (2) θ L → − 1/(2 π −Λ ) 𝒰 ( θ L , θ T ) = π −Λ /2 ≠ 0 � 13