Holographic stress-energy tensor near the Cauchy horizon inside a rotating black hole 3 rd Karl Schwarzschild Meeting, Germany 24 July 2017 Paper: PRD 96(2017) Kengo Maeda : Shibaura Institute Of Technology Collaborators Akihiro Ishibashi: Kindai University Eric Mefford: UCSB

CH is classically unstable due to the infinite blue shift (Penrose) Ⅰ Introduction. What happens inside a rotating (charged) BH? Time Ex.) Charged BH Cauchy Horizon(CH) Space: r Singularity r=0 CH is classically unstable due to the infinite blue shift (Penrose) BH Collapsing star

Strong cosmic censorship conjecture(Penrose 1978) All physically reasonable spacetimes are globally hyperbolic Singularity Singularity CH Initial surface Singularity

Classical instabilities of CH ▶ Linear stabilities CH is unstable against linear perturbations(Chandrasekhar, Hartle, …) ▶ Nonlinear stabilities CH of charged spherically symmetric BH is unstable against non-linear perturbations(Poisson, Israel, Brady and Smith, … ) Null singularity on CH Numerical calculation by Brady and Smith(1995, PRL)

Quantum instabilities of CH ▶ Hiscock(1977, PRD) : Two-dim. free Massless scalar field model Energy-momentum tensor diverges on the CH Singularity Null-null component of the quantum EM tensor negatively diverges on the CH! V Questions What happens in higher dimensions? What happens in strongly coupled field theory?

Ⅱ Holographic approach AdS/CFT duality (Maldacena) Duality D dim. Strongly coupled gauge theory on an asym. curved spacetime D+1 dim. Asym. AdS spacetime Many applications: Quark-Gluon plasma, Condensed matter physics, Hawking radiation, …. Strongly coupling effect is very important!

Black droplet solutions ●　Horizon is disconnected from the bulk planar Schwarzschild AdS horizon ●　Hawking radiation is investigated analytically and numerically AdS BD Sch-AdS Hubeny, Marolf, Rangamani, Santos, Way, Haddad, Fischetti, Figueras, ….. Complicated Solutions Semi-analytic approach is important

Ⅲ Construction of BD solution Application of derivative expansion method (by Haddad 2012) Sch-AdS horizon scale: AdS radius scale: Horizon radius scale on the boundary: We consider for any function g

We perturvatively construct a six-dim. Duality Cohomogeneity-one Black hole with CH: rotating Myers-Perry (MP) BH Cohomogeneity-two rotating Black Droplet solution Six-dim. Five-dim. Sch-AdS We perturvatively construct a six-dim. rotating BD solution by the gradient expansion Black Droplet z

Field Eqs. Metric ansatz: MP solution Inner Cauchy horizon radius

Derivative expansion of the metric: : Formal expansion parameter ●　First order in

The droplet horizon shrinks to zero at the planar Sch-AdS horizon ●　Second order in Metric ansatz: Sch-AdS We obtained five ordinary differential Eqs. for five variables The droplet horizon shrinks to zero at the planar Sch-AdS horizon

Analytic Solution for hzz Singular at Expansion near the CH

IV Holographic Energy-momentum(EM) tensor Regularized action (Emparan et. al. (1999)) Counter terms

Strong cosmic censorship holds! Near the CH Energy densities diverge near the CH! Strong cosmic censorship holds! Null Energy condition is strongly violated near the CH! Hiscock(1977) : Two-dim. free Massless scalar field model Qualitative behavior is same as the 2dim. massless scalar field model!

Numerical calculations

Q. Why Holographic EM-tensor diverges on the CH? Our droplet solution The divergence of holographic Quantum energy near the CH would be quite generic Locally inhomogeneous Classical black string Charged AdS Black brane CH is regular Static perturbation CH is singular K. M., T. Okamura, J. Koga (2012) Thank you