1. 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

2. 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

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

4. 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)

5. 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?

6. 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!

7. 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

8. 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

9. 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

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

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

12. 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

13. Analytic Solution for hzz
Singular at
Expansion near the CH

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

15. 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!

16. Numerical calculations

17. 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