Cosmic censorship in overcharging a black hole

with a charged particle

SI, N. Sago and T. Tanaka: Phys. Rev. D 84 (2011) 124024; arXiv 1108.6207.

Soichiro Isoyama (Yukawa Institute for Theoretical Physics: YITP)

Kyoto University

Overviews of Talk

3.Conclusion.

2.Backreaction effects in overcharging process ✔”Self-energy” of the charged particle ✔Energy loss due to the radiation.

1.Motivation and aim of this work.

✔Overcharging a static charged black hole ✔Cosmic censorship conjecture.

✔The problem in proposed scenario.

Cosmic censorship conjecture ・ A space time singularity is almost inevitable in GR

✔ Indicate the breakdown of space time structure

[Penrose (1969,1979)]

[Hawking and Penrose (1970)]

Need to assure the predictability of GR

No.1: All singularities of gravitational collapse are hidden within black holes, where they cannot be seen by a distant observer.

Cosmic censorship conjecture

No.2: All “physically reasonable space time” is globally hyperbolic (singularity free)

Violation of the censorship ・Neither of the statements have been proved yet.

・ What is worse, we know some counter examples

✔ Critical collapse of the matter field

✔ Spherical symmetric system (Dust collapse) [Joshi (1993)]

[Choptuik (1993)]

✔ Higher dimensions: BH-BH scattering, Gregory-Laflamme instability of a black string

[Okawa+ (2011), Lehner+(2010)]

✔ Overcharging/overspinning a black hole [Hubeny (1999), De felice+(2001),Jacobson+(2010),Saa+(2011)]

Overcharging/spinning a BH

? Kerr-Newman:

Point particle

✔ The particle can fall into the black hole. ✔ Then, saturate the extreme condition:

・We can overcharging (overspinning) the BH if

NB: The 3rd-law of the BH thermodynamics is NOT applicable.

[Wald (1974)]

E.g. Reisner-Nordström (RN) B.H. ・ Radial infall of a charged particle into RN black hole

Charged particle

RN black hole

✔Overcharging condition :

✔Causality condition :

for

✔Absorption condition :

・ We have 3 conditions for overcharging the RN-B.H.

[Hubeny(1999)]

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Overcharging scenarios

✔ Extremal case:

・Find allowed parameter space for a test particle.

[Wald (1974)]

✔ Near extremal case.No.1:

No allowed parameters.

[Cohen+(1974)]

✔ Near extremal case No.2: [Hubeny(1999)]

Overcharging parameters

Back reaction effects ? ・ The analysis is NOT consistent to the test particle limit.

・Absorption condition ・Overcharging condition

The same scaling as the proposed scenario.

Back reaction: perturbations

✔ Need to re-examine the proposed process with the back reaction effects at accuracy.

(Kinematics) (Energetics)

Overviews of Talk

3.Conclusion.

2.Backreaction effects in overcharging process ✔”Self-energy” of the charged particle ✔Energy loss due to the radiation.

1.Motivation and aim of this work.

✔Overcharging a static charged black hole ✔Cosmic censorship conjecture.

✔The problem in proposed scenario.

Radial marginal orbit ・ The particle orbits can be classified into 3 categories

・ Assumption: the marginal orbit exists with self-forces

The marginal orbit

✔ Repulsive orbit (Not absorbed by the B.H.)

✔ Plunge orbit (O.K. if the marginal orbit cannot overcharging the B.H.)

✔ Marginal orbit (The most danger case)

http://maru.bonyari.jp/texclip/texclip.php?s=%5Cbegin%7Balign*%7D%0Ar%0A%5Cend%7Balign*%7Dhttp://maru.bonyari.jp/texclip/texclip.php?s=%5Cbegin%7Balign*%7D%0A0%0A%5Cend%7Balign*%7Dhttp://maru.bonyari.jp/texclip/texclip.php?s=%5Cbegin%7Balign*%7D%0Ar_%2B%0A%5Cend%7Balign*%7Dhttp://maru.bonyari.jp/texclip/texclip.php?s=%5Cbegin%7Balign*%7D%0Ar_0%0A%5Cend%7Bslign*%7D

The double RN (DRN) solution [Alekseev+(2007),Manko(2007)]

✔ Equilibrium configuration in marginal orbit

Why not get “self-energy” from an exact solution?

NB: No Wyel strut on the symmetry axis.

Black Hole

Singularity

✔ An asymptotic flat solution

✔ 5-parameters

・ The particle’s “self-energy” is very problematic in RN.

✔ Static, axis-symmetric solution.

http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}z\end{align*}http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}l\end{align*}http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}z=z_1\end{align*}http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}z=z_2\end{align*}http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}z=z_2 + \sigma_2\end{align*}http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}z=z_2 - \sigma_2\end{align*}http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}(m_2,~e_2)\end{align*}http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}(m_1,~e_1)\end{align*}http://maru.bonyari.jp/texclip/texclip.php?s=%5Cbegin%7Balign*%7D%0Ads%5E2%20%3D%20H%28%5Crho%2C%7Ez%29dt%5E2%20-%20f%28%5Crho%2C%7Ez%29%28d%20%5Crho%5E2%20%2B%20dz%5E2%29%20-%20%5Cfrac%7B%5Crho%5E2%7D%7BH%28%5Crho%2C%7Ez%29%7Dd%20%5Cphi%5E2%0A%5Cend%7Balign*%7Dhttp://maru.bonyari.jp/texclip/texclip.php?s=%5Cbegin%7Balign*%7D%0AA_%7Bt%7D%20%3D%20%5CPhi%28%5Crho%2C%7Ez%29%2C%20%7EA_%7B%5Crho%7D%3DA_%7Bz%7D%3DA_%7B%5Cphi%7D%20%3D%200%2C%0A%5Cend%7Balign*%7D

Features of the DRN solution

・ 5-parameters cannot be independent

・ Total mass and charge are easily read out.

Total mass: Total charge:

✔ Balance condition:

(“Gravitational attractive force” = “electromagnetic repulsive force”)

✔ RN black hole: ✔ Singularity:

http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}\sigma_1^2 := m_1^2 - e_1^2 + 2 e_1 \left( \frac{m_2 e_1 - m_1 e_2}{l + m_1 + m_2 } \right)\end{slign*}http://maru.bonyari.jp/texclip/texclip.php?s=\begin{align*}m_1 m_2 = \left[ e_1 - \frac{\sigma_1^2 - m_1^2 + e_1^2}{2e_1} \right ]\left[ e_2 - \frac{\sigma_2^2 - m_2^2 + e_2^2}{2e_2} \right ]\end{align*}

Constraints on the total mass ・We only need the total energy of the system including particle’s “self-energy” correction.

✔ The RN black hole: , the singularity

・The total energy > the total charge in DRN geometry

✔Assumption:

Balance condition

Mapping between the 2 pictures ・Relate the DRN geometry to the equilibrium point of the marginal orbit

RN black Hole

singularity

RN Black hole

Charged particle

✔ Re-parameterize the 5 DRN parameters w.r.t

・NB.

1 to 1 mapping

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Conclusion: ①

✔ At the equilibrium position, the total mass of the system is always greater

than its charge due to the particle’s “self-energy” contribution.

Overviews of Talk

3.Conclusion.

2.Backreaction effects in overcharging process ✔”Self-energy” of the charged particle ✔Energy loss due to the radiation.

1.Motivation and aim of this work.

✔Overcharging a static charged black hole ✔Cosmic censorship conjecture.

✔The problem in proposed scenario.

The energy loss via radiation ・ Energy loss with radiation to infinity during the plunge stage can be evaluated via perturbation theory.

・ Just need the geodesic motion in RN black hole as a source of the perturbation.

✔ e.g. The quadrupole formula Black hole

✔linear order is enough

http://maru.bonyari.jp/texclip/texclip.php?s=%5Cbegin%7Balign*%7D%0Ar%0A%5Cend%7Balign*%7D

Gauge invariant perturbation [Kodama and Ishibashi (2003, 2004)]

・ Perturbations of the RN-BH is messy. But…

NB. Radial infalling particle only excites scalar(even)-perturbation.

✔ Expand perturbations via the harmonic functions on

Master equations

✔ Summarized in 2 gauge independent variables:

The energy flux formula ・ GW- and EM- radiations are decoupled at infinity.

+ ✔ Only asymptotic solution is relevant

(In: pure-ingoing, up: pure-outgoing modes)

Field reconstruction

・ Solve the master equation with the Green function.

・ The trend of the flux formula is basically determined by

Suppression of the energy loss

✔ Keys: Use identity and do integral by parts

=

Suppression

Regular functions

Oscillating term

decompose

Oscillating term

Conclusion: ②

✔ In the plunge stage, the particle’s energy lose via radiation is always suppressed

Conclusion: ➀+②

✔ The particle cannot satisfy the overcharging condition

as long as following the radial marginal orbit.

Relation to other works [Hod (2008), Zimmerman+(2012)] ・ RN black hole vs. overcharging

[Barausse+(2010,2011)] ・ Kerr black hole vs. overspinning ✔ Conservative self-forces are essential for saving the cosmic censorship conjecture

✔ Neither the absorption nor overcharging condition are satisfied due to the electro-magnetic self-forces

(Consistent with our analysis and expectation)

(Consistent with the importance of “self-energy” correction)

Summary of the talk ・We consider the back reaction effects on the proposed overcharging process, focusing on the marginal orbit

✔ At the equilibrium position, the total mass is always greater than its charge because of “self-energy” correction of the particle.

✔ At the plunge stage, energy loss due to the radiation is negligibly small.

The marginal orbit never overcharges the RN-BH.

A. No. The back reaction effects do prevent the charged black hole from being overcharged, and save the cosmic censorship.

Charged particle

black hole

Q. Is the charged black hole overcharged via charged

particle absorption ?

Final message of the talk

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糸冬 (Fin.)

Back up

The definition of the parameters in DRN

・ Satisfy the local conservation law.

✔ Mass parameters (Komar-like integral)

✔ Charge parameters Interaction terms

Near-singularity metric in DRN geometry

✔Singular RN-geometry + multi-pole perturbations ✔Monopole perturbation starts at

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