1. Testing the cosmic censorship with an extremal black hole
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
  Testing the cosmic censorship with an extremal black hole
Sijie Gao
Beijing Normal University, China
Ref. Sijie Gao and Yuan Zhang, Phys. Rev.D 87, (2013)

2. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
Black hole
singularity
（future null infinity）
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

3. Kerr-Newman black hole (Stationary black hole)
Three parameters:
RN black hole
Kerr black hole
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

4. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
Surface gravity (temperature):
event horizon:
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

5. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
(RN black hole)
extremal black hole
Naked singularity
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

6. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
Cosmic censorship conjecture (Penrose 1969): Naked singularity
does not exist. All singularities should be hidden in black holes.
Gedanken experiments to test the cosmic censorship
conjecture
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

7. In 1974, Wald [Ann.Phys. 83,548(1974)] proved that an extremal Kerr-Newman black hole cannot be overcharged or overspun. In his proof, linear approximation was used.
In 1999, Hubeny [PRD, 59, ] showed that a nearly
extremal RN black hole can be overcharged, leading
to potential violation of the cosmic censorship.
In 2009, Jacobson and Sotiriou (PRL 103, ) obtained similar
Result for nearly extremal Kerr black hole.
Both Hubeny and Jacobson’s results agree with Wald’s in the
extremal limit even without linear approximation, i.e., no violation
for extremal RN black hole ( ) or Kerr black hole ( ).

8. Results so far:
So it seems that cosmic censorship is safe for extremal black holes.
But the extremal Kerr-Newman black hole ( , ) with
non-linear terms had not been checked.

9. Destroying an extremal Kerr-Newman black hole
The metric is written as
The particle’s four-velocity is
The conserved energy and angular momentum are given by
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

11. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
At the horizon of an extremal Kerr-Newman black hole, it becomes
Keeping terms linear to , Eq. (22) becomes
which contradicts Eq.(21), meaning the particle cannot reach the
horizon.
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

12. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
Define So
with
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

13. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

14. The effective potential defined by is always
negavive for , so the particle can be released from
infinity and fall all the way to the horizon.

15. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
We also show that, to violate the cosmic censorship, the allowed
range of is of order This fine tunning
indicates that the back reaction and self-force effects should be
considered.
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

16. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
Conclusions
By considering high order terms, we show that
the horizon of an extremal Kerr-Newman black hole can be destroyed by a test particle.
The apparent violation of the cosmic censorship is generic.
The fine-tuning on the particle’s parameters indicates that the back-reaction effect should be considered and may prevent violation of the cosmic censorship.
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island

17. Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island
Thank you!
Asia Pacific School on Gravitation and Cosmology 2013, Jeju Island