1. Geometric aspects of extremal Kerr Black Hole entropy E M Howard
Questions:
(i) Is it possible to stop the flow of time, or say, go backwards in time? (ii) Is the topology of space dynamical? (ii) From the classical point of view, Can complete knowledge of the initial Bing Bang singularity
the beginning of time) be enough to predict the future?
(iv) Are there other Universes inside black holes?
E M Howard
Macquarie University
Australian Institute of Physics Congress 2014

2. Outline www.scirp.org/journal/PaperDownload.aspx?paperID=28854
Ecaterina Howard, Geometric Aspects of Extremal Kerr Black Hole Entropy
1. A black hole is a classical solution in GR with special properties.
2. It is surrounded by an EH which acts as a one way membrane.
3. Nothing can escape from inside the event horizon to the outside  In classical GR a BH behaves as a perfect black body at zero
T and is an infinite sink of entropy.
unusual nature of the extremal limit explain entropy of extremal BH
Properties: time-independence of the extremal BH, the zero surface gravity, the zero entropy and the absence of a bifurcate Killing horizon  reduce to one single unique feature; true geometric discontinuity as the underlying cause of a vanishing entropy
geometrical disparity between the extreme and near extreme
Kerr geometries, due to the singular nature of extreme regime
Is there any physical principle telling a locally naked singularity how to behave? Is there any rule controlling the entropy?

3. Four laws of Black hole mechanics
The 0th law
k =const. The horizon has constant surface gravity for a stationary black hole.
The 1st law
For perturbations of stationary black holes, the change of energy is related to change of area, angular momentum, and electric charge by
d E=k dA/8πG + Ω d J +Φd Q
The 2nd law
d A >0 The horizon area is, assuming the weak energy condition, a non-decreasing function of time; superseded by Hawking's radiation, causing both the BH mass and the EH area to decrease over time.
The 3rd law
k >0 It is not possible to form a black hole with vanishing surface gravity. k = 0 is not possible to achieve. Extremal black holes have vanishing surface gravity.
J. Bardeen,B. Carter, S. Hawking, CMP,1973

4. Four laws of Black hole thermodynamics
BH exhibits an unusual similarity to a thermodynamic system. If κ is like a temperature, and A is like an entropy, then there is a close analogy:
The 0th law
T=const. on the horizon
temperature of an object in thermal equilibrium is constant
The 1st law
d M= T d S + Ω dJ+Φ d Q
entropy is dimensionless, whereas horizon area is a length squared
The 2nd law
d (SBH +Smatter)>=0 Entropy always increases
The 3rd law
T->0 the area of a black hole is separately non- decreasing, whereas only the total entropy is non-decreasing in thermodynamics.
J. Bekenstein, 1973; S. Hawking, 1974, 1975

5. Semiclassical black holes
Hawking coupled quantum matter fields to a classical black hole, and
showed that they emit black body radiation with a temperature
This implies black holes have an entropy
measure of disorder
Fundamental questions:
What is the origin of black hole entropy?
can a singularity exist in the absence of a horizon?
can a spinning singularity exist without two separate horizons?
Why is the approach to extremality not continuous?
Where should we calculate the entropy: on (or nearby) the horizon or within the black hole (disc) itself?
Is the entropy created immediately after the collapse, or later during the BH evolution? The extremal entropy is independent
of the BH evolution and its internal configuration. Can such a function be purely derived from geometric or topological considerations?

6. Extreme or Near-extreme?
properties of extreme Kerr BH  topological nature of the discrepancy between extreme and non-extreme regimes. features of Kerr BH that distinguish the extremal regime from the near-extremal one?
-EBH not a limit of NEBH due to this discontinuity.
-understand the discontinuity on geometric grounds.
-discontinuous nature of entropy ~ topological nature of the EH
NEBH and EBH topologically different; the switch is not continuous.
-condition for a non-zero entropy: bifurcate Killing horizon.
Hawking, Horowitz (1995): EBH different object from its non-extreme counterpart; Bekenstein-Hawking formula of the entropy fails to describe the entropy of EBH. The EBH must have zero entropy, despite the non-zero area of the event horizon, and can be in thermal equilibrium at arbitrary temperature.
Discontinuous transition non-extremal (NEBH)  extremal (EBH)

7. Extreme or Near-extreme?
The entropy of a self-contained system never decreases –true for classical GR
The area of a black hole’s event horizon can stay the same or increase, but
can never decrease. Area increases when: mass increases, spin decreases
Deep connection to thermodynamics: horizon area ~ entropy
Black hole entropy can be also defined as a measure of the observer’s accessibility to information about the internal configuration hidden behind the event horizon.
The EBH entropy is not the limit of the non-extremal one.
Dual and string theory dual microstate counting predict non-vanishing entropy solutions for EBH, the unusual nature of extremal limit using semi-classical methods is a genuine topological discontinuity  S=0
S=0 (semi-classical solutions), due to degeneracy of the EH.
Spacetime topology – essential role in the explanation EBH solution.
The topology itself of EBH is enough to explain entropy in this regime.

8. Kerr metric EH EBH a=m, no region II; 2 horizons coincide BH Naked!
coordinate singularities

9. No evolution or time reversibility
Conformal diagram
for extreme Kerr
Black holes
i0 =spacelike infinity,
ic =cylindrical end.
The surface S has one asymptotically
flat end i0 and one cylindrical end ic.

10. No evolution or time reversibility
3+1 decomposition

11. No evolution or time reversibility
EFE- time-reversal invariant.
A maximally extended spacetime = BH + its ”time-reverse” case.
NEBH - the extended spacetime
( k≠0) has a bifurcate Killing horizon.
EBH(k=0), no distinct time-reverse equivalent, time-independent everywhere + a single degenerate Killing horizon.
k cannot be reduced to zero within a finite time.

12. No evolution or time reversibility
Extended Schwarzschild - a white hole region becomes the time-reverse of the black hole region. Future EH, separating 2 regions, ≠ past EH.
The 2 regions intersect at the bifurcation 2-sphere.
The time-reverse of one region yields another region.
NEBH - a new region is the time-reverse of the black hole, generating distinct patches in between the two horizons.
EBH does not contain a distinct time-reverse patch but duplicates of the same patch. No distinct time-reverse region, no distinct EH.
No time-reversal: time-independent everywhere
EBH - Killing vector field on the horizon is null on a timelike hypersurface intersecting the horizon and it is spacelike on both sides. EH determined by a Killing vector field whose causal properties change timelike spacelike across EH. Killing horizon is null on a timelike hypersurface surrounding EH.
Killing vector field timelike in a region around EH?!!

13. No thermality
NEBH becomes extremal as the outer horizon approaches the inner one.
The two horizons are in equilibrium at 2 temperatures;
the temperature of the outer horizon approaches zero 
Same flux outside and inside the horizon and  0 on the horizon,
with a vanishing temperature 
finite discontinuity of flux in the extremal limit of NEBH.
No physical smooth transition.
Thermal nature at the horizon  probability flux across the horizon.
No thermality observed at the outer horizon.

14. No thermality Killing vector (Kruskal coordinates)
outgoing probability flux across the horizon
The horizon is located at r = m

15. No thermality Extremal regime: U + fails to be a proper distance.
The discontinuity in U+ is not physical.
The flux needs an infinite proper time to become 0 at the horizon when it is incident either from inside or outside.
Extremal regime:
r+ r- the effective temperature is 0 and the emission probability is 0.
The flux is the same both outside and inside the horizon 
vanishing temperature 
finite flux discontinuity
A finite discontinuity of flux, no thermality at the horizon (not coordinate related).
EBH - the proper radial distance from the horizon to any point close to the horizon
outside or inside, is infinite. No incident particle state to cross the horizon; cannot absorb or emit particle states.
coordinates remove the coordinate
singularity at the horizon.

16. Vanishing Entropy extremal regime no analogy!
-internal configuration can be depicted as the sum of points in the phase space, defined by a number of classical microstates.
-entropy of a system defined as the natural logarithm of internal states count S = lnQ. A microstate Q is F(system’s macrostate). Entropy is F(ALL variables).  relation between (macroscopic) entropy and statistical thermodynamics (number of microscopic states). IF 1 microstate, Q = 1, entropy=0no disorder. No continuous set of classical states, no time evolution. All metric components time independent. Any observer should have complete access to the unique classical state found within the region beyond EH.
An infinitesimal perturbation in mass,
spin and horizon area in Kerr metric:
Bekenstein-Hawking entropy
Hawking temperature:
extremal regime no analogy!
compare with first law of thermodynamics

17. Vanishing Entropy
Let us assume
After the particle crosses the horizon, there is no information about its state.  equally probable for it to exist or not
The minimum entropy change is:
for all states of the particle
The minimum change of BH area is:

18. Vanishing Entropy
A semi-classical picture evaluates the gravitational path integral by employing the euclideanized black hole geometry.
To avoid a conical singularity at the horizon, the period is 2∏  topology of a disc with zero deficit angle in the plane – ok for NEBH
euclideanised metric in n dimensions near EH:
is not 0 for NEBH
EBH - the proper radius diverges, the proper angle 0. Disk topology replaced with an annulus one. The topology of the transverse section in either case is
For a wrong periodicity  singularity at the origin linked to the excess angle, (cone)
The conical excess angle becomes 2∏ and the
topology is that of an annulus.
N’ is the N differentiated with respect to r.
N depends on the proper angle
BH interior completely absent in euclidean case. The euclideanized spacetime is continued to an imaginary value of the radial coordinate near the horizon.
In 2d, the metric near the horizon is

19. Vanishing Entropy
The topology of the Euclidean extremal solution becomes
The periodicity of the Euclidean time is
not fixed. Origin doesn’t exist any more, for any periodicity of the Euclidean time 
no conical singularity is formed.
The origin is not part of the manifold and the contribution from the vicinity of the origin vanishes  ENTROPY IS ZERO
The boundary contributions from the vicinity of the origin for the non-extremal regime are independent of
If BH = microcanonical ensemble, in a Hamiltonian formulation, action I ~ entropy. A dimensional continuation of Gauss-Bonnet theorem to n dim Action
the canonical action is proportional to
 Bekenstein-Hawking entropy S = A=4G.
EBH: r+ is infinitely far away from any point outside the horizon point must be removed from the Euclidean manifold.
Entropy becomes
The extremal BH horizon = infinite proper distance from any stationary observer no conical singularity at manifold.
NEBH X=1 DISC REGULAR AREA LAW
EBH X=0 ANNULUS S=0

20. Conclusions
–not limits of NEBH regime because of the discontinuity  topologically different; discontinuity explained on geometric grounds.
-the inner and outer horizons coincide. k = 0 and no bifurcate Killing horizon.
-the past and future horizons never intersect.
-there is no physical process that can make an EBH out of a NEBH.
-potential barrier close to the horizon that prevents to reach extremal regime.
-NOT the asymptotic limit of physical black holes.
-never behave as thermal objects. T undefined, emission spectrum non-planckian. Zero entropy because of one classical microstate.
-the non-thermal nature - a consequence of the geometric nature of the horizon. -all characteristics of the stress-energy tensor are different
final thermal macrostate can’t be achieved in a smooth continuous way without violating the energy conditions. It cannot be produced through a process involving any finite number of steps without violating the weak energy condition. It cannot be produced through standard gravitational collapse.
-EH present but because the phase space is time independent, it doesn’t hide more than one internal configuration.
-no time-reverse equivalent or a bifurcate Killing horizon
-geometric origin of the entropy  connection between the gravitational entropy, the topological structure of the spacetime and the nature of gravity.

21. Questions?