1. Closed trapped surfaces and Cauchy horizons in gravitational collapse
Australian Mathematical Society 53rd Meeting
Adelaide, 2009
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

2. Outline
-review of causal structure of space-time and the cosmic -study the structure and features of a singularity and understand the spacetime inside a black hole –what happens inside?
-intuitive approach
-overview of the fundamental aspects of relativistic causality and the current status of research in the field.
-brief survey of key concepts in singularity theory
-touch the inherent conundrums and challenging problems, difficulties, unusual features, ambiguities, inconsistencies and major open questions in the field
-under certain physical situations of gravitational collapse spacetime singularities, in the sense of causal geodesic incompleteness, must occur
-fundamental questions concerning marginally trapped surfaces (apparent horizons), in Cauchy data sets for the Einstein equation.
Is there any physical principle telling a locally naked singularity how to behave? Is there any rule controlling the entropy?
Spacetime may exhibit naked singularities for certain values of the spin
trapped surfaces play a central role in the study of spacetime geometry.
Study the null, weak, scalar-curvature singularity along the inner horizon (Riemann divergent) and the asymptotic behaviour

3. Present-day picture
-for bodies of too large a mass, concentrated in too small a volume, unstoppable collapse will lead to a singularity in the structure of space-time.
-the term ‘singularity’ refers to a region where the conventional classical picture of space-time breaks
down
-standard picture of collapse to a BH (Penrose 1978)- the singularities are not visible to observers at a large distance from BH, being ‘shielded’ from view by an absolute EH. Every singularity is surrounded by an EH.  no naked singularities
Space-time diagram
-investigate the interior of regular axisymmetric and stationary BHfor non-vanishing angular momentum the space time can always be extended regularly up to and including an inner Cauchy horizon.
Assumption: weakly asymptotically simple and empty (WASE) space-time
-the final state of gravitational collapse ->spacetime singularities in the causal future of regular initial data, with divergence of curvature invariants and geodesic incompleteness
-save the situation: spacetime singularities are "invisible" to external observers ->"cosmic censorship conjecture“ –unproven and the foremost unsolved GR problem.
Singularity theorems (Hawking & Penrose) ->a spacetime which satisfies suitable energy and causality conditions, and contains a trapped surface, must contain a BH.
Penrose- if the universe posesses a noncompact Caucy surface  a collapsing star must develop a singularity

4. Related issues
-a spacetime is said to have singularities if it possesses inextendible curves with finite affine length.
-inside each BH region there is a Cauchy horizon, beyond which a timelike singularity becomes visible.
-trapped surface formation - intimately connected with the question
of singularities.
-a signal can be sent between two points of M only if they can be joined by a non-spacelike curve
-investigate further the properties of such causal relationships. the study of causal relationships is equivalent to that of the conformal geometry of M i.e. of the set of all metrics g conformal to the physical metric g
-under a conformal transformation of the metric a geodesic curve will not, in general, remain a geodesic curve unless it is null, and even in this case an affine parameter along the curve will not remain an affine parameter.
-mathematical criteria for ‘unstoppable collapse’ = existence of a trapped surface or of a point whose future light cone begins to reconverge in every direction along the cone

5. Horizons and Singularities of the Kerr Metric
Kerr solution - family of exact solutions which represent the stationary axisymmetric asymptotically flat field outside a rotating massive object.
a > 1 -> Event Horizon vanishes -> Naked singularity!
In GR, for Kerr-BH we expect: a < 1
Looking at our metric we find an essential singularity for
Remembering the definitions of our coordinates we find
This corresponds to a ring of radius a
for a2 < M2
A surface of infinite gravitational red shift is defined by
Setting a = 0, or θ = π/2 these reduce to
curvature singularity = ring = real singularity
The intrinsic geometry of the ring is singular
The second singularity =coordinate singularity for
Gives the location of EH
for m->0 flat Minkowski spacetime

6. Overview of Kerr Geometry (a2 < m2)
Essential ring singularity at:
2 disjoint sets
(surfaces of infinite
red shift) at :
2 event horizons at:
KERR BH assumptions:
Non-zero angular momentum
Insignificant charge
Axial symmetry
No-Hair Theorem
For a2 > m2 we find only the essential singularity at r = 0
The naked singularity violates CC hypothesis
The solution for a2 = m2 is unstable
Ergosphere –stationary limit surface
close to the outer EH
nonstationary observers ->frame dragging
delta potential
generalized radius
sigma potential
frame-dragging
frequency
cylindrical radius
black hole mass M
spin parameter a

7. Cross-section of a Kerr black hole
Event horizon at (=0) and angular momentum between
0a<M and
-Kerr solutions = two-parameter family parametrized by M,J
-class of axisymmetric and stationary space–times- E,J conserved
-metric terms-independent of the coordinate time and the axial coordinate
-asymptotically flat; for r  infinity, Kerr metric  flat Minkowski for m → 0
-regularity outside the EH, smooth convex horizon
-dragging of inertial frames-light dragged in the same direction of spin
-Schwarzschild metric for a=0. No Birkhoff
-No-hair theorem-> all stationary BH solutions of Einstein-Maxwell equations are uniquely determined by -3 parameters: M, Q, and J with for extreme BH.
-solutions for which have no EH; their singularities are exposed (naked singularities)

8. Kerr naked singularity
-final fate of a continual collapse - BH or naked singularity, depending on the nature of the regular initial data, from which the collapse develops evolving from an initial spacelike surface.
Singularity: a point at which space-time diverges
Naked singularity
-a singularity that is not inside a BH (not surrounded by an EH), it can be seen by someone outside it.
-the outcome of a continued gravitational collapse when no EH forms hiding the singularity to the asymptotic region.
Kerr metric in Boyer-Lindquist coordinates
The null geodesics are given by the tangent vector components:
First equation shows that null geodesics exist which partially run in a time reversal regime
“Subtle is the Lord ... in theory, naked singularities do form dynamically from regular initial data satisfying the usual energy conditions.” (Penrose)

9. Possible Kerr analytic extension?
In Kerr-Schild coordinates ,the metric takes the form
The function r can in fact be analytically continued from positive to negative values through the interior of the disc x2 + y2 < a2, z = 0, to to obtain a maximal analytic extension of the solution.
r determined implicitly, up to a sign, in terms of x, y, z
attach another plane (x‘, y', z') where a point on the top side of the disc x2 + y2 < a2, z = 0 in the (x, y, z) plane is identified with a point with the same x and у coordinates on the bottom side of the corresponding disc in the (x‘, y', z') plane.
Solution geodesically incomplete at the ring singularity
The metric on the (x’ y', z') region has the same form but with negative values of r. At large negative values of r, the space is again asymptotically flat but with negative mass. The circles {t = constant, r = constant, theta = constant) are closed timelike curves.
The metric extends to a larger manifold.

10. Maximally extended spacetimes
at the border the null energy condition will start to be violated
Kerr black hole
Reissner-Nordstrom BH
Kerr extended diagram
Multiple asymptotic regions and BH regions that contain timelike singularities
Schwarzschild black hole

11. Space-time singularities
define singularities in spacetime
four theorems are given which prove the existence of incompleteness under a wide variety of situations.
The problem of defining whether space-time has a singularity now becomes one of determining whether any singular points have been cut out. One would hope to recognize this by the fact that space-time was incomplete in some sense.
Possible attempt: relation with geodesic incompleteness, as an indication that singular points have been cut out of space-time; associate with a form of curvature singularity.
Analogy with electrodynamics: define a space-time singularity as a point where the metric tensor is undefined or not suitably differentiable. ->WRONG! -> we could cut out points and infer that the remaining manifold represent the whole of space-time, which would then be non-singular
Space-time = pair where the metric g is Lorentzian and suitably differentiable
ensure that no regular points are omitted from the manifold along with the singular points by requiring that could not be extended with the required differentiability.
Define whether space-time has a singularity -> determine whether any singular points have been cut out.

12. What is a singularity? Definition of Singularity
occurrence of singularities
by the existence of incomplete geodesics that can not be extended to infinite values of the affine parameter
Singularity theorems: guarantee formation of a singularity if there is a trapped surface.
I+(p) = set of all points of the spacetime M that can be reached from p by future directed
time like curves /the set of all events that can be influenced by what happens at p.
consider the boundary I+(S) = the future of a set S -this boundary can not be time like.
Singularity theorems:
Regular, generic initial data for reasonable matter will evolve to yield a pathological behavior if gravity becomes sufficiently strong.
2. The nature of the pathology is not predicted,various types are known in special cases.
Definition of Singularity
Popular: spacetime singularity-region in which the curvature becomes unboundedly large.
Define spacetime as the maximal manifold on which the metric is suitably smooth 
A spacetime is singular if it is timelike or null geodesically incomplete, but can not be embedded in a larger spacetime.

13. More definitions (Hawking & Ellis)
Assumption: space-time is time-orientable
suppose that there is a local thermodynamic arrow of time defined continuously at every point of
space-time -> define continuously a division of non-spacelike vectors (at each point) into 2 sets
arbitrarily labeled future- and past-directed.
A set is called achronal (semi-spacelike)
if is empty or if there are no two points of with timelike separation.
= Future set if

14. Necessary conditions Singularity Theorems - three types of conditions
1. Energy condition. -weak, strong or generic
2. Condition on global structure. no closed time like curves
3. Gravity strong enough to trap a region.
Conditions that spacetime must be time like or null geodesically incomplete if different combinations of the three kinds of conditions hold.
3rd condition -->spatial cross section of the universe was closed, for then there
was no outside region to escape to OR closed two surface such that both the ingoing and out going null geodesics orthogonal to it were converging
Closed trapped surface = a compact spacelike two-surface in space-time such that
outgoing null rays perpendicular to the surface are not expanding.

15. Convergence of the geodesics
gravity is an attractive force  conjugate points in spacetime neighbouring geodesics are bent towards each other rather than away
v is affine parameter along a congruence of geodesics,
with tangent vector la which are hypersurface orthogonal
ρis the average rate of convergence of the geodesics, σ measures the shear
Rablalb gives the direct gravitational effect of the matter on the convergence of the geodesics.

16. Energy conditions Generic energy condition
every time like or null geodesic encounters some point where there is some curvature that is not specially aligned with the geodesic
Energy momentum tensor obeys
not satisfied by a number of known exact solutions
=matter cannot travel faster than light
=Weak EC +additional requirement: pressure not exceed energy density. To any obs, the local energy density appears non negative and the local energy flow vector is non-spacelike
energy density T00 mesured by any
obs is non negative in any frame
obeyed by the classical energy momentum tensor of any reasonable matter
Matter always has a converging effect on congruences of null geodesics (null and timelinke convergence condition)
Generic energy condition
each geodesic encounters a region of gravitational focussing
1. The strong energy condition holds.
2. Every timelike or null geodesic contains a point where
there are pairs of conjugate points if one can extend the geodesic far enough in each direction.

17. Global condition
Aim: study of the global causal structure of spacetime
-consider only space-times which do not permit causality violations.
given any spacelike surface, there is a maximal region of space-time (Cauchy development) which can be predicted from knowledge of data on the surface.
spacetime complete  the generators of the boundary
need to have past end points each generator of the
boundary of the future has a past end point on the set  impose global
condition on the causal structure Global hyperbolicity - if two points in it can be joined
by a time-like curve, then there exists a longest such curve between the points. (geodesic)
The causal structure of space-time can be used to define a boundary (edge) to space-time
(represents both infinity and the part of the edge of space-time which is at a finite
distance: singular points)
singularity theorems show that spacetime must be time like or null geodesically incomplete if different combinations of the three kinds of conditions hold.

18. Unpredictability of the Kerr solution
the unknown = Lorentzian metric gμν , the characteristic sets are its light cones.
Schwarzschild: the trapped region coincides with BH and ends in a spacelike singularity
For any point P, the hyperbolic nature of the equations det. the past domain of influence of P, = its causal past J−(P). Uniqueness of the solution at P (modulo the diffeomorphism invariance) follows from a domain of dependence argument that requires that J−(P) have compact intersection with the initial data
spacetime=future inextendible as a manifold with continuous Lorentzian metric
conformal representation
of a 2d cross section
-explicit solutions containing points P’ where the solution is regular; the compactness property fails. nonunique solutions to the initial value problem.
The light-like surface= Cauchy horizon (Cauchy problem posed in its past is insufficient to uniquely determine the solution in its future -> unpredictability
Kerr- no spacelike surface intersects any timelike/null line more than one
-Einstein equations = quasilinear, the geometry of the characteristic set depends strongly on the unknown ->
nonuniqueness ->unpredictability occuring for a family of special solutions of the Einstein equations, Kerr spacetime
unstable scenario ->in gravitational collapse, unpredictability is exceptional, for generic initial data in a certain = strong CC

19. Still unclear…
-Christodoulou proved CC for the spherically symmetric Einstein-scalar field ->trapped regions. A point in a trapped region corresponds to a trapped surface in the 4d space-time manifold
-predictability for the Einstein equations and strong CC
-conditions for predictability are then related to the behavior of the unique solution of the initial value problem on the boundary of this region.
-there always exists a maximal region of spacetime, the maximal domain of development, for which the initial value problem uniquely determines the solution.
-consequences of curvature singularity at inner horizon: the metric tensor is continuous, the Riemann tensor diverges at inner horizon
Weak curvature singularity: although curvature diverges, the metric tensor has a well-defined, continuous, non-singular limit at the singularity (Tipler, 1977)

20. Cosmic Censorship Conjecture
Weak CC
“the generic gravitational collapse of an isolated physical system, starting from perfectly physically reasonable nonsingular initial state, can’t produce spacetime singularities that can be seen from infinity, even though observations from infinity are allowed to continue indefinitely”
Strong CC <=> Global hyperbolicity
“Every inextendible spacetime M which evolves according to classical GR with physically reasonable matter satisfying appropriated energy conditions from a generic nonsingular initial data on a complete spacelike hypersurface, is globally hyperbolic.”
-WCC -any singularity originated in gravitational collapse must be hidden inside an EH. Solutions-either geodesically complete or, if incomplete, end up in curvature singularities. WCCC validity -necessary condition to ensure the predictability of the laws of physics
-without a full description of spacetime singularities, WCC = true in order to assure the predictability of the laws of physics.
Generically, singularities will be hidden inside the horizons of BH.
Naked singularities do not occur in nature.
Hawking: God abhors a naked singularity. A naked singularity should be prohibited by the laws of classical physics
2. Time-like singularities will not occur generically even inside an EH. An observer will only detect a singularity by hitting it.

21. Cauchy horizons Cauchy horizon (CH)
=surface of infinite blueshift with respect to the EH -> dynamical instability (mass inflation) which replaces CH by a null singularity that turns spacelike deep inside the BH
=light-like boundary of the domain of validity of a Cauchy problem (separates closed timelike geodesic and closed spacelike geodesic regions)
-while approaching EH, when stress-energy tensor diverges at the horizon, CH prevents spacetime from developing
Schwarzschild singularity replaced by Cauchy horizon
Under the averaged weak energy condition, CH are unstable.
closed time-like curves that would otherwise be feasible.
geodesic completeness (g-completeness)
- every geodesic can be extended to arbitrary values of its affine parameter.
If one cuts a regular point out of space-time, the resulting manifold is incomplete (null, timelinke and spacelike)
timelike and null g-completeness -minimum conditions for singularity-free space-time -> if a space-time is timelike or null geodesically incomplete ->it has a singularity.
timelike or null incompleteness = indicative of the presence of a singularity
closed trapped surface T = closed (i.e. compact, no boundary) spacelike two-surface such that the two families of null geodesics orthogonal to T are converging at T.
T - in such a strong gravitational field that even the 'outgoing' light rays are dragged back (converging). Nothing can travel faster than light  matter within T is trapped inside a succession of two-surfaces of smaller area
Trapped surface formation is intimately connected with the question of singularities.

22. Relativistic Cauchy developments
Newtonian theory - instantaneous action-at-a-distance; in order to predict events at future points in space-time one has to know the state of the entire universe at the present time and assume boundary conditions at infinity (potential goes to zero etc.)
Relativity - events at different points of
space-time can be causally related only if they can be joined by a non-spacelike curve -> knowledge of the appropriate data on
a closed set would determine events in a region to the future of
= future Cauchy development (domain of dependence of defined as the set of all points such that every past-inextendible non-spacelike curve through p intersects
The future Cauchy development and future Cauchy horizon of a closed set which is partly null and partly spacelike.
is not necessarily connected.

23. Singularity theorems Theorem 1 Theorem 2 (Hawking and Penrose 1970)
matter always has a converging effect on congruences of null geodesics.
Theorem 2 (Hawking and Penrose 1970)

24. Theorem 3 (Hawking 1967))
convergence condition
(null or timelike)
Th 2,3 - the most useful since it can be shown that their conditions are satisfied in a number of physical situations. However it might be that what occurred was not a singularity but a closed timelike curve, violating the causality conditions.
do such causality violations prevent the occurrence of singularities? The 4th theorem shows that they cannot in certain situations. We have to take singularities seriously and it gives us confidence that, in general, causality breakdowns are not the way out.
Theorem 4 (Hawking 1967))

25. Back to the definiton
timelike and null geodesic completeness = minimum conditions for space-time to be considered singularity-free.
If a space-time is timelike or null geodesically incomplete -> it has a singularity.
Timelike geodesic incompleteness physical significance - 2 possibilities:
1. there could be freely moving observers or particles whose histories did not exist after (or before) a finite interval of proper time.
2: infinite curvature or curvature singularity (scalar polynomial curvature singularity)
Both features are objectionable and challenging!
Consequence of the singularity theorems: timelike or null incompleteness are indicative of the presence of a singularity
Prediction of singularities  classical GR is not a complete theory singular points cut out of the spacetime manifold can’t define the field equations there  can’t predict what will come out of a singularity.

26. Where are we?
Singularities are inevitable CCany break down of predictability at these singularities won't affect what happens in the external world (not according to classical GR)
-GR incomplete or CC violated!
– infinite forces are acting
– laws of physics break down
– “The stability of EH require exotic matter
violating the average null energy condition”
- quantum gravity/string theory may help ?
– no problem as long as a singularity is shielded
from the outside world by an EH. Accept CCC?
-unpredictability created by the presence of
singularities
-what forms first, singularity or apparent horizon (trapped surface)?
-CC proposal  thunderbolts (Hawking 1993, Penrose 1978). -‘wave of singularity’
-changes in ideas of determinism and causality
Lenz's Law: The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant.
Newton’s third law of motion.
-if singularities can be observed from the rest of spacetime causality may break down -> physics loses its predictive power.

27. Future research What are the possible solutions?
Are the solutions stable?
-what will be the fate of an infalling observer, while attempting to cross the inner horizon?
-possible future formulation and proof of CCH
– Critical phenomena beyond spherical symmetry?
-conditions for predictability? pathological behaviour
-Are naked singularities stable and generic?
What do they represent? ( BH information paradox, stability of spacetimes)
-What is the extension of spacetime beyond the singularity?
-better insight into the phenomenon of naked singularity
-open questions: naked singularities, causality violating regions, genericity and stability of naked singularities --investigate the stability properties of the collapse models which develop into naked singularities.
-what happens inside the BH?
-which of the unusual features of Kerr spacetime are physically relevant?

28. Thank you