Analytic continuation of Spacetime Metrics

Analytic continuation of
Spacetime Metrics
Katie Howard
Macquarie University
ANITA workshop
Mount Stromlo Observatory
Canberra, 13-14 March 2009

Historical note
The first public use of the term
"black hole".
- appeared in the Phi Beta
Kapper journal "The American
Scholar"
(Vol.37, No.2, Spring 1968,
pp.248) and in the Sigma Xi
journal, "American Scientist"
Vol.56 No.l Spring 1968, pp. 1-
20.
This page was sent to the
authors by John Wheeler with
his handwritten notes.
-first postulated in 1783 by John
Michell
-the term “black hole” coined in
1969
-observational evidence starting
in 1970s

Formulation of the problem
-BH: regions of space from which nothing, not even light, can escape
because gravity is so strong.
-singularity is “clothed” inside EH where cosmic censorship prevails (we
cannot see inside the event horizon)
-can one See what happens "Inside a Black Hole"?
-is it possible for a distant observer to receive information about the BH interior?
-Can we get information about the region lying inside the apparent horizon?
-investigate the interior of regular axisymmetric and stationary BH
-the occurrence of singularities possible breakdown of GR ->QG??
-what are the possible solutions? Are the solutions stable?
-critical phenomena beyond spherical symmetry? conditions for predictability?
-the final state of gravitational collapse ->spacetime singularities in the causal future of regular initial data,
with divergence of curvature invariants and geodesic incompleteness; we expect continued gravitational
collapse leading to a singularity
-save the situation: spacetime singularities are "invisible" to external observers ->"cosmic censorship
conjecture“ –unproven and the foremost unsolved GR problem.
-under certain physical situations of gravitational collapse spacetime singularities, in the sense of causal
geodesic incompleteness, must occur
-we consider some fundamental questions concerning marginally trapped surfaces (apparent horizons), in
Cauchy data sets for the Einstein equation.
-What is the extension of spacetime beyond the singularity?
Black hole interiors
(a) Trapped surfaces
(b) Singularities and inner horizon instability
(c) Einstein-Rosen bridge on analytic extension

Black holes overview
-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 spacetime
breaks down
-standard picture of collapse to a BH (Penrose 1978)- the singularities are not visible to
observers at a large distance from the hole, being ‘shielded’ from view by an absolute EH.
solutions of the vacuum field equations of GR (1915) G mn
= 0
BH types
schwarzschild - 1916 (static, neutral)
reissner-nordstrøm - 1918 (static,
electrically charged)
kerr - 1963 (rotating, neutral)
kerr-newman 1965 (rotating, charged)
all are Petrov type-D space-times
BH mass hidden in (point or ring)
singularity

Exact solutions
any space-time metric can be regarded as satisfying Einstein's field equations
exact solution = space-time in which the field equations are satisfied with T ab the
energy-momentum tensor of some specified form of matter which obeys
postulate of 'local causality' and the some energy conditions. – for empty space
(T ab = 0), electromagnetic field, perfect fluid/space containing an electromagnetic
field and a perfect fluid.
complexity of the field equations -> solutions in spaces of high symmetry
the unknown = Lorentzian metric g μν
the characteristic sets are its light
cones.
μν
,
The most simple and natural problem of physics since Newton:
-initial conditions and laws of evolution given. Can we predict the future?
-GR: appropriate initial conditions given, can we describe the future universe?

Unpredictability of the solutions
-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
For any P, the hyperbolic nature of the equations determines
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
Schwarzschild solution: the trapped
region coincides with BH and terminates
in a spacelike singularity
conformal representation
of a 2d cross section
-explicit solutions that contain 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

Beyond the Event Horizon
(M′, g′ ab
) is said to be an extension of (M, g ab
) if (M, g ab
) can be
isometrically embedded as a proper open subset of (M′, g′ ab
)
-Christodoulou proved CC for the spherically symmetric
Einstein-scalar field system->trapped regions. A point in
a trapped region corresponds to a trapped surface in
the 4d space-time manifold
-conditions for predictability for the Einstein equations
are related to the behavior of the unique solution of
the initial value problem on the boundary of this
region.
conformal representation of the manifold
into 2d Minkowski space
spacetime=future inextendible as a
manifold with continuous Lorentzian metric
-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 welldefined,
continuous, non-singular limit at the singularity (Tipler, 1977)

Maximal space-time
-represent infinity by bounded region
-use a conformal mapping such that the causality is the one of Minkowski spacetime
-suppress 2 dimensions using symmetry ->future null infinity
-find an appropriate condition to ensure that a given space-time (M, g) is maximally extended
necessary condition in order to rule out inessential singularities which can always be created
just by artificial “cutting out” regular regions from a larger space-time.
condition based on the properties of conjugate points along null geodesics
-Choquet-Bruhat (1952)- for any set of initial data, there exists a local solution and there exists
a unique maximal Cauchy development (M, g ab )
-Is (M, g ab ) complete ?
-Can we extend (M, g ab ) ?
(M, g ab ) = complete if every inextendible causal geodesics is complete
-are all maximal developments complete? negative cf. Penrose singularity theorem
-maximal developments arising from non compact initial data containing a closed trapped
surface are incomplete
-initial data suitably close to the Minkowski case gives maximal developments with features
close to Minkowski spacetime
Conformal infinity -> glue together blocks of spacetime -> bring infinity a bit closer.
examine the infinity structure -> new coordinates where the infinities take on finite values

Closed trapped surfaces
Cauchy horizon (CH)
=surface of infinite blueshift with respect to the EH of the BH ->
leads to a dynamical instability, referred to as 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 closed time-like
Schwarzschild singularity replaced by Cauchy horizon
curves that would otherwise be feasible. Under the averaged weak energy condition, CH are unstable.
geodesic completeness (g-completeness)
- every geodesic can be extended to arbitrary values of its affine parameter.
- 3 kinds: timelike, null and spacelike geodesics. If one cuts a regular point out of space-time, the
resulting manifold is incomplete in all three ways ->a spacetime which was complete in one of them would
be complete in the other two.
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. The advantage of taking timelike or null
incompleteness as being indicative of the presence of a singularity is that on this basis one can establish
a number of theorems about their occurrence.
closed trapped surface T
=a closed (i.e. compact, without boundary) spacelike two-surface such that the two families of null
geodesies orthogonal to T are converging at T. One may think of T as being in such a strong gravitational
field that even the 'outgoing' light rays are dragged back and are, in fact, converging. Since nothing can
travel faster than light, the matter within T is trapped inside a succession of two-surfaces of smaller area -
> something must go wrong.
Trapped surface formation is intimately connected with the question of singularities.

Inside the black hole
Assumption: weakly asymptotically simple and
empty (WASE) space-time
Hypothesis: Every singularity is surrounded by
an EH. There are no naked singularities
-extension will be built up from blocks arising naturally
from the metric itself.
-in order to glue these blocks in a reasonable fashion,
we will first compactify each spacetime following
Penrose, as described by Hawking and Ellis
Kerr black hole
Schwarzschild
black hole

Compacted conformal infinity
by choosing advanced and retarded null coordinates v, w defined
by v = t + r, w = t — r , the second (spherical) metric becomes:
Minkowski spacetime
with its conformal infinity
structure imbedded in the
Einstein static universe
with two spatial
dimensions suppressed.
define p and q by tan p = v and tan q = w, introduce a
conformal factor and extend ds 2 to cover the whole manifold ->
Einstein static universe as the cylinder x 2 +y 2 = 1 imbedded in a
3-d Minkowski space.
conformally map Minkowski spacetime into a patch of the
cylinder; the boundary = conformal structure of infinity
two null surfaces
any future-directed timelike geodesic will originate from i - and terminate at i +
all null-like geodesics begin at past null infinity and end on future null infinity
all spacelike geodesics both begin and end at spacelike infinity i 0
the shaded region = conformal to the whole of Minkowski space-time

Minkowski space retarded coordinates
Minkowski spacetime
• the simplest empty space-time in GR or the SR space-time
• mathematically, the manifold R 4 with a flat Lorentz metric.

Revisiting the problem
Time slices, or Cauchy surfaces, intersect all timelike and null geodesics ->
cross-sections of the space, reaching the boundary everywhere at i 0

Penrose diagrams & spacetime patches

The Schwarzschild black hole
The metric has two singularities:
r = 0
r = r s = 2m, r s = Schwarzschild radius
the singularity at r = r s -> coordinate singularity!
metric singular at r = 0, but we know that flat spacetime has no true spacetime singularities
->coordinates are badly behaved at coordinate singularities!

Schwarzschild maximal extension
1. Connection (bridge) in the sense
of Einstein-Rosen between 2
otherwise euclidean spaces
2. Wormhole in the sense of
Wheeler connecting 2 regions in
1 euclidean space
multiple asymptotic regions
and BH regions that contain
timelike singularities
Curves of constant r = hyperbolas
asymptotic to the lines r=2m
In the (u, v) plane, the spacetime
is regular
If a test particle crosses r=2m
into the interior, it can never get
back out but must hit inevitably
the singularity r=0 (curvature
invariants infinite) ->guarantee
of causality non-violation (no
signals sent through the wormhole
faster than c)

Cross-section of a Kerr black hole
Event horizon at (∆=0) and angular momentum between
0≤a 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; exposed singularities (naked)
-subsitute
solution.
->Kerr-Newman metric. for a=0 ->Reissner-Nordstrom

Kerr analytic extension
In Kerr-Schild coordinates ,the metric takes the form
r determined implicitly, up to a sign, in terms of x, y, z
The function r can in fact be analytically
continued from positive to negative values
through the interior of the disc x 2 + y 2 < a 2 , z
= 0, to to obtain a maximal analytic extension
of the solution.
attach another plane (x‘, y', z') where a point
on the top side of the disc x 2 + y 2 < a 2 , 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.

Maximal extension diagrams
Penrose diagram of maximally
extended Reissner-Nordstrom BH
at the border the null energy
condition will start to be violated

Black Hole interiors
-over the last 30 years, BH have been shown to have a number of surprising properties -
directions for further research?
-unforeseen relations between distinct areas of GR, quantum physics and statistical
mechanics. -> deep puzzles at the foundations of physics
-theoretical investigations of BH interiors aim to understand the geometry of spacetime inside
generic BH - significant progress in the last decade.
-singularities inside spherical, (non-)charged BH: null singularity along the Cauchy horizon
precedes a spacelike, central singularity. The singularity along the Cauchy horizon is weak
(tidal distortion of extended bodies is finite there).
The nature of the Cauchy horizon singularity inside rotating BH: the divergence of curvature
dominated by the propagating modes of the gravitational field leading to an intuitive picture
of the singularity as a singular shock wave propagating along the Cauchy horizon.
Proposed BH definition = missing piece of the spacetime that disrupts the predictability
of the laws of physics and whose interior structure depends on the conditions on the event
horizon at very distant future (infinite future) of an observer. The presence of a Cauchy
horizon (light-like signals propagating from the future infinity), generated by infalling matter
hitting the singularity and feeding the BH with information represents the minimal condition
for forming a realistic BH.
BH existence condition - possible only if the spacetime possesses inextendible curves
with finite affine length.

Where are we?
-if singularities can be observed from the rest of spacetime causality may break down ->
physics loses its predictive power. The existence of BH = conundrum
-facts: matter reaches the speed of light, hits the singularity, feeds the BH with information
that can be used to create conditions of predictability of the infinite past.
It may be possible to “feed” the initial conditions of the spacetime metric to be primarily
created.
We may possess the current conditions to “predict” the past initial conditions of the
spacetime topology.
Which comes first? (circular cause and consequence): singularity as a necessary condition
of the metric or matter reaching the singularity causes the spacetime fabric to break.
Our hypothesis could answer the question and close the circle.
– 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
-censorship proposal doesn’t eliminate the possibility of as thunderbolts (Hawking 1993,
Penrose 1978). -‘wave of singularity’
-what forms first, singularity or apparent horizon (trapped surface)?

Questions?