Titles

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Title(s)

• Dynamical Black Holes and Horizons
• Evolving Black Holes and the Growth of Horizons
• Dynamical Black Holes and the Growth of Horizons

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Abstract

• Recently there has been some interest in
  quasi-local definitions of black holes. The key
  features of these definitions will be discussed
  and simple examples will be given in the context
  of fully dynamical, spherically symmetric
  spacetimes. These models give useful insights
  into both black hole accretion and black hole
  evaporation.

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Evolving Black Holes and the Growth of Horizons
Alex Nielsen(Canterbury University)
Berlin, October 9th, 2006
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History

• (1965) Penrose introduces the idea of trapped
  surfaces to complete his singularity proofs.
• (1972) Hawking introduces the notion of event
  horizons, to capture the idea of a black hole.
• (1994) Hayward introduces the notion of trapping
  horizons to extend Penroses idea
• (1997) Ashtekar et al. used isolated horizons to
  compute the entropy of a black hole in Loop
  Quantum Gravity.

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Event Horizons

• Intuitive definition inside an event horizon one
  cannot send signals to infinity.
• Technical definition the past causal boundary of
  future null infinity (if future null infinity
  exists).

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Intuitive Definition of Trapped Surfaces

• Neighbouring light rays must move towards one
  another.

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(Technical points on trapped surfaces)

• A trapped surface is a two dimensional spacelike
  surface whose two null normals have negative
  expansion.
• If one of the expansions is zero we have a future
  marginal surface.
• The trapped region is the region containing
  trapped surfaces.
• The boundary of (a connected component of) the
  trapped region is an apparent horizon.

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Traditional Black Holes

• Intuition is largely based on globally stationary
  solutions of Einsteins equations, such as the
  Schwarzschild solution and the Kerr solution.
• Global Killing vectors are used to define masses
  and angular momentum, either as a boundary term
  at the Killing horizon or at infinity.
• Black holes arent black when quantum field
  effects are considered on the curved spacetime.

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Killing Vectors

• Standing at a fixed spatial position in
  Schwarzschild, four-velocity is

The spacetime looks the same as one moves along
this curve, it is a Killing vector.
To stay on this curve at one must move at
the speed of light, it is a Killing horizon.
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The Area Law for Black Holes

• If the Null Energy Condition is violated the area
  of a black hole can decrease.

or (for a perfect fluid)
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Trapping horizons

• A (future, outer) trapping horizon is a 3
  dimensional hypersurface that can be foliated by
  2 dimensional spacelike surfaces.
• The 2 null normals to the foliation surfaces (n
  and l) have the properties

(trapping boundary)
(future)
(outer)
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An expanding horizon is spacelike
The vector tangent to the horizon but normal to
the foliation
Thus
The change in the area in moving from one
foliation to another
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The second law for trapping horizons
The Raychaudhuri equation for a null congruence
For a hypersurface orthogonal, shear free,
trapping horizon
The vector tangent to the horizon, but normal to
the foliations, can be written
The expansion can be written
Change in areaalong horizon is
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Why Event Horizons

• The event horizon is always a null hypersurface.
• The event horizon does not depend on a choice of
  foliation.
• The event horizon always evolves continuously.
• The event horizon captures the intuitive idea of
  the boundary of what can reach observers at
  infinity.

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Why Local Horizons

• They can be located on hypersurfaces and do not
  require knowledge of the full spacetime.
• They cannot develop in flat spacetime.
• They are independent of the details of any
  asymptotic regions.
• They are independent of the details of physics
  near singularities.

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The way the information gets out seems to be
that a true event horizon never forms, just an
apparent horizon." Hawking, 2004
The supposed information paradox for black holes
is based on the fundamental misunderstanding that
black holes are usefully defined by event
horizons. Understood in terms of locally defined
trapping horizons, the paradox disappears
information will escape from an evaporating black
hole. Hayward, 2005
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Bekenstein-Hawking Entropy
Underlying quantum gravitational degrees of
freedom? What spacetimes/boundary conditions?

• Ashtekar, Baez, Corichi, Krasnov gr-qc/9711031
• Strominger, Vafa hep-th/9601029

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A Metric to Describe Evolving Horizons
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Derivation of spherically symmetric metric

• Killing vectors of spherical symmetry

Gives a general, spherically symmetric metric of
the form
(Technical point, the gradient of the q
coordinate must be spacelike.)
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Spherically Symmetric metrics
A general spherically symmetric metric can be
written in Schwarzschild/curvature coordinates as,
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Painlevé-Gullstrand coordinates
Transforming the time coordinate and
demandingthat gives
where
and
(Technical point, one needs to pick a boundary
condition to fully specify the new coordinates.)
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Radial Null Vectors
Outgoing null geodesic
Ingoing null geodesic
(Technical point, the normalisation is chosen to
give simple formulae later on and maintain some
symmetry between l and n.)
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Surface gravity
We can define a surface gravity by
For the spherically symmetric metric
And thus
(Technical point, this looks like the first law
but it involves partial derivatives, not full
derivatives.)
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The expansions
Expansion of a vector k
For the outgoing null geodesics we get
And for the ingoing,
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Outer horizon condition

• For the spherically symmetric metric in
  Painlevé-Gullstrand coordinates

For the ingoing null geodesic we have
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Evolution of the horizons
For this metric we find
At the horizon this can be rearranged to give
And for the case where the horizon is moving
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Advantages

• For spherically symmetric foliations, apparent
  horizons coincide with trapping horizons.
• The external matter is totally arbitrary.
• Any matter falling across the horizon can be
  timelike, spacelike or null.
• They can include Null Energy Condition violating
  evaporating black holes.

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Disadvantages

• They are restricted to spherical symmetry.
• The foliations and two-surfaces are required to
  be spherically symmetric.
• Some equations are coordinate dependent.
• They are purely classical in the sense of
  containing a classical spacetime metric

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(adapted from HawkingEllis)
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Jumpy Horizons

• The horizon will jump if the number of
  solutions to

changes from one hypersurface to the next. This
will occur if the matter crosses the horizon with
or
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Mass profile of an infinitely thin collapsing
shell
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Time evolution of the function 1-2m/r for the
infinitely thin collapsing mass shell
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Mass profile for finitely thick collapsing shell
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Time evolution of the function 1-2m/r for the
finitely thick collapsing shell
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Conclusions

• Local horizons can reproduce many of the results
  of event horizons and offer some significant
  advantages
• Spherically symmetric spacetimes admit local
  horizons, are simple to work with and can provide
  useful intuition for more general situations.
• However, not all features are ideal and care
  still needs to be taken.

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Vaidya metric

• (Schrodinger Black Hole)

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