SLICE STRETCHING ARISING FOR MAXIMAL SLICING OF SCHWARZSCHILD

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SLICE STRETCHING ARISING FOR MAXIMAL SLICING OF
SCHWARZSCHILD
Bernd Reimann AEI, 8/3/2004
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Aim of this Talk


is to give a brief summary of the following
papers 1. Maximal,gr-qc/0307036, PRD69 (2004)
044006 derivation of zgp maximal slices (for
RN) 2. Late Time Analysis, gr-qc/0401098,
submitted to PRD discussion of the late time
behavior of the maximal slices (for RN) 3. Slice
Stretching (in preparation) origin of slice
sucking wrapping for maximally slicing SS
with vanishing shift, effect of BCs 4. Geodesic
Slicing (in preparation) slice stretching at
the EH of a SS BH for different lapse choices 5.
Shift (planned) avoiding slice stretching with
shift and/or singularity excision for maximally
slicing SS
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Deriving the Maximal Slices
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For the Schwarzschild BH spacetime one can
analytically find the maximal slices by solving
Einsteins equations while demanding the
maximality condition The lapse then arises
from the elliptic equation when specifying 2
BCs for this 2nd-order ODE 1) The lapse is
unity at spatial infinity to measure proper time
there. 2a) odd lapse Schwarzschild metric in
static coordinates is recovered 2b) even lapse
collapse of the lapse at the throat, maximal
slices approach the limiting surface
avoiding the singularity 2c) general lapse can
be constructed as linear combination since the
ODE is linear in ?, slices again approach
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Maximal Slices in Carter-Penrose Diagrams
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odd
even
zgp
infinity
puncture
right EH
left EH
throat
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Collapse of the Puncture Lapse
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left EH throat right EH
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Approaching the Limiting Slice r 3M/2
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(Areal radius obtained as root of angular metric
part)
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Development of a Peak in the Metric
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Note that the peak in g is located in between
throat and right EH
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Late Time Analysis
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• The analytically found 4-metric for evolutions of
  Schwarzschild puncture data is valid for all
  times but difficult to evaluate at late
  times
• Hence it is useful to perform a late time
  analysis in order to discuss the behavior of
  the maximal slices at 5 markers being from left
  to right
• puncture - lefthand EH throat - righthand
  EH infinity

• The idea here is to introduce
  , to look at expansions of ? and discuss
  the limit
• For even and zgp BCs it turns out that ? is
  decaying with time exponentially where a
  fundamental timescale is given by
• (Estabrook et
  al. 1973, Beig 98)

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Slice Stretching for the Puncture Evolution
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From left to right in terms of
Slices penetrate EH to approach r 3M/2
asymptotically
Collapse of the lapse, outward moving shoulder
Puncture
Holds not for the odd but for all other
BCs! Numerically a value of ? ? 0.3 has been
used to locate the EH for excision!
Lefthand EH
Throat
Righthand EH
Infinity
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Slice Stretching evaluated at the markers
2
right EH peak in g throat left EH

• Slice Sucking occurs like
• as measured at throat or EH
• Slice Wrapping takes place as a peak in the
  radial metric component develops like

right EH peak in g throat left EH
The peak in g does not grow exponentially!
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Origin of Slice Stretching
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• Analyzing coordinate transformations which lead
  to Eulerian spatial coordinates z on the maximal
  slices (i.e. zero shift in the metric), in terms
  of ? it turns out that independently of BCs
  certain integrals like
• diverge when approaching the limiting slice rlim
  3M/2.
• Slice stretching is caused by this diverging term
  being picked up at the throat.
• Hence either slice sucking (throat and e.g. EH
  move away from each other) or slice wrapping
  (steepening gradient in between throat and e.g.
  EH) or both effects occur in this limit.
• To discuss these effects in terms of time at
  infinity ?, it is necessary to specify BCs.

as
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Example even/zgp BCs andIsotropic Grid
Coordinates
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left EH throat right EH
zgp
even
zgp
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Make Slice Stretching occur late
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• The idea now is to make slice stretching
  described in terms of ? occur late at large
  values of ?. This can be achieved by demanding
  that ? approaches zero slowly as a function of
  ?.
• Specifying ?(?), however, corresponds to imposing
  BCs as for
• To get better behavior as for even BCs,
  with
  , the lapse gets more favorable the closer it
  is to the odd lapse, the latter being negative in
  the left-hand region.
• Demanding for a numerically favorable lapse in
  addition i.e. ? ? ½, it turns out that the
  average of odd and even lapse is the
  best-possible lapse!

holds
The zgp lapse at late times has ?zgp ½!
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Example One-parameter family of BCs
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• Numerical tests have been performed with
• in the linear combination corresponding to the
  inner BC
• For parameter range note that
• odd BCs
• even BCs
• At late times as predicted
• is observed numerically.

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The Singularity Avoidance as intuitive Argument
for the Origin of Slice Stretching
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Singularity

• constant time slices wrap up around the
  singularity (Slice Wrapping)
• evolution essentially frozen in the
  inner region
• evolution marches ahead outside to cover a large
  fraction of the spacetime
• infalling observers (Slice
  Sucking)
• gt Slice Stretching

Event Horizon
t150
t100
Throat
Throat
t50
t0
Collapsing Star
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Geodesic Slicing vs. Maximal Slicing
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The overall slice stretching is in leading
order proportional to time for both geodesic and
maximal slicing,
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Avoid Slice Stretching I Throat Excision
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• For any shift and arbitrary spatial coordinates z
  by integrating Gzz from the left- to the
  right-hand EH, one finds
• Hence slice stretching has to be present in
  between the limits of integration!?!
• However, if the throat is not part of the slice,
  i.e. if a piece in between left- and
  right-hand EH containing the throat is excised,
  the late time divergence is not present and
  slice stretching is stopped as the evolution
  freezes.
• Throat Excision (a better name than
  singularity excision since the singularity is
  not encountered by maximal slices) works
  numerically as shown e.g. in Anninos et al.
  1995.

as
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Avoid Slice Stretching II Conformal Factor
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• The radial metric component
  appearing in the slice stretching integral can
  be written as
  making use of a conformal factor.
• In particular, if ? has a coordinate singularity,
  the integral
• can diverge while numerically evolved quantities
  freeze.
• For puncture evolutions, with a
  (finite) shift has to act such that throat and
  left-hand EH move towards x 0 like
• For 1log lapse and gamma-freezing shift
  numerical evidence is found in Alcubierre et al.,
  2003.
• But for logarithmic coordinates, with
  there is no chance to avoid slice stretching
  which explains numerical observations made in the
  PhD Thesis of Bernstein Daues!