Area Invariance of Apparent Horizons under Arbitrary Lorentz Boosts

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Area Invariance of Apparent Horizons under
Arbitrary Lorentz Boosts

• Sarp Akcay
• Center for Relativity
• University of Texas at Austin

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Outline

• Motivation
• Apparent Horizons
• Boosted Schwarzschild black hole
• Boosted Kerr black hole
• Conclusions

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Motivation

• A well known result in relativity
• Null surfaces remain null
• Thermodynamic considerations
• Schwarzschild (Sch.) black hole (BH) boosted in
  the z-direction calculated explicitly by Matzner
  in Kerr-Schild (KS) coordinates.
• Generalize to arbitrary boosts for Sch. and Kerr
  BHs in KS coordinates.

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Apparent Horizons

• Outer boundary of a connected component of a
  trapped region (?(l) 0) (Hawking Ellis)
• Outermost marginally trapped surface (?(l) 0
  and ?(n) lt 0)
• 2 dimensional intersection of the event horizon
  (EH) worldtube with t constant hypersurface
• Topologically equivalent to 2-spheres.

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Boosting a Spacetime

• Work with spacetimes that can be cast the metric
  into Kerr-Schild (KS) form
• Admits a Lorentz boost
• Retains the same form under Lorentz boosts
• Horizon appears distorted due to contraction
  (coordinate effect)

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Schwarzschild Spacetime

• Metric in spherical coordinates for a BH of mass
  M
• Metric in KS coordinates
• with H M/ r, r (x2 y2 z2)1/2 and
• lµ (1, x/ r, y/ r, z/ r)

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Boosting the Sch. BH

• Work with boost friendly coordinates r, r- and
  f ? 0, 2p
• r2 r2 r-2
• Given a boost
• ß ß (sin?ß cosfß, sin ?ß sinfß, cos ?ß)
• Kerr-Schild Cartesian coordinates are given by

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Boosting the Sch. BH

• ADM 3 1 split
• AH is intersection of EH
  with a t constant slice
  ? dt 0 in the metric.
• Work with t 0 slice
• t-t and t-i components of the metric drop out

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Boosted Sch. BH

• In these new coordinates, the boosted metric
  becomes
• The following transformations occurred
• r ? ?r, dr ? ?dr
• r2 r2 r-2 ? ?2r2 r-2
• only spatial components left as dt 0

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Boosted Sch. BH

• New coordinate transformation
• ?r r? cos?
• r- r? sin? ? ? 0, p
• with r?2 ?2r2 r-2
• The metric now becomes

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Boosted 2-metric

• Use r 2M ? dr 0 to project down to the
  2-metric
• Since r2 ?2r2 r-2 r?2
• This translates to
• r? 2M ? dr? ?2 r dr r- dr- 0
• which gives

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Kerr Spacetime

• The metric in KS coordinates for a BH of mass M,
  spin a J/ M
• with r4 r2 (x2 y2 z2 a2) a2z2 0
• Same coordinate transformation
• x, y, z ? r, r-, f
• Metric is much more complicated

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Kerr Spacetime

• Metric in the new coordinates on a t
  constant slice
• Look at ?ß 0 and 90

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Boosted the Kerr BH

• Boost in the z-direction i.e. ?ß 0
• We recover the metric in ordinary cylindrical
  coordinates (r ? ?r)
• New spheroidal coordinates
• ?r r? cos? , r- (r?2 a2)1/2sin? , ? ?
  0, p
• ?2r2 r-2 r?2 a2sin2?

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Boosted Kerr 2-metric

• r4 r2 (x2 y2 z2 a2) a2z2 0 yields
• r2 r?2
• r r, dr 0 ? r? r, dr? 0 with r M
  (M2 - a2)1/2
• Putting it all together
• (det)1/2 (r2 a2)sin? d? df
• ? Area 4p(r2 a2)

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Boosted Kerr BH

• Boost in the x-y plane i.e. ?ß 90
• New spheroidal coordinates
• r-cosf r? cos? , f ? 0,
  2p
• ?r (r?2 a2)1/2 sin? cosf
• r- sin f (r?2 a2)1/2 sin? sinf

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Boosted 2-metric

• We still have
• ?2r2 r-2 r?2 a2sin2?
• Which gives (once again)
• r r, dr 0 ? r? r, dr? 0
• Final result

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Conclusion

• Boosted the Sch. BH in an arbitrary direction
• Boosted the Kerr BH along the z-axis and in the
  x-y plane
• Shown the invariance of the area for the
  transformations above
• Next repeat for the Kerr BH in an arbitrary
  direction

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