1. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Area Invariance of Apparent Horizons under Arbitrary Lorentz Boosts Sarp Akcay Center for Relativity University of Texas at Austin

2. MWRM, Nov 16-19, 2006, Washington University, Saint Louis

3. Outline  Motivation  Apparent Horizons  Boosted Schwarzschild black hole  Boosted Kerr black hole  Conclusions

4. MWRM, Nov 16-19, 2006, Washington University, Saint Louis 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.

5. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Apparent Horizons  Outer boundary of a connected component of a trapped region (θ (l) = 0) (Hawking & Ellis)  Outermost marginally trapped surface (θ (l) = 0 and θ (n) < 0)  2 dimensional intersection of the event horizon (EH) worldtube with t = constant hypersurface  Topologically equivalent to 2-spheres.

6. MWRM, Nov 16-19, 2006, Washington University, Saint Louis 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)

7. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Schwarzschild Spacetime  Metric in spherical coordinates for a BH of mass M  Metric in KS coordinates with H = M/ r, r = (x 2 + y 2 + z 2 ) 1/2 and l μ = (1, x/ r, y/ r, z/ r) l μ = (1, x/ r, y/ r, z/ r)

8. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Boosting the Sch. BH  Work with boost friendly coordinates: r ||, r ┴ and φ̃ є [0, 2π] r 2 = r || 2 + r ┴ 2 r 2 = r || 2 + r ┴ 2  Given a boost β = β (sinθ β cosφ β, sin θ β sinφ β, cos θ β ) Kerr-Schild Cartesian coordinates are given by

9. MWRM, Nov 16-19, 2006, Washington University, Saint Louis 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

10. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Boosted Sch. BH  In these new coordinates, the boosted metric becomes  The following transformations occurred r || → γr ||, dr || → γdr || r || → γr ||, dr || → γdr || r 2 = r || 2 + r ┴ 2 → γ 2 r || 2 + r ┴ 2 r 2 = r || 2 + r ┴ 2 → γ 2 r || 2 + r ┴ 2  only spatial components left as dt = 0

11. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Boosted Sch. BH  New coordinate transformation γr || = r cosθ̃ γr || = r cosθ̃ r ┴ = r sinθ̃ θ̃ є [0, π] r ┴ = r sinθ̃ θ̃ є [0, π] with r 2 = γ 2 r || 2 + r ┴ 2  The metric now becomes

12. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Boosted 2-metric  Use r = 2M → dr = 0 to project down to the 2-metric  Since r 2 = γ 2 r || 2 + r ┴ 2 = r 2 This translates to r = 2M → dr = γ 2 r || dr || + r ┴ dr ┴ = 0 r = 2M → dr = γ 2 r || dr || + r ┴ dr ┴ = 0 which gives

13. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Kerr Spacetime  The metric in KS coordinates for a BH of mass M, spin a = J/ M with r 4 – r 2 (x 2 + y 2 + z 2 – a 2 ) – a 2 z 2 = 0  Same coordinate transformation: x, y, z → r ||, r ┴, φ̃  Metric is much more complicated

14. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Kerr Spacetime  Metric in the new coordinates on a t = constant slice  Look at θ β = 0° and 90°

15. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Boosted the Kerr BH  Boost in the z-direction i.e. θ β = 0° We recover the metric in ordinary cylindrical coordinates (r || → γr || )  New spheroidal coordinates γr || = r cosθ̃, r ┴ = (r 2 + a 2 ) 1/2 sinθ̃, θ̃ є [0, π] γr || = r cosθ̃, r ┴ = (r 2 + a 2 ) 1/2 sinθ̃, θ̃ є [0, π] γ 2 r || 2 + r ┴ 2 = r 2 + a 2 sin 2 θ̃ γ 2 r || 2 + r ┴ 2 = r 2 + a 2 sin 2 θ̃

16. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Boosted Kerr 2-metric  r 4 – r 2 (x 2 + y 2 + z 2 – a 2 ) – a 2 z 2 = 0 yields r 2 = r 2 r 2 = r 2  r = r +, dr = 0 → r = r +, dr = 0 with r + = M + (M 2 - a 2 ) 1/2  Putting it all together  (det) 1/2 = ( r + 2 + a 2 )sinθ̃ dθ̃ dφ̃ → Area = 4π( r + 2 + a 2 ) → Area = 4π( r + 2 + a 2 )

17. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Boosted Kerr BH  Boost in the x-y plane i.e. θ β = 90°  New spheroidal coordinates r ┴ cosφ̃ = r cosθ̃, φ’ є [0, 2π] r ┴ cosφ̃ = r cosθ̃, φ’ є [0, 2π] γr || = (r 2 + a 2 ) 1/2 sinθ̃ cosφ’ γr || = (r 2 + a 2 ) 1/2 sinθ̃ cosφ’ r ┴ sin φ̃ = (r 2 + a 2 ) 1/2 sinθ̃ sinφ’ r ┴ sin φ̃ = (r 2 + a 2 ) 1/2 sinθ̃ sinφ’

18. MWRM, Nov 16-19, 2006, Washington University, Saint Louis Boosted 2-metric  We still have γ 2 r || 2 + r ┴ 2 = r 2 + a 2 sin 2 θ̃ γ 2 r || 2 + r ┴ 2 = r 2 + a 2 sin 2 θ̃  Which gives (once again) r = r +, dr = 0 → r = r +, dr = 0 r = r +, dr = 0 → r = r +, dr = 0  Final result

19. MWRM, Nov 16-19, 2006, Washington University, Saint Louis 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

20. MWRM, Nov 16-19, 2006, Washington University, Saint Louis