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Area Invariance of Apparent Horizons under Arbitrary Lorentz Boosts

Area Invariance of Apparent Horizons under Arbitrary Lorentz Boosts. Sarp Akcay Center for Relativity University of Texas at Austin. Outline. Motivation Apparent Horizons Boosted Schwarzschild black hole Boosted Kerr black hole Conclusions. Motivation. A well known result in relativity

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Area Invariance of Apparent Horizons under Arbitrary Lorentz Boosts

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

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

  3. Outline • Motivation • Apparent Horizons • Boosted Schwarzschild black hole • Boosted Kerr black hole • Conclusions MWRM, Nov 16-19, 2006, Washington University, Saint Louis

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

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

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

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

  8. Boosting the Sch. BH • Work with boost friendly coordinates: r||, r┴ and φ̃є[0, 2π] r2 = r||2 + r┴2 • Given a boost β = β (sinθβ cosφβ, sin θβ sinφβ, cos θβ) Kerr-Schild Cartesian coordinates are given by MWRM, Nov 16-19, 2006, Washington University, Saint Louis

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

  10. Boosted Sch. BH • In these new coordinates, the boosted metric becomes • The following transformations occurred r||→ γr||, dr||→ γdr|| r2 = r||2 + r┴2 → γ2r||2 + r┴2 • only spatial components left as dt = 0 MWRM, Nov 16-19, 2006, Washington University, Saint Louis

  11. Boosted Sch. BH • New coordinate transformation γr||= r cosθ̃ r┴ = r sinθ̃θ̃є[0, π] with r2 = γ2r||2 + r┴2 • The metric now becomes MWRM, Nov 16-19, 2006, Washington University, Saint Louis

  12. Boosted 2-metric • Use r = 2M → dr = 0 to project down to the 2-metric • Since r2 = γ2r||2 + r┴2 = r2 This translates to r = 2M →dr = γ2 r|| dr|| + r┴ dr┴= 0 which gives MWRM, Nov 16-19, 2006, Washington University, Saint Louis

  13. 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┴, φ̃ • Metric is much more complicated MWRM, Nov 16-19, 2006, Washington University, Saint Louis

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

  15. Boosted the Kerr BH • Boost in the z-direction i.e.θβ= 0° We recover the metric in ordinary cylindrical coordinates (r||→ γr||) • New spheroidal coordinates γr||= r cosθ̃ , r┴ = (r2 + a2)1/2sinθ̃ , θ̃є[0, π] γ2r||2 + r┴2 = r2 + a2sin2θ̃ MWRM, Nov 16-19, 2006, Washington University, Saint Louis

  16. 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 = (r+2 + a2)sinθ̃ dθ̃ dφ̃ → Area = 4π(r+2 + a2) MWRM, Nov 16-19, 2006, Washington University, Saint Louis

  17. Boosted Kerr BH • Boost in the x-y plane i.e. θβ= 90° • New spheroidal coordinates r┴cosφ̃ = r cosθ̃ , φ’ є[0, 2π] γr|| = (r2 + a2)1/2 sinθ̃ cosφ’ r┴ sinφ̃ = (r2 + a2)1/2 sinθ̃ sinφ’ MWRM, Nov 16-19, 2006, Washington University, Saint Louis

  18. Boosted 2-metric • We still have γ2r||2 + r┴2 = r2 + a2sin2θ̃ • Which gives (once again) r = r+, dr = 0 → r= r+, dr = 0 • Final result MWRM, Nov 16-19, 2006, Washington University, Saint Louis

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

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

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