II Workshop on Black Holes

Transcription

1 II Workshop on Black Holes December 2009 Instituto Superior Técnico, Lisboa Madalena Lemos (CENTRA/IST) Work in progress: E. Berti, V. Cardoso, T. Hinderer, M.L., F. Pretorius, U. Sperhake, N. Yunes

2 Zero frequency limit of gravitational radiation in high-energy collisions Zero Frequency Limit (ZFL) (Weinberg 64 Smarr 77) L. Smarr, Phys. Rev. D 15, 2069 (1977)

3 Zero frequency limit of gravitational radiation in high-energy collisions Zero Frequency Limit (ZFL) (Weinberg 64 Smarr 77) Zero duration infinite acceleration Hard collision incoming and outgoing trajectories with constant velocities L. Smarr, Phys. Rev. D 15, 2069 (1977)

4 Zero frequency limit of gravitational radiation in high-energy collisions Zero Frequency Limit (ZFL) (Weinberg 64 Smarr 77) Zero duration infinite acceleration Hard collision incoming and outgoing trajectories with constant velocities Valid for arbitrary velocities Details of internal structure irrelevant Flat, large memory L. Smarr, Phys. Rev. D 15, 2069 (1977)

5 ZFL: Head-on Collision Free particles, changing abruptly at t=0: z Φ y x θ L. Smarr, Phys. Rev. D 15, 2069 (1977)

6 ZFL: Head-on Collision Free particles, changing abruptly at t=0: z Φ x θ y Center of momentum frame: Independent of ω Introduce a cutoff frequency L. Smarr, Phys. Rev. D 15, 2069 (1977)

7 ZFL: Head-on Collision Radiated Momentum: Momentum is radiated in the direction of motion

8 ZFL: Head-on Collision Radiated Momentum: Momentum is radiated in the direction of motion

9 Numerical Results: Head-on Collision ZFL Point-particle falling into a black hole

10 Numerical Results: Head-on Collision U. Sperhake, V. Cardoso, F. Pretorius, E. Berti, and J.A. Gonzalez, Phys. Rev. Lett., 101, (2008) ZFL ZFL Point-particle falling into a black hole Collision of two equal mass black holes

11 ZFL: Non Head-on Collision Center of momentum frame Rotation frequency: =ξ 2

12 ZFL: Non Head-on Collision Center of momentum frame Rotation frequency: =ξ 2 Energy momentum tensor is not conserved!

13 ZFL: Non Head-on Collision Constraining forces: (Price and Sandberg 73) R. H. Price and V. D. Sandberg, Phys. Rev. D 8, 1640 (1973)

14 ZFL: Non Head-on Collision equal mass collision Radiated energy: Fourier transform:

15 ZFL: Non Head-on Collision equal mass collision Radiated energy: Fourier transform: Jacobi-Anger expansion

16 ZFL: Non Head-on Collision equal mass collision All spectra blow up for Except for Only blows up at Zero frequency limit: Smarr s head-on collision is recovered E. Poisson, Phys. Rev. D47, 1497 (1993).

17 ZFL: Non Head-on Collision equal mass collision All spectra blow up for Except for Only blows up at Zero frequency limit: E. Poisson, Phys. Rev. D47, 1497 (1993).

18 ZFL: Non Head-on Collision extreme mass ratio Center of momentum frame All spectra blow up for Except for Only blows up at

19 ZFL: Non Head-on Collision extreme mass ratio Center of momentum frame All spectra blow up for Except for Only blows up at Zero frequency limit: Smarr s head-on collision is recovered

20 ZFL: Non Head-on Collision extreme mass ratio Center of momentum frame All spectra blow up for Except for Only blows up at Zero frequency limit: Smarr s head-on collision is recovered

21 Numerical Results: Non Head-on Collision Head-on limit spectrum is flat for small frequencies Different behaviour near resonance for numerical data and toy model limit very weakly dependent on consistent with the toy model

22 Numerical Results: Non Head-on Collision l=2 limit very weakly dependent on Head-on case ZFL yields Good agreement with numerical results

23 Numerical Results: Non Head-on Collision l=2 have positive/negative slope for Toy model:

24 Summary Zero-frequency limit is independent of the impact parameter Numerical results weakly dependent on the impact parameter Different small frequency behaviour Not the same behaviour near resonances Future Work Dependence on the constraining forces Particles with structure Higher-dimensions Radiated Momentum Different ways to model the collision Thank you

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27 Small Mω: ZFL: Non Head-on Collision equal mass collision

28 Numerical Results: Non Head-on Collision

29 ZFL: Non Head-on Collision Same structure Zero frequency limit: Smarr s head-on collision is recovered

30 Numerical Results: Non Head-on Collision l=2 have positive/negative slope for Toy model:

31 Numerical Results: Non Head-on Collision Head-on case ZFL yields Good agreement with numerical results

32 Smarr g + μν = ημν hμν z Φ z Φ y y x θ x θ