Shock Wave Collisions. Carlos A. R. Herdeiro Departamento de Física da Universidade de Aveiro

Transcription

1 Shock Wave Collisions Carlos A. R. Herdeiro Departamento de Física da Universidade de Aveiro

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6 Belém do Pará - Aveiro: Cidades Irmãs Iniciativa do Prefeito de Santa Maria de Belém de Grão Pará o Dr. Stélio de Mendonça Maroja Convénio assinado a 12 de Janeiro de 1970 O Dr. Leandro Tocantis, Adido à Embaixada do Brasil em Lisboa, em representação do Encarregado de Negócios, descerra a lápide com o nome de «Rua de Belém do Pará - Cidade Irmã». Publicado em Aveiro e o seu Distrito, Publicação Semestral da Junta Distrital de Aveiro, nº 9, Junho de 1970

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8 Formation of a Common Event Horizon

9 Formation of a Common Event Horizon

10 Ultra-relativistic particle collision:

11 Ultra-relativistic particle collision:

12 Ultra-relativistic particle collision: γ,w ±,Z 0,g,...

13 Ultra-relativistic particle collision: At sufficiently high energies: γ,w ±,Z 0,g,...

14 Ultra-relativistic particle collision: At sufficiently high energies: γ,w ±,Z 0,g,... At energies well above the Planck energy, black hole production sets in, accompanied by the coherent emission of real gravitons (gravitational waves) t Hooft 87;

15 Ultra-relativistic particle collision:

16 Ultra-relativistic particle collision: Riemann 1 s

17 Ultra-relativistic particle collision: Riemann 1 s The corresponding computations should follow completely from the well known laws of general relativity, since any non-trivial quantum field theoretical phenomena are well hidden behind the horizon (on which quantum corrections become small for s large).

18 Ultra-relativistic particle collision: Riemann 1 s The corresponding computations should follow completely from the well known laws of general relativity, since any non-trivial quantum field theoretical phenomena are well hidden behind the horizon (on which quantum corrections become small for s large). Graviton dominance in ultra-high-energy scattering

19 Along the idea of the hoop conjecture Thorne 72 Distance λ = p c E Planck scale r s GE c 4 Energy de Broglie wavelength smaller than Schwarzschild radius suggests black hole formation

20 Along the idea of the hoop conjecture Thorne 72 Distance λ = p c E Planck scale r s GE c 4 Energy de Broglie wavelength smaller than Schwarzschild radius suggests black hole formation This argument is supported by numerical evidence Choptuik, Pretorius 09

21 Two important quantities: 1) INELASTICITY 2) CRITICAL IMPACT PARAMETER

22 Two important quantities: 1) INELASTICITY 2) CRITICAL IMPACT PARAMETER A relevant extension: CONSIDER GENERAL RELATIVITY IN D-DIMENSIONS - TeV-gravity scenarios - D as a parameter

23 Numerical relativity approach:

24 Numerical relativity approach: Energy extracted in a head-on collisions of equal mass black holes (in terms of centre of mass energy) Sperhake, Cardoso, Pretorius, Berti, Gonzales, 08 Witek, Zilhao, Cardoso, Gualtieri, Nerozzi, CH, Sperhake 10 Low energy High energy D= % 14% D=5 D=6 D=7 D=8 D=9 D=10

25 Numerical relativity approach: Energy extracted in a head-on collisions of equal mass black holes (in terms of centre of mass energy) Sperhake, Cardoso, Pretorius, Berti, Gonzales, 08 Witek, Zilhao, Cardoso, Gualtieri, Nerozzi, CH, Sperhake 10 Low energy High energy D= % 14% D= % D=6 D=7 D=8 D=9 D=10

26 Numerical relativity approach: Energy extracted in a head-on collisions of equal mass black holes (in terms of centre of mass energy) Sperhake, Cardoso, Pretorius, Berti, Gonzales, 08 Witek, Zilhao, Cardoso, Gualtieri, Nerozzi, CH, Sperhake 10 Low energy High energy D= % 14% D=5 D=6 D=7 D=8 D=9 D= % Uli s talk Uli s talk

27 (Semi-) Analytical approach: Shock wave collisions

28 Two colliding shock waves t Flat region II Flat region I z u=0 ds 2 AS(E, u = 0)

29 Two colliding shock waves t Flat region II Flat region I z v=0 ds 2 AS(E, v = 0)

30 Two colliding shock waves t Curved region IV Flat region II Flat region I z regions I, II and III v=0 Flat region III ds 2 AS(E, v = 0) + u=0 ds 2 AS(E, u = 0)

31 Two colliding shock waves t Curved region IV Flat region II Flat region I z regions I, II and III v=0 Flat region III ds 2 AS(E, v = 0) + u=0 ds 2 AS(E, u = 0) Determine apparent horizon on past light cone: Penrose 74; Eardley and Giddings 02

32 Two colliding shock waves t Curved region IV u = 8GE ln regions I, II and III r 4GE Flat region II v=0 Flat region III ds 2 AS(E, v = 0) + Flat region I u=0 ds 2 AS(E, u = 0) z v = 8GE ln r 4GE Determine apparent horizon on past light cone: Penrose 74; Eardley and Giddings 02

33 Two colliding shock waves t Curved region IV u = 8GE ln regions I, II and III r 4GE Flat region II v=0 Flat region III ds 2 AS(E, v = 0) + Flat region I u=0 ds 2 AS(E, u = 0) z v = 8GE ln r 4GE Determine apparent horizon on past light cone: Penrose 74; Eardley and Giddings 02 Has the intrinsic geometry of two flat disks at u=0, v=0:

34 Two colliding shock waves t Curved region IV u = 8GE ln regions I, II and III r 4GE Flat region II v=0 Flat region III ds 2 AS(E, v = 0) + Flat region I u=0 ds 2 AS(E, u = 0) z v = 8GE ln r 4GE Determine apparent horizon on past light cone: Penrose 74; Eardley and Giddings 02 Has the intrinsic geometry of two flat disks at u=0, v=0:

35 Two colliding shock waves t Curved region IV u = 8GE ln regions I, II and III r 4GE Flat region II v=0 Flat region III ds 2 AS(E, v = 0) + Flat region I u=0 ds 2 AS(E, u = 0) z v = 8GE ln r 4GE Determine apparent horizon on past light cone: Penrose 74; Eardley and Giddings 02 Has the intrinsic geometry of two flat disks at u=0, v=0:

36 E rad AH bound 2E Eardley and Giddings, 2002

37 E rad AH bound 2E Eardley and Giddings, 2002

38 E rad AH bound 2E Eardley and Giddings, 2002 In D=4, curiously, this bound is the same than the one for two black holes colliding from rest, starting at infinity, obtained from the area theorem.

39 Two colliding shock waves t Curved region IV Flat region II Flat region I z Flat region v=0 III u=0 regions I, II and III ds 2 AS(E, v = 0) + ds 2 AS(E, u = 0) region IV? D Eath and Payne 92 gave a more precise estimate of the energy radiated:

40 Two colliding shock waves t Curved region IV Flat region II Flat region I z Flat region v=0 III u=0 regions I, II and III ds 2 AS(E, v = 0) + ds 2 AS(E, u = 0) region IV? D Eath and Payne 92 gave a more precise estimate of the energy radiated: 1) they moved into a highly boosted frame (velocity v, 1-v<<1):

41 Two colliding shock waves t Curved region IV Flat region II Flat region I z regions I, II and III v=0 Flat region III + u=0 ds 2 AS(e α E, v = 0) ds 2 AS(e +α E, u = 0) region IV? D Eath and Payne 92 gave a more precise estimate of the energy radiated: 1) they moved into a highly boosted frame (velocity v, 1-v<<1): e α = 1 + v 1 v

42 Two colliding shock waves t Curved region IV Flat region II Flat region I z regions I, II and III v=0 Flat region III + u=0 ds 2 AS(e α E, v = 0) ds 2 AS(e +α E, u = 0) region IV? D Eath and Payne 92 gave a more precise estimate of the energy radiated: 1) they moved into a highly boosted frame (velocity v, 1-v<<1): e α = 1 + v 1 v 2) made a perturbative expansion of the metric in region IV, using the ratio of energies as the expansion parameter; they computed the metric to second order

43 Two colliding shock waves t Curved region IV Flat region II Flat region I z regions I, II and III v=0 Flat region III + u=0 ds 2 AS(e α E, v = 0) ds 2 AS(e +α E, u = 0) region IV? D Eath and Payne 92 gave a more precise estimate of the energy radiated: 1) they moved into a highly boosted frame (velocity v, 1-v<<1): e α = 1 + v 1 v 2) made a perturbative expansion of the metric in region IV, using the ratio of energies as the expansion parameter; they computed the metric to second order 3) to this order they computed the news function and obtained that 16,4% of the initial shock waves energy is radiated into gravitational radiation

44 This (16.4%) is in agreement with numerical relativity simulations for high energy head on black hole collisions (14 +/- 3 %):

45 This (16.4%) is in agreement with numerical relativity simulations for high energy head on black hole collisions (14 +/- 3 %): Coincidence? Higher order perturbation theory does not change result? Why should this perturbative method work? Matching in generalised versions? (higher dimensions, including charge,...)

46 Shock waves approach: 1st order perturbation theory Herdeiro, Rebelo and Sampaio, JHEP 07 (2011) 121

47 Shock waves approach: 1st order perturbation theory Herdeiro, Rebelo and Sampaio, JHEP 07 (2011) 121

48 Shock waves approach: 1st order perturbation theory Herdeiro, Rebelo and Sampaio, JHEP 07 (2011) 121

49 Shock waves approach: 1st order perturbation theory Herdeiro, Rebelo and Sampaio, JHEP 07 (2011) 121

50 Shock waves approach: 1st order perturbation theory Herdeiro, Rebelo and Sampaio, JHEP 07 (2011) 121

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55 Coelho, Herdeiro and Sampaio, Phys. Rev. Lett. 108 (2012) st order = D

56 Discussion: 1) Is this fit formula exact in first order perturbation theory? Can one derive it analytically? 2) Is there an analogous simple formula in second order theory? (see M. Sampaio s talk)

57 Discussion: 1) Is this fit formula exact in first order perturbation theory? Can one derive it analytically? 2) Is there an analogous simple formula in second order theory? (see M. Sampaio s talk) Observation: This simple pattern was found by taking D as a parameter and (even if one is not interested in higher dimensional gravity) it gives a new perspective on the four dimensional result.