Self-force from equivalent periodic sources

Transcription

1 Self-force from equivalent periodic sources Barak Kol Hebrew University, Jerusalem Capra 16 Dublin, July 2013 Based on arxiv:1307.xxxx

2 Self-force from equivalent periodic sources Barak Kol Hebrew University, Jerusalem Capra 16 Dublin, July 2013 Outline Based on arxiv:1307.xxxx Orbits and frequencies Equivalent sources Regularization Discussion

3 Issues

4 Issues Regularization MiSaTaQuWa Mode sum Regularization (Barack & Ori) Detweiler-Whiting decomposition

5 Issues Regularization MiSaTaQuWa Mode sum Regularization (Barack & Ori) Detweiler-Whiting decomposition Computational cost

6 Issues Regularization MiSaTaQuWa Mode sum Regularization (Barack & Ori) Detweiler-Whiting decomposition Computational cost Characterization of conservative sector Hinderer-Flanagan Brinholtz-Hadar-BK

7 Goal

8 Goal Trajectory Parameters (E, l, t 0, 0)

9 Goal Trajectory Parameters (E, l, t 0, 0) Obtain the adiabatic flow in the space of trajectories - first order EMR Osculating orbits

10 Orbits and frequencies

11 Orbits and frequencies Quasi periodic frequency spectrum! mn = m + n r r = 2 P t = P P t

12 Orbits and frequencies Quasi periodic frequency spectrum! mn = m + n r r = 2 P t = P P t Frequency requirement - constructive interference in azimuthal direction

13 A stroboscope

14 A stroboscope Imagine a stroboscope flashing every P mn

15 A stroboscope Imagine a stroboscope flashing every P mn The body traces a curve

16 A stroboscope Imagine a stroboscope flashing every P mn The body traces a curve It is equivalent to folding the trajectory over a periodic time coordinate

17 (m,n) equivalent source

18 (m,n) equivalent source ergodic: time average to ensemble average

19 (m,n) equivalent source ergodic: time average to ensemble average equation 0= (r, )=! mn t(r)+m ( (r))

20 (m,n) equivalent source ergodic: time average to ensemble average equation 0= (r, )=! mn t(r)+m ( (r)) parametric form = m = n + ( ) P 2

21 (m,n) equivalent source ergodic: time average to ensemble average equation 0= (r, )=! mn t(r)+m ( (r)) parametric form = m = n + ( ) P 2 winds (-m) times around phi and n times around r

22 Examples (00) (01) (10) (11) (20) 0 Ω r Ω Φ Ω Φ Ω r 2Ω Φ

23 Zero frequency

24 Zero frequency Is in the conservative sector

25 Zero frequency Is in the conservative sector := h i t = q 2 P t r (dr/d ) (z)

26 Zero frequency Is in the conservative sector := h i t = q 2 P t r (dr/d ) (z) Only (phi 0,t 0 ) drift

27 Zero frequency Is in the conservative sector := h i t = q 2 P t r (dr/d ) (z) Only (phi 0,t 0 ) drift The dissipative part 0 :=

28 Solving the field equations

29 Solving the field equations 4 f 2 t =4

30 Solving the field equations 4 f 2 t =4 Frequency domain - natural here, elliptic equations

31 Solving the field equations 4 f 2 t =4 Frequency domain - natural here, elliptic equations Time domain

32 Regularization

33 Regularization Singular source

34 Regularization Singular source Less singular (1d density) for aperiodic motion and equivalent source

35 Regularization Singular source Less singular (1d density) for aperiodic motion and equivalent source Zero freq. sector: surface charge density

36 Regularization Singular source Less singular (1d density) for aperiodic motion and equivalent source Zero freq. sector: surface charge density In frequency space - similar to electrostatics with singular source

37 Electrostatics

38 = S + R Electrostatics

39 Electrostatics = S + R 4 R := 4 [4 + f 1! 2 ] S

40 Electrostatics = S + R 4 R := 4 [4 + f 1! 2 ] S [4 + f 1! 2 ] R =4 R

41 Electrostatics = S + R 4 R := 4 [4 + f 1! 2 ] S [4 + f 1! 2 ] R =4 R Time domain in Progress - generalizing Hadamard s local construction S log

42 Edge

43 Edge Charge density near r min (or r max ) (x, y, z) = 1/2 p x (z) x 0

44 Edge Charge density near r min (or r max ) (x, y, z) = 1/2 p x (z) x 0 Solution =2 1/2 < p w w = x + iz

45 Outgoing radiation and self-force

46 Outgoing radiation and self-force Outgoing radiation 1 (, ) r e i! r +0 e i! r for r!1

47 Outgoing radiation and self-force Outgoing radiation 1 (, ) r e i! r +0 e i! r for r!1 Self-force throughout trajectory: drift in (E, l, t 0, 0)

48 Method summary

49 Method summary Goal: drift in trajectory parameters

50 Method summary Goal: drift in trajectory parameters Equivalent periodic source

51 Method summary Goal: drift in trajectory parameters Equivalent periodic source Conservative is zero frequency

52 Method summary Goal: drift in trajectory parameters Equivalent periodic source Conservative is zero frequency Regularization

53 Method summary Goal: drift in trajectory parameters Equivalent periodic source Conservative is zero frequency Regularization Self-force computed throughout at once

54 Generalizations

55 Generalizations Electromagnetism and gravity: source, waves

56 Generalizations Electromagnetism and gravity: source, waves Rotating BH (Kerr)

57 Generalizations Electromagnetism and gravity: source, waves Rotating BH (Kerr) Higher EMR orders

58 Range of usefulness

59 Range of usefulness Relativistic - or else hierarchy of scales

60 Range of usefulness Relativistic - or else hierarchy of scales For freq. space: few freq. (low eccen.)

61 Range of usefulness Relativistic - or else hierarchy of scales For freq. space: few freq. (low eccen.) Nearly incommensurate - the rational w. smallest denominator within the freq. ratio range

62 Paths for continuation

63 Paths for continuation Gauge choice

64 Paths for continuation Gauge choice Invitation for collaboration: Numerical evaluation

65 Paths for continuation Gauge choice Invitation for collaboration: Numerical evaluation Time domain regularization - local expansion generalizing Hadamard

66 Paths for continuation Gauge choice Invitation for collaboration: Numerical evaluation Time domain regularization - local expansion generalizing Hadamard Formulate conservative sector of EMR

67 Thank you for your attention!

68 Givat Brenner My home Kibbutz (village) Thank you for your attention!