Gravitational Waves in any dimension

Transcript

1 How is an EFT for Gravitational Waves in any dimension like a bead on a string? with S. Hadar and B. Kol O.B., S.H. & B.K., Phys. Rev. D 88, (2013) O.B. & S.H., Phys. Rev. D 89, (2014) O.B., S.H. & B.K., IJMPA 29, 24, 30 (2014) University, Berlin 20 October 2014

2 Outline 1 Motivation 2 A bead on a string 3 Electromagnetism: the AD self-force in any dimension 4 Gravitational Waves in d dimensions 5 Conclusions

3 Everything should be made as simple as possible, but not simpler - Albert Einstein

4 Everything should be made as simple as possible, but not simpler - Albert Einstein Bead + string (elastic eld) Electric charge + EM eld 2-Body system + GR eld

5 Everything should be made as simple as possible, but not simpler - Albert Einstein Ŝ = = ( ) ˆQ G Q Bead + string (elastic eld) Electric charge + EM eld 2-Body system + GR eld

6 Everything should be made as simple as possible, but not simpler - Albert Einstein Ŝ = = ( ) ˆQ G Q, F = δŝ δ^x Bead + string (elastic eld) Electric charge + EM eld 2-Body system + GR eld

7 y(t) T, λ F ext m φ(x, t) [ m ] [ λ S = dt 2 ẏ2 U ext + dtdx 2 φ 2 T ] 2 φ 2 dtq(t) [φ(x = 0) y] x

8 y(t) T, λ F ext m φ(x, t) [ m ] [ λ S = dt 2 ẏ2 U ext + dtdx 2 φ 2 T ] 2 φ 2 dtq(t) [φ(x = 0) y] = φ(x = 0, t) = y(t), x

9 y(t) T, λ F ext m φ(x, t) [ m ] [ λ S = dt 2 ẏ2 U ext + dtdx 2 φ 2 T ] 2 φ 2 dtq(t) [φ(x = 0) y] = φ(x = 0, t) = y(t), φ = [ c 2 2 t 2 x] φ = δ(x) Q(t)/ T, c 2 = T/λ x

10 y(t) T, λ F ext m φ(x, t) [ m ] [ λ S = dt 2 ẏ2 U ext + dtdx 2 φ 2 T ] 2 φ 2 dtq(t) [φ(x = 0) y] = φ(x = 0, t) = y(t), φ = [ c 2 t 2 x] 2 φ = δ(x) Q(t)/ T, c 2 = T/λ { y(t x/c) x > 0 = φ(x, t) = y(t + x/c) x < 0 x

11 y(t) T, λ F ext m φ(x, t) [ m ] [ λ S = dt 2 ẏ2 U ext + dtdx 2 φ 2 T ] 2 φ 2 dtq(t) [φ(x = 0) y] = φ(x = 0, t) = y(t), φ = [ c 2 t 2 x] 2 φ = δ(x) Q(t)/ T, c 2 = T/λ { y(t x/c) x > 0 = φ(x, t) = y(t + x/c) x < 0 = Q = T [( x φ) 0 + ( x φ) 0 ] = 2 Z ẏ, Z = λt = mÿ = U ext + Q = F ext 2 Z ẏ x

12 y(t) T, λ F ext m φ(x, t) [ m ] [ λ S = dt 2 ẏ2 U ext + dtdx 2 φ 2 T ] 2 φ 2 dtq(t) [φ(x = 0) y] = φ(x = 0, t) = y(t), φ = [ c 2 t 2 x] 2 φ = δ(x) Q(t)/ T, c 2 = T/λ { y(t x/c) x > 0 = φ(x, t) = y(t + x/c) x < 0 = Q = T [( x φ) 0 + ( x φ) 0 ] = 2 Z ẏ, Z = λt = mÿ = U ext + Q = F ext 2 Z ẏ x

13 y(t) T, λ F ext m φ(x, t) [ m ] [ λ S = dt 2 ẏ2 U ext + dtdx 2 φ 2 T ] 2 φ 2 dtq(t) [φ(x = 0) y] = φ(x = 0, t) = y(t), φ = [ c 2 t 2 x] 2 φ = δ(x) Q(t)/ T, c 2 = T/λ { y(t x/c) x > 0 = φ(x, t) = y(t + x/c) x < 0 = Q = T [( x φ) 0 + ( x φ) 0 ] = 2 Z ẏ, Z = λt = mÿ = U ext + Q = F ext 2 Z ẏ x

14 Field doubling at the level of the Action An eective action for φ (kinetic term & source term): S = T [ ] 1 dtdx 2 c φ 2 2 φ Z dt ẏ φ(x = 0)

15 Field doubling at the level of the Action An eective action for φ (kinetic term & source term): S = T [ ] 1 dtdx 2 c φ 2 2 φ Z dt ẏ φ(x = 0) Double the eld and the source (Schwinger-Keldysh, Galley) : These reect directed propagation Interpretation: "pulling" mirror / radiation "sink" φ ˆφ, Q ˆQ = δq δy ŷ

16 Field doubling at the level of the Action An eective action for φ (kinetic term & source term): S = T [ ] 1 dtdx 2 c φ 2 2 φ Z dt ẏ φ(x = 0) Double the eld and the source (Schwinger-Keldysh, Galley) : These reect directed propagation Interpretation: "pulling" mirror / radiation "sink" φ ˆφ, Q ˆQ = δq δy ŷ Doubled action: Ŝ [ φ, ˆφ; Q, ˆQ ] = = [ d d δs x δφ ˆφ + δs ] δq ˆQ [ Z dx c ˆφ ( ω 2 + x 2 dω 2π EOM found by varying w.r.t. ˆφ ) ( φ 2iωZ y ˆφ(0)+ŷ ) ] φ(0)

17 Feynman rules Directed propagator (non-hatted hatted) Source vertex x = G ω (x, x) = c ei ω c x x i ω Z x Q ω = iω Z y ω Hatted source vertex ˆQ ω = + iω Z ŷ ω

18 Radiation (x > 0) A bead on a string - results φ ω (x) = x = c y ω e iω x/c = φ(t, x) = y(t x c )

19 Radiation (x > 0) A bead on a string - results φ ω (x) = x = c y ω e iω x/c = φ(t, x) = y(t x c ) Radiation-Reaction eective action dω Ŝ RR = + c.c. = 2π ˆQ ω G ω (0, 0) Q ω + c.c. dω = 2π iω Z ŷ ω y ω + c.c. = 2 Z ŷ ẏ dt

20 Radiation (x > 0) A bead on a string - results φ ω (x) = x = c y ω e iω x/c = φ(t, x) = y(t x c ) Radiation-Reaction eective action dω Ŝ RR = + c.c. = 2π ˆQ ω G ω (0, 0) Q ω + c.c. dω = 2π iω Z ŷ ω y ω + c.c. = 2 Z ŷ ẏ dt Radiation-Reaction (Self-) Force F RR = δŝ RR δŷ = 2 Z ẏ

21 Radiation (x > 0) A bead on a string - results φ ω (x) = x = c y ω e iω x/c = φ(t, x) = y(t x c ) Radiation-Reaction eective action dω Ŝ RR = + c.c. = 2π ˆQ ω G ω (0, 0) Q ω + c.c. dω = 2π iω Z ŷ ω y ω + c.c. = 2 Z ŷ ẏ dt Radiation-Reaction (Self-) Force F RR = δŝ RR δŷ = 2 Z ẏ

22 Radiation (x > 0) A bead on a string - results φ ω (x) = x = c y ω e iω x/c = φ(t, x) = y(t x c ) Radiation-Reaction eective action dω Ŝ RR = + c.c. = 2π ˆQ ω G ω (0, 0) Q ω + c.c. dω = 2π iω Z ŷ ω y ω + c.c. = 2 Z ŷ ẏ dt Radiation-Reaction (Self-) Force F RR = δŝ RR δŷ = 2 Z ẏ damping

23 Electromagnetism in d dimensions S = 1 4Ωˆd+1 F µν F µν d d x A µ J µ d d x (D := d 1, ˆd := d 3, c = 1)

24 Electromagnetism in d dimensions S = 1 4Ωˆd+1 F µν F µν d d x A µ J µ d d x (D := d 1, ˆd := d 3, c = 1) Dierent, more, or both?

25 Electromagnetism in d dimensions S = 1 4Ωˆd+1 F µν F µν d d x A µ J µ d d x (D := d 1, ˆd := d 3, c = 1) Dierent, more, or both? In the right basis, only a little more, and a little dierent!

26 Electromagnetism - intermediate basis (have no fear!) Multipoles = (solid) spherical harmonics, scalar & vector dω A t/r = 2π J t/r dω = 2π A ω t/r x e iωt, A Ω = J t/r ω x e iωt, J Ω = = (k 1 k 2 k l ) STF, x = dω ( 2π dω ( A ω S Ωx +A ω Vℵ x ℵ Ω ) e iωt ) 2π J S ω Ω x +J Vℵ ω xℵ Ω e iωt ( x k 1x k2 x k ) STF l r l Y l m (Ω)

27 Electromagnetism - intermediate basis (have no fear!) Multipoles = (solid) spherical harmonics, scalar & vector dω A t/r = A ω 2π t/r x e iωt dω ( ), A Ω = A ω S 2π Ωx +A ω Vℵ x ℵ Ω e iωt J t/r dω = J t/r ω 2π x e iωt dω (, J Ω = J S ω 2π Ω x ) +J Vℵ ω xℵ Ω e iωt = (k 1 k 2 k l ) STF, x ( = x k 1x k2 x k ) STF l S = 1 dω N r l Y l m (Ω) 2 2π l,ˆd S ω 2l+ˆd+1{[ S ω = drr iωaω r 1 2 r l (rl A ω t ) + cs iωa r 2 ω S A ω t 2 cs r 2 1 r l (rl A ω S ) A ω 2 r ( ω 2 + c s r 2 cv ) A r 4 ω 2 c s Vℵ r ] r l (rl A ω Vℵ ) ]} Ωˆd+1 [A ω r J r ω + Aω t J t ω + csaω S JS ω + csaω Vℵ JVℵ ω + c.c. N l,ˆd = Γ(1 + ˆd/2) 2 l Γ(l ˆd/2) = ˆd!! (2l + ˆd)!!, c s := l(l + ˆd), c v = cs + ˆd 1

28 Electromagnetism - intermediate basis (have no fear!) Multipoles = (solid) spherical harmonics, scalar & vector dω A t/r = A ω 2π t/r x e iωt dω ( ), A Ω = A ω S 2π Ωx +A ω Vℵ x ℵ Ω e iωt J t/r dω = J t/r ω 2π x e iωt dω (, J Ω = J S ω 2π Ω x ) +J Vℵ ω xℵ Ω e iωt = (k 1 k 2 k l ) STF, x ( = x k 1x k2 x k ) STF l S = 1 dω N r l Y l m (Ω) 2 2π l,ˆd S ω 2l+ˆd+1{[ S ω = drr iωaω r 1 2 r l (rl A ω t ) + cs iωa r 2 ω S A ω t 2 cs r 2 1 r l (rl A ω S ) A ω 2 r ( ω 2 + c s r 2 cv ) A r 4 ω 2 c s Vℵ r ] r l (rl A ω Vℵ ) ]} Ωˆd+1 [A ω r J r ω + Aω t J t ω + csaω S JS ω + csaω Vℵ JVℵ ω + c.c. N l,ˆd = Γ(1 + ˆd/2) 2 l Γ(l ˆd/2) = ˆd!! (2l + ˆd)!!, c s := l(l + ˆd), c v = cs + ˆd 1

29 Electromagnetism - intermediate basis (have no fear!) Multipoles = (solid) spherical harmonics, scalar & vector dω A t/r = A ω 2π t/r x e iωt dω ( ), A Ω = A ω S 2π Ωx +A ω Vℵ x ℵ Ω e iωt J t/r dω = J t/r ω 2π x e iωt dω (, J Ω = J S ω 2π Ω x ) +J Vℵ ω xℵ Ω e iωt = (k 1 k 2 k l ) STF, x ( = x k 1x k2 x k ) STF l S = 1 dω N r l Y l m (Ω) 2 2π l,ˆd S ω 2l+ˆd+1{[ S ω = drr iωaω r 1 2 r l (rl A ω t ) + cs iωa r 2 ω S A ω t 2 cs r 2 1 r l (rl A ω S ) A ω 2 r ( ω 2 + c s r 2 cv ) A r 4 ω 2 c s Vℵ r ] r l (rl A ω Vℵ ) ]} Ωˆd+1 [A ω r J r ω + Aω t J t ω + csaω S JS ω + csaω Vℵ JVℵ ω + c.c. N l,ˆd = Γ(1 + ˆd/2) 2 l Γ(l ˆd/2) = ˆd!! (2l + ˆd)!!, c s := l(l + ˆd), c v = cs + ˆd 1

30 Electromagnetism - reduction to gauge invariant elds 1 Scalar eld & source: A r ω = 1 ω 2 cs r 2 Ã ω S : = A ω t iωa ω S ρ S ω : = Jt ω + i ωr l+ˆd+1 Vector eld & source: [ iω r l (rl A t ω ) + c ] s r l+2 (rl A S ω ) Ωˆd+1 Jr ω ( r l+ˆd+1 A ω Vℵ, ρ Vℵ ω := JVℵ ω ) Λ Λ 1 Jr ω, Λ := ω2 r 2 c s

31 Electromagnetism - reduction to gauge invariant elds 2 Scalar (Electric) eld A ω E = ( ) 1 ( l r l 1 (1 Λ) r l (A ω t iωa ω S ) ) ρ A rˆd [ ( ) E ω = x i Λ ] dωˆd+1 rˆd+1 l + ˆd ω rˆd 1 Λ 1 J w( r) n r 2 ρ ω( r) ( ) 0 = N r2l+ˆd+1 l l,ˆd ω l + ˆd 2 + r 2 + 2l + ˆd + 1 r A E ρ A E ω r

32 Electromagnetism - reduction to gauge invariant elds 2 Scalar (Electric) eld A ω E = ( ) 1 ( l r l 1 (1 Λ) r l (A ω t iωa ω S ) ) ρ A rˆd [ ( ) E ω = x i Λ ] dωˆd+1 rˆd+1 l + ˆd ω rˆd 1 Λ 1 J w( r) n r 2 ρ ω( r) ( ) 0 = N r2l+ˆd+1 l l,ˆd ω l + ˆd 2 + r 2 + 2l + ˆd + 1 r A E ρ A E ω r Vector (Magnetic) eld A ω Mℵ = l A ω Vℵ /r ρ A Mℵ ω = 1 l rˆd+2 0 = N l,ˆd r2l+ˆd+1 l + ˆd l J w( r) ( ( r ) ) x ℵ dωˆd+1 ( ω 2 + r 2 + 2l + ˆd + 1 r r ) A Mℵ ρ A Mℵ ω

33 Electromagnetism - reduction to gauge invariant elds 2 More (2 x x ω) elds, but 1d ; Dierent wave equations Scalar (Electric) eld A ω E = ( ) 1 ( l r l 1 (1 Λ) r l (A ω t iωa ω S ) ) ρ A rˆd [ ( ) E ω = x i Λ ] dωˆd+1 rˆd+1 l + ˆd ω rˆd 1 Λ 1 J w( r) n r 2 ρ ω( r) ( ) 0 = N r2l+ˆd+1 l l,ˆd ω l + ˆd 2 + r 2 + 2l + ˆd + 1 r A E ρ A E ω r Vector (Magnetic) eld A ω Mℵ = l A ω Vℵ /r ρ A Mℵ ω = 1 l rˆd+2 0 = N l,ˆd r2l+ˆd+1 l + ˆd l J w( r) ( ( r ) ) x ℵ dωˆd+1 ( ω 2 + r 2 + 2l + ˆd + 1 r r ) A Mℵ ρ A Mℵ ω

34 EM Feynman rules in general d Directed propagator, from solution of wave equation r r = G A E /A Mℵ ret (r, r) = i ω 2l+ˆd M l,ˆd RE/M 1 jα (ωr < ) h+ α (ωr > )δ, R E 1 = l+ˆd, R M ˆd 1 l = 1/RM 1, α = l+ 2, M l,ˆd = π 2 2α+1 N l,ˆd Γ2 (α+1)

35 EM Feynman rules in general d Directed propagator, from solution of wave equation r r = G A E /A Mℵ ret (r, r) = i ω 2l+ˆd M l,ˆd RE/M 1 jα (ωr < ) h+ α (ωr > )δ, R E 1 = l+ˆd, R M ˆd 1 l = 1/RM 1, α = l+ 2, M l,ˆd = π 2 2α+1 N l,ˆd Γ2 (α+1) Electric & Magentic source vertices Q (E) ω = dr j α(ωr )ρ A E ω (r ) = 1 [ d l + ˆd D x x iω j α(ωr ) J ω( x ) x 1 ( ] ρω( x r l+ˆd 1 r l+ˆd j α(ωr )) ) Q (M,ℵ) ω = d D ( x j α(ωr) ( r ) ℵ J w( r)) (k l x 1)

36 EM in general d - results Radiation-Reaction eective action: Ŝ RR = A E + A M = ˆQ G Q

37 EM in general d - results Radiation-Reaction eective action: Ŝ RR = A E + A M = ˆQ G Q = 1 2 dω 2π In even d: = dt, [ ˆQ(E) ω GA E ret (0, 0) Q (E) ω l+ ˆd+1 ( ) 2 l+ˆd ˆd!!(2l+ˆd)!! l ˆQ (E) + ˆQ(M,ℵ) ω 2l+ˆd t Q (E) G A Mℵ ret l2ˆd ˆQ(M) + ] (0, 0) Q (M,ℵ) ω +c.c. 2l+ˆd t Q (M) (l+1)(l+ˆd 1)

38 EM in general d - results Radiation-Reaction eective action: Ŝ RR = A E + A M = ˆQ G Q = 1 2 dω 2π In even d: = dt, [ ˆQ(E) ω GA E ret (0, 0) Q (E) ω l+ ˆd+1 ( ) 2 l+ˆd ˆd!!(2l+ˆd)!! l ˆQ (E) + ˆQ(M,ℵ) ω 2l+ˆd t Q (E) G A Mℵ ret l2ˆd ˆQ(M) + ] (0, 0) Q (M,ℵ) ω +c.c. 2l+ˆd t Q (M) (l+1)(l+ˆd 1) Radiation-Reaction (Self-) Force (on a point-charge in d dimensions) FAD = δŝ d RR = ( ) 2 (d 2) δˆ x q2 (d 1)!!(d 3)!! d 1 t x + (6 1PN terms) +

39 Gravitation - GW from a 2-body system S = 1 16πG d g R d d x 1 h µν T µν d d x, 2 µ T µν = 0, g µν = η µν + h µν ( c = 1, linearized source )

40 Gravitation - GW from a 2-body system S = 1 16πG d g R d d x 1 h µν T µν d d x, 2 µ T µν = 0, g µν = η µν + h µν ( c = 1, linearized source ) The same as EM elds and sources, but dierent Radiation zone: A (E/M) ω (r) h (E/M/T) System zone: h µν (φ, A i, σ ij ) ω (r) (gauge-invariant elds) (NRG elds, Kol & Smolkin '07) these highlight spatial tensor structure, hierarchy in terms of PN ds 2 = e 2φ ( dt A i dx i) 2 e 2φ γ ij dx i dx j

41 Gravitation - 1D reduced action & Feynman rules Goal - like EM: S (E/M/T) = 1 dω 2 2π dr [ r 2l+ˆd+1 N l,ˆd R ɛ l,ˆd h ω ɛ ( ( h ω ɛ Tω ɛ + c.c. )] ω 2 + r 2 + 2l+ˆd+1 r )h ɛ ω r

42 Gravitation - 1D reduced action & Feynman rules Goal - like EM: S (E/M/T) = 1 dω 2 2π dr [ r 2l+ˆd+1 N l,ˆd R ɛ l,ˆd h ω ɛ ( ( h ω ɛ Tω ɛ + c.c. )] ω 2 + r 2 + 2l+ˆd+1 r )h ɛ ω r r ɛ = Gret(r, r) = i G ω 2l+ˆd M l,ˆd Rɛ l,ˆd j α (ωr < ) h + α (ωr > )δ r = Q ɛ ω, = ˆQ ɛ ω

43 Gravitation - special case d = 4 No tensor modes! S (E/M) = 1 dω [ r 2l+2 ( dr 2 2π R ɛ (2l+1)!! hω (E/M) ω 2 + r 2 + 2(l+1) ) r h (E/M) ω r ( )] h ω (E/M) T (E/M) ω + c.c., R ɛ : = l + 2 ( ) ɛ l + 1, ɛ = 1 (E), ɛ = 1 (M) l 1 l

44 Gravitation - special case d = 4 No tensor modes! S (E/M) = 1 dω [ r 2l+2 ( dr 2 2π R ɛ (2l+1)!! hω (E/M) ω 2 + r 2 + 2(l+1) ) r h (E/M) ω r ( )] h ω (E/M) T (E/M) ω + c.c., R ɛ : = l + 2 ( ) ɛ l + 1, ɛ = 1 (E), ɛ = 1 (M) l 1 l r r ɛ = Gret(r, r) = igω2l+1 (2l + 1)!! j l+ 1 (ωr < ) h + 2 l+ 1 (ωr > ) R ɛ 2 = Q ɛ ω, = ˆQ ɛ ω

45 Gravitation - Vertices via zone seperation Work in radiation zone eliminate system zone System zone itself includes non-linear vertices; grouped together (by PN order) Radiation-zone vertices dened as Q := :=

46 Gravitation (4d) - results Ŝ linear = h E + h M = dt l G ( ) l+1 [ (l + 2) (l + 1) (2l + 1)!! (l 1) l ˆQ (E) 2l+1 t Q (E) l + (l + 1) ˆQ (M) 2l+1 t Q (M) ]

47 Gravitation (4d) - results Ŝ linear = h E + h M = dt l G ( ) l+1 [ (l + 2) (l + 1) (2l + 1)!! (l 1) l ˆQ (E) 2l+1 t Q (E) l + (l + 1) ˆQ (M) 2l+1 t Q (M) ] As promised.

48 Gravitation (4d) - results O Ŝ O = G dt 1 5 ˆQ ij E 5 t Q ij E Q ij E : Mass quadrupole = Burke-Thorne potential & self-force

49 Gravitation (4d) - results O, NO Ŝ O = G dt 1 5 ˆQ ij E 5 t Q ij E Q ij E : Mass quadrupole = Burke-Thorne potential & self-force [ Ŝ O+NO = G dt 1 5 ˆQ ij E 5 t Q ij E 4 45 ˆQ ij M 5 t Q ij M + 1 ] ijk ˆQ 189 E 7 t Q ijk E Q ij E : Mass quadrupole (+1PN corrected including rst system zone nonlinear eect: gravitating potential energy, Gm Am B r ) Q ij M : Current quadrupole Q ijk E : Mass octupole

50 Everything should be made as simple as possible, but not simpler - Albert Einstein Ŝ = = ( ) ˆQ G Q, F = δŝ δ^x Bead + string (elastic eld) Electric charge + EM eld 2-Body system + GR eld

51 Everything should be made as simple as possible, but not simpler - Albert Einstein Ŝ = = ( ) ˆQ G Q, F = δŝ δ^x Ŝ = 2 Z ŷẏdt Ŝ= dt l+ ˆd+1 ( ) 2 ˆd!!(2l+ˆd)!! l+ˆd l ˆQ (E) 2l+ˆd t Q l2ˆd ˆQ (M) (E) + 2l+ˆd t Q (M) (l+1)(l+ˆd 1) Ŝ= dt G ( ) l+1 (l+2) (2l+1)!!(l 1) [ l+1 ˆQ (E) l 2l+1 t Q (E) + l ] l+1 ˆQ (M) 2l+1 t Q (M)

52 Conclusions Unied description of dierent physical systems Joint analytical Action formulation of radiation & reaction EM (AD): New results in general dimension GR: Economization of traditional computations Nonlinear eects Coming soon: leading, +1PN, some +2PN in any dimension

53 Questions? Thank you for your attention

54 Image credits K. Thorne (Caltech) & T. Carnahan (NASA GSFC) MIT OpenCourseWare Electrostatic Visualizations Flickr's Flood G, (creative commons license) C. R. Galley, PR 110 (2013) 17, W. D. Goldberger & I. D. Rothstein, PRD 73 (2006)

55 In frequency domain - SAME. EM in non-even d Transforming to time domain = branch-cut = non-locality [( ) ˆQ(t) t 2l+ˆd 1 Q(t) ˆQ(t) 2 H(2l + ˆd ˆd) H(l + 2 ) t 2l+ˆd Q (t) t dt t t 2l+ˆd t Q(t ) Reg Reg.: Detweiler-Whiting decomposition, generalizes Dirac's odd propagator ]

56 Gravitation - Full source vertices [ Q (E) = 1 d 3 x x ( r l r l+2 j (l + 1)(l + 2) l+ 1 (ri t) 2 4 ( r l+1 r l+2 j l+ 1 (ri t)) tt 0a x a +2 j 2 l+ 1 ( (2l + 1)!! = 1 + p=0 (2p)!!(2l + 2p + 1)!! ( (2l + 1)!! p=0 (2p)!!(2l + 2p + 1)!! (2l + 1)!! p=0 (2p)!!(2l + 2p + 1)!! (2l + 1)!! + p=0 (2p)!!(2l + 2p + 1)!! d 3 x Q (M) = 1 l + 2 8p(l + p + 1) (l + 1)(l + 2) ) (T 00 + T aa ) (ri t) 2 t T ab x a x b +r 2 j 2 l+ 1 (ri t) 2 ( 2 t T 00 T aa )] ) [ ] STF d 3 x t 2p T 00 r 2p x ) [ 4p (l + 1)(l + 2) ( p ) [ l + 1 l + 2 [ 2 (l + 1)(l + 2) ] STF d 3 x t 2p T aa r 2p x ] STF d 3 x t 2p+1 T 0a r 2p x a ] STF d 3 x t 2p+2 T ab r 2p x ab { r l 1 ( r l+2 ) ( ( j l+ 1 (ri t) 2 x T 0a)) k l x 1 2 j l+ 1 2 (ri t) 2ɛ k lba tt ac x bc 1} ( (2l + 1)!! = 1 + 2p ) [ d 3 x t 2p r 2p ( Bi-group ( meeting, 2 x T 0a)) ] k STF l Humboldt x 1

57 Origin-normalized Bessel functions From Bessel's functiosn B α {J α, Y α, H ± α }, b α := Γ(α + 1)2 α B α(x) x α They satisfy [ x 2 + 2α + 1 ] x + 1 b α (x) = 0 x jα in the vicinity of the origin x = 0 is given by jα (x) = p=0 ( ) p (2α)!! (2p)!!(2p + 2α)!! x2p The outgoing waves at x are h± := j ± iỹ: h ± α (x) ( i) α+1/2 2α+1/2 Γ(α + 1) π e ±ix x α+1/2

58 Singular origin-normalized Bessel functions Around x = 0 the Bessel function of the 2nd kind for α non-integer is ỹ α (x) = = Γ(α + 1)2α cos(απ)j α (x) J α (x) x α sin(απ) Γ(α + 1)2α ( ) p x 2p [ cos(απ) sin(απ) (2p)!! 2 p 2 α Γ(p + α + 1) 2α x 2α ] Γ(p α + 1) p=0 while for integer α = n, ỹ n (x) = 2(2n)!! n (2m 2)!! π (2n 2m)!! x 2m + 2 ( x ) jn π ln (x) 2 m=1 (2n)!! ( ) k [ψ(k + 1) + ψ(n + k + 1)] π (2k)!!(2n + 2k)!! x2k k=0 where ψ(n + 1) = H(N) γ is the digamma function, dened using the Harmonic numbers H(N) and the Euler-Mascheroni constant γ.

59 Eective Field Theory description of GR For example, Einstein-Infeld-Homann agrangian Figure from W. D. Goldberger and I. D Rothstein, PRD 73 (2006)

60 Hatted variables - interpretation Doubling the elds eectively mirrors them. Hatted elds "pull" from the mirror's surface, absorbing radiation from the original elds. 0 = δŝ δφ(x) = dx δeom φ(x ) δφ(x) ˆφ(x ) ˆρ(x) ˆφ satises an equation for linearized deviations from the background φ with a source ˆρ and reversed propagation. In the absence of its source ˆφ vanishes, namely ˆρ = 0 = ˆφ = 0 φ, ρ are physical observables, unlike ˆφ, ˆρ.

61 Field doubling Problem: Action formalism not compatible with non-conservative eects. Solution: Field Doubling (QFT: in-in/ctp, Schwinger '61; Classical version: Galley 2012, 2013). Ŝ := S[q 1 ] S[q 2 ] Figure from C. R. Galley, PR 110 (2013)

62 Field doubling Keldysh representation: q 1 := q ˆq q 2 := q 1 2 ˆq ˆq is deviation from physical trajectory. Expand: Ŝ := S[q 1 ] S[q 2 ] = δs δq ˆq + 1 δ 3 S 6 δq 3 ˆq In classical setting (physical condition q 1 = q 2 ) only terms linear in ˆq survive.

63 Field doubling In our problem: S[φ, ρ] Ŝ[φ, ˆφ; ρ, ˆρ] := [ δs dx δφ(x) ˆφ(x) + δs ] δρ(x) ˆρ(x) EOM: δŝ δˆx = 0 Propagators are directed (retarded/odd).

64 GR - NO details Ŝ = G dt [ 1 5 ˆQ ij E 5 t Q ij E 4 45 ˆQ ij M 5 t Q ij M + 1 ] ijk ˆQ 189 E 7 t Q ijk E where Q ij E = 2 A=1 [ m A v2 m B 4 r 3 t ( v x) + 11 ] 42 2 t (x x2 i x j 13 ) δij x 2 A A 2 Q ij M [m = ( x v) i x j] TF Q ijk E = A=1 2 A=1 ( m x i x j x k) TF A A