Nonlinear wave-wave interactions involving gravitational waves

Transcription

1 Nonlinear wave-wave interactions involving gravitational waves ANDREAS KÄLLBERG Department of Physics, Umeå University, Umeå, Sweden Thessaloniki, 30/8-5/ p. 1/38

2 Outline Orthonormal frames. Thessaloniki, 30/8-5/ p. 2/38

3 Outline Orthonormal frames. Tetrad bases. Thessaloniki, 30/8-5/ p. 3/38

4 Outline Orthonormal frames. Tetrad bases. Examples. Thessaloniki, 30/8-5/ p. 3/38

5 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Thessaloniki, 30/8-5/ p. 4/38

6 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Overview Thessaloniki, 30/8-5/ p. 5/38

7 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Overview Prerequisites Thessaloniki, 30/8-5/ p. 5/38

8 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Overview Prerequisites Results Thessaloniki, 30/8-5/ p. 5/38

9 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Overview Prerequisites Results Example of application Thessaloniki, 30/8-5/ p. 5/38

10 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Nonlinearly coupled electromagnetic and gravitational waves in vacuum. Thessaloniki, 30/8-5/ p. 6/38

11 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Nonlinearly coupled electromagnetic and gravitational waves in vacuum. Background Thessaloniki, 30/8-5/ p. 7/38

12 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Nonlinearly coupled electromagnetic and gravitational waves in vacuum. Background Our work Thessaloniki, 30/8-5/ p. 7/38

13 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Nonlinearly coupled electromagnetic and gravitational waves in vacuum. Background Our work Results Thessaloniki, 30/8-5/ p. 7/38

14 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Nonlinearly coupled electromagnetic and gravitational waves in vacuum. Background Our work Results Possible applications Thessaloniki, 30/8-5/ p. 7/38

15 Outline Orthonormal frames. Nonlinear coupled Alfvén and gravitational waves. Nonlinearly coupled electromagnetic and gravitational waves in vacuum. Future work. Thessaloniki, 30/8-5/ p. 8/38

16 Tetrads Why? Thessaloniki, 30/8-5/ p. 9/38

17 Tetrads Why? More direct interpretation of physical quantities.= Easier to distinguish coordinate effects from physical processes. Thessaloniki, 30/8-5/ p. 10/38

18 Tetrads Why? More direct interpretation of physical quantities.= Easier to distinguish coordinate effects from physical processes. Spacetime split into space+time, and metric locally Minkowski everywhere.= Greatly simplifies the algebra. (Need not distinguish between co- and contravariant quantities.) Thessaloniki, 30/8-5/ p. 10/38

19 Tetrads How? Thessaloniki, 30/8-5/ p. 11/38

20 Tetrads How? Transformation from coordinate basis, µ to more general basis e a : e a = X µ a µ Thessaloniki, 30/8-5/ p. 12/38

21 Tetrads How? Transformation from coordinate basis, µ to more general basis e a : e a = X µ a µ Dual basis to e a are differential forms ω a (i.e. ω a (e b ) = δ a b ): ω a = ω a µdx µ Thessaloniki, 30/8-5/ p. 12/38

22 Tetrads How? Transformation from coordinate basis, µ to more general basis e a : e a = X µ a µ Dual basis to e a are differential forms ω a (i.e. ω a (e b ) = δ a b ): ω a = ω a µdx µ Tensorial quantities expressed in either basis, e.g.: Vector field: A = A µ µ = A a e a = A a X µ a µ Metric: ds 2 = g µν dx µ dx ν = g ab ω a ω b = g ab ω a µω b νdx µ dx ν Thessaloniki, 30/8-5/ p. 12/38

23 Tetrads How? Tetrad can be chosen so that g ab = η ab = diag( 1, 1, 1, 1) (orthonormality). Thessaloniki, 30/8-5/ p. 13/38

24 Tetrads How? Tetrad can be chosen so that g ab = η ab = diag( 1, 1, 1, 1) (orthonormality). Connection given by Ricci rotation coefficients: Γ abc = 1 2 (g adc d cb g bdc d ca + g cd C d ab ) where C a bc (xµ ) are the commutation functions for the basis {e a }, (i.e. [e a, e b ] = C c ab e c). Thessaloniki, 30/8-5/ p. 13/38

25 Tetrads How? Tetrad can be chosen so that g ab = η ab = diag( 1, 1, 1, 1) (orthonormality). Connection given by Ricci rotation coefficients: Γ abc = 1 2 (g adc d cb g bdc d ca + g cd C d ab ) where C a bc (xµ ) are the commutation functions for the basis {e a }, (i.e. [e a, e b ] = C c ab e c). E.g. covariant derivative: b A a = e b A a Γ c ab A c Thessaloniki, 30/8-5/ p. 13/38

26 Tetrad equations Introduce observer four-velocity V a, = EM-field can be decomposed relative to this into electric and magnetic part: E a = F ab V b, B a = 1 2 ɛ abcf bc Thessaloniki, 30/8-5/ p. 14/38

27 Tetrad equations Introduce observer four-velocity V a, = EM-field can be decomposed relative to this into electric and magnetic part: E a = F ab V b, B a = 1 2 ɛ abcf bc Choose tetrad so that e 0 = V a e a (i.e. V a = δ a 0 ). Thessaloniki, 30/8-5/ p. 14/38

28 Tetrad equations Introduce observer four-velocity V a, = EM-field can be decomposed relative to this into electric and magnetic part: E a = F ab V b, B a = 1 2 ɛ abcf bc Choose tetrad so that e 0 = V a e a (i.e. V a = δ a 0 ). Introduce three vector notation E = (E 1, E 2, E 3 ) etc. and = (e 1, e 2, e 3 ) = Thessaloniki, 30/8-5/ p. 14/38

29 Tetrad equations Maxwell field equations: a F ab = j b, a F bc + b F ca + c F ab = 0, and fluid evolution equations: b T ab = F ab j b, can be written E = ρ + ρ E B = ρ B e 0 E B = j j E e 0 B + E = j B e 0 (γn) + (γnv) = n (µ + p)(e 0 + v )γv = γ 1 p γv(e 0 + v )p where γ = 1/ 1 v i v i, i = 1, 2, 3 +qn(e + v B) + (µ + p)g Thessaloniki, 30/8-5/ p. 15/38

30 Nonlinear coupled Alfvén and gravitational waves (A. Källberg, G. Brodin and M. Bradley, PRD 2004) Thessaloniki, 30/8-5/ p. 16/38

31 Nonlinear coupled Alfvén and gravitational waves (A. Källberg, G. Brodin and M. Bradley, PRD 2004) Self-consistent weakly nonlinear analysis of Einstein-Maxwell system. EMW and GW propagating in strongly magnetized plasma described by multifluid description. Resonant wave coupling = direct interaction with matter magnified compared to other nonlinearities, (e.g. coupling to background curvature). WKB-approximation = Nonlinear Schrödinger equation (NLS). Weak 3D-dependence = Self-focusing and collapse of pulse. Thessaloniki, 30/8-5/ p. 16/38

32 Nonlinear coupled Alfvén and gravitational waves Focus on the direct interaction with matter = consider linearized gravitational wave (nonlinearity comes from resonant response from matter) in basis e 0 = t, e 1 = (1 1 2 h +) x, e 2 = ( h +) y, e 3 = z Linearized Einstein field equations ( 2 t z 2 ) h+ = κ (δt 11 δt 22 ) Background magnetic field, B 0, in 1-direction, wave propagation in 3-direction. Introduce perturbations n = n 0 + δn, B = (B 0 + B x )e 1, E = E y e 2 + E z e 3 and v = v y e 2 + v z e 3. = Thessaloniki, 30/8-5/ p. 17/38

33 Nonlinear coupled Alfvén and gravitational waves Maxwell and fluid equations reduces to ( t + V(B x ) z )B x = 1 2 B 0 t h + (1) ( 2 t 2 z )h + = 2κB 0 B x (2) V(B x ) = 1 (1/2C 2 A ) (B 0/ (B 0 + 2B x )) 3/2 and we have introduced the Alfvén velocity C A (1/ s ω2 p/ω 2 c ) 1/2. LHS of (1-2) are wave operators for compressional Alfvén and gravitational wave respectively. RHS are mutual interaction terms for the wave modes. May be combined to single wave equation: ( t + z )( t + V(B x ) z )B x = κb2 0 2 B x (3) Thessaloniki, 30/8-5/ p. 18/38

34 Nonlinear coupled Alfvén and gravitational waves Linearizing = dispersion relation: (ω k) ( ω k + k 2C 2 A ) = κb2 0 2 For large k we have two distinct modes: "fast" mode, ω k, where most of the energy is gravitational, and "slow" mode, ω k(1 1 2C 2 A ), where energy is mainly electromagnetic. For longer wavelengths, k κb0 2C2 A, modes are not clearly separated, and energy is divided equally between electromagnetic and gravitational form. Thessaloniki, 30/8-5/ p. 19/38

35 Nonlinear coupled Alfvén and gravitational waves Include terms up to 3rd order in amplitude expansion of wave equation = higher harmonic generation. Apply WKB-approximation. Coordinate transformations, z, t ξ, τ. (1) used to relate B and h + Standard NLS-equation for rescaled GW amplitude: ( i τ ± ξ 2 ) h+ = ± h + 2 h+ ± refers to fast and slow mode respectively. Thessaloniki, 30/8-5/ p. 20/38

36 Nonlinear coupled Alfvén and gravitational waves In reality we will not have exact plane wave solutions to linearized Einstein and Maxwell equations. Assume deviation from plane waves small, and apply perturbative treatment. Keeping only lowest order terms, the wave equation (3) is modified to: ( ) ( ) t + z t 2 where 2 2 x + 2 y. t + V(B x ) z t 2 B x = κb2 0 2 B x Thessaloniki, 30/8-5/ p. 21/38

37 Nonlinear coupled Alfvén and gravitational waves Including x- and y-dependence in WKB-ansatz, and keeping terms up to 3rd order in amplitude, one obtains: ( i τ ± ξ 2 ) +Υ 2 h+ = ± h + 2 h+ (4) Same equation as before with a small correction to the linear wave operator. = allows self-focusing of pulse. Thessaloniki, 30/8-5/ p. 22/38

38 Nonlinear coupled Alfvén and gravitational waves Consider long 3D-pulses with shape depending only on (normalized) cylindrical radius, thus neglecting the dispersive term in (4). (4) can be written as cylindrically symmetric NLS equation: ( i τ + 1 r r ( r )) h + = ± h + 2 h+ r Consider case with minus sign (slow, electromagnetically dominated mode) = nonlinearity of focusing type. No (physically relevant) exact solutions known, but approximate variational techniques and numerical work has been done. Thessaloniki, 30/8-5/ p. 23/38

39 Nonlinear coupled Alfvén and gravitational waves Strong enough nonlinearity pulse radius, r p 0 in finite time. Estimated condition of collapse: c2 k 2 r 2 dist C 2 A 1, can be fulfilled within reasonable parameter range = possibility of structure formation of electromagnetic radiation pattern Note that the collapse condition does not contain gravitational parameters, which reflects the EM dominance of the slow mode. However, the process is induced by GW-EMW coupling and thus still has gravitational origin. Systems of interest for this effect are: binary pulsars, quaking stars surrounded by accreting matter, supernovæ etc. Thessaloniki, 30/8-5/ p. 24/38

40 Four wave coupling of EMWs and GWs in vacuum Background Parallel EMWs and GWs do not interact in vacuum. Parallel EMWs and GWs may interact and exchange energy through a medium (EM field, matter, background gravitational field etc.) Propagation on background leads to scattering and wave tail formation. Antiparallelly propagating waves may interact weakly in vacuum, causing polarization rotation, frequency shifting, energy exchange etc. What about four wave coupling in flat spacetime? Thessaloniki, 30/8-5/ p. 25/38

41 Four wave coupling of EMWs and GWs in vacuum (A. Källberg, G. Brodin and M. Marklund, 2004) Thessaloniki, 30/8-5/ p. 26/38

42 Four wave coupling of EMWs and GWs in vacuum (A. Källberg, G. Brodin and M. Marklund, 2004) Resonant wave coupling involving two GWs and two EMWs. Perturbative treatment up to 3rd order in amplitudes. Calculations performed in flat background, but results also valid in the high frequency approximation. Surprisingly simple result for coupling equations. Preliminary solutions to coupling equations and cross-section for incoherent process presented. Thessaloniki, 30/8-5/ p. 26/38

43 Four wave coupling of EMWs and GWs in vacuum Consider Maxwell s equations, keeping only the effective gravitational sources. Can derive the generalized wave equations E α = e 0 j α E ε αβγ e β j Bγ δ αγ e γ ρ E ε αβγ C a β0 e ab γ δ αγ C a βγ e ae β (5) B α = e 0 j α B + ε αβγ e β j Eγ δ αγ e γ ρ B + ε αβγ C a β0 e ae γ δ αγ C a βγ e ab β (6) where e 0 e 0 Consider waves of the form E = E(x µ )e ik µx µ + c.c., amplitude variations are slow compared to exponential part. Thessaloniki, 30/8-5/ p. 27/38

44 Four wave coupling of EMWs and GWs in vacuum No resonant three-wave coupling: Matching conditions = parallel propagation = no interaction. Matching conditions for resonant four-wave interaction: k µ E A + k µ E B = k µ h A + k µ h B. Use center of mass system = ω EA = ω EB = ω ha = ω hb = ω and k hb = k ha, k EB = k EA. Interaction equations of the form E A = C EA h A h B E B E B = C EB h A h B E A h A = C ha E A E B h B h B = C hb E A E B h A Thessaloniki, 30/8-5/ p. 28/38

45 Four wave coupling of EMWs and GWs in vacuum Nonlinear gravitational response to GW calculated from metric ansatz: g µν = η µν + h T µν T + h (2) µν, (TT-gauge: h T 11 T = ht 22 T h +, h T 12 T = ht 21 T h ) = Vacuum Einstein s equations on the form: R (1) ab + R(2) ab = 0 Nonlinear response terms connected to wave perturbations, h +, h, through R (2) ab = 0. From form of evolution equations we see that we need only solve for terms e 2iωt = h (2) 11 = h(2) 22 = h(2) 33 = (h2 + + h 2 )/4 Thessaloniki, 30/8-5/ p. 29/38

46 Four wave coupling of EMWs and GWs in vacuum Largest nonlinear terms in (5) of the form Eh, Bh etc. = induction of nonresonant (ω nr k nr ) EM fields. Total EM field of the form E tot = E A + E B + E nr ; E nr of one order higher in amplitude. Will combine with terms of appropriate frequency/wavenumber in (5) and produce terms resonant with original wave perturbation. Will also enter the energy momentum tensor and contribute to back reaction on GWs. Introduce linear polarization states, E +, E, of EMWs, = Thessaloniki, 30/8-5/ p. 30/38

47 Four wave coupling of EMWs and GWs in vacuum Amplitude evolution equations E A+ = 1 2 ω2 (1 + cos 2 θ)h I E B+ + ω 2 cos θh II E B (7) E A = ω 2 cos θh II E B ω2 (1 + cos 2 θ)h I E B (8) E B+ = 1 2 ω2 (1 + cos 2 θ)h I E A+ ω 2 cos θh II E A (9) E B = ω 2 cos θh II E A ω2 (1 + cos 2 θ)h I E A (10) where H I h A+ h B+ h A h B, H II h A+ h B + h A h B+ Thessaloniki, 30/8-5/ p. 31/38

48 Four wave coupling of EMWs and GWs in vacuum Back reaction on GWs given by: δg 11 δg 22 = κ(δt 11 δt 22 ) δg 12 + δg 21 = κ(δt 12 + δt 21 ) Largest terms in δt ab of the form E A E B with oscillating part e 2iωt = induction of nonresonant GW fields with same temporal variation. Nr fields will combine with GW fields of appropriate frequency/wavenumber through nonlinearities in EE, and form terms resonant with original perturbation. Can separate terms in EE describing energy momentum pseudotensor from metric response to EMW energy momentum tensor = nonresonant GW fields calculated separately. Thessaloniki, 30/8-5/ p. 32/38

49 Four wave coupling of EMWs and GWs in vacuum Expanded metric ansatz: g µν = η µν + h T T µν + h (2) µν + h (nr) µν and corresponding tetrad basis in EE = h A+ = κ(1 + cos 2 θ)e I h B+ + 2κ cos θe II h B (11) h A = 2κ cos θe II h B+ κ(1 + cos 2 θ)e I h B (12) h B+ = κ(1 + cos 2 θ)e I h A+ + 2κ cos θe II h A (13) h B = 2κ cos θe II h A+ κ(1 + cos 2 θ)e I h A (14) where E I E A+ E B+ + E A E B, E II E A+ E B E A E B+ Thessaloniki, 30/8-5/ p. 33/38

50 Four wave coupling of EMWs and GWs in vacuum Considering long pulses, so that we may let 2iω t energy conservation is easily verified. Example: Let E A = E A+ E, E B = E B+ E, h A = h A+ h, h B = h B+ h and put E = Êeiϕ E, h = ĥeiϕ h and rewrite in terms of normalized energy densities, ẼEM E EM /E tot, Ẽ GW E GW /E tot, for the waves = where τ (1+cos2 θ)κe tot ω τ Ẽ EM + sin ΨẼEMẼGW = 0 τ Ẽ GW sin ΨẼEMẼGW = 0 τ Ψ cos Ψ(ẼEM ẼGW ) = 0 t and Ψ 2ϕ h 2ϕ E Thessaloniki, 30/8-5/ p. 34/38

51 Four wave coupling of EMWs and GWs in vacuum ( EEM Rescaled time ( EGW Rescaled time E GW )t=0 104, Ψ t=0 = π 2 E EM )t=0 104, Ψ t=0 = π 2 E EM (t) = E EM (0)e κ(1+cos 2 θ)etot E GW E GW ω t E GW (t) = E GW (0)e κ(1+cos 2 θ)etot t ω E EM E EM Thessaloniki, 30/8-5/ p. 35/38

52 Four wave coupling of EMWs and GWs in vacuum ( EEM Rescaled time ( EGW Rescaled time E GW )t=0 104, Ψ t=0 = π 2 E EM )t=0 104, Ψ t=0 = π ( EEM Rescaled time ( EGW Rescaled time E GW )t=0 104, Ψ t=0 ± π 2 E EM )t=0 104, Ψ t=0 ± π 2 Thessaloniki, 30/8-5/ p. 35/38

53 Four wave coupling of EMWs and GWs in vacuum Time scale for coherent interaction: T coh h 2 ω 1 Cross-section for incoherent interaction: σ L 2 P ω2 T 2 P, T inc T coh /ω 2 T 2 P Collisional frequency: ν = σnc, n photon/graviton number density. Possible applications (to be worked out) Processes in early Universe: thermalization of high frequency GW background, but perhaps not of GWs with longer wavelength. Enables us to put boundaries on energy density of GWs in early universe. Highly exotic astrophysical systems. Relevant process if ν > H. Thessaloniki, 30/8-5/ p. 36/38

54 Future work Calculations for early universe Thessaloniki, 30/8-5/ p. 37/38

55 Future work Calculations for early universe Find effective Lagrangian field theory Thessaloniki, 30/8-5/ p. 37/38

56 Future work Calculations for early universe Find effective Lagrangian field theory 4-wave coupling involving 4 gravitational waves Thessaloniki, 30/8-5/ p. 37/38

57 Future work Calculations for early universe Find effective Lagrangian field theory 4-wave coupling involving 4 gravitational waves Coupling to other types of fields? Thessaloniki, 30/8-5/ p. 37/38

58 Thank You! Thessaloniki, 30/8-5/ p. 38/38