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Gravitational Radiation reading course on “Gravitational Waves” Marcelo Ponce: CCRG, RIT Rochester, NY – October, 2008 mponce «Gravitational Radiation» Outline 1 2 3 4 5 Introduction to GW Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW Non-linear wave solution Detection of GW Resonant detector Generation of GW General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitudes estimates Exact solution of the wave equation Energy carried by Gravitational Waves General considerations The energy flux of a Gravitational Wave Energy lost by a radiating system Example: A Binary Pulsar Summary mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Linearized Einstein equations Basic properties of GW Einstein field eqs. Gµν Rµν 12 gµν R 8πTµν 10 linearly independent eqs. Gravitational Wave as solution for Einstein eqs. is valid under idealized assumption. vacuum and asymptotically flat space-time, and a linearized regime. search for wave-like solutions to Einstein eqs. in a space-time with very modest curvature and with a metric line element which is that of flat space-time but for small derivations on nonzero curvature, gµν ηµν mponce hµν O prhµν sq «Gravitational Radiation» Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations (cont.) Basic properties of GW therefore linearized expressions for Christoffel symbols and Riemann tensor are required Γµαβ 12 gµν pgνα,β Rµν 12 gβν,α gαβ,ν q hµα,να 1 µ h α,β 2 hνα,µα hµν,αα h,µν R gµν Rµν hνα,α µ hµν, αα h,µν mponce ηµν Rµν ù Einstein equations –linearized form– hµα,αν hµ β,α hαβ,µ ηµν hαβ,αβ h,αα 16πTµν «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave asympt. flat, “Lorentz gauge”, ... Basic properties of GW – Linearized Einstein equations (cont.) introduce «trace-free» tensors, defined as h̄µν hµν 21 ηµν h ñ R̄µν Gµν , ¯ h̄µν hµν , h̄ h the linearized Einstein eqs. take a more compact form, h̄µν,αα ηµν h̄αβ,αβ h̄να,α µ 16πTµν D’Alambertian (or wave) operator ù h̄µν,αα gauge freedom inherent to GR, h̄µα,α lh̄µν 16πTµν devoid GGGGGGGGGGGA of matter mponce lh̄µν 0 (“Lorentz” gauge) «Gravitational Radiation» lh̄µν 0 Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave a wave solution to Einstein equations Basic properties of GW – Linearized Einstein equations grav. field weak but not stationary, vacuum – linearized eqs.: 3d-wave eq. plane wave, (solution) h̄µν R BBt 2 2 Aµν eiκα x ∇2 h̄µν α 0 ( η µν κµ κν κν κν 0 Ð condition for having a solution ñ κµ null 4-vector, i.e. tang. to the world-line of a photon, ç κµ Ñ pω, ~κq wave vector (direction of the wave). µ x pλq κµ λ lµ , photon’s path ñ κµ xµ κ0 t ~κ ~x const κµ lµ the photon travels with the GW, staying forever at the same phase 6 the wave itself travels at the speed of light dispersion relation for the wave, ω 2 |~κ|2 ñ wave’s phase velocity = group velocity = 1 ù light speed. mponce «Gravitational Radiation» Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties TT gauge Linearized Einstein equations – gauge condition gauge condition for Einstein linearized eqs, ñ Aµν must be orthogonal to ~κ, r Aµν κν h̄µα,α 0 s 0 using the gauge freedom, it’s possible to change the gauge while remaining within the Lorentz class of gauges, Aµµ 0 and r transverse-traceless (TT) gauge Aµν U ν Aµν κν 0, orthogonality cond. Aµµ 0, infintesimal gauge transf. Aµν U ν 0, global Lorentz frame. mponce 0 s. example: Lorentz frame, Minkowski spacetime background – plane waves in z-direction TT ãÑ only 2 dof, ATT xx and Axy . «Gravitational Radiation» Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties TT gauge Linearized Einstein equations – gauge condition gauge condition for Einstein linearized eqs, ñ Aµν must be orthogonal to ~κ, r Aµν κν h̄µα,α 0 s 0 using the gauge freedom, it’s possible to change the gauge while remaining within the Lorentz class of gauges, Aµµ 0 and r transverse-traceless (TT) gauge Aµν U ν Aµν κν 0, orthogonality cond. Aµµ 0, infintesimal gauge transf. Aµν U ν 0, global Lorentz frame. mponce 0 s. example: Lorentz frame, Minkowski spacetime background – plane waves in z-direction TT ãÑ only 2 dof, ATT xx and Axy . «Gravitational Radiation» Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties TT gauge Linearized Einstein equations – gauge condition gauge condition for Einstein linearized eqs, ñ Aµν must be orthogonal to ~κ, r Aµν κν h̄µα,α 0 s 0 using the gauge freedom, it’s possible to change the gauge while remaining within the Lorentz class of gauges, Aµµ 0 and r transverse-traceless (TT) gauge Aµν U ν Aµν κν 0, orthogonality cond. Aµµ 0, infintesimal gauge transf. Aµν U ν 0, global Lorentz frame. mponce 0 s. example: Lorentz frame, Minkowski spacetime background – plane waves in z-direction TT ãÑ only 2 dof, ATT xx and Axy . «Gravitational Radiation» Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties TT gauge Linearized Einstein equations – gauge condition gauge condition for Einstein linearized eqs, ñ Aµν must be orthogonal to ~κ, r Aµν κν h̄µα,α 0 s 0 using the gauge freedom, it’s possible to change the gauge while remaining within the Lorentz class of gauges, Aµµ 0 and r transverse-traceless (TT) gauge Aµν U ν Aµν κν 0, orthogonality cond. Aµµ 0, infintesimal gauge transf. Aµν U ν 0, global Lorentz frame. mponce 0 s. example: Lorentz frame, Minkowski spacetime background – plane waves in z-direction TT ãÑ only 2 dof, ATT xx and Axy . «Gravitational Radiation» Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties TT gauge Linearized Einstein equations – gauge condition gauge condition for Einstein linearized eqs, ñ Aµν must be orthogonal to ~κ, r Aµν κν h̄µα,α 0 s 0 using the gauge freedom, it’s possible to change the gauge while remaining within the Lorentz class of gauges, Aµµ 0 and r transverse-traceless (TT) gauge Aµν U ν Aµν κν 0, orthogonality cond. Aµµ 0, infintesimal gauge transf. Aµν U ν 0, global Lorentz frame. mponce 0 s. example: Lorentz frame, Minkowski spacetime background – plane waves in z-direction TT ãÑ only 2 dof, ATT xx and Axy . «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Effect on free particles Making sense of the TT gauge Lorentz frame, particle initially at rest, TT-gauge. particle init. in a wave-free region encounters a GW, free d α particle obeys the geodesic eq., Γαµν U µ U ν 0 dτ U ñ particle remains at rest forever regardless of the GW 6 TT-gauge ÞÑ coord.system attached to individual particles. nearby particles, beginning at rest, proper distance between ³ them ³ 1 ∆l |ds2 | 2 |gαβ dxα dxβ |1{2 1 21 hTT xx px 0q ε 6 proper distance does change with time (gral. hTT xx 0). obs: difference between computing a coord-dependent nbr. (position) and a coord-indep. nbr. (proper distance); the effect of the wave is unambiguously seen in the mponce «Gravitational Radiation» coord-indep. nbr. Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Effect on free particles Making sense of the TT gauge Lorentz frame, particle initially at rest, TT-gauge. particle init. in a wave-free region encounters a GW, free d α particle obeys the geodesic eq., Γαµν U µ U ν 0 dτ U ñ particle remains at rest forever regardless of the GW 6 TT-gauge ÞÑ coord.system attached to individual particles. nearby particles, beginning at rest, proper distance between ³ them ³ 1 ∆l |ds2 | 2 |gαβ dxα dxβ |1{2 1 21 hTT xx px 0q ε 6 proper distance does change with time (gral. hTT xx 0). obs: difference between computing a coord-dependent nbr. (position) and a coord-indep. nbr. (proper distance); the effect of the wave is unambiguously seen in the mponce «Gravitational Radiation» coord-indep. nbr. Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Effect on free particles Making sense of the TT gauge Lorentz frame, particle initially at rest, TT-gauge. particle init. in a wave-free region encounters a GW, free d α particle obeys the geodesic eq., Γαµν U µ U ν 0 dτ U ñ particle remains at rest forever regardless of the GW 6 TT-gauge ÞÑ coord.system attached to individual particles. nearby particles, beginning at rest, proper distance between ³ them ³ 1 ∆l |ds2 | 2 |gαβ dxα dxβ |1{2 1 21 hTT xx px 0q ε 6 proper distance does change with time (gral. hTT xx 0). obs: difference between computing a coord-dependent nbr. (position) and a coord-indep. nbr. (proper distance); the effect of the wave is unambiguously seen in the mponce «Gravitational Radiation» coord-indep. nbr. Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Effect on free particles Making sense of the TT gauge Lorentz frame, particle initially at rest, TT-gauge. particle init. in a wave-free region encounters a GW, free d α particle obeys the geodesic eq., Γαµν U µ U ν 0 dτ U ñ particle remains at rest forever regardless of the GW 6 TT-gauge ÞÑ coord.system attached to individual particles. nearby particles, beginning at rest, proper distance between ³ them ³ 1 ∆l |ds2 | 2 |gαβ dxα dxβ |1{2 1 21 hTT xx px 0q ε 6 proper distance does change with time (gral. hTT xx 0). obs: difference between computing a coord-dependent nbr. (position) and a coord-indep. nbr. (proper distance); the effect of the wave is unambiguously seen in the mponce «Gravitational Radiation» coord-indep. nbr. Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Effect on free particles (cont.) Making sense of the TT gauge, «geodesic deviation» eq. of geodesic deviation, ï connecting vector ξ for 2 parts., dτd 2 Uµ ξµ 2 ξα dxµ {dτ ; to the lowest order, ñ Uµ p0, ε, 0, 0q. two particles initially separeted in the x-direction, $ ' & B2 x B t2 ξ 2 1 TT 2 ε t2 hxx ' % B2 y B t2 ξ 2 1 TT 2 ε t2 hxy Rαµνβ UµUν ξβ p1, 0, 0, 0q, two particles initially separeted in the y-direction, B B $ ' & B2 y B t2 ξ 2 1 TT 2 ε t2 hyy B B ' % B2 x B t2 ξ 2 1 TT 2 ε t2 hxy mponce «Gravitational Radiation» B B B B Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Polarization of Gravitational Waves Linear polarization ring of free particles initially at rest in the x y plane, GW in the z-direction. « » polarization hTT xx e 0, hTT xy «» polarization 0 hTT xy e~x b e~x e~y b e~y e mponce TT 0, hTT xx hyy 0 e~x b e~y «Gravitational Radiation» e~y b e~x Linearized Einstein equations Wave solution to Einstein equations Transverse-Traceless gauge Polarization of GW non-linear plane wave Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Non-linear wave solution any wave can be written as superposition of plane waves linearized theory, describes the kind of waves arrive to Earth linear plane wave ÞÑ solution of the non-linear equation null coordinates: ñ ds2 dudv utz dx2 vt dy2 vacuum field equation, z dudv g2 {g 0 ds2 f 2 {f f 2 puqdx2 g2 puqdy2 prescribe an arbitrary function gpuq, and solve for f puq. same freedom as in the linear case g 1 εpuq, near the linear case Ñ f 1 εpaq ñ linear wave eq. mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Resonant detector Detection of Gravitational Waves General Considerations – resonant detector gravitational wave spectrum, unexplored GR carries information that no electromag. rad. can provide. «antenna» monitoring electromag. signals traveling between 2 freely falling particles technical difficulties (small perturb, noise,...) oldest type of detector proposed: resonant oscillator pioneered by J.Weber (1961) ultimate propossal, use of entangled photons through their polarization state... (0810.3911) mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Resonant detector Resonant detector 2 point massive (m) particles, spring (k) and damping (ν) " mx1,00 mx2,00 kpx1 x2 kpx2 x1 l0 q ν px1 x2 q,0 l0 q ν px2 x1 q,0 Damped Harm. Osc. GW passing... 1 ξ,00 2γξ,0 ω02 ξ 0 ! ) 1 free particle remains at rest in TT coordinates xα . assuming motions only due to GW, i.e. ξ Opl0 |hµν |q ! l0 TT-coords. j ñ mx,0j1 101 Fj1 mx,00 Fj Op|hµν |2q x, x,0 , x,00 Ophµν q GGGGGGGGGGGGGGGGGGGA 2 ñ only nongravit. force spring 9 lptq ξ,00 2γξ,0 ω02 ξ 12 l0hTT xx,00 mponce ³ r1 1{2 hTT xx ptqs dt forced, damped harm. osc. «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Resonant detector Resonant detector 2 point massive (m) particles, spring (k) and damping (ν) " mx1,00 mx2,00 kpx1 x2 kpx2 x1 l0 q ν px1 x2 q,0 l0 q ν px2 x1 q,0 Damped Harm. Osc. GW passing... 1 ξ,00 2γξ,0 ω02 ξ 0 ! ) 1 free particle remains at rest in TT coordinates xα . assuming motions only due to GW, i.e. ξ Opl0 |hµν |q ! l0 TT-coords. j ñ mx,0j1 101 Fj1 mx,00 Fj Op|hµν |2q x, x,0 , x,00 Ophµν q GGGGGGGGGGGGGGGGGGGA 2 ñ only nongravit. force spring 9 lptq ξ,00 2γξ,0 ω02 ξ 12 l0hTT xx,00 mponce ³ r1 1{2 hTT xx ptqs dt forced, damped harm. osc. «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Resonant detector resonant detector for sources of GW of fixed freq. (e.g. pulsars or close binary stars or even bursts) assuming hTT xx A cos Ωt, the steady solution ξ R cos pΩt φq osc., avg. energy, resonant,... 21 mpx1,0q2 12 mpx2,0q2 21 kξ2 xEy 81 mR2pω2 Ω2q A Rreson 41 l0 ApΩ{γ q R 12 rpω Ωlq Ω 4Ω γ s{ 1 Ereson 64 ml02 Ω2 A2 pΩ{γ q2 tan φ 2γΩ{pω02 Ω2 q Q ω0 {2γ massive cylindrical bars, ‘spring’ Ñ elasticity of the bar 0 0 2 E 2 2 2 1 2 -stretched along its axis-; bar hited broadside, excite its long. modes noise sources: random, thermal, external vibrations mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Resonant detector resonant detector for sources of GW of fixed freq. (e.g. pulsars or close binary stars or even bursts) assuming hTT xx A cos Ωt, the steady solution ξ R cos pΩt φq osc., avg. energy, resonant,... 21 mpx1,0q2 12 mpx2,0q2 21 kξ2 xEy 81 mR2pω2 Ω2q A Rreson 41 l0 ApΩ{γ q R 12 rpω Ωlq Ω 4Ω γ s{ 1 Ereson 64 ml02 Ω2 A2 pΩ{γ q2 tan φ 2γΩ{pω02 Ω2 q Q ω0 {2γ massive cylindrical bars, ‘spring’ Ñ elasticity of the bar 0 0 2 E 2 2 2 1 2 -stretched along its axis-; bar hited broadside, excite its long. modes noise sources: random, thermal, external vibrations mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Resonant detector resonant detector for sources of GW of fixed freq. (e.g. pulsars or close binary stars or even bursts) assuming hTT xx A cos Ωt, the steady solution ξ R cos pΩt φq osc., avg. energy, resonant,... 21 mpx1,0q2 12 mpx2,0q2 21 kξ2 xEy 81 mR2pω2 Ω2q A Rreson 41 l0 ApΩ{γ q R 12 rpω Ωlq Ω 4Ω γ s{ 1 Ereson 64 ml02 Ω2 A2 pΩ{γ q2 tan φ 2γΩ{pω02 Ω2 q Q ω0 {2γ massive cylindrical bars, ‘spring’ Ñ elasticity of the bar 0 0 2 E 2 2 2 1 2 -stretched along its axis-; bar hited broadside, excite its long. modes noise sources: random, thermal, external vibrations mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation Generation of Gravitational Waves General considerations amplitude of any GW incident on Earth SMALL, hµν amplitude of GW falls off as r1 far from the source Op1q. a distance R from a source of mass M, largest amplitude waves expected are of order M {R (e.g. formation of a 10Md BH in a supernova explosion in a nearby galaxy 1023 m away, is about 1017 ) Goal: to solve wave eq. with source 2 BBt2 ∇2 h̄µν 16πTµν mponce approx. soln.: simplified, but realistic assumptions exact solution «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation approximate calculation of wave generation Generation of GW Sµν pxiqeiΩt 2nd assumption, region (Sµν 0) is small compared with 2π {Ω λGW – «slow-motion assumption» ãÑ soln, h̄µν Bµν pxiqeiΩt ñ ∇2 Ω2 Bµν 16πSµν “pB2{Bt2 ∇2qf g” outside of the source (Sµν 0) ù outgoing radiation determine Aµν ÞÑ source is solution outside the source nonzero only inside a sphere of Z A iΩr Bµν r eiΩr r e radius ε ! 2π {Ω no ingoing wave Ñ Zµν 0 1st assumption, Tµν µν µν mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation approximate calculation of wave generation Generation of GW, determining the outside solution ∇2 Ω2 Bµν 16πSµν h̄µν Bµν eiΩt Bµν Ar eiΩr ³ 2 Ω Bµν d3 x ¤ Ω2 |Bµν |max 4πε3 {3, ù negligible term ³ 2 ¶ d ∇ Bµν d3 x n ∇Bµν dS ÝÑ 4πε2 dr Bµν rε 4πAµν µν * expressions for GW generated by the sources, neglecting terms r2 and r1 that are higher order in εΩ def. Jµν ³ Sµν d3 x; Aµν εÑ0 ÝÑ 4Jµν ñ h̄µν h̄µν are componentes of a single tensor 4Jµν eiΩprtq{r conservation law for T µν , T µν,ν 0 6 T|µν r if Ω 0, J µ0 0 ñ h̄µ0 0 mponce 0 «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation approximate calculation of wave generation Generation of GW, quadrupole approximation for gravitational radiation expression for Jij , for a source in slow motion, T 00 ρ, Newtonian mass density Quadrupole moment tensor of mass distribution, I lm h̄jk ³ T 00 xl xm d3 x Dlm eiΩt 2Ω2Djk eiΩprtq{r 2Ω2Ijk e r Jlk 21 Djk Ω2eiΩprtq{r }Quadrupole approximation for gravitational radiation~ iΩr terms neglected of order r2 but also r1, not dominant in the slow-motion approximation h̄jk,k is of higher order, and this guarantees the gauge condition h̄µν,ν 0 is satisfied at the lowest order in r1 and Ω mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation approximate calculation of wave generation Generation of GW, quadrupole approximation for gravitational radiation expression for Jij , for a source in slow motion, T 00 ρ, Newtonian mass density Quadrupole moment tensor of mass distribution, I lm h̄jk ³ T 00 xl xm d3 x Dlm eiΩt 2Ω2Djk eiΩprtq{r 2Ω2Ijk e r Jlk 21 Djk Ω2eiΩprtq{r }Quadrupole approximation for gravitational radiation~ iΩr terms neglected of order r2 but also r1, not dominant in the slow-motion approximation h̄jk,k is of higher order, and this guarantees the gauge condition h̄µν,ν 0 is satisfied at the lowest order in r1 and Ω mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation approximate calculation of wave generation Generation of GW, quadrupole approximation for gravitational radiation expression for Jij , for a source in slow motion, T 00 ρ, Newtonian mass density Quadrupole moment tensor of mass distribution, I lm h̄jk ³ T 00 xl xm d3 x Dlm eiΩt 2Ω2Djk eiΩprtq{r 2Ω2Ijk e r Jlk 21 Djk Ω2eiΩprtq{r }Quadrupole approximation for gravitational radiation~ iΩr terms neglected of order r2 but also r1, not dominant in the slow-motion approximation h̄jk,k is of higher order, and this guarantees the gauge condition h̄µν,ν 0 is satisfied at the lowest order in r1 and Ω mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation approximate solution for GW generation – TT gauge choosing axes so that the point where the wave is measured, travels in the z-direction, $ TT & h̄zi 0 TT 2 iΩr {r h̄ h̄TT yy Ω p-Ixx -Iyy qe % xx TT 2 iΩr h̄xy 2Ω -Ixy e {r mponce -Ijk Ijk 31 δjkIll Trace-free reduced quadrupole «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation GW generation, approximate solution I examples Examples: 1 waves emitted by a simple oscillator «laboratory oscillator», both masses osc. with ω à only one non-zero component: Ixx mrpx1 q2 px2 q2 s mrp 21 l0 A cos ωtq2 p 12 l0 A cos ωtq2 s const. mA2 cosp2ω tq 2ml0A cosploωoo n tq å ω-term: å 2ω-term: " 4 iωt ß same radiation pattern, -Ixx 3 ml0 Ae TT h̄TT 4mω 2 l Ae2iω prtq {r 2 iωt h̄ 0 -Iyy -Izz 3 ml0 Ae xx yy TT h̄TT 0, h̄ no rad. in x dir. ß radiation in z-direction, xy ij TT TT 2 iω p r t q h̄xx h̄yy 2mω l0 Ae {r ë Total radiation: TT TT h̄xy h̄ij 0, no rad. in x dir. 2 h̄TT xx r2mω l0 A cos ω pr tq ë off-diagonal vanished, 4mω 2 A2 cos 2ω pr tqs{r 1034 {r ë radiation linearly polarized mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation GW generation, approximate solution II examples 2 gravitational source: Binary System, two stars of mass m, idealized as points in circular orbit è orbit equation: m2" {l02 mω2 pl0{2q ñ ω p2m{l03q1{2 x1 ptq 21 l0 cos ωt y1 ptq 12 l0 sin ωt î curves of motion: x2 ptq x1 ptq y2 ptq y1 ptq å Quadrupole tensor: $ & Ixx % Iyy Ixy 14 ml02 cosp2ωtq const. 14 ml02 cosp2ωtq const. 14 ml02 sinp2ωtq å Reduced Quadrupole tensor: " I-xx -Ixy -Iyy 14 ml02 ω 2 e 2iωt 1 2 2 2iωt 4 iml0 ω e mponce î all radiation comes out Ω 2ω, ß along z-direction (perp. to the plane of the orbit) circularly polarized radiation, h̄TT xx h̄TT xy 2 2 2iω prtq {r h̄TT yy 2ml0 ω e 2 2 2iω p 2iml0ω e rtq{r «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation GW generation, approximate solution III examples ß Linear polarization radiation on x-direction: h̄TT yy 1 2 2 2iωprtq h̄TT {r zz ml0 ω e 2 aligned with the orbital plane e.g. pulsar PSR 1913+16, two very compact stars orbiting each other closely, orbital period (Doppler shift) 7h 45min 7s both stars have masses approx. equal to 1.4Md 5 kpc away from the Earth ß its radiation amplitude 1020 at Earth mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation Order-of-magnitudes Estimates MR2, for a system of mass M and size R radiation amplitude about M pΩRq2 {r v2 pM {Rq Djk a collapsing mass moving under its own gravitational forces: v2 φ0 (typical Newtonian potential in the source), M {r φr (Newtonian potential of the source at distance r) ñ h φ0φr 6 wave amplitude À Newtonian potential BUT we detect h not φ spherically symmetric motions do not radiate if T 00 is spherically symmetric ñ I lm 9δ lm and I- lm vanishes. conservation of energy eliminates ‘monopole’ radiation in linearized theory ( charge conservation eliminates monopole radiation in EM) mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations approximate calculation of wave generation Examples, GW generation–approx. soln. order-of-magnitude estimates Exact Solution of the Wave Equation Exact Solution of the Wave Equation Arbitrary Tµν , outgoing-wave outgoing-wave for arbitrary source h̄µν pt, xi q 4 ³ p q d3 y Tµν t R,yi R |xi| r " |yi| y, ñ using conservation of T µν,ν 0 (total enegy and momentum conserved), h̄jk pt, xi q 2r Ijk,00 pt rq , R |xi yi | h̄µν pt, xi q 4 r ³ Tµν pt R, yi qd3 y adopting TT gauge h̄TT xx h̄TT xy mponce 1r r-Ixx,00pt rq -Iyy,00pt rqs 2r -Ixy,00pt rq «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations The energy flux of a Gravitational Wave Energy lost by a radiating system Example: a binary pulsar Energy carried away by Gravitational Waves GW can put energy into things they pass through (detector) carry energy away from their sources effects of the loss of energy by GW can be observed although the GW itself not á harmonic oscillator as a detector of waves: ‘test body’ without influence on the GW è inconsistent! à detector extract energy from the wave è waves must be weaker (‘downstream’ ‘upstream’) ß detector (oscillator) in motion will radiate itself mponce ß waves of two frequencies will be emitted ß waves with same freq. as incident GW (Ω): total downstream wave field Ý amplitude, sum of two waves interfere desctructively, producing a net decrease in «Gravitational Radiation» the dowstream amplitude Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations The energy flux of a Gravitational Wave Energy lost by a radiating system Example: a binary pulsar The energy flux of a Gravitational Wave I energy flux: energy carried by the GW across a surface per unit area per unit time á array of oscillators, filling the plane z 0 ( continuous distribution, σ oscillators per unit area) à incident wave (TT gauge), $ TT ç net energy flux (decreasing), δF & h̄xx A cosrΩpz tqs TT TT 21 σmγΩ2R2 h̄yy h̄xx % ç each oscillator, Ixx ml0 R cospΩt 0, other components ç each oscillator steady state φq ç producing a wave amplitude, oscillation, ξ R cospΩt φq 2 ç energy given to each osc., δ h̄xx 2Ω ml0 R cosrΩpr tq φs{r ç total radiated field due to all the dE{dt ν pdξ {dtq2 mγ pdξ {dtq2 total Ù averaging, steady energy loss, oscs., δ h̄xx 4πσmΩl0 R sinrΩpz ³ tq φs x dEdt y 2π1 p dEdt qdt 12 mγΩ2R2 Ω mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations The energy flux of a Gravitational Wave Energy lost by a radiating system Example: a binary pulsar The energy flux of a Gravitational Wave II energy flux: energy carried by the GW across a surface per unit area per unit time Total Flux F 32π1 Ω2A2 ñ F TTµν y 32π1 Ω2xh̄TT µν h̄ invariant under background Lorentz transformations, but not under gauge changes applies to all polarizations (Lorentz inv.) applies to any waveform (either plane waves or spherical...) not proved energy conservation –GR– (assumption) valid only in linearized theory (‘lowest order’) mponce «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations The energy flux of a Gravitational Wave Energy lost by a radiating system Example: a binary pulsar Energy Lost by a Radiating System general isolated system radiating according a QUADRUPOLE distribution integrating the «total flux» over a sphere surrounding the system net energy loss rate A Ω 2y Iij -I ij 4-I zj-Iz j -Izz at distance r along the z-axis, F 16πr 2 x2j j generalizing for arbitrary locations, n x {r (unit normal vector), Ω6 F 16πr Iij -I ij 4nj nk -Iji -Iki ni nj nk nl -Iij -Ikl y 2 x26 Luminosity L of a source of Gravitational Waves, integral the flux over the sphere of radius r, ³ of 2 L FR sin θdθdφ ñ L 15 Ω6 x-Iij -I ij y 1 Generalization to general time dependence L 5 x;-I ij;-I only for weak gravitational fields and slow velocities. alt.deriv. ij mponce «Gravitational Radiation» y Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties strong sources L is very sensitive to velocity largest velocities, v2 ñ L À pφ0q5 φ0 φ0 À 1, L À 1 c5 {G 3.6 1052 W –luminosity upper limit –. mponce General considerations The energy flux of a Gravitational Wave Energy lost by a radiating system Example: a binary pulsar Ø anything whose Newtonian potential substancially exceeds 1 ì must form a BH: its gravitational field will be so strong that no radiation will escape at all é L 1 is the largest luminosity any source can have ç Quasars, the most luminous class of object known, geom.luminosity À 107. «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties General considerations The energy flux of a Gravitational Wave Energy lost by a radiating system Example: a binary pulsar Example: a Binary Pulsar ç -Iij (reduced quadrupole tensor) for a binary system consisting of two stars of equal mass M in circular orbits a distance l0 , luminosity of the binary system, 4.0p q L ? p q { 8 2 4 6 32 Mω 10 3 5 M l0 ω 534 Mω 10 3 (geom.units) { Newtonian energy of the system L(SI units) cG L(geometrized) for the binary pulsar system, circ.orb., ω 2π {P 7.5049 1013 m1 ñ L 1.71 1029 (geom.units) 5 L, dP{dt 3PL 2E 2 1013 6 106s yr1 eccentricity e 0.617 !!! dE dt mponce defining the orbital radius r 12 l0 , 1 M2 2 2 E 21 Mω 2 r2 2 Mω r 2r 1.11 103m energy radiates in waves change the orbit by decreasing its 2 1 dω 2 1 dP energy, E1 dE dt 3 ω dt 3 P dt dP{dt 7.2 105 s yr1 , p2.30 0.22q 1012 (obs.) «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties Summary GW are fluctuations in curvature that propagate through the universe at speed of light. They are a natural consequence of Einstein’s theory of relativity. GW are transverse and traceless in nature and are produced by changes in the quadrupole moment of a mass distribution. Mass and momentum conservation ensure that there mponce GW carry energy and momentum, which create background curvature. This is equal to the energy and momentum lost by radiation reaction at the source. GW can be lensed and redshifted in the same way as electromagnetic waves, but not readily absorved or dispersed. «Gravitational Radiation» Introduction to GW Detection of GW Generation of GW Energy carried by Gravitational Waves Summary of GW properties References Schutz, Bernard F. { A First Course in General Relativity | . Rezzolla, Luciano Gravitational Waves from Perturbed Black Holes and Relativistic Stars. gr-qc/0302025. Gair, Jonathan R. «Gravitational Waves Astronomy» online course notes @ Cavendish Lab., Cambridge. http://www.ast.cam.ac.uk/jgair/GWAstronomy/ mponce «Gravitational Radiation» Appendixes Appendix i: alternative deduction of generation of GW Appendix ii: «Polarization» – complement Appendix iii: GW Spectrum Appendix iv: }Formal complements~ Generation of GW I alternative approach // EM analogy... Ý classical electrodynamics, energy emitted put by an oscillating electric dipole d Lelect.dip. emitted) (energy 23 q2a2 23 pd:q2; (unit time) d qx Ý Luminosity in GW produced by an oscillating mass-dipole, N point-like particles of mass mA ~d ~d9 °NA1 mA~x9 A ~P : linear momentum conserv. ñ ~d ~p9 0 é Lmassdip. 0 therefore no mass-dipole radiation in GR ( no EM °NA1 mA~xA, radiation from an oscillating electric-monopole) mponce «Gravitational Radiation» Appendixes Appendix i: alternative deduction of generation of GW Appendix ii: «Polarization» – complement Appendix iii: GW Spectrum Appendix iv: }Formal complements~ Quadrupole radiation I Generation of GW - alternative approach // EM analogy... Ý electromagnetic energy emission produced by an oscillating electric quadrupole p ; q pQ;jk Q;jk q where Qjk rpxA qj pxA qk 1 i s, δ p x q p x q is the electric 3 jk A i A Lelect.quad. 1 2 20 Q °N A 1 qA 1 20 quadrupole for a distribution of N charges mponce Ý in close analogy, energy loss due to an oscillating mass quadrupole Lmass.quad. 51 cG5 x;I y2 1G ; ; 2 5 c5 x-Ijk -Ijk y where -Ijk is the trace-less mass quadrupole (or “reduced” mass quadrupole) I jk °N 1 i A³1 mA rpxA qj pxA qk 3 δjk pxA qi pxA q s 1 i ρpxjxk 3 δjk xix qdV «Gravitational Radiation» Appendix i: alternative deduction of generation of GW Appendix ii: «Polarization» – complement Appendix iii: GW Spectrum Appendix iv: }Formal complements~ Appendixes Quadrupole radiation II Generation of GW - alternative approach // EM analogy... A crude estimative of the third derivative of the mass quadrupole is given by, ;-I jk mass of the system in motion size of the system ptime scaleq Lmass quad. cG Mv τ 3 2 MRτ Mvτ 2 2 3 2 so that, 5 Although extremely simplified, this eqs. show that: energy emission in GW is severely suppressed by the coef. G{c5 1059 ÐÑ the conversion of any type of energy into GW is, in general, not efficient. mponce «Gravitational Radiation» Appendixes Appendix i: alternative deduction of generation of GW Appendix ii: «Polarization» – complement Appendix iii: GW Spectrum Appendix iv: }Formal complements~ Quadrupole radiation III Generation of GW - alternative approach // EM analogy... time variation of the mass quadrupole, can become considerable only for very large masses moving at relativisitic speeds. 6 these conditions can not be reached by sources in terrestrial laboratories, but can easily met by astrophisycal compact objects, which therefore become the most promising sources of gravitational radiation. return mponce «Gravitational Radiation» Appendixes Appendix i: alternative deduction of generation of GW Appendix ii: «Polarization» – complement Appendix iii: GW Spectrum Appendix iv: }Formal complements~ Polarization of Gravitational Waves Circular Polarization, Eliptical Polarization, General Polarization Circularly polarized in the x y plane if hTT yy TT . hTT ih xy xx eR e ?2ie eL rotates clockwise hTT xx and e ?2ie rotates counter-clockwise Elliptically polarized with principal axes x and y if TT TT TT hTT xy iahxx , where a is a real number, and hxy hxx . TT If hTT xy αhxx , where α is a complex number (general case for a plane wave), new axes x1 and y1 can be found for which the wave is elliptically polarized with these principal axes. circular and linear polarization are special cases of elliptical. mponce «Gravitational Radiation» Appendixes Appendix i: alternative deduction of generation of GW Appendix ii: «Polarization» – complement Appendix iii: GW Spectrum Appendix iv: }Formal complements~ Gravitational Wave spectrum Frequency of GW in the Universe extends over many decades Frequency of GWs from a system scales as 1/mass mponce «Gravitational Radiation» Appendix i: alternative deduction of generation of GW Appendix ii: «Polarization» – complement Appendix iii: GW Spectrum Appendix iv: }Formal complements~ Appendixes Formulae Generalization for an arbitrary binary system average energy lost rate and angular momentum: 2 3 xdE{dty 325 aµ5pp1m e2Mq7q{2 1 2 5{2 xdL{dty 325 µa7p{2mp1 Meq2q2 average orbital elements: xda{dty 645 aµpp1meMqq{ 1 µpm M q e xde{dty 304 15 a p1e q { 1 µpm M q { xdP{dty 192π 5 a { p1e q { 2 2 7 2 3 73 2 24 e 2 121 2 304 e 2 5 2 4 3 2 5 2 2 7 2 mponce 1 37 4 e 96 73 2 e 24 1 37 4 96 e 73 2 24 e 7 2 e 8 37 4 96 e «Gravitational Radiation»