Chapter 13. Gravitational Waves

Transcription

1 Chapter 13 Gravitational Waves One of the most interesting preditions of the theory of General Relativity is the existene of gravitational waves. The idea that a perturbation of the gravitational field should propagate as a wave is, in some sense, intuitive. For example eletromagneti waves were introdued when the Coulomb theory of eletrostatis was replaed by the theory of eletrodynamis, and it was shown that they transport through spae the information about the evolution of harged systems. In a similar way when a mass-energy distribution hanges in time, the information about this hange should propagate in the form of waves. However, gravitational waves have a distintive feature: due to the twofold nature of g µν, whih is the metri tensor and the gravitational potential, gravitational waves are metri waves. Thus when they propagate the geometry, and onsequently the distane between points, hange in time. Gravitational waves an be studied by following two different approahes, one based on perturbative methods, the seond on the solution of the non linear Einstein equations. The perturbative approah Be gµν 0 a known exat solution of Einstein s equations; it an be, for instane, the metri of flat spaetime η µν, or the metri generated by a Shwarzshild blak hole. Let us onsider a small perturbation of gµν 0 µν aused by some soure desribed by a stress-energy tensor T pert. We shall write the metri tensor of the perturbed spaetime, g µν, as follows g µν = g 0 µν + h µν, 13.1) where h µν is the small perturbation h µν << g 0 µν. It is lear that this assumption is ambiguous, beause we should speify in whih referene frame this is true; however we shall assume that this frame does exists. The inverse metri an be written as g µν = g 0 µν h µν + Oh 2 ), 13.2) where the indies of h µν have been raised with the unperturbed metri h µν g 0 µα g 0 νβ h αβ. 13.3) 157

2 CHAPTER 13. GRAITATIONAL WAES 158 Indeed, with this definition, g 0 µν h µν )g 0 να + h να ) = δ µ α + Oh 2 ). 13.4) In order to find the equations that desribe h µν, we shall write Einstein s equations for the metri 13.1) in the form R µν = 8πG T µν 1 ) 2 g µνtλ λ, 13.5) 4 where T µν is the sum of two terms, one assoiate to the soure that generates the bakground geometry gµν, 0 say Tµν, 0 and one assoiate to the soure of the perturbation Tµν pert. We remind that the Rii tensor R µν is R µν = and that the affine onnetions Γ γ βµ are x α Γα µν x ν Γα µα + Γ α σαγ σ µν Γ α σνγ σ µα, 13.6) Γ γ βµ = 1 2 gαγ [ x µ g αβ + x β g αµ The Γ γ βµ omputed for the perturbed metri 13.1) are Γ γ βµ g µν) = 1 2 [ g 0αγ h αγ] [ x µ g0 αβ + = 1 2 g0αγ [ x µ g0 αβ hαγ [ x µ g0 αβ + x β g0 αµ x β g0 αµ x β g0 αµ ] x α g0 βµ ] x α g0 βµ ] x g α βµ. 13.7) ) x α g0 βµ + x h µ αβ g0αγ [ x µ h αβ + + Oh 2 ) x β h αµ x β h αµ x α h βµ )] x h α βµ ] = Γ γ βµ g 0 ) + Γ γ βµ h) + Oh2 ), 13.8) where Γ γ βµ h) are the terms that are first order in h µν Γ γ βµ h) = 1 2 g0αγ [ x µ h αβ + x β h αµ ] x h α βµ 1 2 hαγ [ x µ g0 αβ + x β g0 αµ When we substitute these expressions of the Γ γ βµ g µν) in the Rii tensor we get ] x α g0 βµ. 13.9) R µν g µν ) = R µν g 0 ) 13.10) + x α Γα µν h) x ν Γα µα h) + Γ ) α σα g 0 Γ σ µν h) + Γ α σα h) Γ ) σ µν g 0 Γ ) α σν g 0 Γ σ µα h) Γ α σν h) Γ ) σ µα g 0 + Oh 2 ) = 8πG T 4 µν 1 ) 2 g µνtλ λ.

3 CHAPTER 13. GRAITATIONAL WAES 159 Sine gµν 0 is by assumption an exat solution of Einstein s equations in vauum R µν g 0 ) = 8πG T 0 4 µν 1 2 g0 µν T ) λ 0 λ ; thus, if we retain only first order terms, the equations for the perturbations h µν redue to x α Γα µν h) x ν Γα µα h) 13.11) + Γ ) α σα g 0 Γ σ µν h) + Γ α σα h) Γ ) σ µν g 0 Γ ) α σν g 0 Γ σ µα h) Γ α σν h) Γ ) σ µα g 0 = 8πG T pert 4 µν 1 ) 2 g µνt pert λ λ. that are linear in h µν ; their solution will desribe the propagation of gravitational waves in the onsidered bakground. 1 This approximation works suffiiently well in a variety of physial situations beause gravitational waves are very weak. This point will be better understood in the next hapter, when we will disuss the generation of gravitational waves. The exat approah The seond approah to the study of gravitational waves seeks for exat solutions of Einstein s equations whih desribe both the soure and the emitted wave, but no solution of this kind has been found so far. Of ourse the non-linearity of the equations makes the problem very diffiult; however, it may be noted that also in eletrodynamis an exat solution of Maxwell s equations appropriate to desribe the eletromagneti field produed by a urrent whih dereases in an eletri osillator due to the emission of eletromagneti waves has never been found, although Maxwell s equations are linear. Exat solutions of Einstein s equations desribing gravitational waves an be found only if one imposes some partiular symmetry as for example plane, spherial, or ylindrial symmetry. The interation of plane waves an also be desribed in terms of exat solutions, and due to the non-linearity of the equations of gravity it is very different from the interation of eletromagneti waves. In the following we shall use the perturbative approah to show that a weak perturbation of the flat spaetime satisfies the wave equation A perturbation of the flat spaetime propagates as a wave and a small pertur- Let us onsider the flat spaetime desribed by the metri tensor η µν bation h µν, suh that the resulting metri an be written as g µν = η µν + h µν, h µν << ) The affine onnetions 13.8) omputed for the metri 13.12) give Γ λ µν = 1 [ 2 ηλρ x h µ ρν + x h ν ρµ ] x h ρ µν + Oh 2 ) ) 1 Notie that the right-hand side of eq.13.11) is a partiular ase of the Palatini identity.

4 CHAPTER 13. GRAITATIONAL WAES 160 Sine the metri g 0 µν η µν is onstant, Γ λ µνg 0 ) = 0 and the right-hand side of eq ) simply redues to Γ α µν Γα µα + Oh 2 ) 13.14) x α x { ν [ ]} = 1 2 F h µν + 2 x λ x µ hλ ν + 2 x λ x ν hλ µ The operator F is the D Alambertian in flat spaetime Einstein s equations 13.5) for h µν finally beome 2 x µ x ν hλ λ + Oh 2 ). F = η αβ x α x = 2 β 2 t ) { [ 2 ]} F h µν x λ x µ hλ ν + 2 x λ x ν hλ µ 2 x µ x ν hλ λ = 16πG T pert 4 µν 1 ) 2 η µνt pert λ λ ) As already disussed in hapter 8, the solution of eqs ) is not uniquely determined. If we make a oordinate transformation, the transformed metri tensor is still a solution: it desribes the same physial situation seen from a different frame. But sine we are working in the weak field limit, we are entitled to make only those transformations whih preserve the ondition h µ ν << 1. If we make an infinitesimal oordinate transformation x µ = x µ + ɛ µ x), 13.17) where ɛ µ ɛ is an arbitrary vetor suh that µ is of the same order of h x ν µν, it is easy to hek that sine x α x β g µν = g α β x µ x = η ν αβ + h α β ) xα x β x µ x, 13.18) ν then h µ ν = h µν ɛ µ x ɛ ν ν x ) µ In order to simplify eq ) it appears onvenient to hoose a oordinate system in whih the harmoni gauge ondition is satisfied, i.e. g µν Γ λ µν = ) Let us see why. This ondition is equivalet to say that, up to terms that are first order in h µν, the following equation is satisfied 2 2 g µν Γ λ µν = 1 2 ηµν η λk { hkµ x ν Sine the first two terms are equal we find + h kν x µ h } µν x k = 1 2 ηλk {h ν k,ν + h µ k,µ h ν ν,k} g µν Γ λ µν = ηλk {h µ k,µ 1 2 hν ν,k } q.e.d.

5 CHAPTER 13. GRAITATIONAL WAES 161 x µ hµ ν = 1 2 x ν hµ µ ) Using this ondition the term in square brakets in eq ) vanishes, and Einstein s equations redue to a simple wave equation supplemented by the ondition 13.21) F h µν = 16πG 4 Tµν 1η ) 2 µνt λ λ h µ x µ ν = 1 h µ 2 x µ, ν 13.22) to hereafter, we omit the supersript pert to indiate the stress-energy tensor assoiated to the soure of the perturbation). If we introdue the tensor h µν h µν 1 2 η µνh λ λ, 13.23) eqs ) beome { F hµν = 16πG T 4 µν h µ x µ ν = 0, and outside the soure where T µν = 0 { F hµν = 0 h µ x µ ν = ) 13.25) Thus, we have shown that a perturbation of a flat spaetime propagates as a wave travelling at the speed of light, and that Einstein s theory of gravity predits the existene of gravitational waves. As in eletrodynamis, the solution of eqs ) an be written in terms of retarded potentials h µν t, x) = 4G Tµν t x-x, x ) 4 x-x d 3 x, 13.26) and the integral extends over the past light-one of the event t, x). This equation represents the gravitational waves generated by the soure T µν. We may now ask how eqs ) and 13.24) should be modified if, instead of onsidering the perturbation of a flat spaetime, we would onsider the perturbation of a urved bakground. For example, suppose g 0 µν is the Shwarzshild solution for a non rotating blak hole. In this ase, it is possible to show that, by a suitable hoie of the gauge, the Einstein equations written for ertain ombinations of the omponents of the metri tensor, an be redued to a form similar to eqs ). However, sine the bakground spaetime is now urved, the propagation of the waves will be modified with respet to the flat ase. The urvature will at as a potential barrier by whih waves are sattered and the final equation will have the form F Φ x µ )Φ = 16πG 4 T µν, 13.27)

6 CHAPTER 13. GRAITATIONAL WAES 162 where Φ is the appropriate ombination of metri funtions, F is the d Alambertian of the flat spaetime and is the potential barrier generated by the spaetime urvature. In other words, the perturbations of a sperially symmetri, stationary gravitational field would be desribed by a Shroedinger-like equation! A omplete aount on the theory of perturbations of blak holes an be found in the book The Mathematial Theory of Blak Holes by S. Chandrasekhar, Oxford: Claredon Press, 1984) How to hoose the harmoni gauge We shall now show that if the harmoni-gauge ondition is not satisfied in a referene frame, we an always find a new frame where it is, by making an infinitesimal oordinate transformation x λ = x λ + ɛ λ, 13.28) provided F ɛ ρ = hβ ρ x 1 h β β β 2 x. ρ 13.29) Indeed, when we hange the oordinate system Γ λ = g µν Γ λ µν transforms aording to equation 8.63), i.e. where, from eq ) Γ λ = xλ x ρ Γρ g ρσ 2 x λ x ρ x σ, 13.30) x λ x = ρ δλ ρ + ɛλ x. ρ If g µν = η µν + h µν see footnote after eq )) Γ ρ = η ρk {h µ k,µ 1 2 hν ν,k } ; 13.31) moreover [ )] g ρσ 2 x λ x = g ρσ λ x ρ x σ x ρ x + ɛλ = 13.32) σ x σ [ )] [ g ρσ δ λ x ρ σ + ɛλ η ρσ 2 ɛ λ ] = x σ x ρ x σ F ɛ λ, therefore in the new gauge the ondition Γ λ = 0 beomes Γ λ = η ρk [ ] [ δρ λ + ɛλ h µ x ρ k x µ 1 2 If we neglet seond order terms in h eq.13.33) beomes Γ λ = η λk [ h µ k x µ 1 2 h ν ] ν x k F ɛ λ = ) h ν ] ν x k F ɛ λ = 0.

7 CHAPTER 13. GRAITATIONAL WAES 163 Contrating with η λα and remembering that η λα η λk = δ k α we finally find F ɛ α = hµ α x 1 h ν ν µ 2 x. α This equation an in priniple be solved to find the omponents of ɛ α, whih identify the oordinate system in whih the harmoni gauge ondition is satisfied Plane gravitational waves The simplest solution of the wave equation in vauum 13.25) is a monoromati plane wave h µν = R { A µν e ikαxα}, 13.34) where A µν is the polarization tensor, i.e. the wave amplitude and k is the wave vetor. By diret substitution of 13.34) into the first equation we find F hµν = η αβ ) e ik γxγ = η αβ [ x γ ] ik x α x β x α γ = 13.35) x β eikγxγ η αβ [ ikγ δ γ βe ] ikγxγ = η αβ [ ikβ e ] ikγxγ = x α x α = η αβ k α k β e ikγxγ = 0, η αβ k α k β = 0, thus, 13.34) is a solution of 13.25) if k is a null vetor. In addition the harmoni gauge ondition requires that x h µ µ ν = 0, 13.36) whih an be written as η µα x h µ αν = ) Using eq ) it gives η µα x A ανe ikγxγ = 0 η µα A µ αν k µ = 0 k µ A µ ν = ) This further ondition expresses the orthogonality of the wave vetor and of the polarization tensor. Sine h µν is onstant on those surfaes where k α x α = onst, 13.39) these are the equations of the wavefront. It is onventional to refer to k 0 is the frequeny of the waves. Consequently as ω, where ω ω k =, k) ) Sine k is a null vetor k 0 ) 2 + k x ) 2 + k y ) 2 + k z ) 2 = 0, i.e ) ω = k 0 = k x ) 2 + k y ) 2 + k z ) 2, 13.42) where k x, k y, k z ) are the omponents of the unit 3-vetor k.

8 CHAPTER 13. GRAITATIONAL WAES The T T -gauge We now want to see how many of the ten omponents of h µν have a real physial meaning, i.e. what are the degrees of freedom of a gravitational plane wave. Let us onsider a wave propagating in flat spaetime along the x 1 = x-diretion. Sine h µν is independent of y and z, eqs ) beome as before we raise and lower indies with η µν ) i.e. hµ ν is an arbitrary funtion of t ± x, and ) 2 2 t + 2 h µ 2 x 2 ν = 0, 13.43) x µ h µ ν = ) Let us onsider, for example, a progressive wave h µ ν = h µ ν [χt, x)], where χt, x) = t x. Being t h µ ν = h µ ν χ = h µ ν, χ t χ x h µ ν = h µ ν χ = 1 h µ ν, 13.45) χ x χ eq ) gives x h µ µ ν = 1 h t ν t + h x ν x = 1 [ ht ν χ h ] x ν = ) This equation an be integrated, and the onstants of integration an be set equal to zero beause we are interested only in the time-dependent part of the solution. The result is h t t = h x t, ht y = h x y, 13.47) h t x = h x x, ht z = h x z. We now observe that the harmoni gauge ondition does not determine the gauge uniquely. Indeed, if we make an infinitesimal oordinate transformation x µ = x µ + ξ µ, 13.48) from eq ) we find that, if in the old frame Γ ρ = 0, in the new frame Γ λ = 0, provided namely, if ξ µ satisfies the wave equation η ρσ 2 x λ = 0, 13.49) x ρ xσ F ξ µ = ) Thus, we an use the four funtions ξ µ to set to zero the following four quantities h t x = h t y = h t z = h y y + h z z = )

9 CHAPTER 13. GRAITATIONAL WAES 165 From eq ) it then follows that h x x = h x y = h x z = h t t = ) The remaining non-vanishing omponents are h z y and h y y h z z. These omponents annot be set equal to zero, beause we have exhausted our gauge freedom. From eqs ) and 13.52) it follows that h µ µ = h t t + h x x + h y y + h z z = 0, 13.53) and sine h µ µ = h µ µ 2h µ µ = h µ µ, 13.54) it follows that h µ µ = 0, hµ ν h µ ν, 13.55) i.e. in this gauge h µν and hµν oinide and are traeless. Thus, a plane gravitational wave propagating along the x-axis is haraterized by two funtions h xy and h yy = h zz, while the remaining omponents an be set to zero by hoosing the gauge as we have shown: h µν = h yy h yz 0 0 h yz h yy ) In onlusion, a gravitational wave has only two physial degrees of freedom whih orrespond to the two possible polarization states. The gauge in whih this is learly manifested is alled the T T -gauge, where T T - indiates that the omponents of the metri tensor h µν are different from zero only on the plane orthogonal to the diretion of propagation transverse), and that h µν is traeless How does a gravitational wave affet the motion of a single partile Consider a partile at rest in flat spaetime before the passage of the wave. We set an inertial frame attahed to this partile, and take the x-axis oinident with the diretion of propagation of an inoming T T -gravitational wave. The partile will follow a geodesi of the urved spaetime generated by the wave d 2 x α dτ + dx µ dx ν 2 Γα µν dτ dτ du α dτ + Γα µνu µ U ν = ) At t = 0 the partile is at rest U α = 1, 0, 0, 0)) and the aeleration impressed by the wave will be ) du α = Γ α 00 = 1 dτ 2 ηαβ [h β0,0 + h 0β,0 h 00,β ], 13.58) t=0)

10 CHAPTER 13. GRAITATIONAL WAES 166 but sine we are in the T T -gauge it follows that ) du α dτ t=0) = ) Thus, U α remains onstant also at later times, whih means that the partile is not aelerated neither at t = 0 nor later! It remains at a onstant oordinate position, regardeless of the wave. We onlude that the study of the motion of a single partile is not suffiient to detet a gravitational wave Geodesi deviation indued by a gravitational wave We shall now study the relative motion of partiles in the gravitational field produed by a gravitational wave. Consider two neighbouring partiles A and B, and hoose a referene frame with origin oinident with the position of the partile A x λ A = 1, 0, 0, 0) ) We shall assume that the two partiles are initially at rest with respet to this frame, and that a plane-fronted gravitational wave reahes them at some time t = 0, propagating along the x-axis. We shall also assume that we are in the T T -gauge, so that the only non-vanishing omponents of the wave are those on the y, z)-plane. Be τ = t the proper time of the partile A. Sine the two partiles are initially at rest, they will remain at a onstant oordinate position even later, when the wave arrives. However, sine the metri hanges, the proper distane between them will hange. For example if the partile B is initially at some point on the y-axis yb l = ds = g yy 1 yb 2 dy = 1 + h T T yy 1 2 dy onstant ) 0 Another way of studying the effet of the passage of the wave, is by means of the equation of geodesi deviation. Be δx µ the vetor whih separates the two partiles, i.e. initially The equation of geodesi deviation 0 δx µ = 0, x B, y B, z B ). d 2 δx λ dτ 2 written in the gauge we have hoosen, beomes = R λ dx β νβµ dτ dx ν dτ δxµ 13.62) d 2 dτ 2 δxλ = R λ 00µδx µ ) If the gravitational wave is due to a perturbation of the flat metri, as disussed in this hapter, the metri an be written as g µν = η µν + h µν, and the Riemann tensor R iklm = 1 2 ) g im 2 x k x + 2 g kl l x i x 2 g il m x k x 2 g km ) m x i x l + g np Γ n klγ p im Γ n kmγ p il),

11 CHAPTER 13. GRAITATIONAL WAES 167 after negleting terms whih are seond order in h µν, beomes R iklm = 1 2 ) h im 2 x k x + 2 h kl l x i x 2 h il m x k x 2 h km + Oh 2 ); 13.65) m x i x l onsequently R i00m = ) h im x 0 x + 2 h 00 0 x i x 2 h i0 m x 0 x 2 h 0m = 1 m x i x 0 2 ht im,00 T, 13.66) beause in the T T -gauge h i0 = h 00 = 0. i and m an assume only the values 2 and 3, i.e. they refer to the y and z omponents. It follows that and the equation of geodesi deviation 13.63) beomes For R λ 00m = η λi R i00m = 1 2 ηλi 2 h T T im 2 t 2, 13.67) d 2 dt 2 δxλ = 1 2 ηλi 2 h T T im t 2 δx m ) t 0 the two partiles are at rest relative to eah other, and onsequently δx j = δx j 0, with δx j 0 = onst, t ) Sine h µν is a small perturbation, when the wave arrives the relative position of the partiles will hange only by infinitesimal quantities, and therefore we put δx λ t) = δx λ 0 + δx λ 1t), t > 0, 13.70) where δx λ 1 t) has to be onsidered as a small perturbation with respet to the initial position δx λ 0. Substituting 13.70) in 13.68), remembering that δx λ 0 is a onstant and retaining only terms of order Oh), eq ) beomes This equation an be integrated and the solution is d 2 dt 2 δxλ 1 = 1 2 ηλi 2 h T T ik t 2 δx k ) δx λ = δx λ ηλi h T T ik δx k 0, 13.72) whih learly shows the tranverse nature of the gravitational wave; indeed, using the fat that if the wave propagates along x only the omponents h 22 = h 33, h 23 = h 32 are different from zero, from eqs ) we find δx 0 = δx η00 h T T 0k δx k 0 = ) δx 1 = δx η11 h T T 1k δx k 0 = δx1 0 δx 2 = δx η22 h T T 2k δx k 0 = δx h T T 22 δx ht T 23 δx0) 3 δx 3 = δx η33 h T T 3k δx k 0 = δx h T T 32 δx ) ht T 33 δx 3 0.

12 CHAPTER 13. GRAITATIONAL WAES 168 Thus, the partiles will be aelerated only in the plane orthogonal to the diretion of propagation. Let us now study the effet of the polarization of the wave. Consider a plane wave whose nonvanishing omponents are we omit in the following the supersript T T ) h yy = h zz = 2R { A + e iωt x )}, 13.74) h yz = h zy = 2R { A e iωt x )}. Consider two partiles loated, as indiated in figure 13.1) at 0, y 0, 0) and 0, 0, z 0 ). Let us onsider the polarization + first, i.e. let us assume A + 0 and A = ) Assuming A + real eqs ) give h yy = h zz = 2A + os ωt x ), h yz = h zy = ) If at t = 0 ωt x ) = π 2, eqs ) written for the two partiles for t > 0 give 1) z = 0, y = y h yy y 0 = y 0 [1 + A + os ωt x )], 13.77) 2) y = 0, z = z h zz z 0 = z 0 [1 A + os ωt x )]. After a quarter of a period os ωt x ) = 1) After half a period os ωt x ) = 0) 1) z = 0, y = y 0 [1 A + ], 13.78) 2) y = 0, z = z 0 [1 + A + ]. 1) z = 0, y = y 0, 13.79) 2) y = 0, z = z 0. After three quarters of a period os ωt x ) = 1) 1) z = 0, y = y 0 [1 + A + ], 13.80) 2) y = 0, z = z 0 [1 A + ]. Similarly, if we onsider a small ring of partiles entered at the origin, the effet produed by a gravitational wave with polarization + is shown in figure 13.2). Let us now see what happens if A 0 and A + = 0 : h yy = h zz = 0, h yz = h zy = 2A os ωt x ) )

13 CHAPTER 13. GRAITATIONAL WAES 169, Figure 13.1:

14 CHAPTER 13. GRAITATIONAL WAES 170, Figure 13.2:

15 CHAPTER 13. GRAITATIONAL WAES 171 Comparing with eqs ) we see that a generi partile initially at P = y 0, z 0 ), when t > 0 will move aording to the equations y = y h yz z 0 = y 0 + z 0 A os ωt x ), 13.82) z = z h zy y 0 = z 0 + y 0 A os ωt x ). Let us onsider four partiles disposed as indiated in figure 13.3) 1) y = r, z = r, 13.83) 2) y = r, z = r, 3) y = r, z = r, 4) y = r, z = r. As before, we shall assume that the initial time t = 0 orresponds to ωt x ) = π 2. After a quarter of a period os ωt x ) = 1), the partiles will have the following positions 1) y = r[1 A ], z = r[1 A ], 13.84) 2) y = r[ 1 A ], z = r[1 + A ], 3) y = r[ 1 + A ], z = r[ 1 + A ], 4) y = r[1 + A ], z = r[ 1 A ]. After half a period os ωt x ) = 0, and the partiles go bak to the initial positions. After three quarters of a period, when os ωt x) = 1 1) y = r[1 + A ], z = r[1 + A ], 13.85) 2) y = r[ 1 + A ], z = r[1 A ], 3) y = r[ 1 A ], z = r[ 1 A ], 4) y = r[1 A ], z = r[ 1 + A ]. The motion of the partiles is indiated in figure 13.3). It follows that a small ring of partiles entered at the origin, will again beome an ellipse, but rotated at 45 0 see figure 13.4)) with respet to the ase previously analysed. In onlusion, we an define A + and A as the polarization amplitudes of the wave. The wave will be linearly polarized when only one of the two amplitudes is different from zero.

16 CHAPTER 13. GRAITATIONAL WAES 172, Figure 13.3:

17 CHAPTER 13. GRAITATIONAL WAES 173, Figure 13.4:

18 Chapter 14 The Quadrupole Formalism In this hapter we will introdue the quadrupole formalism whih allows to estimate the gravitational energy and the waveforms emitted by an evolving physial system desribed by the stress-energy tensor T µν. We shall solve eq ) under the following assumption: we shall assume that the region where the soure is onfined, namely x i < ɛ, T µν 0, 14.1) is muh smaller than the wavelenght of the emitted radiation, λ GW = 2π. This implies that ω 2π ω >> ɛ ɛ ω << v typial <<, i.e. the veloities typial of the physial proesses we are onsidering are muh smaller than the speed of light; for this reason this is alled the slow-motion approximation. Let us onsider the first equation in 13.24) where F hµν = 16πG 4 T µν, 14.2) h µν = h µν 1 2 η µνh and F = [ 1 2 ] 2 t By Fourier-expanding both h µν and T µν T µν t, x i ) = h µν t, x i ) = + + T µν ω, x i )e iωt dω, 14.3) h µν ω, x i )e iωt dω, i = 1, 3 eq. 14.2) beomes [ where 2 + ω2 2 ] h µν ω, x i ) = KT µν ω, x i ) 14.4) K = 16πG ) 174

19 CHAPTER 14. THE QUADRUPOLE FORMALISM 175 We shall solve eq. 14.4) outside and inside the soure, mathing the two solutions aross the soure boundary. The exterior solution Outside the soure T µν = 0 and eq. 14.4) beomes [ ] 2 + ω2 h 2 µν ω, x i ) = ) In polar oordinates, the Laplaian operator 2 is 2 = 1 [ r 2 ] + 1 [ sin θ ] + r 2 r r r 2 sin θ θ θ 1 r 2 sin 2 θ 2 φ 2. We shall onsider the simplest solution of this equation, i.e. one whih does not depend on φ and θ: h µν ω, r) = A µνω) r e i ω r + Z µνω) r e i ω r. This solution represents a spherial wave, with an ingoing part e i ω r ), and an outgoing e i ω r ) part; indeed, substituting in the seond eq. 14.3) h µν ω, x i ) by e ±i ω r the result of the integration over ω gives a funtion of t r ) respetively. Sine we are interested only in the wave emitted from the soure, we shall set Z µν = 0, and onsider the solution h µν ω, r) = A µνω) r e i ω r. 14.7) This is the solution outside the soure and on its boundary, where T µν vanishes as well. A µν is the wave amplitude to be found by solving the equations inside the soure. The wave equation [ 2 + ω2 2 The interior solution ] h µν ω, x i ) = KT µν ω, x i ) 14.8) an be solved for eah assigned value of the indies µ, ν, therefore in the integration of eq. 14.8) the funtions h µν and T µν will be onsidered as salar funtions. To solve eq. 14.8) let us integrate over the soure volume [ 2 + ω2 2 ] h µν ω, x i )d 3 x = K T µν ω, x i )d 3 x. The first term an be developed as follows 2 hµν ω, x i ) d 3 x = div[ hµν ] d 3 x = S hµν ) k dsk 14.9) where hµν is the gradient of the salar funtion h µν, S is the surfae surrounding the soure volume, and we have applied Gauss theorem to the vetor hµν. Using eq. 14.7)

20 CHAPTER 14. THE QUADRUPOLE FORMALISM 176 the surfae integral an be approximated as follows S hµν ) k dsk 4π ɛ 2 d dr = 4π ɛ 2 [ A µν r 2 e i ω r + A µν r iω A µν r if we keep the leading term and disard terms of order ɛ, we find and eq. 14.8) beomes The seond term 4π A µν + satisfies the following inequality 2 hµν ω, x i ) d 3 x ω 2 2 ) e i ω r ) e i ω r ] 4π A µν ω), h µν ω, x i ) d 3 x = K ω 2 2 h µν ω, x i ) d 3 x r=ɛ r=ɛ ; T µν ω, x i ) d 3 x ) ω 2 2 h µν ω, x i ) d 3 x < h µν max ω πɛ3, 14.11) where h µν max is the maximum reahed by h µν in the volume, and sine the right-hand side of eq ) is of order ɛ 3 it an be negleted. Consequently eq ) beomes 4πA µν ω) = K T µν ω, x i ) d 3 x 14.12) i.e. A µν ω) = 4G T 4 µν ω, x i ) d 3 x. Thus, the solution of the wave equation inside the soure gives the wave amplitude A µν ω) as an integral of the stress-energy tensor of the soure over the soure volume. Knowing A µν ω) we finally find h µν ω, r) = 4G 4 or, by the inverse Fourier transform r ei ω r T µν ω, x i ) d 3 x, 14.13) h µν t, r) = 4G 1 T 4 µν t r r, xi ) d 3 x ) This is the gravitational signal emitted by the soure. The integral in 14.14) an be further simplified, but in the meantime note that:

21 CHAPTER 14. THE QUADRUPOLE FORMALISM 177 1) The solution 14.14) for h µν automatially satisfies the seond eq ), i.e. the harmoni gauge ondition x h µ µ ν = 0. To prove this, we first notie that the solution 14.14) is equivalent to the expression 13.26) h µν t, x) = 4G 4 T µν t x-x, x ) x-x d 3 x ; 14.15) indeed, sine then By defining the following funtion x < ɛ, and r ɛ, 14.16) r x x-x ) g x x ) 4G [ )] 1 4 x-x δ t t x-x, 14.18) where x = t, x) and x = t, x ), eq ) an be written as a four-dimensional integral as follows h µν x) = T µν x ) g x x ) d 4 x, 14.19) Ω where Ω I, and I is the time interval to be taken suh that g x x ) vanishes at the extrema of I; this happens if I is so large that, for all x, the expression t x-x is inside I; indeed, from the definition 14.18) g is different from zero only for t = t x-x. Sine g is a funtion of the differene x x ), then Consequently, x [g x x )] = µ x [g x x )] ) µ x h µν x) = T µν x ) µ Ω x g x x ) d 4 x = µ Ω T µν x ) x g x x ) d 4 x ) µ Integrating by parts, and using the fat that T µν = 0 on the boundary of and g = 0 on the boundary of I, we have x h µν x) = g x x ) µ Ω x T µν x ) d 4 x = ) µ beause the stress-energy tensor satisfies the onservation law T µν,ν = 0. Q.E.D. 2) In order to extrat the physial omponents of the wave we still have to projet h µν on the TT-gauge. 3) Eq ) has been derived on two very strong assumptions: weak field g µν = η µν +h µν ) and slow motion v typial << ). For this reason that expression has to be onsidered as an estimate of the emitted radiation by the system, unless the two onditions are really satisfied.

22 CHAPTER 14. THE QUADRUPOLE FORMALISM The Tensor irial Theorem In order to simplify the integral in eq ) we shall use the onservation law that T µν satisfies see hapter 7) T µν x ν = 0, 1 T µ0 t = µk T, µ = 0, 3, k = 1, ) xk Let us integrate this equation over the soure volume, assuming the index µ is fixed 1 t T µ0 d 3 x = T µk x k d3 x. By Gauss theorem, the integral over the volume is equal to the flux of T µk aross the surfae S enlosing that volume, thus the right-hand-side beomes T µk x k d3 x = T µk ds k. S By definition, on S T µν = 0 and onsequently the surfae integral vanishes; thus 1 t From eq ) it follows that T µ0 d 3 x = 0, h µ0 = onst, µ = 0, 3, T µ0 d 3 x = onst ) and sine we are interested in the time-dependent part of the field we shall put h µ0 = 0, µ = 0, ) Indeed, we shall show that in the TT-gauge h µ0 = 0.) We shall now prove the Tensor- irial Theorem whih establishes that 1 2 T 00 x k x n d 3 x = 2 T kn d 3 x, k, n = 1, ) 2 t 2 Let us onsider the spae-omponents of the onservation law 14.23) T n0 x 0 = ni T, i, n = 1, 3; xi multiply both members by x k and integrate over the soure volume 1 t = = S T n0 x k d 3 x = T ni x k) x i d 3 x T ni x k) ds i + T ni x i x k d 3 x ni xk T x i T nk d 3 x, d 3 x

23 CHAPTER 14. THE QUADRUPOLE FORMALISM 179 remember that xk x i = δ k i ). As before 1 t S T ni x k) ds i = 0, therefore T n0 x k d 3 x = T nk d 3 x. Sine T nk is symmetri we an rewrite this equation in the following form 1 2 t T n0 x k + T k0 x n) d 3 x = Let us now onsider the 0 omponent of the onservation law multiply by x k x n and integrate over 1 t = = S 1 T 00 0i T + = 0, i = 1, 3 t x i T 00 x k x n d 3 x = T 0i x k x n) x i d 3 x T 0i x k x n) ds i + T 0i x i the first integral vanishes and this equation beomes 1 t T 00 x k x n d 3 x = If we now differentiate with respet to x 0 we find t 2 T 00 x k x n d 3 x = 1 and using eq ) we finally find t 2 T 00 x k x n d 3 x = 2 x k x n d 3 x 0i xk T x i T nk d 3 x ) ) x n + T 0i x k xn x i T 0k x n + T 0n x k) d 3 x t T 0k x n + T 0n x k) d 3 x. T 0k x n + T 0n x k) d 3 x, d 3 x T kn d 3 x, k, n = 1, ) The left-hand-side of this equation is the seond time derivative of the quadrupole moment tensor of the system q kn t) = 1 2 whih is a funtion of time only. Thus, in onlusion T 00 t, x i ) x k x n d 3 x, k, n = 1, 3, 14.29) T kn t, x i ) d 3 x = 1 2 d 2 dt 2 qkn t).

24 CHAPTER 14. THE QUADRUPOLE FORMALISM 180 By using eqs ) and 14.25) we finally find h µ0 = 0, µ = 0, 3 h ik t, r) = 2G [ d 2 4 r dt 2 qik t r ] ) ) This is the gravitational wave emitted by a gravitating system evolving in time. It an be omposed of masses or of any form of energy, beause mass and energy are both soures of the gravitational field. NOTE THAT 1) G s 2 /g m : this is the reason why gravitational waves are extremely weak!! 3) In order to make the physial degrees of freedom expliitely manifest we still have to transform to the TT-gauge 4) These equations have been derived on very strong assumptions: one is that T µν,ν = 0, i.e. that the motion of the bodies is dominated by non-gravitational fores. However, and remarkably, the result 14.30) depends only on the soures motion and not on the fores ating on them. 5) Gravitational radiation has a quadrupolar nature. A system of aelerated harged partiles has a time-varying dipole moment d EM = i q i r i and it will emit dipole radiation, the flux of whih depends on the seond time derivative of d EM. For an isolated system of masses we an define a gravitational dipole moment d G = i m i r i, whih satisfies the onservation law of the total momentum of an isolated system d dt d G = 0. For this reason, gravitational waves do not have a dipole ontribution. It should be stressed that for a spherial or axisymmetri distribution of matter or energy) the quadrupole moment is a onstant, even if the body is rotating. Thus, a spherial or axisymmetri star does not emit gravitational waves; similarly a star whih ollapses in a perfetly spherially symmetri way has a vanishing d2 q ik and does not emit gravitational waves. To produe these dt 2 waves we need a ertain degree of asymmetry, as it ours for instane in the non-radial pulsations of stars, in a non spherial gravitational ollapse, in the oalesene of massive bodies et.

25 CHAPTER 14. THE QUADRUPOLE FORMALISM How to transform to the TT-gauge The solution 14.30) desribes a spherial wave far from the emitting soure. Loally, it looks like a plane wave propagating along the diretion of the unit vetor orthogonal to the wavefront n α = 0, n i ), i = 1, ) where n i = xi r ) In order to express this waveform in the TT-gauge we shall make an infinitesimal oordinate transformation x µ = x µ + ɛ µ and hoose the vetor ɛ µ whih satisfies the wave equation F ɛ µ = 0, so that the harmoni gauge ondition is preserved, as explained in hapter 14. The onditions we will impose on the perturbed metri are h αβ δαβ = 0, vanishing trae h αβ n β = 0, trasverse wave ondition. It should be mentioned that these onditions imply that h µ0 = 0, µ = 0, 3 as required in eq ). Indeed, given the wave-vetor k µ = ost, rn i ) we know by eq ) that k µ h µν = 0, i.e. k 0 h 0ν + rn i h iν = 0. The seond term vanishes beause of the trasverse wave ondition, therefore h 0ν = 0. We remind here that, as shown in eq ), in the TT-gauge h µν and h µν oinide. To hereafter, we shall work in the 3-dimensional eulidean spae with metri δ ij. Consequently, there will be no differene between ovariant and ontravariant indies. As a first step, we define the operator whih projets a vetor onto the plane orthogonal to the diretion of n P jk δ jk n j n k ) Indeed, it is easy to verify that for any vetor j, P jk k is orthogonal to n j, i.e. P jk k )n j = 0, and that P j kp k l l = P j l l ) Note that P jk = P kj, i.e. P jk is symmetri. The projetor is transverse, i.e. Then, we define the transverse traeless projetor: n j P jk = ) P jkmn P jm P kn 1 2 P jkp mn ) ) 0 whih extrats the transverse-traeless part of a tensor. In fat, substituting the 2 definition 14.36), it is easy to see that it satisfies the following properties

26 CHAPTER 14. THE QUADRUPOLE FORMALISM 182 P jklm = P lmjk P jklm = P kjlm P jklm = P jkml and P jkmn P mnrs = P jkrs ; 14.37) it is transverse: n j P jkmn = n k P jkmn = n m P jkmn = n n P jkmn = 0 ; 14.38) it is traeless: δ jk P jkmn = δ mn P jkmn = ) Sine h jk and h jk differ only by the trae, and sine the projetor P jklm extrats the traeless part of a tensor eq ), the omponents of the perturbed metri tensor in the TT-gauge an be obtained by applying the projetor P jkmn either to h jk or to h jk h TT jk = P jkmn h mn = P jkmn hmn ) By applying P on h jk defined in eq ) we get h TT µ0 = 0, µ = 0, 3 [ d 2 h TT jk t, r) = 2G 4 r dt 2 QTT jk t r ] ) 14.41) where Q TT jk P jkmn q mn 14.42) is the transverse traeless part of the quadrupole moment. Sometimes it is useful to define the redued quadrupole moment Q jk whose trae is zero by definition, i.e. and from eq ), it follows that Q jk q jk 1 3 δ jkq m m 14.43) δ jk Q jk = 0, 14.44) Q TT jk = P jkmn q mn = P jkmn Q mn )

27 CHAPTER 14. THE QUADRUPOLE FORMALISM Gravitational wave emitted by a harmoni osillator Let us onsider a harmoni osillator omposed of two equal masses m osillating at a frequeny ν = ω with amplitude A. Be l 2π 0 the proper length of the string when the system is at rest. Assuming that the osillator moves on the x-axis, the position of the two masses will be { x1 = 1l 2 0 A os ωt x 2 = + 1 l A os ωt The 00-omponent of the stress-energy tensor of the system is y z x T 00 = 2 n=1 p 0 δx x n ) δy) δz); and sine v <<, γ 1 p 0 = m, it redues to T 00 = m 2 2 n=1 δx x n ) δy) δz); the xx-omponent of the quadrupole moment q ik t) = 1 2 T 00 t, x n ) x i x k dx 3 is [ q xx = q xx = m δx x 1 ) x 2 dx δy) dy δz) dz 14.46) ] + δx x 2 ) x 2 dx δy) dy δz) dz = m [ [ ] 1 x ] x2 = m 2 l A2 os 2 ωt + 2Al 0 os ωt = m [ ost + A 2 os 2ωt + 2Al 0 os ωt ],

28 CHAPTER 14. THE QUADRUPOLE FORMALISM 184 where we have used the trigonometri expression os 2α = 2 os 2 α 1. The zz-omponent of the quadrupole moment is [ q zz = = m δx x 1 ) dx δy) dy δz) z 2 dz ] + δx x 2 ) dx δy) dy δz) z 2 dz = 0 beause z 2 δz) dz = 0. Sine the motion is onfined to the x-axis, all remaining omponents of q ij vanish. Let us now ompute the redued quadrupole moment Q ij = q ij 1 δ 3 ij q k k; sine q k k = η ik q ik = η xx q xx = q xx we find Q xx = q xx 1 3 q xx = 2 3 q xx 14.47) Q yy = Q zz = 1 3 q xx Q xy = 0. We shall ompute, as an example, the wave emerging in the z-diretion; in this ase n = x 0, 0, 1) and r P jk = δ jk n j n k = By applying to Q ij the transverse-traeless projetor P jkmn onstruted from P jk, we find Q TT xx = P xm P xn 1 ) 2 P xxp mn Q mn 14.48) = P xx P xx 1 ) 2 P xx 2 Q xx 1 2 P xxp yy Q yy = 1 2 Q xx Q yy ), Q TT xy = P xm P yn 1 ) 2 P xyp mn Q mn = P xx P yy Q xy = 0, Q TT zz = P zm P zn 1 ) 2 P zzp mn Q mn = 0. Using these expressions it is easy to show that eqs ) beome h TT µ0 = 0 h TT zi = 0, h TT xy = 2G d 2 4 z dt Q 2 xy = 0 h TT xx = h TT yy = G d 2 4 z dt Q 2 xx Q yy ), 14.49) and using eqs ) and 14.46) h TT xx = h TT yy = G [ d 2 4 z dt q xxt z ] 2 ), 14.50) = 2Gm [2A 4 z ω2 2 os 2ωt z ) + Al 0 os ωt z ] ).

29 CHAPTER 14. THE QUADRUPOLE FORMALISM 185 Thus, radiation emitted by the harmoni osillator along the z-axis is linearly polarized. If, for instane, we onsider two masses m = 10 3 kg, with l 0 = 1 m, A = 10 4 m, and ω = 10 4 rad/s, the term [2A 2 os 2ωt] is negligible, and the dominant term is at the same frequeny of the osillations: h TT xx 2Gm 4 z ω2 Al 0 os ωt z ), z whih is, as expeted, very very small. It should be notied that due to the symmetry of the system, the wave emitted along y will be the same. To find the wave emitted along x, we hoose n = 1, 0, 0) and use the same proedure: no radiation will be found How to ompute the energy arried by a gravitational wave In order to evaluate how muh energy is radiated in gravitational waves by an evolving system, we need to define a tensor that properly desribes the energy ontent of the gravitational field. Our effort will not be ompletely suessful, sine we will be able to define a quantity whih behaves like a tensor only under linear oordinate transformations. However, this pseudo-tensor will be useful for the purpose we have in mind The stress-energy pseudotensor of the gravitational field In Chapter 7 we have shown that the stress-energy tensor of matter satisfies a divergeneless equation T µν ;ν = ) If we hoose a loally inertial frame LIF), the ovariant derivative redues to the ordinary derivative and eq ) beomes T µν x ν = ) We shall now try to find a quantity, η µνγ, suh that T µν = x γ ηµνγ ; 14.53) In this way, if we impose that η µνγ is antisymmetri in the indies ν and γ, the onservation law 14.52) will automatially be satisfied. The problem now is: an we find the expliit expression of η µνγ? From Einstein s equations we know that T µν = 4 R µν 1 ) 8πG 2 gµν R ; 14.54)

30 CHAPTER 14. THE QUADRUPOLE FORMALISM 186 sine we are in a loally inertial frame, the Riemann tensor, whose generi expression is R γαδβ = 1 [ 2 ] g γβ 2 x α x + 2 g αδ δ x γ x 2 g γδ β x α x 2 g αβ 14.55) β x γ x δ +g σρ Γ σ αδ Γ ρ γβ Γσ αβγ ρ γδ), redues to the term in square brakets sine all Γ σ αδ s vanish; it follows that in this frame the Rii tensor beomes R µν = g µα g νβ R αβ = g µα g νβ g γδ R γαδβ 14.56) = 1 2 gµα g νβ g γδ 2 ) g γβ x α x + 2 g αδ δ x γ x 2 g γδ β x α x 2 g αβ. β x γ x δ By using this equation, after some umbersome alulations eq ) beomes T µν = { } 4 1 [ g) g µν g αβ g µα g νβ)] ) x α 16πG g) x β The term in parentheses is antisymmetri in the indies ν and α and it is the quantity we were looking for: If we now introdue the quantity η µνα = 4 1 [ g) g µν g αβ g µα g νβ)] ) 16πG g) x β ζ µνα = g)η µνα = 4 [ g) g µν g αβ g µα g νβ)], 14.59) 16πG x β sine we are in a loally inertial frame x β 1 g) = 0, therefore we an write eq ) as ζ µνα x α = g)t µν ) This equation has been derived in a LIF, where all first derivatives of the metri tensor ζ vanish, but in any other frame this will not be true and the differene µνα g)t µν will x α not be zero, but a quantity whih we shall all g)t µν i.e. g)t µν = ζµνα x α g)t µν ) t µν is symmetri beause both T µν and ζµνα x α are symmetri in µ and ν. The expliit expression of t µν an be found by substituting in eq ) the definition of ζ µνα given in eq ), and T µν omputed in terms of the Rii tensor from eq ) in an arbitrary frame i.e. starting from the full expression of the Riemann tensor given in eq ): after some areful manipulation of the equations it is possible to show that t µν = 4 { 2Γ δ αβ Γ σ δσ Γ δ ασγ σ βδ Γ δ αδγ βσ) σ g µα g νβ g µν g αβ) 16πG + g µα g βδ Γ ν ασγ σ βδ + Γ ν βδγ σ ασ Γ ν δσγ σ αβ Γ ν αβγ σ δσ) + g να g βδ Γ µ ασγ σ βδ + Γ µ βδγ σ ασ Γ µ δσγ σ αβ Γ µ αβγ σ δσ) + g αβ g δσ Γ µ αδγ ν βσ Γ µ αβγ ν δσ) }

31 CHAPTER 14. THE QUADRUPOLE FORMALISM 187 This is the stress-energy pseudotensor of the gravitational field we were looking for. Indeed we an rewrite eq ), valid in any referene frame, in the following form and sine ζ µνα is antisymmetri in µ and α g) t µν + T µν ) = ζµνα x α, 14.62) x µ ζ µνα x α = 0, and onsequently x [ g) µ tµν + T µν )] = ) This equation expresses a onservation law, beause, as explained in hapter 7, it has the form of a vanishing ordinary divergene of the quantity [ g) t µν + T µν )]. Sine t µν when added to the stress-energy tensor of matter or fields) satisfies a onservation law, and sine it vanishes only in a loally inertial frame where gravity is suppressed, we interpret t µν as the entity that ontains the information on the energy and momentum arried by the gravitational field. Thus eq ) expresses the onservation law of the total energy and momentum. Unfortunately, t µν is not a tensor; indeed it is a ombination of the Γ s that are not tensors. However, as the Γ s, it behaves as a tensor under linear oordinate transformations The energy flux arried by a gravitational wave Let us onsider an emitting soure and the assoiated 3-dimensional oordinate frame O x, y, z). Be an observer loated at P = x1, y1, z1) as shown in figure Be r = x12 + y1 2 + z1 2 its distane from the origin. The observer wants to detet the wave oming along the diretion identified by the versor n = r. As a pedagogial tool, let us r onsider a seond frame O x, y, z ), with origin oinident with O, and having the x -axis aligned with n. Assuming that the wave traveling along x diretion is linearly polarized and has only one polarization, the orresponding metri tensor will be g µ ν = t) x ) y ) z ) [1 + h TT + t, x )] [1 h TT + t, x )] The observer wants to measure the energy whih flows per unit time aross the unit surfae orthogonal to x, i.e. t 0x, therefore he needs to ompute the Christoffel symbols i.e. the derivatives of h TT µ ν. Aording to eq ) the metri perturbation has the form h TT t, x ) = onst ft x ), and sine the only derivatives whih matter are those with x respet to time and x. h TT t ḣtt = onst x f,

32 CHAPTER 14. THE QUADRUPOLE FORMALISM 188 y y P x n z x z Figure 14.1: A binary system lies in the z-x plane. An observer loated at P wants to detet the energy flux of gravitational waves emitted by the system. h TT h TT = onst x x 2 f + onst x f 1 onst x f = 1 ḣtt, where we have retained only the dominant 1/x term. Thus, the non-vanishing Christoffel symbols are: Γ 0 y y = Γ0 z z = 1 2 ḣtt + Γ y 0y = Γz 0z = 1 2 ḣtt ) Γ x y y = Γx z z = 1 2 ḣtt + Γ y y x = Γz z x = 1 2 ḣtt +. By substituting the Christoffel symbols in t µν we find t 0x = de GW dtds = 3 dh TT t, x ) 2 ). 16πG dt If both polarizations are present g µ ν = t) x ) y ) z ) [1 + h TT + t, x )] h TT 0 0 h TT t, x ) [1 h TT t, x ) + t, x )],

33 CHAPTER 14. THE QUADRUPOLE FORMALISM 189 and t 0x = de GW dtds = = 3 32πG 3 dh TT + t, ) 2 x ) dh TT + 16πG dt dh TT jk t, ) 2 x ). jk dt t, x ) dt ) ) This is the energy per unit time whih flows aross a unit surfae orthogonal to the diretion x. However, the diretion x is arbitrary; if the observer is loated in a different position and omputes the energy flux he reeives, he will find formally the same eq ) but with h TT jk referred to the TT-gauge assoiated with the new diretion. Therefore, if we onsider a generi diretion r = rn t 0r = 2 ) dh TT 2 jk t, r) ) 32πG jk dt In General Relativity the energy of the gravitational field annot be defined loally, therefore to find the GW-flux we need to average over several wavelenghts, i.e. de GW dtds = t 0r = 3 32πG ) dh TT 2 jk t, r). jk dt Sine h TT µ0 = 0, µ [ = 0, 3 h TT 2G d 2 ik t, r) = 4 r dt 2 QTT ik t r ) ] by diret substitution we find de GW dtds = G 8π 5 r 2 jk... Q TT jk t r )) ) If we now want to make expliit the dependene of the flux on the propagation diretion n, we an express Q TT jk in terms of the projetor P jkmn as in eq ) de GW dtds = G 8π 5 r 2 jk... P jkmn Q mn t r )) ) From this formula we an now ompute the gravitational luminosity L GW = de GW, i.e. the dt gravitational energy emitted by the soure per unit time L GW = = G π degw dtds ds = degw dtds r2 dω 14.69) dω jk... P jkmn Q mn t r )) 2,

34 CHAPTER 14. THE QUADRUPOLE FORMALISM 190 where dω = d os θ)dφ is the solid angle element. This integral an be omputed by using the properties of P jkmn : jk Pjkmn... Q mn ) 2 = jk P jkmn... Q mn P jkrs... Q rs = 14.70) = P mnjk P jkrs Q mn Q rs = P mnrs Q rs jk [ = δ mr n m n r ) δ ns n n n s ) 1 ]... 2 δ... mn n m n n ) δ rs n r n s ) Q mn Q rs. If we expand this expression, and remember that Q mn δ mn Q mn = δ rs Q rs = 0 beause the trae of Q ij vanishes by definition, and n m n r δ ns Q mn Q rs = n n n s δ mr beause Q ij is symmetri, at the end we find jk... Q mn Q rs Pjkmn... Q mn ) 2 =... Q rn... Q rn 2n m... Q ms... Q sr n r n mn n n r n s... Q mn... Q rs ) This expression has to be substituted in eq ), and the integrals to be performed over the solid angle are: 1 4π dωn i n j, and Let us ompute the first. In polar oordinates, the versor n an be written as 1 4π dωn i n j n r n s ) n i = sin θ os φ, sin θ sin φ, os θ) ) Thus, for parity reasons 1 dωn i n j = 0 when i j ) 4π Furthermore, sine there is no preferred diretion in the integration isotropy), it must be For instane, 1 4π dω n 2 1 = and onsequently dωn 3 ) 2 = 1 4π dω n 2 2 = dω n π dωn i n j = onst δ ij ) d os θdφ os 2 θ = 1 2π 1 dφ d os θ os 2 θ = 1 4π 0 1 3, 14.76) 1 4π dωn i n j = 1 3 δ ij )

35 CHAPTER 14. THE QUADRUPOLE FORMALISM 191 The seond integral in 14.72) an be omputed in a similar way and gives Consequently 1 4π dωn i n j n r n s = 1 15 δ ijδ rs + δ ir δ js + δ is δ jr ) ) π dω Q rn Q rn 2n m and, finally, the emitted power is... Q ms = Q 5 rn Q sr n r n mn n n r n s ) Q mn Q rs Q rn 14.79) L GW = G 3... Q 5 5 kn t r )... Q kn t r ) ) k,n= Gravitational radiation from a rotating star In this setion we shall show that a rotating star emits gravitational waves only if its shape deviates from axial symmetry. Consider an ellipsoid of uniform density ρ. Its quadrupole moment is q ij = and it is related to the inertia tensor I ij = by the equation ρ x i x j dx 3, i = 1, 3 ρ r 2 δ ij x i x j ) dx 3 q ij = I ij + δ ij Tr q, where Tr q q m m. Consequently, the redued quadrupole moment an be written as Q ij = q ij 1 3 δ ijtr q = I ij 1 ) 3 δ ijtr I. Let us first onsider a non rotating ellipsoid, with semiaxes a, b,, volume = 4 πab, and 3 equation: x1 a ) 2 + x2 b ) 2 ) x3 2 + = 1. The inertia tensor is I ij = M 5 b a a 2 + b 2 = I I I 3, where I 1, I 2, I 3 are the prinipal moments of inertia.

36 CHAPTER 14. THE QUADRUPOLE FORMALISM 192 Z a b Y X Let us now onsider an ellipsoid whih rotates around one of its prinipal axes, for instane I 3, with angular veloity 0, 0, Ω). What is its inertia tensor in this ase? Be {x i } the oordinates of the inertial frame, and {x i} the oordinates of a o-rotating frame. Then, x i = R ij x j, where R ij is the rotation matrix R ij = os ϕ sin ϕ 0 sin ϕ os ϕ , with ϕ = Ωt. For instane, a point at rest in the o-rotating frame, with oordinates x i = 1, 0, 0), has, in the inertial frame, oordinates x i = os Ωt, sin Ωt, 0), i.e. it rotates in the x y plane with angular veloity Ω. Sine in the o-rotating frame {x i } in the inertial frame {x i } it will be I ij = I ij = R ik R jl I kl = RI R T ) ij I I I 3, = I 1 os 2 ϕ + I 2 sin 2 ϕ sin ϕ os ϕi 2 I 1 ) 0 sin ϕ os ϕi 2 I 1 ) I 1 sin 2 ϕ + I 2 os 2 ϕ I 3.

37 CHAPTER 14. THE QUADRUPOLE FORMALISM 193 It is easy to hek that Tr I = I 1 + I 2 + I 3 = onstant. The quadrupole moment therefore is Q ij = I ij 1 ) 3 δ ijtr I = I ij + onstant Using os 2ϕ = 2 os 2 ϕ 1, et., the quadrupole moment an be written as Q ij = I 2 I 1 2 os 2ϕ sin 2ϕ 0 sin 2ϕ os 2ϕ onstant Sine I 1 = M 5 b2 + 2 ), and I 2 = M a 2 ), if a, b are equal, the quadrupole moment is onstant and no gravitational wave is emitted. This is a generi result: an axisymmetri objet rigidly rotating around its symmetry axis does not radiate gravitational waves. In realisti ases, a b, and I 1 I 2 ; however the differene is expeted to be extremely small. It is onvenient to express the quadrupole moment of the star in terms of a dimensionless parameter ɛ, the oblateness, whih expresses the deviation from axisymmetry ɛ a b a + b)/2. It is easy to show that Indeed, I 2 I 1 I 3 = ɛ + Oɛ 3 ). a b = 1 ɛa + b), 14.81) 2 thus I 2 I 1 = a2 b 2 a + b)a b) = = 1 + b)2 ɛa I 3 a 2 + b2 a 2 + b 2 2 a 2 + b ) 2 On the other hand, from 14.81) we have a b) 2 = Oɛ 2 ) = a 2 + b 2 2ab, 14.83) therefore and I 2 I 1 I 3 2ab = a 2 + b 2 + Oɛ 2 ) 14.84) = 1 2 ɛa2 + b 2 + 2ab a 2 + b 2 = ɛ + Oɛ 3 ) )