0.1 Perturbed Einstein s equations

Transcription

1 1 0.1 Perturbed Einstein s equations After all these general preliminaries, we now turn to General Relativity and work out the corresponding linearized theory. For this purpose, we write down the Einstein s field equations: where G ab is the Einstein s tensor: G ab = 8πT ab, (0.1.1) G ab = R ab 1 2 g abr, (0.1.2) and imagine that we have a one parameter family of exact solutions g ab (x; λ), T ab (x; λ), coinciding, for λ = 0, with a given background solution g ab (x), T ab (x): g ab (x; 0) = g ab (x), T ab (x; 0) = T ab (x). (0.1.3) According to eq.(??), we define the perturbations γ ab (x) and τ ab (x) as: γ ab (x) = dg ab (x; 0), dλ τ ab(x) = dt ab (x; 0). (0.1.4) dλ In order to shorten our notation, we shall use the superscript ˇ to denote derivatives with respect to λ, evaluated at λ = 0. In this way, the above definitions of the perturbations γ ab and τ ab are written as: γ ab (x) = ǧ ab (x), τ ab (x) = Ťab(x). (0.1.5) Following the procedure discussed above, we can derive the equations satisfied by the perturbations by plugging the one-parameter family of exact solutions into the field equations (0.1.1), taking a derivative with respect to λ and then setting λ = 0. We start by computing the linearized form of the Einstein s tensor: Ǧ ab = Řab 1 2 R γ ab g ab Rcd g ce g df γef 1 2 g ab g cd Řcd, (0.1.6) where in the third term of the r.h.s. we made use of the equation: ǧ cd = g ce g df γef, (0.1.7) that can be easily obtained by deriving the equation g ab (λ)g bc (λ) = δ a c with respect to λ and then taking λ = 0. In order to obtain a simple expression

2 2 for Řab, we consider, for each λ, the derivative operator, λ a, compatible with the metric g ab (λ), and we compare it with the derivative operator a, compatible with the background metric. As it is well known, the difference λ a a is characterized by a tensor field Cab c (λ), whose expression in terms of g ab (λ) can be obtained from the compatibility condition: 0 = λ a g bc (λ) = a g bc (λ) C d ab(λ)g dc (λ) C d ac(λ)g bd (λ). (0.1.8) By inverting this equation, one obtains the following expression for C c ab (λ): C c ab(λ) = 1 2 gcd (λ){ a g bd (λ)+ b g ad (λ) d g ab (λ)}. (0.1.9) Upon deriving the above equation with respect to λ and using the fact that g cd = 0, we find : a Č c ab = 1 2 g cd { a γ bd + b γ ad d γ ab }. (0.1.10) Consider now the expression of the Ricci tensor R ab (λ) in terms of C c ab (λ): R ac (λ) = Rac 2 [a Cb]c b (λ) + 2Ce c[a (λ)cb b]e (λ), (0.1.11) Upon taking a derivative with respect to λ and recalling that C c bd= 0, we get: Ř ac = 2 [a Č b b]c = = 1 2 g bd a c γ bd 1 2 g bd b d γ ac + g bd b (c γ a)d (0.1.12). In order to simplify our notations, we shall drop from now on the superscipt from the background quantities, in all the equations that involve the perturbations. Moreover, we assume that the raising and lowering of indices is done using the background metric g ab. In this way, a denotes g ab b and γ ab = g ac g bd γcd (and so γ ab is not the inverse of γ ab! In fact γ ab may not even be invertible.). Using these conventions, the above equation is written as: Ř ac = 1 2 a c γ 1 2 b b γ ac + d (c γ a)d, (0.1.13) where γ = g cd γ cd. (0.1.14)

3 0.2. GAUGE INVARIANCE OF THE PERTURBATIVE EQUATIONS 3 Using eq.(0.1.13) into eq.(0.1.6), we get the following expression for the linearized Einstein s tensor: Ǧ ab = 1 2 c c γ ab + c (a γ b)c 1 2 g ab c d γ cd 1 2 R γ ab g abr cd γ cd, where we introduced, for convenience, the field: (0.1.15) γ ab = γ ab 1 2 g abγ. (0.1.16) The linearized field equations, can then be written as Ǧ ab = 8πτ ab. (0.1.17) They constitute, in general, a system of ten coupled, linear, partial differential equations in the unknown twenty fields γ ab, and τ ab. The left hand side has the form of a generalized wave operator, but, for arbitrary backgrounds g ab, the coefficients are functions of the space-time point. It is clear then that the solution of these equations is in general very hard, and a complete analysis will be possible only in special instances, for example in the case of backgrounds with a high degree of symmetry. Most of these Lectures will deal, in fact, with the simplest case, that of a flat background, when an explicit solution can be easily found. 0.2 Gauge invariance of the perturbative equations We wish to examine now the invariance properties of eqs. (0.1.17). First of all, we observe that they are written in a generally covariant form. Indeed, under a coordinate transformation not depending on λ: x µ = X µ (x), (0.2.1) the background g ab and the perturbations γ ab, τ ab transform as tensors. While this is obvious for g ab, the case of γ ab and τ ab requires a little proof. Just recall that, for each λ, the coordinate components g µν of the metric g ab (x; λ) transforms as: g µν (x; λ) = g ρσ(x (x); λ) µ X ρ (x) ν X σ (x). (0.2.2) The tensor transformation law of γ ab (x) easily follows upon taking a deriva- right tive of the above equation with respect to λ and then setting λ = 0. In

4 4 the case of τ ab, one can proceed in the same way. The general covariance of eqs.(0.1.17) is then clear, in view of the tensor character of all the quantities that occur in them. A case of special interest occurs when there exist coordinate tranformations that leave unchanged the coordinate components of the background g ab. They represent symmetries of the perturbed equations and, when no sources are present, for each one-parameter family of such diffeomorphisms, it will be possible to find, following the Noether procedure, a constant of the motion of the homogeneous version of eqs.(0.1.17). Besides these obvious symmetries, eqs.(0.1.17) possess another remarkable invariance, which is a consequence of the general covariance of the exact field equations (0.1.1). To see this, consider, rather than a fixed diffeomorphism, as we did above, a one-parameter family of diffeomorphisms reducing to the identity for λ = 0: x µ = X µ (x; λ), X µ (x; 0) = x µ. (0.2.3) If, for each value of λ, we transform g µν (λ) and T µν (λ) with the corresponding coordinate transformation, we will obtain a new one-parameter family of exact solutions of the field equations g µν(x ; λ), T µν(x ; λ) reducing, for λ = 0, to the old background g µν (x), T µν (x). Of course, the new family of solutions is physically equivalent to the old one because, for each value of λ, the corresponding fields just differ by a diffeomorphism. This implies that the perturbations γ µν(x) = ǧ µν(x) and τ µν(x) = Ť µν(x) should be regarded as physically equivalent to γ µν (x) and τ µν (x) respectively (notice that the arguments of the transformed perturbations coincide with those of the untransformed ones because, for λ = 0, x µ and x µ coincide). Since the one-parameter family of diffeomorphisms (0.2.3) leaves the background quantities unchanged, the transformed perturbations γ µν and τ µν will automatically be solutions of the same linearized equations (0.1.17), satisfied by the untrasformed perturbations. In order to determine the relation connecting γ µν to γ µν, let us write down the transformation law connecting g µν to g µν : g µν (x; λ) = g ρσ(x (x; λ); λ) µ X ρ (x; λ) ν X σ (x; λ). (0.2.4) Notice that g µν has an explicit λ dependence as well as an implicit one, due to dependence on λ of the change of coordinates (0.2.3). Let us now define the vector field ξ a of components: ξ µ (x) = ˇX µ (x). (0.2.5)

5 0.2. GAUGE INVARIANCE OF THE PERTURBATIVE EQUATIONS 5 Upon deriving equation (0.2.4) with respect to λ and then taking λ = 0, we get: γ µν (x) = (γ µν +ξ ρ ρ g µν +g µρ ν ξ ρ +g ρν µ ξ ρ )(x) = (γ µν + ξ g µν )(x), (0.2.6) where ξ g ab denotes the Lie derivative of g ab along ξ a. Upon using the identity (??), we can rewrite this equation, in a coordinate free way, as : γ ab = γ ab 2 (a ξ b). (0.2.7) For later use, we write down also the transformation law of the field γ ab : γ ab = γ ab 2 (a ξ b) + g ab c ξ c, (0.2.8) that can be easily derived from eq.(0.2.7). A similar computation gives the following transformation law for τ ab : τ ab = τ ab ξ T ab. (0.2.9) If the background is a vacuum solution, T ab = 0 and we see that gauge transformations leave τ ab unchanged. The transformations (0.2.7), (0.2.9) are called infinitesimal diffeomorphisms or gauge transformations, due to the resemblance of eqs.(0.2.7) with the transformation law of the electromagnetic potential A a under local gauge transformations: A a(x) = A a (x) + a α(x), (0.2.10) with the vector field ξ a playing the rôle of the gauge parameter α(x). We can take advantage of the gauge invariance of equations (0.1.17) to simplify them. Indeed, we can show that, for any field γ ab of (0.1.17), there exists another one γ ab, related to γ ab by a gauge transformation of the form (0.2.8), (0.2.9) and then physically equivalent to it, satisfying the condition: b γ ab = 0. (0.2.11) This is called the Lorentz gauge, because of its resemblance with the analogous condition adopted in electromagnetism. Eqs.(0.2.11) constitute four conditions to be satisfied everywhere and it is not surprising that we can fulfil them by suitably choosing the four components of the arbitrary vector field ξ a. If we apply the operator b to both sides of eq.(0.2.8) and impose that γ ab satisfies the gauge condition (0.2.11), we find that ξ a must satisfy the equation: b b ξ a + R ab ξ b = b γ ab (x). (0.2.12)

6 6 This is an inhomogeneous wave equation, with a source term provided by b γ ab (x), and it can be shown that, under suitable fall-off conditions for the source term, it always admits solutions. It is then legitimate, without loosing solutions, to assume the validity of the Lorentz gauge. In order to take the maximum advantage from this gauge choice, it is convenient to interchange the order of the covariant derivatives in the second term in the l.h.s. of eq.(0.1.17), using the identity: c (a γ b)c = (a c γ b)c + R c ab d γ cd + R c (a γ b)c. (0.2.13) Thanks to the Lorentz gauge, the first term on the r.h.s of the above equation and the third term in the l.h.s. of eq.(0.1.17) drop and we are left with: 1 2 c c γ ab + R c ab d γ cd + R c (a γ b)c 1 2 R γ ab g abr cd γ cd = 8πτ ab. (0.2.14) The above equations have to be solved, of course, together with eqs.(0.2.11). We observe that, in empty backgrounds, the last three terms on the l.h.s. are zero, while the term involving the Riemann tensor is obviously present, unless the background is also flat.

7 Chapter 1 The post-minkowskian approximation In this Chapter we initiate the study of eqs.(0.2.14). Since we live in a region characterized by weak gravitational fields, it is natural to regard the space-time around us as a flat space-time, perturbed by small gravitational fields and matter sources. Under such circumstances, we can hope to obtain a rather accurate picture of the gravitational fields that surround us and of their interactions with our measuring instruments by means of the perturbative equations derived in the previous Chapter, taking as background a flat space-time. Since the zeroth order approximation is provided by a minkowskian space time, the resultant expansion of the field equations is also called post-minkowskian. The perturbative equations, written in the Lorentz gauge, eqs.(0.2.14), become very simple in this case: in a minkowskian coordinate system {x µ } for the background, they reduce to a set of inhomogeneous decoupled d Alambert equations: subject to the gauge conditions: 1.1 Vacuum fields. ρ ρ γ µν = 16πτ µν, (1.0.1) ν γ µν = 0. (1.0.2) The case we wish to examine first is when τ ab = 0, which physically corresponds to vacuum perturbations of the gravitational field in the absence of matter. 7

8 8 CHAPTER 1. THE POST-MINKOWSKIAN APPROXIMATION For the purpose of studying this type of solutions, it is convenient to introduce the Fourier transform ˆ γ µν (t, k) (t x 0 ) of the perturbation: ˆ γ µν (t, k) = 1 8π 3 d 3 x γ µν (t, x)e i k x, (1.1.1) where k x = k α x α. Since γ µν is real ˆ γ µν (t, k) satisfies the hermiticity conditions: ˆ γ µν (t, k) = ˆ γ µν (t, k). (1.1.2) Upon multiplying eqs.(1.0.1) by exp( i k x) and integrating over all space, after taking the time derivatives outside the integral and after two integration by parts in the terms involving the space derivatives we find that ˆ γ µν (t, k) must satisfy the following ordinary differential equations: Its most general solution is: d 2ˆ γ µν (t, k) dt 2 + k 2ˆ γ µν (t, k) = 0. (1.1.3) ˆ γ µν (t, k) = Āµν( k)e i k t + B µν ( k)e i k t. (1.1.4) Notice that the hermiticity conditions (1.1.2) imply that: Ā µν ( k) = B µν ( k)). (1.1.5) Upon inverting the Fourier transform (1.1.1) and using eqs. (1.1.4) and (1.1.5), we can write the most general solution of eq.(1.0.1) as: γ µν (t, x) = d 3 kāµν( k)e ikx + h.c., (1.1.6) where kx = k µ x µ and the four-vector k µ is defined as: {k µ } = { k, k}. (1.1.7) We have yet to impose on the solutions (1.1.6) the gauge conditions (1.0.2). They are satisfied if and only if: k ν Ā µν ( k) = 0, (1.1.8) for all k. Since k 0 is always different from zero, the above conditions allow, for each µ, to express Āµ0( k) in terms of Ā µα ( k). This implies that, for

9 1.1. VACUUM FIELDS. 9 each k, we have the freedom to choose at will the six components, Ā αβ ( k), of Ā µν ( k) with space indices. At a closer inspection, though, one realizes that not all of these choices correspond to physically distinct solutions. This is so because there exist residual gauge transformations that preserve the Lorentz gauge conditions (1.0.2). According to eqs.(0.2.12), we see that a gauge transformation with gauge parameters ξ a satisfying the homogeneous wave equation: b b ξ a = 0 (1.1.9) will not spoil the Lorentz gauge conditions. Consider this equation in our minkowskian coordinate system: µ µ ξ ν = 0. (1.1.10) It is clear that it admits an infinite number of solutions, labelled, say, by the arbitrary set of Cauchy data ξ ν (0, x), 0 ξ ν (0, x) on the initial surface t = 0. We shall show now that by means of these residual gauge conditions it is possible to impose on γ µν (x) also the following four gauge conditions, the so-called radiation conditions : γ 0α (x) = 0, (1.1.11) γ(x) = 0. (1.1.12) To prove it, we observe that γ(x) and γ 0α (x) both satisfy the wave equation (1.0.1), and so their values at any time t are uniquely determined by their initial data on the surface t = 0. This implies that, if by means of the residual gauge transformations (1.1.10) we can make their initial data vanish, they will vanish identically at all times. According to eqs.(0.2.8), this will be the case if the arbitrary functions ξ ν (0, x) and 0 ξ ν (0, x) are chosen such as to satisfy the following equations: ( 2 ξ 0 / t + ξ ) = γ, (1.1.13) 2[ 2 ξ 0 + ( ξ/ t)] = γ/ t, (1.1.14) ξ α / t + ξ 0 / x α = γ 0α α = 1, 2, 3 (1.1.15) 2 ξ α + ( ) ξ0 x α = γ 0α α = 1, 2, 3. (1.1.16) t t It can be shown that the above equations determine uniquely the values of ξ µ and of their time derivatives on the initial surface t = 0, thus proving that

10 10 CHAPTER 1. THE POST-MINKOWSKIAN APPROXIMATION the radiation gauge (1.1.11), (1.1.12) can always be achieved. In fact, more can be said. In view of the Lorentz conditions (1.0.2) and of eqs.(1.1.12) and recalling that, thanks to eq.(1.1.11), γ µν = γ µν, we have that: anf then eqs.(1.0.1) imply 0 γ 00 = α γ 0α = 0, (1.1.17) 2 γ 00 = 0. (1.1.18) If we impose the condition that γ 00 is bounded at infinity, the above two equations implies that γ 00 (x) is indeed constant, and thus can be gauged away, without spoiling any of the gauge conditions. Putting it all together, the Lorentz and the radiation gauge conditions imply for the Fourier coefficients Āµν( k) the following equations: k β Ā αβ ( k) = 0 (1.1.19) Ā α α( k) = 0 (1.1.20) Ā 0ν ( k) = 0. (1.1.21) Thus, while all components of Āµν( k) with at least one index equal to zero vanish, the remaining six components having both indices of space kind are subjected to four independent linear conditions, which leave us with two independent degrees of freedom, for each wave-vector k. In order to have a better characterization of these two modes, for each k, we introduce two normalized polarization three-vectors e (s), s = 1, 2 orthogonal to k and to each other: e (s) e (s ) = δ ss, e (s) k = 0. (1.1.22) Upon substituting the following partition of the identity δ α β = k β k α + e (1) β e(1)α + e (2) β e(2)α (1.1.23) into the equation Āαβ = δ γ αδ δ βāγδ, one finds that an arbitrary symmetric three-tensor Āαβ can be decomposed as: where e + αβ = e(1) Ā αβ = h + e + αβ + h e αβ + h tre tr αβ + D s e (s) (α k β) + Dk α k β (1.1.24) α e (1) β e(2) α e (2) β, e αβ = e(1) α e (2) β + e(2) α e (1) β, etr αβ = e (1) α e (1) β + e(2) α e (2) β. (1.1.25)

11 1.1. VACUUM FIELDS. 11 Upon inserting eq.(1.1.24) into eqs.(1.1.19), we find that D s and D must vanish, while eq.(1.1.20) imply that h tr must be zero. We conclude that the most general solution of vacuum Einstein s equations, representing perturbations of a flat space-time, can be written in the Lorentz-radiation gauge as γ (T T ) µν (t, x) = 1 2 δα µδ β ν d 3 k [h (T T ) + ( k)e + αβ ( k) + h (T T ) ( k)e αβ ( k)] e ikx + h.c., (1.1.26) and its elementary modes e +, e are two purely spatial, transverse and traceless (TT) modes. The superscripts (T T ) in eqs.(1.1.26) are just intended to remind us the gauge in which they are written. For suitable choices of the amplitudes h (T T ) + ( k), h (T T ) ( k), supported in a small region of k s around a certain wave-vector k 0, these solutions will have the form of wavepackets of finite extent in the space of coordinates x µ, propagating in the direction of k 0 with the group velocity v g : v g = dk0 d k k= k 0 = 1. (1.1.27) One would then say that the linearized theory predicts the existence of gravitational waves propagating in space-time at the speed of light. This statement is quite naive, though, for it is based on a simple-minded interpretation of the perturbation γ ab like something analogous to, say, an electromagnetic field propagating in a Minkowski space-time. This analogy is however misleading, for the simple reason that, while it is possible to measure the electromagnetic field, there is no coordinate-independent way to separate the total metric η ab + γ ab into a perturbation γ ab plus a background. Our use of such a splitting was only a matter of mathematical convenience, but has no intrinsic physical meaning. This is to say that it is physically incorrect to think of γ ab as a field propagating into a minkowskian background, and one should rather consider η ab + γ ab as a single entity. If one does so, there is not anymore an ambient space, flat or not, with respect to which one can say that anything is propagating. We should look for other, coordinate independent, ways to probe the physical meaning of the solutions (1.1.26) as gravitational waves. Such a way is provided, for example, by studying their effect on a system of test particles, and this is the subject of the next paragraph.

12 12 CHAPTER 1. THE POST-MINKOWSKIAN APPROXIMATION 1.2 The quadrupole formulae We want to study now the linearized field equations in the presence of sources, in order to learn how gravitational waves are generated. For this task too, it is convenient to adopt the Lorentz gauge and to work in a coordinate system {x µ } in which the flat background takes the form of a Minkowsky metric. The equations we have to solve are then (1.0.1) and (1.0.2): ρ ρ γ µν = 16πτ µν, (1.2.1) ν γ µν = 0, (1.2.2) and we recall that the stress-energy tensor of the source, τ µν, has to satisfy the ordinary conservation law ν τ µν = 0, (1.2.3) as a consequence of the linearized Bianchi identities. As it is well known, a particular solution of eqs.(1.2.1) is provided by the so-called retarded potential : γ µν (t, x) = 4 d 3 x τ µν(t x x, x ) x x. (1.2.4) We can check that, thanks to the conservation law (1.2.3), it satisfies the Lorentz gauge condition: ν γ µν = 4 = 4 d 3 x { t τ µ0 (t x x, x ) x x d 3 x { ( t τ µ0 + ατ µα )(t x x, x ) x x + x α ( τ µα (t x x, x ) ) } x x = x α ( τ µα (t x x, x ) ) } x x = 0, (1.2.5) where the vanishing of the first fraction in the second line is due to the conservation law satisfied by τ µν, while the vanishing of the second term is due to the fact that it is total derivative. The general solution of eqs.(1.2.1) and (1.2.2) can be obtained by adding to eq.(1.2.4) the most general solution of the associated homogeneous problem, whose expression, in the T T gauge, was given in eqs.(1.1.26). In order to have a better understanding of the features of the retarded solution (1.2.4), we start by taking the time Fourier-transform of (1.2.4): γ µν (ω, x) = dt γ µν (t, x)e iωt. (1.2.6)

13 1.2. THE QUADRUPOLE FORMULAE 13 Upon substituting to γ µν (t, x) the expression (1.2.4), interchanging the order of integration over t and x and then performing, in the time integral, the change of variables t = t x x, we find that: γ µν (ω, x) = 4 d 3 x τ µν( x, ω) x x e iω x x, (1.2.7) where τ µν ( x, ω) is the time Fourier transform of the stress energy tensor, defined as: τ µν ( x, ω) = dtτ µν (t, x)e iωt (1.2.8) We now make two assumptions, which are analogous to those that, in electromagnetism, allow to derive the multipole expansion of the retarded potential: a) at all times t, the source is fully contained inside a three-sphere centered in the origin of the coordinate system, and having a radius L much smaller than the coordinate distance R = x of the point where the field is observed: L R 1 ; (1.2.9) b) the Fourier transform τ µν ( x, ω) is significantly different from zero only for frequencies ω satisfying, everywhere inside the source, the following slow motion condition: ω x 1. (1.2.10) Condition (a) allows one to expand 1/ x x in series of the small quantities x i /R: 1 x x = 1 ( ( )) L 1 + o, (1.2.11) R R while condition (b), together with (a), allows to take the following low frequency expansion in the phase of the exponential occurring in (1.2.7): e iω x x = e iωr (1 iωn α x α 1 2 ω2 n α n β x αx β + )(1+o(ωL 2 /R) ), (1.2.12) where n α = x α /R and the dots stand for higher multipole moments. Upon substituting the above expansions in eq.(1.2.7), and keeping only the terms that decrease like 1/R, we get: γ µν (ω, x) = 4 eiωr R d 3 x τ µν ( x, ω)(1 iωn α x α 1 2 ω2 n α n β x αx β + ). (1.2.13)

14 14 CHAPTER 1. THE POST-MINKOWSKIAN APPROXIMATION It is easy to compute the Fourier antitrasform of this expression: γ µν (t, x) = 4 R n αn β 2 t 2 { d 3 x τ µν (t, x ) + n α t d 3 x τ µν (t, x )x αx β + d 3 x τ µν (t, x )x α+ } t =t R. (1.2.14) The virtue of the above formula is that the integrands depend only on the distribution of the stress-energy tensor inside the source at the retarded time coordinate t R. We shall devote the rest of the paragraph to the study of this formula. The first distinction to make, is that between radiative and non-radiative terms. A radiative contribution is one whose effects, on a test particle placed at large distances from the source, decrease no faster than 1/R, while a non-radiative contribution is one whose effects decrease like 1/R 2 or faster. By looking now at eq.(1.2.14), it is clear that the only case in which the derivatives of γ µν (t, x) can decrease like 1/R is when the terms inside the curly brackets depend non-trivially on the retarded time t, but a closer inspection reveals that some of them are indeed constant in t. To see it, we have to recall that the stress-energy tensor of the source τ µν satisfies the flat-space conservation law ν τ µν = 0, and thus its four momemtum: P µ = d 3 xτ µ0 (t, x) (1.2.15) and its angular-momentum M µν = d 3 xτ 0[µ (t, x)x ν] (1.2.16) are constant in time. If we now consider the first term inside the curly brackets of eq.(1.2.14), for ν = 0, we recognize in it precisely the four momentum P µ of the source. The freedom to perform a finite Lorentz transformation in the linearized equations (such a coordinate transformation preserves not only the form of the minkowskian background η µν, but also the Lorentz gauge conditions) can be exploited to make the three momentum P α of the source equal to zero. After this is done, P 0 represents the total mass M of the source and then the corresponding non-radiative contribution to γ 00 (t, x) is nothing but the newtonian potential term. This is not all: for µ = ν = 0, also the second term inside the brackets of eq.(1.2.14) can be seen to be

15 1.2. THE QUADRUPOLE FORMULAE 15 independent on t, because: d 3 x t τ 00 x α = d 3 x βτ 0β x α = d 3 x β(τ 0β x α )+ d 3 xτ 0α = P α. (1.2.17) The conclusion is that the first radiative terms in (1.2.14) are: the third one, for γ 00, the second one, for γ 0α, and the first one for γ αβ. What we now show is that they can be all expressed in terms of the second moments q αβ of the energy-density τ 00 : q αβ (t ) = d 3 x τ 00 (t, x )x αx β. (1.2.18) This is obvious for γ 00 ; as for γ 0α, we first notice that the second term in (1.2.14) can be rewritten as: d 3 x τ 0α x β = d 3 x τ 0(α x β) + d 3 x τ 0[α x β]. (1.2.19) The second term is equal to M αβ and is constant in time, and so, when in (1.2.14), we derive it with respect to t, it vanishes. As for the first, we can reexpress it in terms of q αβ, in the following way: d 3 x τ 0(α x β) = d 3 x τ 0γ x (β γx α) = = = d 3 x γ(τ 0γ x α x β ) d 3 x τ 0(β x α) + t d 3 x τ 0(β x α) d 3 x τ 00 x α x β = d 3 x γτ 0γ x α x β = d 3 x τ 0(α x β) + t q αβ. (1.2.20) We finally consider the first term of (1.2.14), for γ αβ. We use the same tricks to write (the twice of it) as: 2 d 3 x τ αβ = d 3 x τ γδ γ δ(x α x β ) = d 3 x γτ γδ δ(x α x β ) = = t d 3 x τ 0δ δ(x α x β ) = t d 3 x δτ 0δ x α x β = 2 t qαβ, (1.2.21) where we have not shown explicitely the vanishing terms containing total divergences. Putting everything together, we can write the leading radiative contributions in eq.(1.2.14) as: γ 00 (t, x) = 2 R n αn β q αβ (t ret ), (1.2.22)

16 16 CHAPTER 1. THE POST-MINKOWSKIAN APPROXIMATION γ 0α (t, x) = 2 R n β q αβ (t ret ), (1.2.23) γ αβ (t, x) = 2 R q αβ(t ret ), (1.2.24) where each dot over q αβ denotes a derivative with respect to its argument and t ret denotes the retarded time t R. We now prove that by means of a gauge transformation on the above expressions, we can gauge away the four components of γ µν (x) with at least one time index. Let us write down first the components of γ µν (x), which, according to eqs.(0.1.16) are equal to γ µν 1/2η µν γ: γ 00 (x) = 1 R (n αn β + δ αβ ) q αβ (t ret ), (1.2.25) γ 0α (x) = 2 R n β q αβ (t ret ), (1.2.26) γ αβ (x) = 1 R [2δ αγδ βδ + δ αβ (n γ n δ δ γδ )] q γδ (t ret ). (1.2.27) Now we search for a vector field ξ µ such that For µ = 0, we get: γ 0µ = γ 0µ 2 (0 ξ µ) = 0 (1.2.28) γ 00 = 1 R (n αn β + δ αβ ) q αβ (t ret ) t ξ 0 = 0. (1.2.29) A solution of this equation is obviosly: ξ 0 = 1 2R (n αn β + δ αβ ) q αβ (t ret ). (1.2.30) Now, we determine ξ α by requiring that γ 0α vanishes: γ 0α = 2 R n β q αβ (t ret ) α ξ 0 t ξ α = 0. (1.2.31) Substituting in this equation the above expression for ξ 0, we find: ξ α = 1 ( ) 1 R [1 2 n α(n β n δ +δ βδ ) q βδ (t ret ) 2n β q αβ (t ret )] q βδ (t ret ) α 2R (n βn δ + δ βδ ) (1.2.32) We have to compute now the space-space components γ αβ in the new gauge. Before doing it, we notice that the last term in the expression of ξ α, for large.

17 1.2. THE QUADRUPOLE FORMULAE 17 R, is of order 1/R 2 and thus generates a contribution of at least this order to the transformed γ αβ. Since we are interested only in radiative effects, which are of order 1/R, we can neglect this contribution. Thus, we find: γ αβ = γ αβ 2 (α ξ β) = = 2 R [ Q αβ (n αn β +δ αβ ) Q γδ n γ n δ (n α n γ Qγβ +n β n γ Qγα )](t ret )+o(1/r 2 ), where Q αβ are the traceless quadrupole moments of the source: (1.2.33) Q αβ = q αβ 1 3 δ αβq δδ. (1.2.34) This expression can be written in a more compact form, upon introducing the projector P αβ : P αβ = δ αβ n α n β, (1.2.35) that projects three-vectors onto the plane orthogonal to n α. With its help, and neglecting the non-radiative 1/R 2 piece, we can rewrite eq.(1.2.33) as : γ αβ(t, x) = 2 R (P αγp βδ 1 2 P αβp γδ ) Q γδ (t ret ), (1.2.36) It can be easily verified that γ αβ (t, x) is traceless, and obviously satisfies the transversality condition: n α γ αβ(t, x) = 0. (1.2.37) For this reason, we shall denote γ αβ T ) (t, x) as γ(t αβ (t, x), where the (T T ) superscript stands for transverse and traceless. Eq.(1.2.36) is called the quadrupole radiation formula and provides the leading contribution to the emission of gravitational waves from a far and slow-moving source. Its outstanding feature is that the production of gravitational radiation is proportional to the second time derivative of the traceless-quadrupole moment of the source. This makes a big difference with respect to the emission of electromagnetic radiation, whose leading contribution is proportional to the second time derivative of the electric dipole moment of the source. It is also interesting to observe that a spherically symmetrical distribution of matter, however varying in time, will not emit either, according to eq.(1.2.36), because its traceless quadrupole moment Q αβ is equal to zero at all times. In fact, the absence of radiation in this case is an exact result, because of the Birkhoff theorem, which states that all spherically symmetric space-times

18 18 CHAPTER 1. THE POST-MINKOWSKIAN APPROXIMATION are static. This is not different from electromagnetism, where the Gauss theorem implies that the electric field of a spherically symmetric distribution of charge, outside the distribution, is necessarily equal to the Coulomb field for the total charge, and thus no radiation is possible.