Gravitational waves from generalized newtonian sources

I review the elementary theory of gravitational waves on a Minkowski background and the quadrupole approximation. The modified conservation laws for energy and momentum keeping track of the gravitational-wave flux are presented. The theory is applied to two-body systems in bound and scattering states subject to newtonian gravity generalized to include a 1/r3 force allowing for orbital precession. The evolution of the orbits is studied in the adiabatic approximation. From these results I derive the conditions for capture of two bodies to form a bound state by the emission of gravitational radiation.


Introduction and Overview
The existence of gravitational waves is now well-established from both direct and indirect observations. [1][2][3][4] A completely new field of astronomy is opening up which will no doubt have an impact also on other branches of astronomy and astrophysics such as dynamics and evolution of stars and galaxies. The supermassive black holes in the centers of galaxies, and possibly intermediatemass black holes in stellar clusters, will by the relatively large curvature they create in the surrounding space enhance the emission of gravitational waves from massive objects on trajectories passing close to them, whether these are on bound or open orbits. The emission of gravitational waves can even lead to the capture of objects originally in open orbits to end up in a bound state.
Apart from these radiative phenomena involving very massive black holes, the emission of gravitational waves also affects more common binary star systems like the well-known close binary neutron stars, the recently discovered binary black holes and presumably systems containing white dwarfs. [5] No doubt radiation has an impact on three-and many-body systems, especially on their stability. Detailed investigations of close binary star systems using high-order post-newtonian expansions of the Einstein equations of General Relativity have been carried out with great success; for a review see e.g. [6]. The inspiral and merger of extreme mass-ratio binaries involving a very massive black hole has also been studied directly in the background geometry of the DOI: 10.1002/prop.201800083 black hole. [7][8][9][10][11] Whenever these theoretical investigations can be compared with data they seem to describe the dynamics of these systems very well, thereby also confirming General Relativity to be the best available theory for gravitational interactions. [12] The study of radiation from two-body scattering has been addressed as well, [13] although no corresponding observations have been announced so far.
Even though they may carry large amounts of energy and momentum, the deformations of space-time created by gravitational waves are extremely small. For example a flux of monochromatic gravitational waves with a frequency of 100 Hz and an extreme intensity of 1 W/m 2 will create spatial deformations of less than 1 part in 10 19 , the diameter of a proton over a distance of 1 km. This testifies as to the extreme stiffness of space and explains both why it is so difficult to create gravitational waves and to observe them. It also implies that most potential sources of gravitational waves are weak and many move on close-to-stationary almost-newtonian orbits.
This review is devoted to gravitational radiation from such weak or very weak sources. They produce the most abundant, though maybe not the most spectacular, form of gravitational waves in the universe and may eventually become relevant to a wide range of astronomical and astrophysical observations. To lowest order their description and propagation involve straightforward applications of linear field theory in Minkowski spacetime. This also provides the starting point for many more elaborate and precise calculations.
We will begin by recapturing in fairly standard fashion the wave equation for gravitational waves, its gauge invariance and its implications for the propagation and polarization states of gravitational waves. We address the quadrupole nature of the waves and the associated sources, and explain how dynamical mass quadrupole motion generates the simplest and most common weak gravitational waves. Next we derive the modification of the conservation laws for energy, momentum and angular momentum by taking account of gravitational radiation. We present equations for the transport of energy and angular momentum by gravitational waves, keeping track of the anisotropic dependence on directions.
This theory is then applied to systems of massive objects moving on generalized newtonian orbits, either in bound states or on open scattering trajectories. The generalization includes the effects of possible 1/r 3 forces causing orbital precession, which may result e.g. from many-body or post-newtonian interactions. We calculate the evolution of orbital parameters due to emission of gravitational radiation and their relations. We finish by www.advancedsciencenews.com www.fp-journal.org establishing which binary scattering orbits are turned into bound states by emission of radiation.

The Wave Equation
Weak gravitational waves are dynamical fluctuations of the spacetime metric about flat Minkowski geometry. [14][15][16] Thus we can split the full space-time metric as where κ is the positive root of G being the newtonian constant of gravity and c the speed of light in vacuum. This endows h μν with the standard dimensions of a bosonic tensor field. Up to non-linear corrections the tensor field is postulated to satisfy the field equation where = η μν ∂ μ ∂ ν is the d'Alembertian and the inhomogeneous term T μν on the right-hand side represents the sources of the field. By factoring out the constant κ this tensor has the dimensions of energy per unit of volume or force per unit of area. In this treatise we always use the flat Minkowski metric η μν with signature (−, +, +, +) and its inverse η μν to raise and lower indices on components of mathematical objects like vectors and tensors. The motivation for postulating this field equation comes from the physical properties of the tensor field h μν implied by its structure. First note that defining the linear Ricci tensor the trace of which reads the field equation takes the form This is the linearized version of Einstein's gravitational field equation in a flat background. Note also that and as a result the inhomogeneous field Equation (6) is seen to imply a conservation law for the source terms: As the energy-momentum tensor of matter and radiation has the required physical dimensions and satisfies the condition (8) in Minkowski space it is the obvious source for the tensor field. As all physical systems possess energy and momemtum this explains the universality of gravity 1 . An observation closely related to (7) is that the linear Ricci tensor is invariant under gauge transformations By such gauge transformations one can straightforwardly eliminate four components of the field to reduce the number of independent components from ten to six. To achieve such a reduction in pratice the standard procedure is to impose the De Donder condition This condition reduces the linear Ricci tensor and its trace to the expressions and therefore the field equation turns into the inhomogeneous wave equation It is then convenient to redefine the field components by which transform under gauge transformations as After implementing the De Donder condition the field is divergence-free and satisfies the inhomogeneous wave equation: Finally a second gauge transformation can be made without changing the De Donder condition provided the parameter satisfies itself the homogeneous wave equation: Such a residual gauge transformation can be made in particular on free fields to remove the trace of the tensor field: in agreement with the Equations (15) provided ξ ν = 0 and T λ λ = 0. It follows automatically that the same condition holds for the original tensor field: h λ λ = 0. Removal of the trace reduces the number of independent components of free fields to five, equal www.advancedsciencenews.com www.fp-journal.org to the dimension of the irreducible spin-2 representation of the rotation group, However, as dynamical free wave fields propagate on the light cone and have only transverse polarization states, the actual number of independent dynamical components of gravitational wave fields is two. This will be discussed in the following.

Solutions of the Inhomogeneous Wave Equation
The inhomogeneous linear wave Equation (15) has many solutions: to a given solution one can always add any solution of the homogeneous equation representing free gravitational waves. Free gravitational waves can therefore appear as a background to gravitational wave signals from specific sources.
In the absence of such a background the standard causal solution for sources localized in a finite region of space is the retarded solution where the integration volume S r can be taken to be a large sphere of radius r = |x| containing the finite region of the sources where T μν = 0 in its center. To evaluate the field by performing the integration is difficult in practice for any realistic type of sources. In order to make progress it makes sense to consider the situation in which the waves are evaluated at large distance from the sources: the radius r of the sphere is taken to be much larger than any typical dimension of the sources. For example we evaluate the waves emitted by a binary star system of orbital extension d at a distance r d. Under this assumption one can expand the integral expression on the right-hand side of (18) in inverse powers of r keeping only terms which do not fall off faster than 1/r. This results in the simpler integral Another simplification is possible as it is straightforward to show that for localized sources these solutions have no dynamical time components: The second equality on the first line follows from energymomentum conservation, whilst the last equality uses Gauss' theorem to convert the volume integral to a surface integral over the corresponding normal component of the energy-momentum tensor,r being the radial unit vector pointing out of the spherical surface ∂ S r . Finally the localization of the sources in a finite region near the center of the sphere guarantee the vanishing of the energy-momentum tensor on the boundary. We infer that the time components may represent static newtonian fields, but they cannot contribute to the flux of dynamical waves across the boundary of the sphere.
As concerns dynamical fields we are therefore left with the spatial components of the outgoing wave solutions (19): In empty space far from the sources the expression on the righthand side actually represents an exact formal solution of the wave equation. Now this solution was obtained by imposing the De Donder condition (15); in addition, as argued after (17), in this region one can always find a local gauge transformation of the fields that makes them traceless. For the solution at hand this implies that after such a gauge transformation and A detailed discussion of the necessary gauge transformations is presented in appendix A. Tensor fields obeying these conditions of are called transverse and traceless (T T) and satisfy h T T i j = h T T i j . We will take these properties for granted in what follows and omit the T T in the notation. Combining the above requirements the outgoing wave fields far from the source must then be represented in the T T-gauge by an expression of the form where the spatial symmetric 3-tensor I is traceless: I kk = 0. Writing u ≡ t − r, agreement of this expression with the result (21) up to gauge transformations is obtained by taking With the help of energy-momentum conservation the integral can be rewritten in terms of the quadrupole moment of the total energy density T 00 of the sources: The proof is easier in backward fashion; first notice that as then perform two partial integrations with respect to x to reobtain (25), observing that the full energy-momentum tensor is supposed to vanish at the boundary ∂ S r . Finally considering non-relativistic sources in the center-ofmass frame, the energy density is dominated by the mass-density ρ(x, t), which allows us to replace the integral in (26) by the components of the mass quadrupole moment and write explicitly: Thus we get the final expression for the wave field h i j for nonrelativistic sources in the T T-gauge: For the dynamical (non-Newtonian) metric fluctuations δg μν = g μν − η μν , recalling Equations (1) and (2) this result implies that

Conservation Laws and Gravitational-Wave Fluxes
Free radiation fields (always taken in the T T-gauge) define conserved currents of energy, momentum and angular momentum; [15,16] in the conventions of the previous sections Subject to the field equations and gauge conditions these quantities satisfy the continuity equations where Applying them to the free fields (29) these expressions determine the flux of energy, momentum and angular momentum carried by outgoing gravitational waves far from the source region. First, integration over a large sphere around the center of mass of the source and using Gauss' theorem gives the change in total energy, momentum and angular momentum of gravitational waves in terms of surface integrals Next, on the spherical surface ∂ S r the surface element of integration taken in polar co-ordinates (r, θ, ϕ) is Evaluating the integrands on the right-hand side in Equations (34) while restoring factors of c then results in differential fluxes As usual overdots denote derivatives with respect to time t. The integrands themselves represent the anisotropic angular distribution of fluxes. The spherical surface integrals can be performed taking note that the quadrupole moments depend only on retarded time u = t − r , and that the angular integrals can be evaluated using the averaging procedure whilst Fortschr. Phys. 2019, 67, 1800083 www.advancedsciencenews.com

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This results in [14][15][16][17] Note that the total flux of linear momentum vanishes by symmetry (in the present approximation) as it involves only products of odd numbers ofr i integrated over a full spherical surface, whereas the integrands of the energy and angular momentum contain even numbers of outward spherical unit vectors.

Generalized Newtonian 2-Body Forces
In the following we will apply the results to systems of masses moving under the influence of mutual newtonian forces, considering two-body systems interacting via a central potential. The classical description of such systems simplifies greatly, first as one can effectively reduce it to a single-body system by separating off the center-of-mass (CM) motion; second as angular momentum conservation implies the relative motion to be confined to a two-dimensional plane. Of course, the emission of gravitational radiation introduces limitations to these simplifications, but as long as the rate of energy and angular-momentum loss by the system is small the orbits will change only gradually and one can evaluate the effect of gravitational-wave emission in terms of adiabatic changes in the orbital parameters. In this section we first discuss non-disspiative motion; the effects of gravitational wave emission will be analysed afterwards. Let the bodies have masses m 1 and m 2 and positions r 1 and r 2 . To make maximal use of the simplifications we work in the CM frame in which In terms of the relative separation vector r = r 2 − r 1 the positions w.r.t. the CM are and Newton's third law of motion implies that where μ is the reduced mass and F (r ) is the magnitude of the central force acting on the masses. As usual r andr represent the modulus and unit direction vector of the separation. In the absence of dissipation the energy and angular momentum of the system are conserved. In the CM frame these quantities can be written as and L = μr ×ṙ.
Angular momentum being a conserved vector, the relative motion takes place in the plane perpendicular to L, which we take to be the equatorial plane θ = π/2. Then and In the following we will always orient the orbit such that the motion is counter-clockwise and therefore ≥ 0. The orbit is represented by the parametrized curve r (ϕ) such thaṫ the prime denoting a derivative w.r.t. ϕ. Newton's law of central force (40) then takes the form This result is tailored to suit Newton's original program of finding the law of force corresponding to a given orbit. [18] We will demonstrate it for the particular case of precessing conic sections: ellipses, parabolae and hyperbolae; these orbits are parametrized by Here ρ is known as the semi-latus rectum; e is the eccentricity: e = 0 for circles, 0 < e < 1 for precessing ellipses, e = 1 for similar parabolae and e > 1 for hyperbolae. Finally the number n determines the rate of precession. For circles this is of course irrelevant. For precessing ellipses the apastra occur for where k is an integer; thus the apastron shift is ϕ = 2π (1 − n)/n per turn. For precessing parabolae n determines the angle over which the directrix turns during the passage of the two bodies, i.e. the asymptotic scattering angle due to precession, also measuring Similarly for hyperbolae it determines the angle between the incoming and outgoing asymptotes: Fortschr. Phys. 2019, 67, 1800083 www.advancedsciencenews.com www.fp-journal.org

Substitution of the expression (47) into Equation (46) leads to the result
the sum of an inverse square and an inverse cube force. Identifying the inverse square term with newtonian gravity and introducing an inverse cubic force with strength βμ: we find with M = m 1 + m 2 the total mass of the two-body system. Such a force follows from a potential The eccentricity is determined by the radial velocity when the system is at the semi-latus rectum ϕ = π/2n, r = ρ: Evaluating the total energy at the semi-latus rectum and observing it is a constant of motion then tells us that This confirms that for e 2 < 1 the orbits are bound, whilst for e 2 ≥ 1 the orbits are open. Obviously the total angular momentum is by definition Note that taking the first-order result for relativistic precession in Schwarzschild space-time with innermost circular orbit R isco = 6GM/c 2 one gets

Gravitational Waves from Two-Body Systems
In this section and the following we address the emission of gravitational radiation by the two-body systems described in section 5.
As announced we treat this as a form of adiabatic dissipation changing the orbital parameters (ρ, e, n) of the system. This applies only to systems in which no head-on collisions or mergers involving strong gravity effects take place; these require more powerful methods of computation. [6] To compute the amplitude h i j from Equation (29) for point masses on the quasi-newtonian orbits (47) we must first determine the components of the quadrupole moment and their derivatives. For a two-body system in the CM frame they read wherer is the orbital unit vector in the equatorial plane defined in (43). We explicitly factor out the three-tensor arrayR with compo-nentsR i j describing the angular dependence of the orbits used in computing the quadrupole moments: They have simple algebraic properties In addition their derivatives are Fortschr. Phys. 2019, 67, 1800083 www.advancedsciencenews.com www.fp-journal.org It follows that Using these results and the ones in appendix B it is now straightforward to establish expressions for the quadrupole moment and its derivatives: More generally we can write for the n-th derivative where the coefficients Q (n) E ,M,N can be read off from the expressions (66) or computed by taking still higher derivatives. These results can now be used to evaluate the amplitude h i j (x, t); the expression (29) for the amplitude is equivalent to Note that the direction of the observer is given by the polar unit vector which is distinct from the orbital unit vectorr; then the amplitude in three-tensor notation takes the form M M ·r + Q (2) N N ·r r T Er · E ·r + Q Mr · M ·r + Q (2) Nr · N ·r .
To evaluate this expression use E ·r = 1 3 (sin θ cos φ, sin θ sin φ, −2 cos θ) , and r · E ·r = sin 2 θ − 2 3 ,r · M ·r = sin 2 θ cos 2(φ − ϕ), The simplest case is that of circular orbits with r = 0 and = ωr 2 , where ω is the constant angular velocity such that ϕ(t) = ωt. Then Note that the frequency of the gravitational waves is twice that of the orbital motion, which is a direct consequence of their quadrupole nature.

Radiative Energy Loss
The first Equation (36) describes the energy flux of gravitational waves per unit of spherical angle as a function of the direction specified by the unit vectorr. Equations (66) specify the quadrupole moments and their derivatives for two-body systems Fortschr. Phys. 2019, 67, 1800083 www.advancedsciencenews.com www.fp-journal.org in generalized newtonian orbits (47). To evaluate the differential energy flux these quadrupole moments are to be substituted into the energy flux equation. First we compute It follows that and r · Q (3) 2 ·r = μ 2 6 r 8 Finallŷ Inserting the coefficients taken from Equation (66): the general result is For purely Keplerian orbits this result was derived in [20]. Using the results from appendix B for the generalized newtonian orbits (47) the expressions for the quantities A and B take the form The intensity distribution of gravitation radiation emitted by a bound binary system in elliptical orbit, precessing and nonprecessing, is illustrated for a particular choice of parameters in appendix C. After integrating the result (82) over all angles the standard result (39) for the total energy loss becomes Substitution of the expressions (83) then results in In the simplest case, that of a circular orbit with e = 0, n = 1, r = ρ and with angular velocity given by this result reduces to the well-known expression The last result has been cast in terms of the dimensionless compactness parameter 2GM/c 2 ρ, defined as the ratio of the Schwarzschild radius for the combined system and the actual orbital scale characterized by ρ. For non-precessing orbits for which n = 1, 2 = GMρ, the rate of energy loss is The expression (85) can also be used to compute the total energy lost by the two-body system in a definite period between times t 1 and t 2 , e.g. between two periastra for bound orbits, or during the total passage of two objects in an open orbit: Fortschr. Phys. 2019, 67, 1800083 www.advancedsciencenews.com www.fp-journal.org where we have introduced the integration variable ψ = nϕ. Now substitute (84) for the energy change and use Recalling that n 2 2 = GMρ and expanding the integrand transforms the expression to The adiabatic approximation implies that we treat the parameters e and n in this interval as constants; then it is straightforward to perform the integrations. For a bound orbit with succesive periastra at ψ 1 = 0 and ψ 2 = 2π the total energy lost per period to gravitational waves is 1 + e 2 24 n 6 + 12n 4 − 120n 2 + 180 + e 4 96 n 6 + 216n 4 − 720n 2 + 540 + 5e 6 16 In particular for non-precessing orbits with n = 1: For the simplest case, a circular orbit with e = 0: On the other hand, for open orbits with e ≥ 1 and asymptotic values of the azimuth (ψ 1 , ψ 2 ) satisfying cos ψ 1 = cos nϕ 1 = 1 e , sin ψ 1 = 1 e e 2 − 1, the result of the integral (90) in a somewhat hybrid notation is with coefficients I 0 = 12 (π − ψ 1 ) , I 1 = −24n 2 + 72 sin ψ 1 , For non-precessing orbits with n = 1 the expression simplifies as I 5 = I 6 = 0. The simplest case is the parabolic orbit with e = 1, n = 1 and ψ 1 = 0, resulting in These results are based on the generalized newtonian approximation. Results for scattering in the Effective One-Body formalism to all orders in v/c have been obtained in ref. [19].

Radiative Loss of Angular Momentum
The gravitational waves emitted by a system of masses in motion not only carry away energy, they also change the system's angular momentum. The last Equation (36) quantifies the directional angular momentum loss per unit of time of a non-relativistic system in terms of the change in the mass quadrupole. In this section we compute the angular momentum lost by a quasi-newtonian twobody system as we did for the energy in the previous section.
www.advancedsciencenews.com www.fp-journal.org After substitution of Equations (66), (67) in the expression (36) for the differential flux of angular momentum we get Er · E ·r + Q The total loss of angular momentum obtained by integration over all angles as given by the result (39) is According to the expansion (67) and the multiplication rules (63) the only antisymmetric contribution to the product of Q (2) and Q (3) comes from which has only a non-vanishing J xy = −J yx = 1 component. As the only non-trivial component of orbital angular momentum is M z this is as expected. Using the results of appendix B it follows that For circular orbits with r = ρ, e = 0 and n = 1 this reduces to and for other non-precessing orbits Following a procedure similar to the treatment of energy we can compute the change in angular momentum in a fixed period of time between precessing angles ψ 1,2 : It follows that for a bound state the angular momentum lost per period between successive periastra ψ 1 = 0 and ψ 2 = 2π is For n = 1 this becomes: for circular motion just take e = 0. Next considering open orbits with asymptotic directions as in (94) Equation (102) takes the form with coefficients www.advancedsciencenews.com www.fp-journal.org − n 2 − 5 2 sin ψ 1 cos ψ 1 − sin 3 ψ 1 cos ψ 1 .
In particular for parabolic orbits with e = n = 1 and ψ 1 = 0: In ref. [13] a similar result was derived for small-angle scattering in purely newtonian gravity with β = 0.

Evolution of Orbits
The flux of energy and angular momentum carried by gravitational waves as expressed by Equations (34) can be determined only if all components of the wave signal are known. With present interferometric detectors this is barely possible by combining the signals received by at least three instruments at different locations. However, the loss of energy and angular momentum by sources such as binary star systems is observable and allows the gravitational-wave flux to be reconstructed as in the well-known case of the binary pulsar systems. Therefore it is of some practical use to evaluate the orbital changes due to the emission of gravitational radiation by such systems. Here as in the previous sections we consider non-relativistic two-body systems, either in bound orbit or on scattering trajectories. In the adiabatic approximation on which our calculations are based the orbits of two-body systems in the CM frame are parametrized by the expression (47). We take the orbital parameters (ρ, e, n) to be slowly changing functions of time; they would be constant in the absence of gravitational radiation. According to Equations (56) and (57) the orbital energy and angular momentum are expressed in terms of these parameters by For comparison with observational data of bound orbits it is sometimes convenient to consider the (possibly precessing) semi-major axis of the orbit related to the semi-latus rectum by This quantity is also related to the precession parameter by It follows that for bound orbits the orbital parameter changes are related to change in orbital energy and angular momentum by As these parameters are related by (109) the changes in ρ and in eccentricy e are related as well: Also for constant β: Now by equating the change in energy and orbital angular momentum to the amount of energy E and angular momentum M z carried away by gravitational waves we can relate the change in orbital parameters to these parameters themselves. In particular according to Equations (91) and (103) during a period between to succesive periastra the orbital parameters change by Furthermore from these results we can determine the period of the orbit between periastra and its evolution. The period itself is This is the appropriate generalization of Kepler's third law for precessing orbits, which holds provided the period T is taken to be that between two periastra. From this it follows that the rate of change of the period is Fortschr. Phys. 2019, 67, 1800083 www.advancedsciencenews.com www.fp-journal.org and the relative change per turn is This amounts to a generalization of the Peter-Matthews Equation [20] dT dt Next we consider open orbits. These we will characterize in terms of ρ and e directly with rates of change determined by (108) and (111) 1 e 2 − 1 This results in −2n 2 (n 2 − 1)(1 − e 2 ) ρ 4 r 4 + n 2 (n 2 + 2) ρ 5 r 5 − 4(n 2 − 1) −2n 2 n 4 + 12(n 2 + 2)(e 2 − 1) ρ 5 r 5 + n 2 (n 2 − 12) + 96(n 2 − 1)(e 2 − 1) ρ 6 r 6 The corresponding changes over the complete orbit are and The total energy change in such an open orbit is given by Finally one can determine for which open orbits the loss of energy by gravitational radiation results in a bound orbit, at least in lowest-order approximation. Such a capture process happens when the initial energy is positive and the final energy is negative: As the semi-latus rectum ρ must be greater than the Schwarzschild radius of the system, the quantity on the left-hand side must be definitely larger than one, and as μ < M it follows that e 2 − 1 must be small, i.e. the orbit must be close to parabolic.

Appendix A: The Transverse Traceless Gauge
In this appendix we explain in more detail how starting from an arbitrary solution of the field Equations (3) for the massless tensor field one can reach the T T-gauge (24) in the far-field region. We will do this in the hamiltonian formulation in which spaceand time components of the fields are considered separately. In this formulation the space-components h i j and their conjugate momentum fields π i j satisfy field equations which are first-order in time derivatives. In contrast the time components represent auxiliary fields N = −h 00 and N i = h 0i acting as Lagrage multipliers to impose constraints: time-independent field equations restricting the allowed field configurations of the space components. The full set of dynamical equations for these fields read The constraints imposed by the auxiliary fields are Together these equations are fully equivalent to the covariant field Equations (3). Our analysis will show that the split in dynamical space-and non-dynamical time components is in full agreement with the properties of the causal solutions (18) Observe that h i j changes only by terms depending on ξ i , whilst the change of π i j is determined only by ξ . Clearly the transformations of the auxiliary fields (N, N i ) suffice to remove these components by takinġ This results in N = N i = 0 and π i j =ḣ i j − δ i jḣ kk , constrained by Now note that the choice of gauge parameters (128) does not fix these transformations completely: one can still make residual gauge transformations with parameters (ξ , ξ i ) subject to the conditionṡ To see how these can be used, first note that combining the second field Equation (129) with the first constraint (130) results iṅ π j j = κ T j j + T 00 .
This condition is invariant under the residual gauge transformations, and therefore in empty space where T j j = T 00 = 0 the trace π j j is seen to be constant in time and can be removed by a timeindependent gauge transformation: ξ = − 1 4 π j j t=0 ⇒ π j j = π j j + 4 ξ = 0.
In view of the first Equation (129) this also implies that at all timesḣ j j = 0 and therefore h j j is time-independent. In empty space the first constraint (130) then asserts that also ∂ i ∂ j h i j is time-independent. Next the residual gauge parameters ξ i can be used to restrict the field combination First it can be removed from the initial configuration by taking In combination with the first constraint (130), and knowing that h j j and ∂ i ∂ j h i j themselves are constant in time, this implies that in empty space at all times. Finally one can still make one more residual gauge transformation, with harmonic parameters (ξ , ξ i ) satisfying These transformations can be used to remove the trace of the field at t = 0 and therefore at all times: Finally as the second constraint (130) in empty space requires we also find that by combining with (135) and (138) ∂ j h j i = ∂ j h j i t=0 = 0.
In conclusion, we have proved that we can find local gauge transformations such that in empty space any solution of the field equation can be transformed to the T T-gauge ∂ j h j i = h j j = 0, by the gauge transformations specified in (128), (133), (135) and (138). The vanishing of the trace also implies that in the T Tgauge h i j = h i j . We close this section by noting that the hamiltonian field Equations (125), (126) follow directly from the action with hamiltonian density As is to be expected, in the T T-gauge this hamiltonian reduces to the energy density (31).