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Relativistic Closed-Form Hamiltonian for Many-Body Gravitating Systems in the Post-Minkowskian Approximation

The Hamiltonian for a system of relativistic bodies interacting by their gravitational field is found in the post-Minkowskian approximation, including all terms linear in the gravitational constant. It is given in a surprisingly simple closed form as a function of canonical variables describing the bodies only. The field is eliminated by solving inhomogeneous wave equations, applying transverse-traceless projections, and using the Routh functional. By including all special relativistic effects our Hamiltonian extends the results described in classical textbooks of theoretical physics. As an application, the scattering of relativistic objects is considered.

The Hamiltonian for a system of relativistic bodies interacting by their gravitational field is found in the post-Minkowskian approximation, including all terms linear in the gravitational constant. It is given in a surprisingly simple closed form as a function of canonical variables describing the bodies only. The field is eliminated by solving inhomogeneous wave equations, applying transverse-traceless projections, and using the Routh functional. By including all special relativistic effects our Hamiltonian extends the results described in classical textbooks of theoretical physics. As an application, the scattering of relativistic objects is considered. Introduction.-The problem of motion of bodies under gravitational interaction has been the central issue in general relativity from its birth. As early as 1916, Einstein, Droste, Lorentz, and de Sitter started to develop post-Newtonian (PN) approximation methods. These are based on the weak-field limit in which the metric is close to the Minkowski metric and the assumption that the typical velocity v in a system divided by the speed of light c is very small, v/c ∼ ε. The deviation from the flat metric can be characterized by the Newtonian potential Φ; so for a binary system, for example, Φ/c 2 ∼ v 2 /c 2 ∼ ε 2 . In an appropriate limit (as ε → 0), the PN approximation yields Newton's equations. In 1916 Einstein also worked out "the linearized approximation" to general relativity in which the flat-space wave equations for the deviations h µν from Minkowski metric with an energy-momentum tensor T µν of matter as the source are considered. This work marks the beginning of the post-Minkowskian (PM) approximation methods: the weakness of the gravitational field is assumed -the deviations of the metric from flat are small, but no assumption about slowness of motion is made. In an appropriate limit, when a suitable parameter, usually identified with Newton's gravitational constant G, is approaching zero, the PM approximation yields equations of special relativity. The "historical" period culminated in 1938 when Einstein, Infeld, and Hoffmann, by formal expansions in c −1 , investigated the n-th iterated field equations of the PN scheme.
Since the 1950s numerous investigations of both PN and PM approximations appeared; see, for example, refined comprehensive reviews [1,2]. Because of the evidence that the orbit of the binary pulsar PSR 1913+16 decays as a consequence of the emission of gravitational waves, the most promising candidates for the detectors such as LIGO, VIRGO, and GEO600 became binary neutron stars or black holes. This led to new studies of higher-order equations of motion. Extensive reviews summarizing these developments were prepared recently [3,4].
The Hamiltonian methods have been widely used in general relativity in such problems as perturbation of black holes, dynamics of anisotropic cosmological models, etc. In the problem of motion of gravitating systems the Hamiltonian yields equations of motion, expressions for the mechanical energy and angular momentum; in the case of binaries, the result for the last stable circular orbit follows from ∂ 2 H/∂ r 2 = 0. The Hamiltonian approach to the PN approximation was initiated by Kimura and Toiya [5] and developed recently in the work of Schäfer, Jaranowski and Damour (see [4] for references). The canonical formalism of Arnowitt, Deser, and Misner (ADM) [6] is commonly used.
Before we enter into details we wish to indicate why our final result-the closed form Hamiltonian (11) including all terms linear in G-is of importance in the theory of gravity and can be relevant in other branches of theoretical physics and astrophysics: (i) Since PM approximation can be restricted to slow motions, the Hamiltonian describes PN approximations to any order in 1/c when terms linear in G are considered. When considering only two particles and terms up to the second order ( v 2 /c 2 ) in center-of-mass system we precisely recover the Hamiltonian given by Landau and Lifshitz [7], except for the last term which is ∼ G 2 . Moreover, we checked that, after a suitable canonical transformation, it yields precisely all the terms linear in G in 3PN approximation of [8]; and we calculated corresponding terms in 4PN order which have not been given thus far. These will be published elsewhere. (ii) An electromagnetic counterpart of our Hamiltonian was derived by completely different methods by Kennedy [9]. The first terms in the expansion in c −2 of the Kennedy Hamiltonian correspond to the well-known Darwin Lagrangian (see e.g. [7,10]) but Kennedy's treatment goes beyond Darwin's approximation. Our procedure shows how the Darwin Hamiltonian or Lagrangian can be generalized in both gravitational and electromagnetic cases. The Darwin Lagrangian or Hamiltonian have many applications in both classical and quantum domains (see, e.g., [10]). The extension for two point charges considered up to order c −4 (included) has been given by the Golubenkov-Smorodinsky Lagrangian, which is described in [7] and discussed in [11]. (iii) As our Hamiltonian can describe particles with ultrarelativistic velocities or with zero rest mass, it is well suited for treating gravitational scattering of such objects. These are closely related to gravitational scattering of shock waves. At high energies in quantum scattering processes, the interaction is well approximated by the classical collision at the speed of light, as first shown by the influential work by 't Hooft (see [12] for detailed description and methods). (iv) Because of its simplicity, our Hamiltonian yields a convenient starting point in investigations of gravitational lensing by extended gravitationally interacting objects since it provides a unified description of both gravitating bodies and deflected photons or neutrinos. (v) Our method of deriving the Hamiltonian by using the Routhian and the induced canonical transformation, etc. should be of interest in other branches of theoretical physics.
The Hamiltonian approach to the PM approximation was first undertaken in 1986 [13]. Following the ADM canonical formalism, the gravitational field is described by h T T i j , the transverse-traceless part of 1, 2, 3), and by conjugate momentaπ i j T T . The system of bodies located at x a , a = 1, ...N, with rest masses m a and momenta p a , has the energy and linear momentum densities γ 1 2 T µν n µ n ν = ∑ a g i j p ai p a j + m 2 , g i j is inverse to g i j , n ν is a unit timelike normal to hypersurface x 0 = const, and T µν is the energy-momentum tensor of the matter system. Hereafter, we call its constituents the "particles", but they may well represent neutron stars or black holes. This is substantiated by "general relativity's adherence to the strong equivalence principle": black holes and other bodies obey the same laws of motion as test bodies; see, e.g., [12].
It is convenient to choose four coordinate conditionsπ ii = 0 and g i j = 1 + 1 8 φ 4 δ i j + h T T i j . The standard ADM Hamiltonian (cf. [6]), H = (1/16πG) dS i (g i j,i − g j j,i ), then becomes, using the Gauss theorem, H = −(1/16πG) d 3 x∆φ . The integrand ∆φ = ∂ i ∂ i φ can be expressed in terms of x a , p a , h T T i j andπ i j T T from the constraint equations. By expansions in powers of G and after adopting suitable regularization procedures of integrals involved (see the Appendix in [13]), one can determine the Hamiltonian. The Hamiltonian worked out in [13] includes terms ∼ G 2 .
Here we start from the same Hamiltonian neglecting, however, terms ∼ G 2 . Still, the form of the Hamiltonian is quite complicated [see Eq. (1)]: rather than just quantities associated with particles it involves field variables, non-local TT projections, and integrals. The main purpose of this Letter is to show that due to somewhat magical simplifications the fields h T T i j , assuming they are generated just by particles, can be expressed entirely in terms of particles' variables, and after two canonical transformations, the Hamiltonian can be cast into the closed form (11). It is local in particles' canonical variables, involving no field variables. It is exact up to terms linear in G, and within the same accuracy it yields equations of motion for particles moving with possibly ultrarelativistic speeds, including those with zero rest mass. (1) where m a = m 2 a + p 2 [This is Eq. (18) in [13]. However, although we also put c = 1, we keep explicitly G and do not put 16πG = 1; the "geometric" momenta π i j T T are thus different from the trueπ i j T T = (16πG) −1 π i j T T . Notice that the regularization is also needed in the terms containing h T T i j , π i j T T , and h T T i j (x = x a ).] The equations of motion for particles are standard Hamilton equations. The Hamilton equations for the field arė here the variational derivatives and the TT-projection operator appear. These equations imply the equations for gravitational field in the first PM approximation to be the following wave equations: As is usual in the system of particles and field, the field is source of particles' accelerations and particles are sources of the field. Once O(G 2 ) terms can be neglected, however, this coupling simplifies. Since both the field and the accelerationṡ p a are proportional to G, the changes of the field due to the accelerations of particles are of the order O(G 2 ). Therefore, we can assume the field to be generated by unaccelerated particles; it is thus given only by their instantaneous positions and velocities. If such a field is used in equations of motion of particles, the gravitational interaction becomes an "action at a distance". Given the linearity of the field equations (3), we can write is the contribution to the field at point x generated by the particle moving at x b with velocityẋ b . Because the d'Alembertian operator commutes with the TT-projection operator, the field from each particle involves the solution of the scalar wave equation with point source -the boosted static spherical field ∼ 1/r. Denoting x − x a = n a |x − x a | and cosθ a = n a .ẋ a /|ẋ a |, the solution of (3) can thus be written as The action of δ T T i jkl in equation (4) consists of two steps: first, one has to solve the Poisson equation twice and then evaluate a number of partial derivatives. The first step is feasible due to the form of the boosted spherical potential. It can be shown to satisfy the relation (here v = |ẋ a |) Hence, instead of working out ∆ −2 , we can evaluate 8 + 7v d/dv + v 2 d 2 /dv 2 −1 , i.e., the elliptic partial differential equation of the fourth order can be simplified into an inhomogeneous linear second order ordinary differential equation. It has a unique solution which is regular at v = 0. Even though this solution is quite complicated and contains logarithmic terms such as ln cos θ a , they cancel out when partial derivatives are combined. After somewhat lengthy calculations (the details of which will be given elsewhere), we find the field of a moving source here y = y b ≡ 1 −ẋ 2 b sin 2 θ b . As indicated by the symbol O(m bẋb − p b ), the last expression gets simplified by usinġ x b = p b /m b and we anticipate that later O(m bẋb − p b )G will turn into terms ∼ O(G 2 ).
In order to later suppress field degrees of freedom, we shall turn to the Routh functional (see, e.g., [8] Dropping out the total time derivatives in (9) means a canonical transformation, but the new canonical coordinates will not be denoted by primes. In fact, another change of coordinates has to follow since the Hamiltonian (10) is a function ofẋ a . We define new momenta by putting p ′ ai = p ai − ∂ H/∂ẋ ai , and then eliminateẋ a by introducing new Hamiltonian H ′ (x a , p ′ a ) = H(x a , p a (p ′ a ),ẋ a (p ′ a )) −∑ bẋbi (p ′ a ) ∂ H/∂ẋ bi . Since ∂ H/∂ẋ a ∼ G, the only change in the Hamiltonian which is linear in G comes into the kinetic term ∑ m a from the last change of momenta. This change is exactly cancelled by the sum ∑ bẋbi (p ′ a ) ∂ H/∂ẋ bi in