Now we want to exhibit a transformation of the particles dynamical variables - or contact transformation, as it is called in the jargon - which transforms the 3PN harmonic-coordinates Lagrangian (174) into a new Lagrangian, valid in some ADM or ADM-like coordinate system, and such that the associated Hamiltonian coincides with the 3PN Hamiltonian that has been obtained by Damour, Jaranowski, and Schäfer [95]. In ADM coordinates the Lagrangian will be “ordinary”, depending only on the positions and velocities of the bodies. Let this contact transformation be and , where and denote the trajectories in ADM and harmonic coordinates, respectively. For this transformation to be able to remove all the accelerations in the initial Lagrangian up to the 3PN order, we determine [103] it to be necessarily of the form

(and idem ), where is a freely adjustable function of the positions and velocities, made of 2PN and 3PN terms, and where represents a special correction term, that is purely of order 3PN. The point is that once the function is specified there is a unique determination of the correction term for the contact transformation to work (see Ref. [103] for the details). Thus, the freedom we have is entirely coded into the function , and the work then consists in showing that there exists a unique choice of for which our Lagrangian is physically equivalent, via the contact transformation (175), to the ADM Hamiltonian of Ref. [95]. An interesting point is that not only the transformation must remove all the accelerations in , but it should also cancel out all the logarithms and , because there are no logarithms in ADM coordinates. The result we find, which can be checked to be in full agreement with the expression of the gauge vector in Equation (169), is that involves the logarithmic terms together with many other non-logarithmic terms (indicated by dots) that are entirely specified by the isometry of the harmonic and ADM descriptions of the motion. For this particular choice of the ADM Lagrangian reads Inserting into this equation all our explicit expressions we find The notation is the same as in Equation (174), except that we use upper-case letters to denote the ADM-coordinates positions and velocities; thus, for instance and . The Hamiltonian is simply deduced from the latter Lagrangian by applying the usual Legendre transformation. Posing and , we get [139, 140, 141, 95, 103]http://www.livingreviews.org/lrr-2006-4 |
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