Higher Order Post-Newtonian Dynamics of Compact Binary Systems in Hamiltonian Form
The Hamiltonian formalism developed by Arnowitt, Deser, and Misner (ADM) is used to derive and discuss the higher order post-Newtonian dynamics and motion of compact binary systems in general relativity including proper rotation of the components. Various
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Abstract The Hamiltonian formalism developed by Arnowitt, Deser, and Misner (ADM) is used to derive and discuss the higher order post-Newtonian dynamics and motion of compact binary systems in general relativity including proper rotation of the components. Various explicit analytic Hamiltonians will be presented for the conservative and dissipative dynamics, the latter resulting from gravitational radiation damping. Explicit analytic expressions for the orbital motion will be given.
1 Introduction The convincing explanation of the anomalous perihelion advance of Mercury, the first success of general relativity, has been right a post-Newtonian (PN) achievement. So PN approximation techniques are part of the application of the Einstein theory from the very beginning. Since 1915, the year of the birth of general relativity, a lot of work has been invested into the exploration of the theory in terms of PN approximation schemes. Typical for all these PN schemes is the Newtonian theory as starting point and the expansion of the structure of general relativity, in the weakfield and slow-motion regime, in inverse powers of the speed of light c in the form of terms containing powers of v 2 /c2 and Gm/r c2 , where v and m respectively denote a typical velocity and a typical mass of the bodies in question and r a typical relative distance; G is the Newtonian gravitational constant, e.g. [1–3]. The nPN order of approximation means the nth power (v 2 /c2 )n . For bound systems, the virial terms guarantees v 2 /c2 ∼ Gm/r c2 . Starting with the level where radiation reaction enters for the first time, half-integer n enter. The first radiation-reaction level is of 2.5PN order. If the internal dynamics of a body is decoupled from its external one, the internal dynamics must not have weak gravitational fields to participate in the external weak-field and slow-motion dynamics; so neutron stars and black holes can be objects of PN systems. Famous is the derivation of the 1PN dynamics of n-body G. Schäfer (B) Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany e-mail: [email protected] © Springer International Publishing Switzerland 2015 D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, DOI 10.1007/978-3-319-18335-0_18
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systems with strong self interaction published by Einstein, Infeld, and Hoffmann (EIH) in 1938, [4]. Here the technique of closed surface integrals around the bodies has been applied, themselves located in the weak field regime. Thinking in terms of Gauss’s theorem it is understandable why with the help of Dirac delta functions the same result could have been achieved later on; however, with much less amount of computational work. The interested reader may compare the two different techniques in the recent computations of the 3PN binary dynamics, where Itoh and Futamase made use out of an EIH-type approach [5, 6] whereas Damour et al. [7] and Blanchet et al. [8] applied the Dir
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