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The Post-Newtonian Approximation

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General Relativity

Part of the book series: Graduate Texts in Physics ((GTP))

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Abstract

This chapter is devoted to the study of the motion of multiple isolated bodies under their mutual gravitational interaction and of the accompanying emission of gravitational radiation, a subject of great astrophysical interest. Only approximation methods, that make use of some small parameters for sufficiently separated bodies are treated. Before starting with this, we address some conceptual issues, in particular the notion of asymptotic flatness. Afterward, we treat in detail only the first post-Newtonian approximation, but an outline of the general strategies of approximation methods is included. As an application of the post-Newtonian approximation, the Einstein–Infeld–Hoffmann equations will be derived. These are, for instance, needed for the study of binary systems of two neutron stars. Binary pulsar data provide the best available tests of GR. For the analysis of the impressive data, an accurate theoretical analysis of the pulsar arrival times is needed. One of the main sections is devoted to this. We present the current results coming from several binary pulsars, in particular those from the first double pulsar that is by now the best laboratory for testing GR or alternative theories of gravity.

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Notes

  1. 1.

    This is a function (like u in Sect. 4.8.1) whose level surfaces are null hypersurfaces opening up toward the future.

  2. 2.

    This term is due to Brillouin and Levi-Civita.

  3. 3.

    The term ∂ 2 ϕ/∂t 2 does not contribute to M because it equals \(-\frac{1}{4}\boldsymbol{\nabla}\cdot(\partial\xi/\partial t)\) (see (6.35)), and hence vanishes upon integration.

  4. 4.

    Note that since the prefactor in (6.80) approaches zero as α⟶0, only an infinitesimal neighborhood of z a contributes to the integral.

  5. 5.

    Let \(\mathcal {L}(q,\dot{q},\lambda)\) be a Lagrangian which depends on a parameter λ. The canonical momenta are \(p=\partial\mathcal{L}/\partial\dot{q}\) and it is assumed that these equations can uniquely be solved for \(\dot{q}=\phi(q,p,\lambda)\) for every λ. Now \(H(p,q,\lambda)=p \phi(q,p,\lambda)-\mathcal{L}(q,\phi(q,p,\lambda),\lambda)\) and hence

    $$\frac{\partial H}{\partial\lambda}=p\frac{\partial\phi}{\partial\lambda}-\frac{\partial \mathcal{L}}{\partial\dot{q}}\frac{\partial\phi}{\partial\lambda}- \frac{\partial\mathcal {L}}{\partial\lambda}=-\frac{\partial\mathcal{L}}{\partial\lambda}. $$
  6. 6.

    In the perturbation terms we can replace p 2 by

    $$\frac{2 m_1m_2}{m_1+m_2} \biggl(E+\frac{G m_1m_2}{r} \biggr). $$
  7. 7.

    The method is similar to the one used in Sect. 6.4.

  8. 8.

    The following readily identifiable nomenclature is used for pulsars: The prefix PSR (abbreviation of “pulsar”) is followed by a four-digit number indicating right ascension (in 1950.0 coordinates). After that a sign and two digits indicate degrees of declination. If the pulsar is a member of a binary system, this is sometimes indicated by the letter B after the prefix PSR.

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  1. Mathematisch-Naturwiss. Fakultät, Institut für Theoretische Physik, Universität Zürich, Zürich, Switzerland

    Norbert Straumann

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Straumann, N. (2013). The Post-Newtonian Approximation. In: General Relativity. Graduate Texts in Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5410-2_6

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