The purpose of these lectures was to give a survey of Einstein’s theory of gravitation. Since special astrophysical topics have been covered by other lecturers and the principal mathematical tools of general relativity were also considered separately, I tried to concentrate on the general structure of the theory and the connection between physical and mathematical concepts and ideas; examples and applications have been mentioned briefly only.

Survey of General Relativity Theory

Dixon's approach to describe the dynamics of extended bodies in metric theories of gravity is elaborated. The exact, general relation between the center-of-mass 4-velocity and the 4-momentum is derived. Quasirigid bodies are defined, and their equations of motion are shown to be determinate for a given metric. Multipole approximations are considered, and the physical meaning of quasirigidity is investigated by establishing an approximate connection with continuum mechanics.

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Dynamics of Extended Bodies in General Relativity Center-of-Mass Description and Quasirigidity

An energy condition, in the context of a wide class of spacetime theories (including general relativity), is, crudely speaking, a relation one demands the stress-energy tensor of matter satisfy in order to try to capture the idea that “energy should be positive”. The remarkable fact I will discuss in this paper is that such simple, general, almost trivial seeming propositions have profound and far-reaching import for our understanding of the structure of relativistic spacetimes. It is therefore especially surprising when one also learns that we have no clear understanding of the nature of these conditions, what theoretical status they have with respect to fundamental physics, what epistemic status they may have, when we should and should not expect them to be satisfied, and even in many cases how they and their consequences should be interpreted physically. Or so I shall argue, by a detailed analysis of the technical and conceptual character of all the standard conditions used in physics today, including examination of their consequences and the circumstances in which they are believed to be violated.

A Primer on Energy Conditions

We compute the next-to-next-to-leading order spin–orbit contributions in the total energy flux emitted in gravitational waves by compact binary systems. Such contributions correspond to the post-Newtonian order 3.5PN for maximally spinning compact objects. Continuing our recent work on the next-to-next-to-leading spin–orbit terms at the 3.5PN order in the equations of motion, we obtain the spin–orbit terms in the multipole moments of the compact binary system up to the same order within the multipolar post-Newtonian wave generation formalism. Our calculation of the multipole moments is valid for general orbits and in an arbitrary frame, the moments are then reduced to the center-of-mass frame and the resulting energy flux is specialized to quasi-circular orbits. The test-mass limit of our final result for the flux agrees with the already known Kerr black hole perturbation limit. Furthermore, the various multipole moments of the compact binary reduce in the one-body case to those of a single-boosted Kerr black hole. We briefly discuss the implications of our result for the gravitational wave flux in terms of the binary’s phase evolution, and address its importance for the future detection and parameter estimation of signals in gravitational wave detectors.

Next-to-next-to-leading order spin–orbit effects in the gravitational wave flux and orbital phasing of compact binaries

We investigate the dynamics of spinning binaries of compact objects at the next-to-leading order in the quadratic-in-spin effects, which corresponds to the third post-Newtonian order (3PN). Using a Dixon-type multipolar formalism for spinning point particles endowed with spin-induced quadrupoles and computing iteratively in harmonic coordinates the relevant pieces of the PN metric within the near zone, we derive the post-Newtonian equations of motion as well as the equations of spin precession. We find full equivalence with available results. We then focus on the far-zone field produced by those systems and obtain the previously unknown 3PN spin contributions to the gravitational-wave energy flux by means of the multipolar post-Minkowskian (MPM) wave generation formalism. Our results are presented in the center-of-mass frame for generic orbits, before being further specialized to the case of spin-aligned, circular orbits. We derive the orbital phase of the binary based on the energy balance equation and briefly discuss the relevance of the new terms.

https://iopscience.iop.org/article/10.1088/0264-9381/32/19/195010/pdf

Quadratic-in-spin effects in the orbital dynamics and gravitational-wave energy flux of compact binaries at the 3PN order

There is considerable freedom in the definition of multipole moments of the energy-momentum tensor of an extended body in general relativity. By studying the corresponding Newtonian theory we obtain guidelines which enable us to choose the most suitable definitions in the relativistic theory. In this way we find two sets, the complete moments and the reduced moments, of which the latter are the most natural choice for studying the dynamics of extended bodies. Expressions as explicit integrals are give for both sets, and the multipole equations of motion of the body are given in a form exact to all orders. Proofs of the relativistic results will appear elsewhere.

The definition of multipole moments for extended bodies

A new (3+1)-dimensional decomposition of the Einstein gravitational field equations is obtained for a general spacetime. The metric is taken in the form

$$ds^2 = e^{ - 2u} k_{ab} (dx^a + \xi ^a dt)(dx^b + \xi ^b dt) - c^2 e^{2u} dt^2 $$

and the resulting equations treatkab as the metric in the space-like hypersurfacest=constant. It is shown that this decompostion forms a more convenient starting point for slow motion approximations than does their usual 4-dimensional formulation. This is illustrated by a derivation of the first post-Newtonian approximation to the field equations, the simplicity there resulting fromkab being still flat to this order.

Slow-motion approximations in general relativity I. Decomposition of the field equations

When the field equations of general relativity are expanded in powers of a small parameter, the general covariance of the exact theory implies a corresponding gauge invariance of the equations obtained in the expansion. In a slow motion expansion, the derivation of this gauge transformation is complicated by the fact that the time coordinate is singled out for special treatment. In a previous paper, a new (3 + 1)-dimensional decomposition of the field equations was obtained which is particularly suitable as a starting point for slow motion approximations. The present paper gives a systematic method, again using covariant techniques throughout, for obtaining the corresponding gauge transformations to arbitrarily high accuracy. The calculations are explicitly carried out as far as is required in the 2 1/2-post-Newtonian approximation.

Slow motion approximations in general relativity II. Gauge transformations

A mass skeleton is defined for a static extended body in a gravitational field. It is a scalar-valued distribution on a tangent space, and is equivalent to that part of the reduced multipole moment structure which describes the mass density of the body. An explicit form is given for this distribution in terms of the mass density and the scalar potential of the field. It is deduced that the mass skeleton and the scalar potential are not completely independent. The smoothness of the mass distribution imposes certain weak restrictions on those scalar potentials which are compatible with a given mass skeleton.

W. G. Dixon

Papers

The definition of multipole moments for extended bodies

Slow-motion approximations in general relativity I. Decomposition of the field equations

Slow motion approximations in general relativity II. Gauge transformations

Multipole structure of static continuous matter distributions in general relativity