Modified gravity theories have received increased attention lately due to combined motivation coming from high-energy physics, cosmology, and astrophysics. Among numerous alternatives to Einstein's theory of gravity, theories that include higher-order curvature invariants, and specifically the particular class of $f(R)$ theories, have a long history. In the last five years there has been a new stimulus for their study, leading to a number of interesting results. Here $f(R)$ theories of gravity are reviewed in an attempt to comprehensively present their most important aspects and cover the largest possible portion of the relevant literature. All known formalisms are presented---metric, Palatini, and metric affine---and the following topics are discussed: motivation; actions, field equations, and theoretical aspects; equivalence with other theories; cosmological aspects and constraints; viability criteria; and astrophysical applications.

f ( R ) theories of gravity

Extended Theories of Gravity

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the postNewtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many nonlinear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail eects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary’s orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.

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Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries.

When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely. In the present paper we study the solutions of the gravitational field equations which describe this process. In I, general and qualitative arguments are given on the behavior of the metrical tensor as the contraction progresses: the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened, and can escape over a progressively narrower range of angles. In II, an analytic solution of the field equations confirming these general arguments is obtained for the case that the pressure within the star can be neglected. The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day; an external observer sees the star asymptotically shrinking to its gravitational radius.

On Continued Gravitational Contraction

Pulsars are presently believed to be rotating neutron stars with frozen-in magnetic fields. Because of the high density of neutron stars, general relativistic effects are important since they effect both the structure and stability of such stars. Besides this, the magnetic field outside the star is also affected. Instead of falling of asr
(2+l) as in flat space, it is shown that each magnetic multipole varies as a hypergeometric function of radius. A closed form of these hypergeometric functions is given in terms of Legendre functions of the second kind. If the mass of a neutron star exceeds about 2.4m
⊙, the star becomes unstable and coliapses. For a quasistatically collapsing body, it is shown that the magnetic field seen by a distant observer vanishes as the radius approaches the gravitational radius.

Gravitational collapse of magnetic neutron stars

A generally covariant set of electromagnetic field equations and associated constitutive relations is developed to deal with electromagnetic radiation in arbitrarily moving media. The equations are sufficiently general to include dispersive as well as nonisotropic media. Several special cases are investigated to illustrate the method and to demonstrate the consistency of the formulation.

Electromagnetic Radiation in Accelerated Systems

By means of a formal solution to the Einstein gravitational field equations a slow motion expansion in inverse powers of the speed of light is developed for the metric tensor. The formal solution, which satisfies the deDonder coordinate conditions and the Trautman outgoing radiation condition, is in the form of an integral equation which is solved iteratively. A stress-energy tensor appropriate to a perfect fluid is assumed and all orders of the metric needed to obtain the equations of motion and conserved quantities to the 21/2post-Newtonian approximation are found. The results are compared to those obtained in another gauge by S. Chandrasekhar. In addition, the relation of the fast motion approximation to the slow motion approximation is examined.

Equations of hydrodynamics in general relativity in the slow motion approximation

Approximate equations of motion for multiparticle systems in an expanding Einstein-deSitter universe are derived from the Einstein-Maxwell field equations using the Einstein-Infeld-Hoffmann surface integral method. At the Newtonian level of approximation one finds that, in comoving coordinates, both the Newtonian gravitational and Coulomb interactions in these equations are multiplied by the inverse third power of the scale factor $R(t)$ appearing in the Einstein-deSitter field and they acquire a cosmic "drag" term. Nevertheless, both the period and luminosity size of bound two-body systems whose period is small compared to the Hubble time are found to be independent of $t$.

Multiparticle dynamics in an expanding universe.

We study the divergent integrals that occur in all post-Newtonian (PN) treatments of radiation reaction in slow-motion, gravitationally bound systems in general relativity. The PN methods implicitly assume that the near-zone metric has a valid asymptotic expansion in powers of the small velocity parameter $\ensuremath{\epsilon}$. We first show explicitly (for the gauge to be used here) that a PN-approximation method leads to a divergent integral at 4-PN order. This divergence arises from the second iteration. Matching arguments are then used to calculate a near-zone term of $O(\mathrm{ln}\ensuremath{\epsilon})$ larger than 4-PN order. On the basis of this calculation and several previous model problems, we argue that the PN divergences signify the breakdown of the PN power-series assumptions, rather than a breakdown of the near and wave zones. Our results suggest that the PN calculations in fact give correct answers at least up to the orders at which divergences appear. The nonanalytic term of $O(\mathrm{ln}\ensuremath{\epsilon})$ beyond 4-PN order arises in the near zone via matching to the wave-zone expansion when we include terms of $O({\ensuremath{\epsilon}}^{3})$ beyond linearized order. We also solve the wave-zone equations at $O({\ensuremath{\epsilon}}^{6})$ beyond linearized order and analyze the inner expansion of the solutions. Matching gives rise to a nonanalytic term in the wave zone at $O({\ensuremath{\epsilon}}^{11}\mathrm{ln}\ensuremath{\epsilon})$, i.e., at $O({\ensuremath{\epsilon}}^{6}\mathrm{ln}\ensuremath{\epsilon})$ beyond linearized order. A straining technique is used in the wave-zone expansion to give a sufficiently accurate approximation to the null surfaces near past and future null infinity. The lowest-order strained solution at first appears to contribute a large, anomalous, time-odd piece to the reaction potential. However, after analyzing the contribution of higher-order wave-zone terms, we obtain agreement with the Burke reaction potential. Our results thus strongly support the usual quadrupole formula.

James L. Anderson

Papers

Gravitational collapse of magnetic neutron stars

Electromagnetic Radiation in Accelerated Systems

Equations of hydrodynamics in general relativity in the slow motion approximation

Multiparticle dynamics in an expanding universe.

Divergent integrals of post-Newtonian gravity: Nonanalytic terms in the near-zone expansion of a gravitationally radiating system found by matching