We consider $f(R,T)$ modified theories of gravity, where the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar $R$ and of the trace of the stress-energy tensor $T$. We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the stress-energy tensor. Generally, the gravitational field equations depend on the nature of the matter source. The field equations of several particular models, corresponding to some explicit forms of the function $f(R,T)$, are also presented. An important case, which is analyzed in detail, is represented by scalar field models. We write down the action and briefly consider the cosmological implications of the $f(R,{T}^{\ensuremath{\phi}})$ models, where ${T}^{\ensuremath{\phi}}$ is the trace of the stress-energy tensor of a self-interacting scalar field. The equations of motion of the test particles are also obtained from a variational principle. The motion of massive test particles is nongeodesic, and takes place in the presence of an extra-force orthogonal to the four velocity. The Newtonian limit of the equation of motion is further analyzed. Finally, we provide a constraint on the magnitude of the extra acceleration by analyzing the perihelion precession of the planet Mercury in the framework of the present model.

$f(R,T)$ gravity

The Creation of the Universe as a Quantum Phenomenon

We generalize the f(R) type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the matter Lagrangian L
 m
 . We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the energy-momentum tensor. The equations of motion for test particles can also be derived from a variational principle in the particular case in which the Lagrangian density of the matter is an arbitrary function of the energy density of the matter only. Generally, the motion is non-geodesic, and it takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equation of motion is also considered, and a procedure for obtaining the energy-momentum tensor of the matter is presented. The gravitational field equations and the equations of motion for a particular model in which the action of the gravitational field has an exponential dependence on the standard general relativistic Hilbert–Einstein Lagrange density are also derived.

f(R,L m ) gravity

The research on viscous cosmological models is reviewed and carried further Inflationary cosmological models of Bianchi type-I with shear, bulk, and nonlinear viscosity are studied The inflation field energy is represented by a two-components cosmic fluid consisting of a vacuum fluid and a Zel'dovich fluid It is shown that there exist models in which the viscous Zel'dovich fluid removes the initial singularity of the corresponding viscosity free models The mean expansion of a Zel'dovich dominated unvierse is found to be independent of shear viscosity and anisotropy Bulk viscosity and shear viscosity cause exponential decay of anisotropy, while nonlinear viscosity causes power-law decay of anisotropy

Viscous Inflationary Universe Models

A new method of variation of the gravitational Lagrangian is proposed. This method leads in a simple and straightforward way to the canonical description of the gravitational field dynamics in a finite volume V with boundary. No boundary terms are neglected or subtracted ad hoc. Two different forms of gravitational quasi-local energy are derived. Each of them is equal to the field Hamiltonian, corresponding to a specific way of controlling the field boundary data. They play the role of the “internal energy” and the “free energy” respectively. A relation with the boundary formula governing the thermodynamics of black holes is discussed.

A Simple Derivation of Canonical Structure and Quasi-local Hamiltonians in General Relativity

A uniform cosmological model filled with fluid which possesses pressure and second viscosity is developed. The Einstein equations can be integrated exactly. One solution is the steady-state cosmology, but this is unstable. Other solutions start from the steady-state one in the infinite past but expand more and more slowly as viscosity dies out. At any finite proper time in the past the curvature is finite. Some comments on possible origins of this viscosity are given.

Big-Bang Model Without Singularities

Exact plane-wave solutions of the Einstein equations which correspond to monochromatic superpositions of plane waves in the linearized theory are studied. The metric involves solutions of the Mathieu equation, and it is shown that it can be interpreted in terms of harmonics of the linearized solution. (AIP)

Generation of harmonics in gravitational waves

The general topic of geometric unified field theories is discussed in the first section. Some reasons are given for pursuing such theories, and some criticisms are considered. The second section develops the fundamental equations of a purely affine theory which is invariant under projective transformations of the affine connection. This theory is a generalization of that of Schr\"odinger. Possible identifications for the space-time metric are considered in Sec. III. Sections IV and V deal with the limits of pure gravitation and electrodynamics. In the symmetric limit, Einstein's vacuum equations with cosmological term are recovered. The theory also contains a generalized electrodynamic set of equations which is very similar to the Born-Infeld set. In the weak-field approximation, a finite mass must be attributed to the photon. The problem of motion for charges is discussed here, and it is argued that criticisms of unified field theories because of a supposed inability to produce the Lorentz force law are probably not justified. Three more speculative sections deal with possible explanations of nuclear forces, the spin-torsion relation, and particle structure.

George L. Murphy

Papers

Big-Bang Model Without Singularities

Generation of harmonics in gravitational waves

Affine-Projective Field Laws

Longitudinal photons in the fireball