In this paper, we consider a theory of gravity with a metric-dependent torsion namely the F(R,T) gravity, where R is the curvature scalar and T is the torsion scalar. We study the geometric root of such theory. In particular we give the derivation of the model from the geometrical point of view. Then we present the more general form of F(R,T) gravity with two arbitrary functions and give some of its particular cases. In particular, the usual F(R) and F(T) gravity theories are particular cases of the F(R,T) gravity. In the cosmological context, we find that our new gravitational theory can describe the accelerated expansion of the Universe.

FRW cosmology in F ( R , T ) gravity

We present a new covariant and gauge-invariant perturbation formalism for dealing with spacetimes having spherical symmetry (or some preferred spatial direction) in the background, and apply it to the case of gravitational wave propagation in a Schwarzschild black-hole spacetime. The 1 + 3 covariant approach is extended to a '1 + 1 + 2 covariant sheet' formalism by introducing a radial unit vector in addition to the timelike congruence, and decomposing all covariant quantities with respect to this. The background Schwarzschild solution is discussed and a covariant characterization is given. We give the full first-order system of linearized 1 + 1 + 2 covariant equations, and we show how, by introducing (time and spherical) harmonic functions, these may be reduced to a system of first-order ordinary differential equations and algebraic constraints for the 1 + 1 + 2 variables which may be solved straightforwardly. We show how both odd- and even-parity perturbations may be unified by the discovery of a covariant, frame- and gauge-invariant, transverse-traceless tensor describing gravitational waves, which satisfies a covariant wave equation equivalent to the Regge–Wheeler equation for both even- and odd-parity perturbations. We show how the Zerilli equation may be derived from this tensor, and derive a similar transverse-traceless tensor equation equivalent to this equation. The so-called special quasinormal modes with purely imaginary frequency emerge naturally. The significance of the degrees of freedom in the choice of the two frame vectors is discussed, and we demonstrate that, for a certain frame choice, the underlying dynamics is governed purely by the Regge–Wheeler tensor. The two transverse-traceless Weyl tensors which carry the curvature of gravitational waves are discussed, and we give the closed system of four first-order ordinary differential equations describing their propagation. Finally, we consider the extension of this work to the study of gravitational waves in other astrophysical situations.

Covariant perturbations of Schwarzschild black holes

An example from a perfect fluid FRW space‐time is presented to show that a conformal Killing vector (CKV) need not map fluid flow lines into fluid flow lines. Kinematic properties of the Lie derivative along a CKV of timelike and spacelike unit vectors are derived and applied to the fluid unit four‐velocity vector. Dynamic properties of special conformal Killing vectors (SCKV) in a fluid with anisotropic pressure and vanishing energy flux are obtained using Einstein’s field equations. It is shown that a SCKV maps both fluid flow lines and integral curves of na into themselves, where na is the unit spacelike vector of anisotropy. The relation between the anisotropic pressure components and the energy density is considered. By means of an example from a radiationlike viscous fluid FRW space‐time it is shown that the dynamic results depend crucially on the vanishing of the energy flux vector. The extension of the dynamic results to a fluid with arbitrary stress tensor and zero energy flux vector is examined.

Kinematic and dynamic properties of conformal Killing vectors in anisotropic fluids

Viscous heat‐conducting fluid and anisotropic fluid space‐times admitting a special conformal Killing vector (SCKV) are studied and some general theorems concerning the inheritance of the symmetry associated with the SCKV are proved. In particular, for viscous fluid space‐times it is shown that (i) if the SCKV maps fluid flow lines into fluid flow lines, then all physical components of the energy‐momentum tensor inherit the SCKV symmetry; or (ii) if the Lie derivative along a SCKV of the shear viscosity term ησab is zero then, again, we have symmetry inheritance. All space‐times admitting a SCKV and satisfying the dominant energy condition are found. Apart from the vacuum pp‐wave solutions, which are the only vacuum solutions that can admit a SCKV, the energy‐momentum tensor associated with these space‐times is shown to admit at least one null eigenvector and can represent either a viscous fluid with heat conduction or an anisotropic fluid. No perfect fluid space‐times can admit a SCKV. These SCKV space‐t...

Special conformal Killing vector space‐times and symmetry inheritance

We present a new covariant and gauge-invariant perturbation formalism for dealing with spacetimes having spherical symmetry (or some preferred spatial direction) in the background, and apply it to the case of gravitational wave propagation in a Schwarzschild black hole spacetime. The 1+3 covariant approach is extended to a `1+1+2 covariant sheet' formalism by introducing a radial unit vector in addition to the timelike congruence, and decomposing all covariant quantities with respect to this. The background Schwarzschild solution is discussed and a covariant characterisation is given. We give the full first-order system of linearised 1+1+2 covariant equations, and we show how, by introducing (time and spherical) harmonic functions, these may be reduced to a system of first-order ordinary differential equations and algebraic constraints for the 1+1+2 variables which may be solved straightforwardly. We show how both the odd and even parity perturbations may be unified by the discovery of a covariant, frame- and gauge-invariant, transverse-traceless tensor describing gravitational waves, which satisfies a covariant wave equation equivalent to the Regge-Wheeler equation for both even and odd parity perturbations. We show how the Zerilli equation may be derived from this tensor, and derive a similar transverse traceless tensor equivalent to this equation. The so-called `special' quasinormal modes with purely imaginary frequency emerge naturally. The significance of the degrees of freedom in the choice of the two frame vectors is discussed, and we demonstrate that, for a certain frame choice, the underlying dynamics is governed purely by the Regge-Wheeler tensor. The two transverse-traceless Weyl tensors which carry the curvature of gravitational waves are discussed.

Covariant Perturbations of Schwarzschild Black Holes

Necessary and sufficient conditions are derived for space‐time to admit a spacelike conformal motion with symmetry vector parallel to a unit spacelike vector field na. These conditions are expressed in terms of the shear and expansion of the spacelike congruence generated by na and in terms of the four‐velocity of the observer employed at any given point of the congruence. It is shown that either the expansion or the rotation of this spacelike congruence must vanish if Dna/dp =0, where  p denotes arc length measured along the integral curves of na, and also that there exist no proper spacelike homothetic motions with constant expansion. Propagation equations for the projection tensor and the rotation tensor are derived and it is proved that every isometric spacelike congruence is rigid. Fluid space‐times are studied in detail. A relation is established between spacelike conformal motions and material curves in the fluid: if a fluid space‐time admits a spacelike conformal Killing vector parallel to na and ...

Spacelike conformal Killing vectors and spacelike congruences

The theory of spacelike congruences in general relativity is briefly reviewed and the physical interpretation of the rotation tensorR a b , the expansion E, and the shear tensorS a b , of the curves is discussed. It is proved that if the unit tangent vector to any curve of the congruence is everywhere orthogonal to the 4‐velocity field u a of a self‐gravitating fluid, then observers comoving with the fluid can be employed along a curve of the congruence if and only if the curves are material curves in the fluid. A congruence of vortex lines is studied in detail. Starting from the Ricci identity for u a and using Einstein’s equations, general expressions in terms of the kinematic quantities and fluid variables are derived for R a b , C, and S a b for a vortex congruence. It is found that E and S a b depend explicitly on the gravitational field only through the magnetic part of the Weyl tensor, and R a b only through a term proportional to the total energy flux q a derived from Einstein’s equations. With the aid of Maxwell’sequations, properties of congruences of magnetic field lines, electric field lines, and a certain combination of vortex and magnetic field lines are determined. For a congruence of magnetic field lines in an electrically conducting fluid and assuming the magnetohydrodynamic approximation of vanishing electric field, it is proved that for a comoving observer, R a b =0 if and only if the conduction current in the fluid is orthogonal to the magnetic field. The propagation equations for R a b , E, and S a b along a curve of a spacelike congruence are considered. These equations are developed in full for the special case of a congruence of material curves in a fluid. The divergence, or constraint, equation for the rotation vector is also derived. Where appropriate, corresponding results in Newtonian gravitation theory are given for comparison.

On spacelike congruences in general relativity

The properties of fluid space‐times that admit a Ricci collineation vector (RCV) parallel to the fluid unit four‐velocity vector ua are briefly reviewed. These properties are expressed in terms of the kinematic quantities of the timelike congruence generated by ua. The cubic equation derived by Oliver and Davis [Ann. Inst. Henri Poincare 30, 339 (1979)] for the equation of state p=p(μ) of a perfect fluid space‐time that admits an RCV, which does not degenerate to a Killing vector, is solved for physically realistic fluids. Necessary and sufficient conditions for a fluid space‐time to admit a spacelike RCV parallel to a unit vector na orthogonal to ua are derived in terms of the expansion, shear, and rotation of the spacelike congruence generated by na. Perfect fluid space‐times are studied in detail and analogues of the results for timelike RCVs parallel to ua are obtained. Properties of imperfect fluid space‐times for which the energy flux vector qa vanishes and na is a spacelike eigenvector of the aniso...

Ricci collineation vectors in fluid space‐times

Two relativistic analogs of Ferraro's law of isorotation for a fluid with infinite electrical conductivity are given. In the first case it is assumed that the flow is shear-free withHa=0 and it is shown that £(H) ωa=0. In the second case it is assumed that there exists a space-like Killing vector field parallel toHa, and it is proved thatω,aHa=0.

Analogs in general relativity of Ferraro's law of isorotation

By assuming that Maxwell's electromagnetic field equations are valid in a Riemann-Cartan space-time and by using a set of rules to transform from Riemannian kinematics to Riemann-Cartan kinematics, the kinematic aspects of magnetohydrodynamics in a Riemann-Cartan space-time are examined. If the electric conductivity of the fluid is infinite, then the magnetic field conservation laws still hold, but torsion affects the physical interpretation of the equation for proper charge density. A result, based on the Ricci identity foru a and the first Bianchi identity, and describing differential rotation of a charged fluid in a Riemann space-time, is extended to a Riemann-Cartan space-time. The kinematic role played by torsion in this result is examined.

D. P. Mason

Papers

Spacelike conformal Killing vectors and spacelike congruences

On spacelike congruences in general relativity

Ricci collineation vectors in fluid space‐times

Analogs in general relativity of Ferraro's law of isorotation

Magnetohydrodynamics in a Riemann-Cartan space-time