Modified Gravity and Cosmology

Extended Theories of Gravity

A method is described for constructing, from any source‐free solution of Einstein's equations which possesses a Killing vector, a one‐parameter family of new solutions. The group properties of this transformation are discussed. A new formalism is given for treating space‐times having a Killing vector.

A Method for Generating Solutions of Einstein's Equations

The authors analyse the behaviour of homogeneous and isotropic solutions to a gravity theory that arises from the variation of an arbitrary analytic function of the space-time scalar curvature. Such a theory generalises Einstein's general relativity wherein this function is linear in the curvature. They prove conditions for the existence and stability of the general relativistic de Sitter and Friedmann solutions within the general theory, prove necessary and sufficient conditions for the existence of cosmological singularities and particle horizons and analyse the asymptotic behaviour of ever-expanding Universe models. The conditions under which Minkowski space-time and Schwarzschild space-time are stable is investigated and their instability, together with the pathological behaviour of certain cosmological models, traced back to the non-minimality of the stationary action giving rise to the field equations. The significance of these results for quantum theories of gravity and the 'inflationary' model of the early Universe is discussed.

https://iopscience.iop.org/article/10.1088/0305-4470/16/12/022/pdf

The stability of general relativistic cosmological theory

Physical acceptability of isolated, static, spherically symmetric, perfect fluid solutions of Einstein's equations

It is known that the equations for particles of spin ⩾ 3/2 become inconsistent when electromagnetic interactions are introduced by the usual rules. An analogous problem is that of the effect of the introduction of gravitational fields on the consistency of the equations. The « natural generalization » of the usual field equations for particles of spin 3/2 is here considered in detail, and it is shown that the equations are compatible if and only if the Riemann space in which the spinor field is contemplated is an Einstein space. This result is discussed.

https://link.springer.com/content/pdf/10.1007/BF02859608.pdf

On the compatibility of relativistic wave equations for particles of higher spin in the presence of a gravitational field

Field equations which arise from (second order) non-linear Lagrangians are generally of the fourth differential order. If the Lagrangian is scale-invariant the equations admit arbitrary Einstein spaces as solutions, but they will also possess solutions other than these. Excepting the case of conform-invariant Lagrangians, no such additional solutions appear to be known. This paper therefore deals with what may in aV4 be regarded as the simplest (scale-invariant) Lagrangian,viz. the square of the Gaussian curvatureR. The resulting equations for aspherically symmetric static field whose Gaussian curvature does not vanish everywhere are reduced to a single ordinary non-linear second order equation of deceptively simple appearance. The first few terms of a solution of this equation in the form of a sequence of ascending polynomials are obtained explicitly, two constants of integrations being involved in it. An examination of the corresponding solutions of the original equations shows that these cannot represent spaces asymptotically flat at infinity, a result which is then strengthened by a more direct method. Finally the condition of spherical symmetry is relaxed, and it is shown quite generally that there exist no static, sufficiently often differentiable solutions of the field equations generated byR2 which are asymptotically flat but whose Ganssian curvature does not vanish everywhere.

On the gravitational field equations arising from the square of the gaussian curvature

On a previous occasion it was shown that the « natural generalization » to a Riemann spaceV
4 of a certain set of flat-space free-field equations for particles of spinS=3/2 is internally consistent if and only if theV
4 is an Einstein space. It is now shown that, this case apart, all equations for particles of spinS⩾3/2 which may be said to conform to a « strong principle of equivalence » are compatible if and only if theV
4 is of constant Riemannian curvature. The corresponding second-order wave equations in such a space are written down. Certain modified first-order equations for the caseS=2 which involve the curvature tensor explicitly are shown to be consistent in an Einstein space.

H. A. Buchdahl

Papers

On the compatibility of relativistic wave equations for particles of higher spin in the presence of a gravitational field

On the gravitational field equations arising from the square of the gaussian curvature

On the compatibility of relativistic wave equations in riemann spaces

RECIPROCAL STATIC SOLUTIONS OF THE EQUATIONS Gμv = 0

The hamiltonian derivatives of a class of fundamental invariants