Is Spacetime Non-metric?

If one assumes higher dimensions and that dimensional reduction from higher dimensions produces scalar-tensor theory and also that Palatini variation is the correct method of varying scalar-tensor theory then spacetime is nonmetric. Palatini variation of Jordan frame lagrangians gives an equation relating the dilaton to the object of non-metricity and hence the existence of the dilaton implies that the spacetime connection is more general than that given soley by the Christoffel symbol of general relativity. Transferring from Jordan to Einstein frame, which connection, lagrangian, field equations and stress conservation equations occur are discussed: it is found that the Jordan frame has more information, this can be expressed in several ways, the simplest is that the extra information corresponds to the function multiplying the Ricci scalar in the action. The Einstein frame has the advantages that stress conservation implies no currents and that the field equations are easier to work with. This is illustrated by application to Robertson-Walker spacetime.

An Alternative Theory on the Spacetime of Non-inertial Reference Frame

In present paper, we have proposed an alternative theory on the spacetime of non-inertial reference frame (NRF) which bases on the requirement of general completeness (RGC) and the principle of equality of all reference frames (PERF). The RGC is that the physical equations used to describe the dynamics of matter and/or fields should include the descriptions that not only the matter and/or fields are at rest, but also they move relative to this reference frame, and the structure of the spacetime of reference frame has been considered. The PERF is that any reference frame can be used to describe the motion of matter and/or fields. The spacetime of NRF is inhomogeneous and deformed caused by the accelerating motion of the reference frame. The inertial force is the manifestation of deformed spacetime. The Riemann curvature tensor of the spacetime of NRF equals zero, but the Riemann-Christoffel symbol never vanishs no matter what coordinate system is selected in the NRF. The physical equations satisfied the RGC remain covariance under the coordinate transformation between the reference frames. Mach’s principle is incorrect. The problem of spacetime of NRF can be solved without considering gravitation.

A TORSIONAL TOPOLOGICAL INVARIANT

Curvature and torsion are the two tensors characterizing a general Riemannian space–time. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space–time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying space–time is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.

IX.—Harmonic Riemannian Spaces

In this paper we consider a new class of Riemannian spaces which arise in the theory of the solution of the tensor generalisation of Laplace's equation ∇2V = o. To obtain this generalisation Beltrami's second differential parameter is defined in terms of the metricof the associated n-dimensional Riemannian space by the usual formulæwhere denotes the Christoffel symbol . The generalised Laplace's equation is then Δ2V = o. For simplicity the quadratic differential form (1.1) is taken to be positive definite, which involves no essential loss of generality.

Solutions of Laplace's equation in an n-dimensional space of constant curvature

This paper is a sequel to an earlier one containing a tensor formulation and generalisation of well-known solutions of Laplace's equation and of the classical wave-equation. The partial differential equation considered waswhere is the Christoffel symbol of the second kind, and the work was restricted to the case in which the associated line-elementwas that of an n-dimensional flat space. It is shown below that similar solutions exist for any n-dimensional space of constant positive or negative curvature K.

II.—On the Equations of Motion of a Single Particle

§ 1. When solved for the second derivatives, the Lagrangian equations of motion for a system in which there are no extraneous forces have the formor, disregarding the parameter t, being the second Christoffel symbol † of the matrix associated with the kinetic energy. If there are extraneous forces Fk and, denoting t by xn+1, we add to the set of equations, the equations of motion are