Variational Principles for Natural Divergence-free Tensors in Metric Field Theories

Let $T^{ab}=T^{ba}=0$ be a system of differential equations for the components of a metric tensor on $R^m$. Suppose that $T^{ab}$ transforms tensorially under the action of the diffeomorphism group on metrics and that the covariant divergence of $T^{ab}$ vanishes. We then prove that $T^{ab}$ is the Euler-Lagrange expression some Lagrangian density provided that $T^{ab}$ is of third order. Our result extends the classical works of Cartan, Weyl, Vermeil, Lovelock, and Takens on identifying field equations for the metric tensor with the symmetries and conservation laws of the Einstein equations

Second-Order, Biconformally Invariant Scalar-Tensor Field Theories in a Four-Dimensional Space

In this paper I shall consider field theories in a space of four-dimensions which have field variables consisting of the components of a metric tensor and scalar field. The field equations of these scalar-tensor field theories will be derivable from a variational principle using a Lagrange scalar density which is a concomitant of the field variables and their derivatives of arbitrary, but finite, order. I shall consider biconformal transformations of the field variables, which are conformal transformations which affect both the metric tensor and scalar field. A necessary and sufficient condition will be developed to determine when the Euler-Lagrange tensor densities are biconformally invariant. This condition will be employed to construct all of the second-order biconformally invariant scalar-tensor field theories in a space of four-dimensions. It turns out that the field equations of these theories can be derived from a linear combination of (at most) two second-order Lagrangians, with the coefficients in that linear combination being real constants.Comment: 24 page

The Multiverse and Cosmic Procreation via Cofinsler Spaces -- or -- Being and Nothingness

In this paper I shall consider a scalar-scalar field theory with scalar field phi on a four-dimensional manifold M, and a Lorentzian Cofinsler function f on T*M. A particularly simple Lagrangian is chosen to govern this theory, and when f is chosen to generate FLRW metrics on M the Lagrangian becomes a function of phi and its first two time derivatives. The associated Hamiltonian is third-order, and admits infinitely many vacuum solutions. These vacuum solutions can be pieced together to generate a multiverse. This is done for those FLRW spaces with k>0. So when time, t, is less than zero we have a universe in which the t=constant spaces are 3-spheres with constant curvature k. As time passes through zero the underlying 4-space splits into an infinity of spaces (branches) with metric tensors that describe piecewise de Sitter spaces until some cutoff time, which will, in general, be different for different branches. After passing through the cutoff time all branches will return to their original 4-space in which the t=constant spaces are of constant curvature k, but will remain separate from all of the other branch universes. The metric tensor for this multiverse is everywhere continuous, but experiences discontinuous derivatives as the universe branches change between different de Sitter spaces. Some questions I address using this formalism are: what is the nature of matter when t<0; what happens to matter as time passes through t=0; and what was the universe doing before the multiple universes came into existence at t=0? The answers to these questions will help to explain the paper's title. I shall also briefly discuss a possible means of quantizing space, how inflation influences the basic cells that constitute space, and how gravitons might act.Comment: 37 pages, no figure