A Combined Entropy/Phase-Field Approach to Gravity

Terms related to gradients of scalar fields are introduced as scalar products into the formulation of entropy. A Lagrange density is then formulated by adding constraints based on known conservation laws. Applying the Lagrange formalism to the resulting Lagrange density leads to the Poisson equation of gravitation and also includes terms being related to curvature of space. The formalism further leads to terms possibly explaining nonlinear extensions known from modified Newtonian dynamics approaches. The article concludes with a short discussion of the presented methodology and provides an outlook on other phenomena, which might be tackled using this new approach.

A CANONICAL APPROACH TO THE EINSTEIN–HILBERT ACTION IN TWO SPACETIME DIMENSIONS

The canonical structure of the Einstein–Hilbert Lagrange density [Formula: see text] is examined in two spacetime dimensions, using the metric density [Formula: see text] and symmetric affine connection [Formula: see text] as dynamical variables. The Hamiltonian reduces to a linear combination of three first-class constraints with a local SO (2, 1) algebra. The first-class constraints are used to find a generator of gauge transformations that has a closed off-shell algebra and which leaves the Lagrangian and det (hμν) invariant. These transformations are distinct from diffeomorphism invariance, and are gauge transformations characterized by a symmetric matrix ζμν.

Radiative corrections in a nonlocal field theory

We extend the notion of a scalar field [Formula: see text] to that of a field Φ(Xμ(σ)) so that the space-time point Xμ(σ) depends on a parameter [Formula: see text]. A straightforward generalization of the [Formula: see text] interaction is considered (viz: a [Formula: see text] theory). Radiative corrections in both cases can be evaluated using a technique involving quantum mechanical path integrals. For the [Formula: see text] model, this involves the classical Lagrange density [Formula: see text] (viz: that of a particle in the "proper time gauge") while for the [Formula: see text] model the Lagrange density [Formula: see text] (viz: that of a string in the "conformai gauge") must be considered. The two-point function is examined in both cases.

The variational derivative of degenerate Lagrange densities

The variational derivative of Lagrange densities which are functions of the metric tensor and its first and second derivatives is considered. This tensor is generally of fourth order in the derivatives of the metric tensor. If derivatives of any order are not present in the variational derivative then the Lagrange density is said to be degenerate in these derivatives. An explicit expression for the variational derivative of Lagrange densities which are degenerate in third and/or fourth derivatives is displayed in tensor form.