Discontinuous Metrics Research Articles

On the local and global properties of geodesics in pseudo-Riemannian metrics

The paper is a study of geodesics in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at such points that leads to a curious phenomenon: geodesics cannot pass through such a point in arbitrary tangential directions, but only in certain directions said to be admissible (the number of admissible directions is generically 1 or 3). Secondly, we study the global properties of geodesics in pseudo-Riemannian metrics possessing differentiable groups of symmetries. At the end of the paper, two special types of discontinuous metrics are considered.

The semi-classical ergodic Theorem for discontinuous metrics

The semi-classical ergodic Theorem for discontinuous metrics

Generalized Regularly Discontinuous Solutions of the Einstein Equations

The physical consistency of the match of piecewise-$C^0$ metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value differential geometry framework on a hypersurface is introduced, and corresponding compatibility conditions are deduced. Examples of generalized boundary layers, gravitational shock waves and thin shells are studied.

On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics

We provide a qualitative description of curves minimizing theenergy functional on a Riemannian manifold whose metricis discontinuous along a hypersurface $\Sigma$.Such a study is motivated by the variational description ofrefraction phenomena.

Area Regge calculus and discontinuous metrics

Taking the triangle areas as independent variables in the theory of Regge calculus can lead to ambiguities in the edge lengths, which can be interpreted as discontinuities in the metric. We construct solutions to area Regge calculus using a triangulated lattice and find that on a spacelike or timelike hypersurface no such discontinuity can arise. On a null hypersurface however, we can have such a situation and the resulting metric can be interpreted as a so-called refractive wave.

Regge calculus from discontinuous metrics

Regge calculus is considered as a particular case of the more general system where the linklengths of any two neighbouring 4-tetrahedra do not necessarily coincide on their common face. This system is treated as that one described by metric discontinuous on the faces. In the superspace of all discontinuous metrics the Regge calculus metrics form some hypersurface defined by continuity conditions. Quantum theory of the discontinuous metric system is assumed to be fixed somehow in the form of quantum measure on (the space of functionals on) the superspace. The problem of reducing this measure to the Regge hypersurface is addressed. The quantum Regge calculus measure is defined from a discontinuous metric measure by inserting the δ-function-like phase factor. The requirement that continuity conditions be imposed in a “face-independent” way fixes this factor uniquely. The term “face-independent” means that this factor depends only on the (hyper)plane spanned by the face, not on it's form and size. This requirement seems to be natural from the viewpoint of existence of the well-defined continuum limit maximally free of lattice artefacts.

The methods of gluing manifolds in general relativity

Some areas of modern theoretical physics such as modern cosmology contain different manifolds which must be glued together along a common boundary. These boundaries can be spacelike, timelike, or lightlike hypersurfaces. This paper shows how this gluing for different hypersurfaces is possible. Two different approaches are considered and the extent to which these approaches are equivalent are discussed. In particular, we will construct a distributional approach for dynamics of lightlike hypersurfaces in general relativity. Since Einstein’s equations are nonlinear PDEs, for discontinuous metrics such as signature changing metrics, product of distributions are unavoidable. To glue two different manifolds which admit signature change, we consider this problem in the context of Colombeau’s new theory of generalized functions. Some examples are given for clarification.

Einstein equations and the joining of discontinuous metrics

We consider the problem of joining of metrics when these are not continuous across the joining (hyper-) surface. We confine ourselves to static, spherically symmetric metrics which join without requiring gradients of a δ-function in the energy-momentum tensor. It is found that a surface tension is always associated in cases where the metrics are discontinuous. In some cases, the joined metrics satisfy Einstein equations (in the sense of distributions) while, in others, the surface tension associated with the limiting discontinuous metric depends on the interpolating functions used to produce it suggesting sensitivity to short distance effects beyond general relativity.

Riemannian Manifolds with Discontinuous Metrics and the Dirichlet Integral

Consider a relatively compact region Ω of a Riemann surface R. The term Dirichlet integral over Ω, DΩ(·), is used for the variation whose Euler-Lagrange equation is Δu = 0 on Ω and the term energy integral over Ω, , is used for the variation with Euler-Lagrange equation