We provide several examples of the use of geometric reasoning with three-dimensional spacetime diagrams, rather than algebraic manipulations using three-dimensional Lorentz transformations, to analyze problems in special relativity.
The process of complexification is used to classify Lie algebras and identify their Cartan subalgebras. However, this method does not distinguish between real forms of a complex Lie algebra, which can differ in signature. In this paper, we show how Cartan decompositions of a complexified Lie algebra can be combined...
We investigate a number of simple toy models to explore interesting relationships between dynamics and typicality. We start with an infinite model that has been proposed as an illustration of how nonergodic dynamics can produce interesting features that are suggestive for cosmological applications. We consider various attempts to define the...
The degenerate nature of the metric on null hypersurfaces makes it difficult to define a covariant derivative on null submanifolds. Recent approaches using decomposition to define a covariant derivative on null hypersurfaces are investigated, with examples demonstrating the limitations of the methods. Motivated by Geroch’s work on asymptotically flat spacetimes,...
A definition is suggested for affine symmetry tensors, which generalize the notion of affine vectors in the same way that (conformal) Killing tensors generalize (conformal) Killing vectors. An identity for these tensors is proven, which gives the second derivative of the tensor in terms of the curvature tensor, generalizing a...
We discuss Einstein’s field equations in the presence of signature change using variational methods, obtaining a generalization of the Lanczos equation relating the distributional term in the stress tensor to the discontinuity of the extrinsic curvature. In particular, there is no distributional term in the stress tensor, and hence no...
We examine some of the subtleties inherent in formulating a theory of spinors on a manifold with a smooth degenerate metric. We concentrate on the case where the metric is singular on a hypersurface that partitions the manifold into Lorentzian and Euclidean domains. We introduce the notion of a complex...
We derive conditions for rotating particle detectors to respond in a variety of bounded spacetimes and
compare the results with the folklore that particle detectors do not respond in the vacuum state appropriate to
their motion. Applications involving possible violations of the second law of thermodynamics are briefly
addressed.
We contrast the two approaches to ‘‘classical’’ signature change used by Hayward with the one used by us (Hellaby and Dray). There is (as yet) no rigorous derivation of appropriate distributional field equations. Hayward’s distributional approach is based on a postulated modified form of the field equations. We make an...
A parametric manifold can be viewed as the manifold of orbits of a (regular) foliation of a manifold by means of a family of curves. If the foliation is hypersurface orthogonal, the parametric manifold is equivalent to the one‐parameter family of hypersurfaces orthogonal to the curves, each of which inherits...
A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) one‐form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, all based on a nonstandard action...
The divergence theorem in its usual form applies only to suitably smooth vector fields. For vector fields which are merely piecewise smooth, as is natural at a boundary between regions with different physical properties, one must patch together the divergence theorem applied separately in each region. We give an elegant...
The divergence theorem as usually stated cannot be applied across a change of signature unless it is reexpressed to allow for a finite source term on the signature change surface. Consequently all conservation laws must also be ‘‘modified,’’ and therefore insistence on conservation of matter across such a surface cannot...
We consider the (massless) scalar field on a two-dimensional manifold with metric that changes signature from Lorentzian to Euclidean. Requiring a conserved momentum in the spatially homogeneous case leads to a particular choice of propagation rule. The resulting mix of positive and negative frequencies depends only on the total (conformal)...
In four dimensions, two metrics that are conformally related define the same Hodge dual operator on the space of two‐forms. The converse, namely, that two metrics that have the same Hodge dual are conformally related, is established. This is true for metrics of arbitrary (nondegenerate) signature. For Euclidean signature a...
Anderson and DeWitt considered the quantization of a massless scalar field in a spacetime whose spacelike hypersurfaces change topology and concluded that the topology change gives rise to infinite particle and energy production. We show here that their calculations are insufficient and that their propagation rule is unphysical. However, our...
A unified, self‐contained treatment of Wigner D functions, spin‐weighted spherical harmonics, and monopole harmonics is given, both in coordinate‐free language and for a particular choice of coordinates.
We compare two independent generalizations of the usual spherical harmonics, namely monopole harmonics and spin‐weighted spherical harmonics, and make precise the sense in which they can be considered to be the same. By analogy with the spin‐gauge language, raising and lowering operators for the monopole index of the monopole harmonics...