[This paper is part of the Focused Collection on Upper Division Physics Courses.] Partial derivatives are used in a variety of different ways within physics. Thermodynamics, in particular, uses partial derivatives in ways that students often find especially confusing. We are at the beginning of a study of the teaching...
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...
Several studies in recent years have demonstrated that upper-division students struggle with the
mathematics of thermodynamics. This paper presents a task analysis based on several expert attempts
to solve a challenging mathematics problem in thermodynamics. The purpose of this paper is twofold.
First, we highlight the importance of cognitive task...
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 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.
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...
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)...
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 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 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...