A friend, who focuses primarily on experimental particle physics, recently asked me an interesting question about gravity. Specifically, he asked how the presence of electromagnetic fields impacts the gravitational field. Applying some modern-physics reasoning, he proposed that electromagnetic fields should exert gravitational influence because photons have momentum that can be viewed as mass in special relativity, and should interact gravitationally. I found this idea interesting, if a bit interpretive, and answered with the precise formulation of the impact of the electromagnetic field on the curvature tensor.I’ve spent time focusing on the best chalkboards and chalks on my tools page, but -until recently- I’ve not spent much time thinking about the last part of the process – erasing. At the suggestion of several colleagues, I played around with several different methods of erasing chalk marks to find which is most effective. The results were decisive. This post outlines the results, and attempts to present a simple test as justification (though my own testing was more extensive).What does type (r, s) mean? I’d like to discuss the notation of the tensor type, commonly denoted $(r, s)$ as it relates to the tensor product. Specifically, the ordering of the vector spaces and dual vector spaces involved in the product. The order matters since tensors are typically categorized by the number of vectors and dual vectors they require as arguments. To avoid ambiguity, for a given tensor $T$, I will denote the number of vector arguments as $n_v$ and the number of dual vector arguments as $n_d$.The introduction to tensor products and tensor algebras is often riddled with rigor, in which a mathematician would delight but a programmer would despair. I find myself in the intersection of these camps and while I appreciate notation, a simpler introduction is possible using functional programming concepts.
Tensors are defined and introduced in two equivalent ways. The first way, called the “expansion coefficient” (or array) style of introducing tensors relies on many indices and iterates over the n dimensions of some array (n-dimensional generalization of a matrix). 