Definition
The Lanczos tensor can be defined in a few different ways. The most common modern definition is through the Weyl–Lanczos equations, which demonstrate the generation of the Weyl tensor from the Lanczos tensor. These equations, presented below, were given by Takeno in 1964. The way that Lanczos introduced the tensor originally was as a Lagrange multiplier on constraint terms studied in the variational approach to general relativity. Under any definition, the Lanczos tensor exhibits the following symmetries:
The Lanczos tensor always exists in four dimensions but does not generalize to higher dimensions. This highlights the specialness of four dimensions. Note further that the full Riemann tensor cannot in general be derived from derivatives of the Lanczos potential alone. The Einstein field equations must provide the Ricci tensor to complete the components of the Ricci decomposition.
Read more about this topic: Lanczos Tensor
Famous quotes containing the word definition:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)