A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity by Peter Collier

Author:Peter Collier
Language: eng
Format: mobi, epub, pdf
Tags: Science & Technical
Publisher: Incomprehensible Books
Published: 2013-12-29T06:00:00+00:00

where are the components of in spherical coordinates.

Using the primed/unprimed convention for new/original coordinate systems, we'll call the gradient with components , and the new gradient with components .

(5.3.2) tells us that the definition of a one-form is given by

which we can rewrite for our particular one-form as

As mentioned above, the term represents a transformation matrix, in this case a 3 × 3 transformation matrix because there are three coordinates x, y, z and three α coordinates r, θ, ϕ.

So, we can represent our mysterious transformation matrix as

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