Introduction to Riemannian Geometry - Covariant & Contravariant Vectors

We start here (GR - 03) to think a little about ‘Curvature’. Initially, this means thinking not so much about what it is, but what it is not, and following that with a hint about what it might entail. The discussion then moves on to thinking about different one-dimensional ‘stretched’ measuring systems, and their effect on two types of vectors - named ‘Contravariant’ and ‘Covariant’. Simple one-dimensional conversion equations are derived for transforming measurements made in one Frame of Reference to and from another possibly ‘stretched’ Frame of Reference for those two types of vectors. That result will be developed in later videos when thinking about more than one dimension, and thinking about the different types of components of vectors.

This video is part of a series of videos on General Relativity (GR-01 to GR-20), which has been created to help someone who knows a little bit about “Newtonian Gravity” and “Special Relativity” to appreciate both the need for “General Relativity”, and for the way in which the ‘modelling’ of General Relativity helps to satisfy that need – in the physics sense.

The production of these videos has been very much a ‘one man band’ from start to finish (‘blank paper’ to ‘final videos’), and so there are bound to be a number of errors which have slipped through. It has not been possible, for example, to have them “proof-watched” by a second person. In that sense, I would be glad of any comments for corrections ……. though it may be some time before I get around to making any changes.

By ‘corrections and changes’ I clearly do not mean changes of approach. The approach is fixed – though some mistakes in formulae may have been missed in my reviewing of the final videos, or indeed some ‘approximate explanations’ may have been made which were not given sufficient ‘qualification’. Such changes (in formulae, equations and ‘qualifying statements’) could be made at some later date if they were felt to be necessary.

This video (and channel) is NOT monetised