The Metric Tensor and Curved Spaces - (Transformations from one Frame to Another)
After a short introduction looking back at the last video (GR - 05) and the measurement of a flat two-dimensional space with three kinds of Metric Tensor (‘Cartesian’, ‘Angled’ and ‘Polar’), this video (GR - 06) investigates a two dimensional CURVED space - namely that of the surface of a sphere, and discusses a suitable Metric Tensor for it.
There then follows a brief discussion on Transformations from one Frame of Reference to another, and whether in some cases such transformations are even possible. This leads to an attempt to provide a fairly simple ‘explanation’ as to what the Metric Tensor is all about - if you like, a shorthand way of ‘conceptualising’ it. This shorthand is then applied to the various Metric Tensors previously discussed to try to appreciate its usefulness.
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.
REPORTED CORRECTION
"Correction: 36:22 The box next to the diagram says that theta is x1-tilde and phi is x2-tilde, and so the bottom right hand corner of this Metric Tensor ought to have x1-tilde and not x2-tilde (it should be r-squared-sine-squared-x1-tilde)"
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