The Metric Tensor and Flat Spaces - (Differential Arc Length)
Building on the ideas of the last video, this video (GR - 05) looks at various ways of measuring a flat two-dimensional space. In other words, it looks at different Frames of Reference - and their Metric Tensors. Crucial to this, is the idea of ‘differential vectors’ and, in particular, the ‘arc length’ or distance between two infinitesimally close points in a space. Metric Tensors for Cartesian Co-ordinates, ‘Angled’ Co-ordinates and Polar Co-ordinate are then developed and compared. Finally, there is a tentative look at a curved one-dimensional space, before the next video (GR - 06) which will look at curved two dimensional-spaces and how they might be measured (their Frames of Reference).
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: 39:12 The angle ‘alpha’ here should be shown as being between dx1 and dx2 (which is the angle between the axes) and NOT (as shown) between dx1 and dS"
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