Geodesics & Tangent Vectors - The Equation of a Geodesic
This video (GR - 08) attempts to give some explanation as to what is meant by the term “Geodesic”, and the basic requirements that a path should satisfy in order to be designated a ‘geodesic'.
After a general introduction about those requirements, it discusses what is meant by the ‘Tangent Vector’, and then proceeds to evaluate an expression for that. Having found such an expression, the discussion moves on to applying that to the requirements for a ‘geodesic’ path. This ends up with what we can call “The Equation of a Geodesic” - an important equation for later use in formulating Einstein’s Field Equations (Video GR - 19).
Finally, this video has a short, but important, section on the Transformation of Vectors between different Frames of Reference. This had been looked at for one dimension in Video GR - 03, but is here extended to two (and more) dimensions. It is from this discussion that one can appreciate why the two types of components of vectors are called Contravariant and Covariant.
Once again, in this video (GR - 08) there is a fair amount of repetitious working through the ‘Einstein Summation Convention’ for those coming at it for the first time.
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: 1.01:14 The red independent variables within the brackets of the blue V1-tilde function should obviously read V1, V2 and not V1, V1 as shown"
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