The Stress-Energy-Momentum Tensor – The Continuity Equation
In moving towards a simplified derivation of Einstein’s Field Equations, this video (GR - 18) sets out to explain the ‘Stress-Energy-Momentum Tensor’.
It starts by discussing the idea of ‘Four Vectors’, and then takes what can be called the ‘Energy-Momentum Four Vector’ (a rank one tensor) and manipulates it into a rank two tensor, which we call the ‘Stress-Energy-Momentum Tensor’. The various components of this tensor (16 of them in four-dimensional space-time) are discussed in some detail in order to try and get a ‘feel’ for what they can mean - in terms of ‘flux’, ‘flow’, ‘stress’ and ‘pressure’.
Finally, this video looks at the ‘Continuity Equation’ as applied to the ‘Stress-Energy-Momentum Tensor’, and the sense in which that equation must follow from the conservation of energy and momentum applied to the ‘fluxes’ and ‘flows’ which it describes.
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: 05:18 Around this time there is a 'typo' in one of the diagrams which keeps being repeated for a while - namely the fact that the THIRD part of the scalar product ought to be x3x3 and NOT (as shown) x2x2 "
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