The Metric Tensor and Flat Spaces - (Differential Arc Length)
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- Опубликовано: 16 окт 2024
- 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.
39:12 Correction - 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
This video (and channel) is NOT monetised
I'm currently doing some PhD research work in the direction of information geometry (statistical manifolds), and this GR playlist is a heaven-sent! Have tried 3 different differential geometry books, but nothing at all as clear as this. In fact, I don't think I've even come across more basic mathematical concept videos as clear as this: so nicely paced and thoroughly explained, not a single logical step skipped or left to be filled in by the viewer. Absolutely brilliant.
This is a great series, and in a few lectures he has made things clear that have been fuzzy for years. I will note that the angle alpha shown above should be between the lower dotted line and dx^2.
These are excellent videos. I've viewed all the GR series up to this one and I'm enjoying them a lot. You have a special talent to explain this topic very clearly. Thanks so much for all the effort that went into these videos. Greetiings from Argentina.
thank you so much for these videos on GR. the amount of work you have put in to making them clear and engaging must have been phenomenal.
I am in love with his presentation. Probably the best explanation of tensor and related concepts so far.
Loving your presentations; slow and steady!
Did catch a graphical error @ 39:13 - The angle alpha is shown between the vector dS and dX1 when it should be shown as between the dx1 & dx2 vectors.
Hi Eddie, I have been dabbling with general relativity for many years as an interest. I recently stumbled across your videos, and in the space of a few weeks you have clearly explained many concepts which is something no one else on RUclips has managed to do. I also find your voice easy to listen to. So, a big thank you for spending the time to share your knowledge and understanding with us all. Best, Andrew.
Bob - well spotted ..... I never noticed that one! I *may* correct it in the future, although this would mean a new URL because (as far as I know) RUclips don't allow you to overlay videos with added comments or corrections. I may therefore simply leave your comment as the correction. Anyway, thanks again. Best ..... Eddie
Hershey - There is already a “Correction” note for this in the RUclips “Video details”, but I can’t yet get RUclips to display it at the appropriate time in the video
Just what I needed. I Lucked upon your lecture series. Thank you.
Fantastic explanation.
40:01 Since δ = 180 deg -α, as mentioned earlier at 39:30, then since cos(180 deg -α) = cosα, why isn't cosδ = cosα? Why the minus sign?
Beautiful lecture
Binge worthy!!!
MARAVILLOSO.
The metric tensor is defined by a vector base, not an orthonormal base, isn’t it?
3:46
Why do you have row vectors equating column vectors when using a change of basis matrix from contravariant to covariant vectors?
Çok güzel ve detaylı anlatımlar yapmışsınız hocam Teşekkür ederim dersler için. Ama Tensörlerde bir karmaşa hakim. Net çözümleri anlamak zor. Bilhassa indislerle ilgili bir çok soru işareti var. Mesela n indeksinin sabit olması gerekir ama değişiyor. g11 g12 n=1 n=2 m=1 bu hatalı değilmi. İkinci satırda g21 g22 n=1 n=2 m=2 Böyle bir şey olabilirmi. Aslında Metrik Tensörler boş bir uğraştan başka bir şey değiller. Sayısal değerler vererek çözümler yapılsa Einstein'ın Alan Denklemlerini anlamak bir kaç saat sürer. Teşekkürler.
That is at the video time 39:37.
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But, but… the basis vectors of a frame should have modules of unity, or shouldn’t? in the latter case, they are not versors…
11:37
0:22