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September 19

Metric tensor

This is a basic differential geometry question, though the context is that I just watched a video about general relativity. The lecturer describes spacetime as a curved surface. It's locally flat, so each point p has an associated matrix M_p. Then if you move away from p by some small vector ds, you can transform it by M_p, giving M_pds = (dx,dy,dz,dt). The distance then is sqrt(dx²+dy²+dx²-dt²) with the -dt² due to the Lorentzian signature.

Later though he calls M a metric tensor, and metric tensor says the tensor takes two vector inputs and produces a scalar output.

What is the deal here? Since the flatness is only local, to find the distance between arbitrary points p and q, first you have to find a geodesic and then integrate along it, amirite? Where does the tensor come in, that requires two vector inputs instead of just one?

Later he talks about contravariant and covariant vectors, which I think means column vectors and row vectors. I couldn't tell what that was about either, but I'll make another try at it later.

Thanks. 2601:644:8581:75B0:4C10:1C2B:E139:C42B (talk) 01:10, 19 September 2025 (UTC)[reply]

The metric tensor is essentially the inner product or dot product (not sure where those links will wind up). To use it to get the length of a vector, you take the dot product of the vector when itself (and then take the square root). --Trovatore (talk) 01:16, 19 September 2025 (UTC)[reply]

Thanks, that helps. I'll probably have to watch the whole video again, and it's a series, so it could be time consuming. Wow. 2601:644:8581:75B0:4C10:1C2B:E139:C42B (talk) 04:24, 19 September 2025 (UTC)[reply]

You might be talking about index raising and index lowering. I'm afraid the linked article (with the whimsical name "musical isomorphism", with which I was hitherto unfamiliar) is not going to be much use to you at the present time, but you can keep it in mind if you learn a whole bunch more about differential geometry. I (or someone else here) might be able to give you a gentler introduction to it, but I'd have to think about how. --Trovatore (talk) 04:28, 19 September 2025 (UTC)[reply]

Thanks. I have some familiarity with tensors from linear algebra, so maybe I can get through that article. I think the differential geometry used in the video I saw is not too bad, and I'm just confused by a single probably-minor point. Of course later videos in the series will be harder, but GR is looking less scary than it's reputed to be. 2601:644:8581:75B0:4C10:1C2B:E139:C42B (talk) 07:49, 19 September 2025 (UTC)[reply]