Re:The Geometric Tool That Solved Einstein's Relativity Problem
by senderista on Monday August 12, 2024 @07:24PM
(#41230506)
That abstract approach tends to be how mathematicians view the tensor product (there is also a categorical construction), but I don't find it very helpful for understanding what tensors do, or why they are useful in physics. With the "multilinear map" definition, taking the tensor product T of tensors U and V just means evaluating U and V respectively on the arguments of T and multiplying their outputs. Extend this definition by linearity and you have the tensor product of spaces of tensors.
But these things are vectors, so you could write e.g. v = a⋅x+b⋅y, and then you want e.g. (a⋅x+b⋅y)⊗w = ax⊗w + by⊗w, and so on.
So in some sense, the quotient space construction[1] gives a better "why". It says
* I want to multiply vectors in V and W. So let's just start by writing down that "v times w" is the symbol "v⊗w", and I want to have a vector space, so take the vector space generated by all of these symbols.
* But I also want that (v_1+v_2)⊗w = v_1⊗w + v_2⊗w
* And I also want that v⊗(w_1+w_2) = v⊗w_1 + v⊗w_2
* And I also want that (sv)⊗w = s(v⊗w) = v⊗(sw)
And that's it. However you want to concretely define tensors, they ought to be "a way to multiply vectors that follows those rules". Quotienting is a generic technique to say "start with this object, and add this additional rule while keeping all of the others".
Another way to say this is that the tensor algebra is the "free associative algebra": it's a way to multiply vectors where the only rules you have to reduce expressions are the ones you needed to have.
[0] https://www.youtube.com/live/mqt1f8owKrU?t=500
[1] https://en.wikipedia.org/wiki/Tensor_product#As_a_quotient_s...