Today I questioned the claim that "SR is way easier and more beautiful to understand with tensors". I think that knowing about vector spaces, their duals and bilinear forms (i.e. bilinear functions) is enough to formulate the theory and I was genuinely curious to know whether some parts of the theory can really be formulated more conveniently using tensors. The question was immediately closed.
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$\begingroup$ What is the difference between knowing about tensor products and knowing about vector spaces, their duals, and multilinear forms? $\endgroup$– WillOJun 27, 2022 at 2:59
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$\begingroup$ @WillO Well, you need to know about vector spaces in order to define tensor products, so I would consider tensor products an advanced subject. As far as multilinear forms are concerned, I think I have seen other ways to prove the existence of tensor products (yes, I like to think of tensor products as being defined up to an isomorphism), so I don't think that we need multilinear forms to discuss tensor products. $\endgroup$– FilippoJun 27, 2022 at 6:25
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$\begingroup$ I'd have said that a tensor product is, by definition, the universal recipient of a multilinear form (and hence, yes, defined only up to isomorphism). But I suppose there are other ways to think about it. $\endgroup$– WillOJun 27, 2022 at 6:52
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$\begingroup$ @WillO Yes, you are totally right, I should have thought about that. Of course we need to know about both vector spaces and multilinear forms in order to define tensor products. But I'd say that this proves my point that tensor products are an advanced subject :D However, I also realized that if we consider how physicists use the term "tensor", then SR is indeed all about tensors, even we do not invoke tensor products. Do you agree? $\endgroup$– FilippoJun 27, 2022 at 7:08
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$\begingroup$ I'm afraid I don't know enough about how physicists use the term "tensor" to have an opinion on this! $\endgroup$– WillOJun 27, 2022 at 7:11
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$\begingroup$ @WillO I think you are just humble and actually know more than I do XD Anyways, my opinion is supported by this article. When I studied physics, I thought that understanding tensor products would allow me to understand tensors (and hence theoretical physics in general, as the term appears everywhere). But it didn't. I think you rather need to understand the basics of differential geometry and exterior algebras! In view of how physicists use the term "tensors" I even believe that it only makes sense to talk about tensors given a manifold. Do you have an opinion on this? $\endgroup$– FilippoJun 27, 2022 at 12:40
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While I thought the close reason as opinion-based was reasonably obvious, I've now left a comment making it explicit:
Whether one formulation of a theory is "easier" or "more beautiful" than another is a matter of subjective opinion, not an objective question about physics.
answered Jun 16, 2022 at 18:55
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$\begingroup$ Thank you for the feedback, I see that my question was badly formulated. I was actually wondering if there are any genuine applications of tensor products to SR. Maybe I'll edit the question. $\endgroup$– FilippoJun 16, 2022 at 19:43
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$\begingroup$ I guess that marks a difference between physics and math, where beauty is intrinsic to a result's value. $\endgroup$ Jul 12, 2022 at 7:15