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Rasalhague

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When we hear something referred to as a “vector,” we usually take this is a statement that it not only transforms as a vector, but also that it adds as a vector. But we have already seen in section 2.3.1 on page 56 that even collinear velocities in relativity do not add linearly; therefore they clearly cannot add linearly when dressed in the clothing of four-vectors. We've also seen in section 2.5.3 that the combination of non-collinear boosts is noncommutative, and is generally equivalent to a boost plus a spatial rotation; this is also not consistent with linear addition of four vectors. At the risk of beating a dead horse, a four-velocity's squared magnitude is always 1, and this is not consistent with being able to add four-velocity vectors.

http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html

Since the "tensors" of relativity are defined with respect to the tangent spaces of a pseudo-Riemannian manifold, which include velocity vectors (i.e. timelike tangent vectors), this might be taken to suggest (carpet-from-under-feet-ingly) that none of the objects called tensors in relativity are, strictly speaking, tensors. Other sources seem pretty confident that they are though... Perhaps the answer is that addition is defined for tangent vectors, in terms of their role as directional derivative operators, but--when applied to a pair of (similarly oriented) timelike tangent vectors--just doesn't happen to represent the composition of those velocities. Is that the idea?

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