Nonlinear Addition of Four-Vectors in Relativity: Exploring the Inconsistencies

[Rasalhague]

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?

 

[PAllen]

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?

So far as I know, vectors in GR behave as expected: the sum and difference of vectors are vectors. However, 4-velocity (equiv. derivative relative to proper time) is a unit vector. The sum of two unit vectors *is* a vector, but it is obviously not a unit vector and is not a velocity. To 'add' velocities in a general way, you have velocity vector for some particle, you have another velocity vector represening a velocity relative to that particle. You apply the Lorentz transform from the particle velocity back to your chosen reference frame *on* the relative velocity vector, to get it in your reference frame. That is, 'velocity addition', in the sense of compounding velocity vectors is a sequence of coordinate transformations rather than a vector addition. A transformed vector is a vector; a sum of vectors is a vector; but you have to know which has meaning for your problem.

 

[Dale]

You actually hit on the key concept already, the tangent space. The tangent space is a vector space, so elements of the tangent space are in fact vectors fully equipped with the requisite vector addition and scalar multiplication operations.

Now, velocity 4-vectors are not just tangent vectors, but unit tangent vectors. The set of all velocity 4-vectors therefore does not form a vector space, any more than the set of all unit 3-vectors would form a vector space. The velocity 4-vectors are simply a subset of the space of all tangent 4-vectors, which does form a vector space, just like the set of all unit 3-vectors is simply a subset of R3 which is a vector space.

Edit: PAllen was faster!

 

[bcrowell]

I'm the one who wrote the quoted material in the OP. I don't have any objection to what PAllen and DaleSpam said, but I would put a somewhat different slant on it. When we say that something in physics "adds," we mean not just that you *can* add it, but also that the sum has some physical interpretation. From Newtonian mechanics, we expect that adding velocity vectors correctly represents A's motion relative to B in terms of A's motion relative to some third object C, and C's motion relative to B. That doesn't hold for velocity four-vectors.

 

[Rasalhague]

Thanks, all.