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

• Rasalhague
In summary, the vector addition in relativity is not the same as the addition you are familiar with in everyday life. The sum of two velocity vectors in relativity does not always represent the velocity of the two vectors combined.
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?

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Rasalhague said:
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.

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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!

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.

Thanks, all.

## 1. What is the concept of nonlinear addition of four-vectors in relativity?

The nonlinear addition of four-vectors in relativity refers to the mathematical operation of combining four-vectors, which are quantities that have both magnitude and direction in four-dimensional space-time. In special relativity, the addition of four-vectors is nonlinear, meaning that it does not follow the same rules as vector addition in classical mechanics.

## 2. What are the inconsistencies that arise when exploring nonlinear addition of four-vectors in relativity?

When exploring the nonlinear addition of four-vectors in relativity, inconsistencies arise due to the fact that the operation does not follow the familiar rules of vector addition. This can lead to paradoxes and contradictions, such as the famous twin paradox, where one twin ages slower than the other due to differences in their relative velocities.

## 3. Why is it important to study the nonlinear addition of four-vectors in relativity?

Studying the nonlinear addition of four-vectors in relativity is important for understanding the fundamental principles of special relativity and its implications for the behavior of objects in high-speed motion. It also helps to uncover the limitations of classical mechanics and how they are overcome in the theory of relativity.

## 4. How is the nonlinear addition of four-vectors in relativity different from vector addition in classical mechanics?

In classical mechanics, vector addition follows the rules of commutativity and associativity, meaning that the order in which vectors are added does not affect the result, and that parentheses can be used to group vectors in any way without changing the result. However, in special relativity, the addition of four-vectors is nonlinear, meaning that the order and grouping of vectors can affect the final result.

## 5. Are there any practical applications of the concept of nonlinear addition of four-vectors in relativity?

Yes, the concept of nonlinear addition of four-vectors has practical applications in areas such as particle physics, where the behavior of high-speed particles is governed by the principles of special relativity. It is also used in the design of technologies such as GPS systems, which rely on precise calculations of time dilation and relativistic effects to accurately measure distances and locations.

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