"Introduction to Electrodynamics" by Griffiths has the following lines:
A vector is any set of three components that transforms in the same manner as a displacement when you change coordinates.
How do I interpret these lines?
Edit: He added a few more lines to this.
There are three common ways of understanding vectors:
- Algebraic
- Geometric
- Transformational
The algebraic definition is common in mathematics where vectors are defined as those objects satisfying a short list of axioms. This is a coordinate-free definition.
The geometric definition is common in physics, and in particular, in mechanics, where a vector is defined as a directed magnitude. It is also a coordinate free definition.
The transformational definition is common in general relativity where it is used to help introduce tangent vectors. It is closely bound up with the set of all bases of the vector space.
Unlike the first two definitions, it is not coordinate-free. In fact, it is as far from the first definitions as it is possible to be since we need to consider the set of all conponents with respect to all bases!
We say it is an operational definition since when we actually come to measure a vector, we don't actually measure the vector itself directly. Instead, we first set up a basis, and measure the components of the vector with respect to this basis.
Now, if we are presented a list of 3 numbers, how do we know they come from a vector in a 3d space? Well, we ask the observer to measure the components of the object wrt to the set of all bases a d we should find a certain relationship between them. That is, if we transform a basis from one to another, then the components will transform contravariantly. In fact, there is another kind of transformational vector that transforms covariantly. This is why it is common to divide transformational vectors into contravariant and covariant vectors.
It's actually not straight-forward to formalise the definition of a transformational vector. It seems as though a functor of some kind can be involved involving the category of all bases of a space. It can be done via the category of elements and the Grothendieck construction.
Given the complexity of the transformational definition, we have to ask why use it? This is because not all vector spaces present themselves to us in the same way as the space around us, the 3d spatial vector space. For example, take for instance colour or isospin space. Here, we can't directly see a colour or isospin vector in the same way we can see a spatial vector. We can only set up bases and take components. Hence, the transformational definition is most natural here.
You can understand this with some examples.
In three-dimensional space you can define a displacement vector with components $(\delta x,\delta y,\delta z)$. You can have other vectors such as velocity, momentum, force, acceleration. To understand the difference between a vector and a non-vector it helps to have some examples of non-vectors.
Mass $m$, charge $q$ and temperature $T$ are non-vectors and this is obvious. But how about the quantity $(m, q, T)$? It has three components. Is it a vector? The answer is no, because it does not have the direction-and-size nature of a vector. One can plot a three-dimensional graph with mass, charge and temperature along the axes, but there is no way to define a kind of 'rotation' which could transform mass into a combination of charge and temperature.
In search of more likely candidates for a vector, you can try things like $(yz, zx, xy)$. It's not so easy now to see whether or not that is a vector.
So the way this is formalized mathematically is to say that a quantity is correctly called a 'vector' when the following happens: if we write it in terms of components, and then rotate the coordinate system, then if the components change by the same mathematical rule as do the components $(\delta x, \delta y, \delta z)$ of a displacement in space, then we have a vector.
The change in the components of the displacement vector $(\delta x, \delta y, \delta z)$, when the coordinate axes are rotated, can be obtained by using a rotation matrix $R$. If the new set of components of some other quantity is correctly obtained by using that same rotation matrix $R$, then the other quantity is a vector.
We'll use the Einstein summation convention here
Let us change coordinates from $x$ ($A$) to $y$ ($B$), with unit vectors $\vec{e}_{\mu}$(in $A$) and $\vec{\epsilon}_{\mu} $ (in $B$),with $\mu=0,1,2,3$ .
The displacement vector in the 2 frames are $dx^{\mu}\vec{e}_{\mu}$ and $dy^{\nu}\vec{\epsilon}_{\nu}$.Let us assume the displacements are related by:
$$dy^{\nu} = S^{\nu}_{\mu}dx^{\mu} \tag{*}$$
Suppose we have a vector $\vec{V}$, written as $V^{\mu}\vec{e}_{\mu}$ (in $A$) and $V^{'\nu}\vec{\epsilon}_{\nu}$ (in $B$). Then what the text means is that, the components of the vector in $A$ and $B$ are also related by the same relationship as $(*)$. This can be shown as follows:
$$dy^{\nu}\vec{\epsilon}_{\nu}=S^{\nu}_{\mu}dx^{\mu} \vec{\epsilon}_{\nu} = dx^{\mu} (S^{\nu}_{\mu}\vec{\epsilon}_{\nu})$$
Since the displacement vector in the 2 frames are equal, we must have: $$S^{\nu}_{\mu}\vec{\epsilon}_{\nu} = \vec{e}_{\mu}$$ $$\Longrightarrow \vec{\epsilon}_{\nu} = P^{\mu}_{\nu}\vec{e}_{\mu}\tag{1}$$
Where the matrix $P$ is the inverse of matrix $S$, thus satisfying the relation $P^{\mu}_{\lambda}S^{\nu}_{\mu}=\delta^{\nu}_{\lambda}$.
Substituting relation $(1)$ in the vector $V^{'\nu}\vec{\epsilon}_{\nu}$, we have: $$V^{'\nu}\vec{\epsilon}_{\nu} = (V^{'\nu} P^{\mu}_{\nu})\vec{e}_{\mu}$$
Since the vector $\vec{V}$ are same in both frames, we now get: $$V^{'\nu} P^{\mu}_{\nu}=V^{\mu}$$ Using $P^{\mu}_{\nu}S^{\lambda}_{\mu}=\delta^{\lambda}_{\nu}$, it becomes: $$V^{'\nu}=S^{\nu}_{\mu}V^{\mu}$$
Which is the same relationship between the displacement in A and B
Thus, we have shown that under a coordinate transformation, the components of a vector must transforms in the same manner as the displacement
What Griffiths is calling a vector is, strictly speaking, a Euclidean vector in three-dimensional space or a four-vector in four-dimensional spacetime.
The components of a Euclidean vector (or a four-vector) are a way of representing it by giving its projections along three (or four) co-ordinate axes or the angles between the vector and certain axes. The definition is saying that if a vector is independent of the co-ordinate system that we are using (which is important if it represents a physically meaningful quantity) then its components must change in a specific way if we change the co-ordinate system. This allows us to tell if two different representations (one in Cartesian co-ordinates and another in spherical co-ordinates, for example) actually refer to one and the same underlying vector.
Euclidean vectors and four-vectors are special cases of the more general mathematical concept of a vector in linear algebra.
