Mathematical Interlude: Covariant Components

I want to pause here for a couple of mathematical points about transformations from one set of coordinates to another. Let’s suppose we have a space described by two sets of coordinates X^μ and (X′)^μ. These could be Art’s and Lenny’s spacetime coordinates, but they needn’t be. Let’s also consider an infinitesimal interval described by dX^μ or d(X′)^μ. Ordinary multivariable calculus implies the following relation between the two sets of differentials:

Einstein wrote many equations of this form. After a while he noticed a pattern:

Whenever he had a repeated index in a single expression—the index ν on the right side of the equation is such a repeated index—it was always summed over. In one of his papers on general relativity, after several pages, he apparently got tired of writing the summation sign and simply said that from now on, whenever an expression had a repeated index he would assume that it was summed over. That convention became known as the Einstein summation convention. It is completely ubiquitous today, to the point where no one in

physics even bothers to mention it. I’m also tired of writing ∑_ν, so from now on we will use Einstein’s clever convention, and Eq. 4.15 becomes

If the equations relating X and x′ are linear, as they would be for Lorentz transformations, then the partial derivatives are constant coefficients.

Let’s take the Lorentz transformations

as an example. Here is a list of the four constant coefficients that would result from Eq. 4.16:

If we plug these into Eq. 4.16 we get the expected result,

This, of course, is the perfectly ordinary Lorentz transformation of the

components of a 4-vector.

Let’s abstract from this exercise a general rule for the transformation of 4-vectors. Going back to Eq. 4.16, let’s replace 4-vector components d(X′)^μ and dX^ν with (A′)^μ and A^ν. These represent the components of any 4-vector A in frames related by a coordinate transformation. The generalization of Eq. 4.16 becomes

where ν is a summation index. Eq. 4.19 is the general rule for transforming the components of a 4-vector. For the important special case of a Lorentz transformation, this becomes

My real reason for doing this was not to explain how dX^μ or A^μ transforms, but to set up the calculation for transforming ∂_μϕ. These objects—there are four of them—also form the components of a 4-vector, although of a slightly different kind than the dX^μ. They obviously refer to the coordinate system X but can be transformed to the X′ frame.

The basic transformation rule derives from calculus, and it’s a multivariable generalization of the chain rule for derivatives. Let me remind you of the ordinary chain rule. Let ϕ(x) be a function of the coordinate X, and let the primed coordinates X′ also be a function of X. The derivative of ϕ with respect to X′ is given by the chain rule,

The multivariable generalization involves a field ϕ that depends on several independent coordinates X^μ and a second set of coordinates (X′)^ν. The generalized chain rule reads

or using the summation convention, and the shorthand notation in Eq. 4.14,

Let’s be more general and replace ∂_μϕ with A_μ so that Eq. 4.21 becomes

Take a moment to compare Eqs. 4.19 and 4.22. I’ll write them again to make it easy to compare—first Eq. 4.19 and then Eq. 4.22:

There are two differences. The first is that in Eq. 4.19 the Greek indices on A appear as superscripts while in Eq. 4.22 they appear as subscripts. That hardly seems important, but it is. The second difference is the coefficients: In Eq. 4.19 they are derivatives of X′ with respect to X, while in Eq. 4.22 they are derivatives of X with respect to X′.

Evidently there are two different kinds of 4-vectors, which transform in different ways: the kind that have superscripts and the kind with subscripts.

They are called contravariant components (superscripts) and covariant components (subscripts), but since I always forget which is which, I just call them upper and lower 4-vectors. Thus the 4-vector dX^μ is a contravariant or upper 4-vector, while the spacetime gradient ∂_μϕ is a covariant or lower 4-vector.

Let’s go back to Lorentz transformations. In Eq. 4.18, I wrote down the coefficients for the transformation of upper 4-vectors. Here they are again:

We can make a similar list for the coefficients in Eq. 4.22 for lower 4-vectors.

In fact, we don’t have much work to do. We can get them by interchanging the primed and unprimed coordinates, which for Lorentz transformations just means interchanging the rest and moving frames. This is especially easy since it only requires us to change the sign of the velocity (remember, if Lenny moves with velocity v in Art’s frame, then Art moves with velocity −v in Lenny’s frame). All we have to do is to interchange primed and unprimed coordinates and at the same time reverse the sign of v.

Here then are the transformation rules for the components of covariant (lower) 4-vectors:

We’ve already seen examples such as X^μ, displacement from the origin. The differential displacement between neighboring points, dX^μ, is also a 4-vector.

If you multiply 4-vectors by scalars (that is, by invariants), the result is also a 4-vector. That’s because invariants are completely passive when you transform. We’ve already seen that the proper time dτ is invariant, and therefore the quantity dX^μ/dτ, which we call 4-velocity, is also a 4-vector:

When we say “U^μ is a 4-vector,” what do we actually mean? We mean that its behavior in other reference frames is governed by the Lorentz transformation.

Let’s recall the Lorentz transformation for the coordinates of two reference frames whose relative velocity is v along the x axis:

If a complex of four quantities (consisting of a time component and three space components) transforms in this way, we call it a 4-vector. As you know, differential displacements also have this property:

Table 4.1 summarizes the transformation properties of scalars and 4-vectors.

We use the slightly abstract notation A^μ to represent an arbitrary 4-vector. A⁰ is

the time component, and each of the other components represents a direction in space.

An example of a field with these properties would be a fluid that fills all of spacetime. At every point in the fluid there would be a 4-velocity as well as an ordinary 3-velocity. We could call this 4-velocity U^μ (t, x). If the fluid flows, the velocity might be different in different places. The 4-velocity of such a fluid can be thought of as a field. Because it’s a 4-velocity, it’s automatically a vector and would transform in exactly the same way as our prototype 4-vector A^μ. The values of U’s components in your frame would be different from their values in my frame; they would be related by the equations in Table 4.1. There are lots of other examples of 4-vectors, and we won’t try to list them here.

Table 4.1: Field Transformations. Greek index μ takes values 0, 1, 2, 3, which correspond to t, x, y, z in ordinary (3 + 1)-dimensional spacetime. In nonrelativistic physics, ordinary Euclidean distance is also considered a scalar.

If you take the four components of a 4-vector, you can make a scalar out of them. We already did this when we constructed the scalar (dτ)² from the 4-vector dX^μ:

(dτ)² = (dt)² − (dx)² − (dy)² − (dz)².

We can follow the same procedure with any 4-vector. If A^μ is a 4-vector, then the quantity

(A⁰)² − (A^x)² − (A^y)² − (A^z)²

is a scalar for exactly the same reasons. Once you know that the A^μ components transform the same way as t and x, you can see that the difference of the squares of the time component and the space component will not change under a Lorentz transformation. You can show this using the same algebra we used with (dτ)².

We’ve seen how to construct a scalar from a 4-vector. Now we’ll do the opposite, that is, construct a 4-vector from a scalar. We do this by differentiating the scalar with respect to each of the four components of space and time. Together, those four derivatives form a 4-vector. If we have a scalar ϕ, the quantities

are the components of a (covariant, or lower) 4-vector.