Lorentz Transformation of the Electromagnetic Field

In Section 6 of his famous paper "On the Electrodynamics of Moving Bodies", Albert Einstein worked out the transformation of the electromagnetic field as measured by two observers moving relative to each other with some unvarying velocity. He obtained that result from applying the Lorentz Transformation for spatio-temporal coordinates to the third and fourth of Maxwell痴 Equations as one observer has them and then comparing the result to the same equations as the other observer has them, applying the basic relativistic criterion that the Lorentz Transformation applied to electric and magnetic fields must leave Maxwell痴 Equations unchanged; specifically, both observers must use the same mathematical form of the equations. However, a closer look at what Einstein did reveals that he already had the transformation equations for the electromagnetic field and merely substituted them into his equations to verify that they did, indeed, represent the electromagnetic field痴 analogue of the Lorentz Transformation of spatio-temporal coordinates, which he had deduced from his two postulates of Relativity.

So now we have before us the task of working out the Lorentz Transformation of the electromagnetic field from scratch. We want to start with equations that describe the relationships among the components of the electric and magnetic fields calculated with respect to the usual rectangular Cartesian grid of spatial components augmented by the temporal component (represented here with lower-case letters). We call the inertial frame occupied and marked by that grid the L-frame. We also have a U-frame, on whose grid observers use upper-case letters to represent measurements of distance and duration. Both of the grids of the L-frame and the U-frame share a common x-axis and the U-frame moves in the positive x-direction at the speed v relative to the L-frame. The Lorentz Transformation of L-frame coordinates into U-frame coordinates thus takes the form,

(Eq地s 1)

in which we have the Lorentz factor between the two frames as

(Eq地 2)

and β=v/c.

For convenience we can represent Equations 1 as a matrix multiplication,

(Eq地 3)

in which we have used the Einstein convention of summing over repeated indices and we have represented the Lorentz Transformation by the matrix,

(Eq地 4)

In order to use that matrix we need to represent the distance-duration
4-vector as x_{j}={x,y,z,ict}.

We start by transforming the electromagnetic potential.
For a charged particle immersed in an electromagnetic field we have {**p**,iE/c}={q**A**,iqφ/c},
which means that the potential, Φ={**A**,iφ/c},
represents a proper 4-vector. If the particle touches two events separated by a
minuscule displacement δx={δ**x**,icδt},
then the product ΦAδx
remains invariant under a Lorentz Transformation. If the U-frame observers use
primed variables to represent their measurements of the electromagnetic field
and the L-frame observers use unprimed variables, then we have

(Eq地 5)

which gives us

(Eq地 6)

That equation tells us that, as a proper 4-vector, the electromagnetic potential transforms in accordance with the standard Lorentz Transformation. Thus we have the electromagnetic-potentials analogue of Equations 1,

(Eq地s 7)

Next we want to look at the forcefields that come from the potential fields. In preparation for taking that step we need to look at how the Lorentz Transformation applies to the process of partial differentiation.

For a differential change in displacement we have the transformation between two coordinate grids as

(Eq地 8)

Comparing that equation with Equation 3 tells us that

(Eq地 9)

Next we differentiate a scalar function, f(X_{i})=f(x_{j}),
that remains invariant under a Lorentz Transformation and get

(Eq地 10)

Because we used a function not subject to the Lorentz Transformation, we can therefore infer the operator relation

(Eq地 11)

so now we know that we can represent the four-dimensional differentiation operator, {L, i∂/c∂t}, as a proper 4-vector.

From the appendix we get the tensor representation of the electromagnetic field, which we can expand into a explicit matrix,

(Eq地 12)

When we multiply that matrix by the column vector representing the current associated with a charged particle the first three rows yield the three components of the force exerted upon the particle and the fourth row yields a description of the rate at which the field alters the particle痴 energy.

We know how to apply the Lorentz Transformation matrix to a description of the 4-potential and to the 4-differentiation that converts it into a description of the electromagnetic field, so now we have the means to apply the Lorentz Transformation to the electromagnetic field directly. We have

(Eq地 13)

In going from the second line to the third line we exploit the fact that the
differentiation operator does not affect the Lorentz Transformation matrix and
the fact that commuting a matrix with a vector changes the matrix into its
transposed version. We also made use of the fact that
Λ_{ik}=Λ_{jm}=Λ.
So now we can rewrite Equation 13 in a more explicit form;

(Eq地 14)

In that equation I have assumed that the two inertial frames occupied by our
observers move only in the x-direction at the speed
β=V_{x}/c.
In drawing out the transformed matrix I also made us of the fact that

(Eq地 15)

in the upper right and lower left corners of the matrix. If we now compare the electromagnetic field matrix in the first line, term by term, with the matrix in the third line, we get the Lorentz Transformation of the electromagnetic field:

(Eq地s 16)

And, of course, multiplying those components by the appropriate components of a 4-current gives us the electromagnetic 4-force acting on the particle carrying the electric charge of the 4-current. And thus we get the same equations that Einstein used in Section 6 of his first Relativity paper.

Appendix: Electromagnetic Force

Given the 4-potential, we want to derive an explicit description of the electromagnetic field and the force that it exerts upon an electrically charged particle. We begin the derivation by restating the principle of least action,

(Eq地 A-1)

which leads to the Euler-Lagrange Equations,

(Eq地 A-2)

For the purpose of applying those equations we devise the Lagrangian function by adding the negative of the relativistic Lagrangian of a free particle to the Lagrangian created from taking the inner product of the 4-potential and the 4-velocity,

(Eq地 A-3)

Substituting that expression into Equation A-2 then gives us

(Eq地 A-4)

Rearranging that equation and working through the vector identities transforms that equation into

(Eq地 A-5)

In that equation we see that the rate at which the particle痴 linear momentum changes (the force acting on the particle) equals the sum of the electrostatic force and the Lorentz force.

Next we want to put Equation A-2 into tensor form so that we can derive the tensor form of the electromagnetic field. In this derivation we have the 4-Lagrangian as

(Eq地 A-6)

in which we sum the subscripts for the values i=1, 2, 3, and 4.

Let痴 go back to the beginning, the principle of least action,

(Eq地 A-7)

In that equation we have T representing the proper time, the elapse of time measured in the inertial frame in which the endpoints of the measured action coincide in space. Because the processes of variation (a kind of differentiation) and integration commute with each other, we rewrite the left side of that equation as

(Eq地 A-8)

As usual, the middle term in the third line vanishes because it gives us the function in parentheses evaluated between the endpoints of the action, where the variation of location goes to zero. Because I left the inertial reaction term (the first term on the right side of Equation A-3) out of my Lagrangian, the expression in square brackets on the fourth line does not go to zero, but, rather, equals the rate at which the linear momentum of the forced particle changes; in other words, it represents the force exerted by the electromagnetic field.

It may seem strange, even illegitimate, to include a
variation in time, δx_{4},
in that derivation. But a proper relativistic treatment of the principle of
least action demands such an inclusion. To understand such a variation imagine
that the force acting on the particle varies with the elapse of time: the
variation then represents the particle coming to a given value of the force
sooner or later than it does on the true path.

We also have the total time derivative operator as

(Eq地 A-9)

Noting that the 4-velocity in that expression represents the motion of an inertial frame and using the Lagrangian function from Equation A-6, we write the Euler-Lagrange expression from Equaton A-8 as

(Eq地 A-10)

In that expression the lower case eff represents the force that the field
exerts upon the particle. The derivatives of the 4-velocity go to zero, because
the 4-velocity represents the motion of an inertial frame, which the particle
occupies, however temporarily. In going from the second line to the third I
interchanged the indices in the second term on the right so that I could factor
out the 4-velocity and thereby represent the 4-force as the product of the
4-current (qu_{j}) associated with the particle and the electromagnetic
force,

(Eq地 A-11)

which we use in the transformation above. In form that description looks like a four-dimensional analogue of the curl of a vector.

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