Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

Why the electromagnetic four-potential is Lorentz covariant

Originally published at 狗和留美者不得入内. You can comment here or there.

Definition and proof of covariance of four-current

Proving that the electromagnetic four-potential is Lorentz covariant rests on the assumption that the four-current given by

[公式]

is Lorentz covariant, which can be demonstrated through charge conservation expressed in continuity equation form as

[公式]

where [公式] is the four-gradient, which holds in every reference frame. We now use the fact that the gradient operator with respect to any Lorentz covariant four-vector [公式] is a Lorentz covariant four-vector which contravaries with [公式] , which was proved in [1]. Applying this to reference frame corresponding to [公式] , we have

[公式]

and consequently the statement of the Lorentz invariant corresponding to charge conservation as given in [公式] implies

[公式]

which holds if and only if

[公式]

Definition and proof of covariance of four-potential

As elaborated in [2], under the Lorentz gauge condition that [公式] or

[公式]

Maxwell’s equations can be simplified as, in SI units,

[公式]

or even simpler than that

[公式]

wherein the d’Alembertian operator

[公式]

One can read in Chapter 14 of Landau-Lifshitz shorter course book 1 on classical mechanics and electromagnetism [3] the process of deriving the solution to [公式] , which is

[公式] where

[公式] is the retarded time and

[公式] is the volume element.

Given that the four-potential by definition is [公式] and [公式] , [公式] can be rewritten as

[公式] Since the prime coordinate [公式] is typically used to denote a Lorentz transformation, we replace the prime in [公式] with the overline, that is [公式] becomes [公式] . [公式] we then rewrite as

[公式] In order to demonstrate Lorentz covariance of the four-potential, we desire to show that

[公式] holds for an arbitrary Lorentz transform with the associated transformation tensor [公式] such that

[公式]

Moreover, [公式] will transform to [公式] per the same rule

From now on we set [公式] in [公式] for simplicity. We will use [公式] or [公式] to denote the space coordinates. Moreover, we set

[公式]

Then, substituting [公式] into both sides of [公式] gives

[公式]

and

[公式] The integrals in [公式] can be viewed as a summation of infinitesimal components the corresponding infinitesimal volume elements of which, in addition to holding equal value, are disjoint. Moreover, that set of infinitesimal volume elements upon any Lorentz transformation remain disjoint. Therefore it suffices to prove that the differential forms within the integral in [公式] and [公式] are equal. By the Lorentz covariance of four-current,

[公式] which simplifies our desired sufficient condition to

[公式] We note how if we prove Lorentz covariance restricted to the simple case where only one space coordinate is transformed, we have proven Lorentz covariance in general since the Lorentz group is generated by the set of transformations where only one space coordinate is changed. This means we can without loss of generality consider only one velocity [公式] along the first coordinate, the Lorentz transform corresponding to which as derived in [1] as [公式] is

[公式]

where [公式] .

Thus [公式] equates to

[公式]

Moreover, length contraction, which tells us that the ratio between the length in the observer frame and proper length (also called rest length) is [公式] , upon application to the purely spatial distances occurring in [公式] . Disregarding indices [公式] ,the points [公式] by [公式] corresponds to

[公式]

and thereby upon subtraction of the two equations in [公式] , we get

[公式]

Similarly, through

[公式]

we get

[公式]

We note how by definition of rest frame, in the rest frame, the length between any two different spatial coordinates is measured to be the same even if the two spatial coordinates are measured at different times. However, in an observer frame, the measured spatial coordinate of an implicit point object moving along with the observed frame differs with respect to the time coordinate of the rest frame per

[公式] and thus, to truly and meaningfully measure the length between those two point objects in an observer frame, one must perform the two coordinate measurements at the same time of the observer frame. In [公式] , the primed coordinate corresponds the observer frame, and the unprimed coordinate corresponds to the rest frame.

In [公式] , the [公式] is the length at the same time [公式] in frame [公式] between two implicit point objects which at time [公式] are located at [公式] spatially. Moreover, we have assumed arbitrary Lorentz transformation to another frame [公式] moving relative to [公式] only along the first spatial coordinate. Note how the two point objects at time [公式] in frame [公式] are only constrained in their spatial coordinate, and there are no constraints in how their spatial coordinates vary with respect to [公式] . Then, for simplicity, we assume that both point objects move along the frame [公式] , which means by definition that [公式] , the spatial coordinates of the two respective point objects in the [公式] frame, are constant with respect to time and thus can be treated as constants [公式] . One observes that the spacetime coordinates associated with the two point objects are [公式] in [公式] , and in [公式] , they transform to the coordinates given in [公式] , which differ in the time component, which, as we just mentioned, does not affect the spatial coordinate at all. Here, [公式] is the rest frame, and [公式] is the observer frame, and by length contraction,

[公式]

which is equivalent to the first equation in [公式] .

Since only one spatial coordinate changes, [公式] is equivalent to

[公式]

which via substitution of [公式] is equivalent to the equality

[公式]

Thus, the Minkowski norm of the electromagnetic four-potential at the same point in spacetime is the same regardless of reference frame. This shows that the four-potential is indeed Lorentz covariant.

References

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