March 13th, 2021

How the Lorentz force law directly implies the antisymmetric electromagnetic field tensor

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

In electromagnetism there is the Lorentz force law

[公式] It is easy to see that this corresponds to a linear transformation of a vector of four components three of which are the three velocity components which can be represented by

[公式] In [1], we defined relativistic velocity or the velocity four-vector as

[公式]

where

[公式]

If follows to define four-acceleration as [公式] , wherein [公式] is proper time. In calculating explicitly this quantity, we shall denote [公式] . It was shown in [1] that [公式] per time dilation in different frame. Moreover,

[公式]

Thus,[公式]

We now define four-momentum as [公式] , where [公式] is the rest mass of the object, a quantity which is invariant with respect to reference frame. Next, we define four-force as [公式] . By [公式] in [1], [公式] is a constant; differentiating it with respect to time, we obtain that [公式] , noting that this is equivalent to

[公式]To conform with four-velocity, we first modify [公式] to

[公式]

We want the result vector in [公式] to actually be the four-force. From [公式] , we know that the [公式] th component of the four-force upon setting [公式] for simplicity should be equal to

[公式]

From this we modify [公式] accordingly to

[公式]

The [公式] matrix in [公式] represents the contents of the electromagnetic field tensor. We multiply it by the matrix corresonding to the Minkowski metric

[公式]

to make it anti-symmetric, which the result being

[公式]

We define the rank two covariant tensor by

[公式]

Using [公式] to denote four-momentum and [公式] to denote four-velocity, [公式] expressed in Einstein notation is (using the metric signature [公式] ),

[公式]

Moreover, the definition [公式] suggests the following antisymmetric formula:

[公式]

The reader is welcome to verify it for [公式] index corresponding to [公式] with the definition

[公式]

References