# 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

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

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

where

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

Thus,

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

To conform with four-velocity, we first modify to

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

From this we modify accordingly to

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

to make it anti-symmetric, which the result being

We define the rank two covariant tensor by

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

Moreover, the definition suggests the following antisymmetric formula:

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

References

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