New Transformation Equations and the Electric Field Four-vector

12. The Electric Field Tensor

We now define the electric field tensor, , using the components of the electric field four-vector , from (11.1), as

The electric field tensor, , can be used to construct the electric field equations by writing

where the are the components of the current density four-vector from (10.9). The equations (12.2) are analogous to Maxwell's electromagnetic field equations. Both the homogeneous and source equations are included in the single equation (12.2). However, (12.2) contains terms related to the time component of the electric field, , unlike Maxwell's equations.

The equations (12.2) reduce to the equations

In a similar manner, we can write the equations for the force density four-vector, , which are analogous to the Lorentz force density equations. However, we will need to use a variation of the electric field tensor (12.1). This variation, , is defined as

Using (12.4) we can write the force density four-vector as

Equations (12.5) include terms related to the time component of the electric field four-vector, unlike the Lorentz equations. The left-hand side of (12.5) can be put, entirely, in terms of the electric field by subsituting from (12.2) into (12.5). After changing dummy indices, we have

The energy-momentum tensor, , can be written in terms of the electric field tensor, , and its variation, , as

We can also write (12.7) as

where the are the components of the electric field four-vector, and is the four-dimensional permutation symbol. If we now define the tensor, , as

we can express (12.7) as

The energy-momentum tensor, , is symmetric, but instead of containing the components of the Poynting vector in the fourth row and column (actually, the Poynting vector terms are included, but they sum to zero), as in the conventional electromagnetic energy-momentum tensor, we have terms including the time component of the electric field four-vector, .

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