New Transformation Equations and the Electric Field Four-vector

The Electric Field Tensor

If we now define the electric field tensor, , using the components of from Eqs. (102), as

we can include both of the field equations, Eqs. (68), in a single equation by writing

where the terms not containing repeated indices are associated with the homogeneous field equations and the remaining terms are related to the inhomogeneous field equations.

In a similar manner, both of the force equations, Eqs. (77), can be combined by writing

where the are the components of the current four-vector, the are the components of the momentum four-vector, and the are the components of the force four-vector. As before, the terms separate into the homogeneous and inhomogeneous force equations.

The energy-momentum tensor, , can be written in terms of Eq. (104) as

which can also be separated into homogeneous and inhomogeneous equations. If we now define the tensor as

we can express Eq. (107) as

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