Electric Field 4-Vector

Let F be the Faraday tensor and U the 4-velocity of an observer. The quantity E defined as

is referred to as the electric field 4-vector as measured by the observer whose 4-velocity is U. is the 1-form corresponding to the 4-velocity U where g is the metric tensor. In component for this becomes

To see that this is a 4-vector we can examine its transformation properties. The components of the Faraday tensor F and the 1-form transform as

The components of the electric field 4-vector transforms as

Substituting the definition from Eq. (1) we finally obtain

Since the components of E transform in the same was as the components of a 4-vector it follows that E is a 4-vector.