In relativity, objects like electric and magnetic fields do not explicitly exist, but rather the EM field is carried by the 4 dimensional vector potential, a relativistic 4-vector. This transforms according the Lorentz transformations.

In non-relativist mechanics energy and momentum are distinct entities.

In relativistic mechanics, they are connected by being the time and three space components of the same four-vector.

If a coordinate system is moving with velocity v in the x direction relative to the original system, they transform thusly:

t' = (t - v*x/c2)*L
x' = (x - v*t)*L
y' = y
z' = z

where L = (1 - (v/c)2) is the Lorentz factor. The EM four-potential, or relativistic vector potential, where the time term is the ordinary potential and the other three are the components of the standard 3-dimensional vector potential.

Energy is the time component of a four-vector of which the space components are the three momenta in the x,y, & z directions.