New Transformation Equations and the Electric Field Four-vector

Energy and Momentum

We can derive an equation for the energy of a particle from the equation for the transformation of charge. The equation for transformation of charge, Eq. (64) is

Squaring Eq. (118), we get

Now, multiplying both sides of Eq. (120) by the square of the static electric potential, , we have

Remembering, from Eq. (87), that the potential four-vector for a single moving point charge is , we can write Eq. (121) as

Taking the square root of both sides of Eq. (122), we get

or, in terms of the total energy, , and proper mass, ,

Noting that the spatial part of the four-momentum is , where , we can write Eq. (124) as

We notice from Eq. (125) that the total energy, , of a particle can be either positive or negative. In addition, from Eq. (121) we see that, when , the total energy due to translatory motion is zero. However, a particle may possess energy due to its rotation in addition to the energy of its translational motion, even though the magnitude of its spatial four-velocity, , is the speed of light, . This additional energy, , due to the frequency of rotation, , is described by the Planck relation, , where is Planck's constant.

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