Page:The Meaning of Relativity - Albert Einstein (1922).djvu/64

The second term of this expansion corresponds to the kinetic energy of the material particle in classical mechanics.

Equations of Motion of Material Particles. From (43) we obtain, by differentiating by the time $l$, and using the principle of momentum, in the notation of three-dimensional vectors,

$\mathbf {K} ={\frac {d}{dl}}\left({\frac {m\mathbf {q} }{\sqrt {1-q^{2}}}}\right)$

(46)

This equation, which was previously employed by H. A. Lorentz for the motion of electrons, has been proved to be true, with great accuracy, by experiments with $\beta$-rays.

Energy Tensor of the Electromagnetic Field. Before the development of the theory of relativity it was known that the principles of energy and momentum could be expressed in a differential form for the electromagnetic field. The four-dimensional formulation of these principles leads to an important conception, that of the energy tensor, which is important for the further development of the theory of relativity.

If in the expression for the 4-vector of force per unit volume,

$K_{\mu }=\phi _{\mu \nu }J_{\nu }$

using the field equations (32), we express $J_{\mu }$ in terms of the field intensities, $\phi _{\mu \nu }$, we obtain, after some transformations and repeated application of the field equations (32) and (33), the expression

$K_{\mu }=-{\frac {\delta T_{\mu \nu }}{\delta x_{\nu }}}$

(47)