Page:The Meaning of Relativity - Albert Einstein (1922).djvu/65

where we have written ^[1]

$T_{\mu \nu }=-{\frac {1}{4}}\phi _{\alpha \beta }^{2}\delta _{\mu \nu }+\phi _{\mu \alpha }\phi _{\nu \alpha }.$

(48)

The physical meaning of equation (47) becomes evident if in place of this equation we write, using a new notation,

$\left.{\begin{aligned}\mathbf {k} _{x}=&-{\frac {\delta p_{xx}}{\delta x}}-{\frac {\delta p_{xy}}{\delta y}}-{\frac {\delta p_{xz}}{\delta z}}-{\frac {\delta (\mathbf {i} b_{x})}{\delta (\mathbf {i} l)}}\\&\vdots \\\mathbf {i} \lambda =&-{\frac {\delta (\mathbf {is} _{x})}{\delta x}}-{\frac {\delta (\mathbf {is} _{y})}{dy}}-{\frac {\delta (\mathbf {is} _{y})}{dy}}-{\frac {\delta (\mathbf {is} _{z})}{dz}}-{\frac {\delta (-\eta )}{\delta (\mathbf {i} l)}}\end{aligned}}\right\}$

(47a)

or, on eliminating the imaginary,

$\left.{\begin{aligned}\mathbf {k} _{x}=&-{\frac {\delta p_{xx}}{\delta _{x}}}-{\frac {\delta p_{xy}}{dy}}-{\frac {\delta p_{xz}}{dz}}-{\frac {\delta b_{x}}{\delta l}}\\&\vdots \\\lambda =&-{\frac {\delta \mathbf {s} _{x}}{\delta x}}-{\frac {\delta \mathbf {s} _{y}}{\delta y}}-{\frac {\delta \mathbf {s} _{z}}{\delta z}}-{\frac {\delta \eta }{\delta l}}\end{aligned}}\right\}$

(47b)

When expressed in the latter form, we see that the first three equations state the principle of momentum; $p_{xx}\cdots p_{zx}$ are the Maxwell stresses in the electromagnetic field, and $(b_{x},b_{y},b_{z})$ is the vector momentum per unit volume of the field. The last of equations (47b) expresses the energy principle; \mathbf{s} is the vector flow of energy, and $\eta$ the energy per unit volume of the field. In fact, we get from (48) by introducing the well-known expressions for the components of the field intensity from electrodynamics,

1. To be summed for the indices $\alpha$ and $\beta$.