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

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The physical meaning of equation (47) becomes evident if in place of this equation we write, using a new notation, {{MathForm2|(47a)| $$ \left. \begin{align} \mathbf{k}_x = & -\frac{\delta p_{xx}}{\delta x} - \frac{\delta p_{xy}}{\delta y} - \frac{\delta p_{xz}}{\delta z} - \frac{\delta(\mathbf{i}b_x)}{\delta(\mathbf{i}l)} \\ & \vdots \\ \mathbf{i}\lambda = & -\frac{\delta(\mathbf{is}_x)}{\delta x} - \frac{\delta(\mathbf{is}_y)}{dy} - \frac{\delta(\mathbf{is}_y)}{dy} - \frac{\delta(\mathbf{is}_z)}{dz} - \frac{\delta(-\eta)}{\delta(\mathbf{i}l)} \end{align} \right\} $$}} or, on eliminating the imaginary, {{MathForm2|(47b)| $$ \left. \begin{align} \mathbf{k}_x = & -\frac{\delta p_{xx}}{\delta_x} - \frac{\delta p_{xy}}{dy} - \frac{\delta p_{xz}}{dz} - \frac{\delta b_x}{\delta l} \\ & \vdots \\ \lambda = & -\frac{\delta\mathbf{s}_x}{\delta x} - \frac{\delta\mathbf{s}_y}{\delta y} - \frac{\delta\mathbf{s}_z}{\delta z} - \frac{\delta\eta}{\delta l} \end{align} \right\} $$}}

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,