Page:The Meaning of Relativity - Albert Einstein (1922).djvu/32

is not a vector but a tensor. On account of their skew-symmetrical character there are not nine, but only three independent equations of this system. The possibility of replacing skew-symmetrical tensors of the second rank in space of three dimensions by vectors depends upon the formation of the vector

$A_{\mu }={\frac {1}{2}}A_{\sigma \tau }\delta _{\sigma \tau \mu }.$

If we multiply the skew-symmetrical tensor of rank 2 by the special skew-symmetrical tensor $\delta$ introduced above, and contract twice, a vector results whose components are numerically equal to those of the tensor. These are the so-called axial vectors which transform differently, from a right-handed system to a left-handed system, from the $\Delta x_{\nu }$. There is a gain in picturesqueness in regarding a skew-symmetrical tensor of rank 2 as a vector in space of three dimensions, but it does not represent the exact nature of the corresponding quantity so well as considering it a tensor.

We consider next the equations of motion of a continuous medium. Let $\rho$ be the density, $u_{\nu }$ the velocity components considered as functions of the co-ordinates and the time, $X_{\nu }$ the volume forces per unit of mass, and $p_{\nu \sigma }$ the stresses upon a surface perpendicular to the $\sigma$-axis in the direction of increasing $x_{\nu }$. Then the equations of motion are, by Newton's law,

$\rho {\frac {du_{\nu }}{dt}}=-{\frac {\delta p_{\nu \sigma }}{\delta x_{\sigma }}}+\rho X_{\nu }$

in which ${\frac {du_{\nu }}{dt}}$ is the acceleration of the particle which at