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

20 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

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,

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