Page:The Meaning of Relativity - Albert Einstein (1922).djvu/34

22 given above, the equation is co-variant with respect to orthogonal transformations in space (rotational transformations); and the rules according to which the quantities in the equation must be transformed in order that the equation may be co-variant also become evident.

The co-variance of the equation of continuity,

requires, from the foregoing, no particular discussion.

We shall also test for co-variance the equations which express the dependence of the stress components upon the properties of the matter, and set up these equations for the case of a compressible viscous fluid with the aid of the conditions of co-variance. If we neglect the viscosity, the pressure, $$p$$, will be a scalar, and will depend only upon the density and the temperature of the fluid. The contribution to the stress tensor is then evidently

in which $$\delta_{\mu\nu}$$ is the special symmetrical tensor. This term will also be present in the case of a viscous fluid. But in this case there will also be pressure terms, which depend upon the space derivatives of the $$u_\nu$$. We shall assume that this dependence is a linear one. Since these terms must be symmetrical tensors, the only ones which enter will be

(for $$\frac{\delta u_\alpha}{\delta x_\alpha}$$ is a scalar). For physical reasons (no slipping)