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

From Wikisource
Jump to navigation Jump to search
This page has been proofread, but needs to be validated.
22
THE MEANING OF RELATIVITY

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,

(17)

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, , 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 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 . 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 is a scalar). For physical reasons (no slipping)