Page:The principle of relativity (1920).djvu/222

 whether the theories of the electro-magnetic field and the gravitational-field together, will form a sufficient basis for the theory of matter. The general relativity postulate can teach us no new principle. But by building up the theory it must be shown whether electro-magnetism and gravitation together can achieve what the former alone did not succeed in doing.

§19. Euler's equations for frictionless adiabatic liquid.

Let p and ρ, be two scalars, of which the first denotes the pressure and the last the density of the fluid; between them there is a relation. Let the contravariant symmetrical tensor

T^{αβ} = -g^{αβ} p + ρ dx_α/ds dx_β/ds (58)

be the contra-variant energy-tensor of the liquid. To it also belongs the covariant tensor

(58a) T_{μν} = -g_{μν} p + g_{μα} dx_α/ds g_{μβ} dx_β/ds ρ

as well as the mixed tensor

(58b) T^α_σ = -δ^α_σ p + g_{σβ} dx_β/ds dx_α/ds ρ.

If we put the right-hand side of (58b) in (57a) we get the general hydrodynamical equations of Euler according to the generalised relativity theory. This in principle completely solves the problem of motion; for the four