Page:The principle of relativity (1920).djvu/221

 Physically the appearance of the second term on the left-hand side shows that for matter alone the law of conservation of impulse and energy cannot hold; or can only hold when g^{μν}'s are constants; i.e., when the field of gravitation vanishes. The second member is an expression for impulse and energy which the gravitation-field exerts per time and per volume upon matter. This comes out clearer when instead of (57) we write it in the Form of (47).

(57a) [part]T_{σ}^α/[part]x_{α} = -Γ_{σβ}^α T_{α}^β.

The right-hand side expresses the interaction of the energy of the gravitational-field on matter. The field-equations of gravitation contain thus at the same time 4 conditions which are to be satisfied by all material phenomena. We get the equations of the material phenomena completely when the latter is characterised by four other differential equations independent of one another.

D. THE "MATERIAL" PHENOMENA.

The Mathematical auxiliaries developed under 'B' at once enables us to generalise, according to the generalised theory of relativity, the physical laws of matter (Hydrodynamics, Maxwell's Electro-dynamics) as they lie already formulated according to the special-relativity-theory. The generalised Relativity Principle leads us to no further limitation of possibilities; but it enables us to know exactly the influence of gravitation on all processes without the introduction of any new hypothesis.

It is owing to this, that as regards the physical nature of matter (in a narrow sense) no definite necessary assumptions are to be introduced. The question may lie open