Page:The Meaning of Relativity - Albert Einstein (1922).djvu/129

facts, it is necessary to introduce energy terms, not contained in Maxwell's theory, so that the single electric particles may hold together in spite of the mutual repulsions between their elements, charged with electricity of one sign. For the sake of consistency with this fact, Poincare has assumed a pressure to exist inside these particles which balances the electrostatic repulsion. It cannot, however, be asserted that this pressure vanishes outside the particles. We shall be consistent with this circumstance if, in our phenomenological presentation, we add a pressure term. This must not, however, be confused with a hydrodynamical pressure, as it serves only for the energetic presentation of the dynamical relations inside matter. In this sense we put

$T_{\mu \nu }=g_{\mu \alpha }g_{\nu \beta }\sigma {\frac {dx_{\alpha }}{ds}}{\frac {dx_{\beta }}{ds}}-g_{\mu \nu }p.$

(122)

In our special case we have, therefore, to put

${\begin{aligned}T_{\mu \nu }&=\gamma _{\mu \nu }p{\text{ (for }}\mu {\text{ and }}\nu {\text{ from }}1{\text{ to }}3{\text{)}}\\T_{44}&=\sigma -p\\T&=-\gamma ^{\mu \nu }\gamma _{\mu \nu }p+\sigma -p=\sigma -4p.\end{aligned}}$

Observing that the field equation (96) may be written in the form

$R_{\mu \nu }=-\kappa \left(T_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }T\right)$

we get from (96) the equations,

${\begin{aligned}+{\frac {2}{a^{2}}}\gamma _{\mu \nu }&=\kappa \left({\frac {\sigma }{2}}-p\right)\gamma _{\mu \nu }\\0&=-\kappa \left({\frac {\sigma }{2}}+p\right).\end{aligned}}$