Über die mögliche lineare Form von L ORENTZ ‐kovarianten Gravitationstheorien

L ORENTZ ‐covariant theories of gravitation which fulfil E INSTEIN 's weak principle of equivalence and which contain a pure Newtonian theory as an approximation are tensortheories with the linear approximative form\documentclass{article}\pagestyle{empty}\begin{document}$$ g\mu = - x(\alpha T\mu + [1 - \alpha]\eta \mu vT) $$\end{document}for the field equations. In the case of E INSTEIN 's strong principle of equivalence the exact field equations must be the general relativistic E INSTEIN ‐equations (or the bimetrical E INSTEIN ‐R OSEN ‐equations). This follows from the dynamical equations and the B IANCHI identity according to J ÁNOSSY and T REDER . However, from N EWTON 's axiom of reaction together with the weak principle of equivalence results that the strong principle of equivalence must be valid for the linear approximation of the field equations with sources. Therefore, the linear approximation of all physically meaningful Lorentz‐covariant theories of gravitation is given by the linearized E INSTEIN ‐equations (with H ILBERT ‐conditions):\documentclass{article}\pagestyle{empty}\begin{document}$$ g\mu = - 2x(T\mu v - \frac{1}{2}\eta \mu T) $$\end{document} , that is by the ansatz α = 2. The main point of our arguments is L AUE 's postulate of the self‐consistency of perfect static systems of isolated gravitational masses. In the lowest order of approximation this self‐consistency is only possible if the gravitational matter‐tensor is identical with the special‐relativistic energy‐momentum‐tensor T μ v . L AUE 's postulate is fulfilled exactly for the general relativistic field equations according to the theorems of B IRKHOFF , T OLMAN and E INSTEIN and P AULI .