Quantum-Theory Restrictions on the General Theory of Relativity

The limitations set by the uncertainty principle on the measurement of the curvature of space are determined by passing a small test particle around a geodesic triangle. In measuring the curvature of space around a Schwarzschild-solution particle of finite size and physically realized density, it is found that the mass must be at least of the order of ${10}^4$ kilograms, or the curvatures cannot be measured with an accuracy equal to the order of magnitude of the terms in the defining Eqs. $G_{\mathrm{ik}}=0\mathrm{}$. For mass points, the lower limit is ${10}^{-5\mathrm{}}$ g. For smaller masses, the curvatures can only be measured with less accuracy and only over large regions of space. Similar limitations apply to alternative laws of gravitation involving higher derivatives of the metric. It is concluded that in any theory which attempts to unite quantum theory with the general theory of relativity, the relation of the metric to the energy momentum tensor, $G_{\mathrm{ik}}-\frac{G_{\mathrm{ik}}G}{2}=-K{T}_{\mathrm{ik}}$, must appear only in the large and in a statistical sense, i.e., for large regions of space and large numbers of elementary particles.