On the possibility of cosmological gravimetry in general relativity

Abstract

In the framework of the parametrized post-Newtonian (PPN) formalism, we substantiate an idea according to which we can measure the value of the cosmological gravitational potential $\Phi$ at the location of the Solar System, which is formed by all the matter of the Universe (including dark matter). This paradoxical result is based on the fact that the PPN formulation of general relativity is not invariant with respect to the transformation $U({\bf r})\,\rightarrow\,U({\bf r})+C$, where $U({\bf r})$ is a gravitation potential and $C$ is an arbitrary real constant (this is due to the nonlinearity of Einstein equations, first of all). Starting from the cosmological description of the Universe within the framework of the standard general relativity (i.e., using initial PPN parameters $\gamma_0=\beta_0=1$ in the cosmological reference frame), we show that from the viewpoint of the local (laboratory) observer there is a renormalization of the PPN parameters $\gamma$ and $\beta$: $|\gamma-1|\sim |\Phi|/c^2$ and $|\beta-1|\sim |\Phi|/c^2$. This leads to the estimates: $|\beta-1|>10^{-6}$ and $|\gamma-1|>10^{-6}$ for gravitational experiments in the Solar System. In addition, we have found an unique post-Newtonian model, which is invariant with respect to the transformation $U({\bf r})\,\rightarrow\,U({\bf r})+C$ and therefore is insensitive to the cosmological potential $\Phi$. The most unusual is that for this model we always have $\gamma=\beta=1$ from the viewpoints of the cosmological and any laboratory observers.