On the possibility of cosmological gravimetry in general relativity

TL;DR

This paper explores the possibility of measuring the cosmological gravitational potential within the Solar System using PPN formalism, revealing how local parameters are affected by the universe's matter distribution and identifying a special model insensitive to this potential.

Contribution

It demonstrates that PPN parameters are renormalized by the cosmological potential and introduces a unique model invariant under potential shifts, unaffected by cosmological influences.

Findings

PPN parameters are renormalized by the cosmological potential.

Estimated deviations of PPN parameters are greater than 10^{-6}.

A special post-Newtonian model is invariant and insensitive to the cosmological potential.

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.