Measuring Space-Time Geometry over the Ages

TL;DR

This paper develops an exact, non-perturbative formalism for measuring space-time geometry through observational data over time, revealing that only a fraction of curvature information can be obtained within practical timescales.

Contribution

It introduces a novel observational framework for space-time geometry that does not rely on assumptions like Einstein's equations, enabling direct measurement from observations.

Findings

15% of curvature info accessible without long-term data

35% of curvature info accessible with decades-long observations

Most curvature details require centuries of data collection

Abstract

Theorists are often told to express things in the "observational plane". One can do this for space-time geometry, considering "visual" observations of matter in our universe by a single observer over time, with no assumptions about isometries, initial conditions, nor any particular relation between matter and geometry, such as Einstein's equations. Using observables as coordinates naturally leads to a parametrization of space-time geometry in terms of other observables, which in turn prescribes an observational program to measure the geometry. Under the assumption of vorticity-free matter flow we describe this observational program, which includes measurements of gravitational lensing, proper motion, and redshift drift. Only 15% of the curvature information can be extracted without long time baseline observations, and this increases to 35% with observations that will take decades. The rest would likely require centuries of observations. The formalism developed is exact, non-perturbative, and more general than the usual cosmological analysis.