TITLE:
Ultra-Relativistic Particle Acceleration as an Observational Probe of Stationary Solutions of Einstein’s Field Equations. I. Applications to the Taub-NUT Family of Metrics
AUTHORS:
Charles H. McGruder III
KEYWORDS:
General Relativity, Stationary Spacetimes, Gravitational Repulsion, Radial Coordinate Acceleration, Cosmic Rays, Neutrinos, Taub-NUT Metrics
JOURNAL NAME:
Journal of Modern Physics,
Vol.17 No.9,
September
22,
2026
ABSTRACT: One of the central objectives of observational General Relativity is to determine which exact solution of Einstein’s field equations most accurately describes the gravitational field associated with a specific physical system. Because Einstein’s field equations admit many stationary solutions, the appropriate spacetime geometry cannot be assumed a priori; it must ultimately be established by comparing the observable predictions of candidate solutions with astronomical observations of the specific physical system under investigation. Here we develop an observational methodology for determining which stationary solution of Einstein’s field equations is applicable to a specific compact astrophysical object and for constraining the associated metric parameters, using the radial coordinate acceleration of outward-moving relativistic particles as the theoretical basis. Beginning with a general expression for radial coordinate acceleration in a static radial metric sector, we apply the formalism to the Taub-NUT, dyonic Lorentzian Taub-NUT-AdS, Taub-NUT anti-de Sitter, and Kerr-Taub-NUT geometries. In the Kerr-Taub-NUT case the analysis is restricted to outward radial motion along the symmetry axis; no claim is made concerning general non-axial Kerr-Taub-NUT geodesics. For each spacetime, we derive the radial acceleration expression, determine the conditions for outward motion,
dr/
dt
>0
, with positive radial coordinate acceleration,
d
2
r/
d
t
2
>0
, and evaluate the weak-field ultra-relativistic limit. The resulting acceleration expressions depend differently upon mass, NUT charge, electric charge, asymptotic curvature, and rotation. The radial acceleration is not itself directly measured by cosmic-ray or cosmic-ray neutrino observations; rather, the methodology relates metric-dependent propagation to observable particle quantities together with independently determined source properties. This publication, Paper I, establishes this mathematical and observational methodology. Quantitative applications to cosmic-ray neutrinos and cosmic-ray protons are undertaken in Papers II and III.