Weak-Field Consistency

A central requirement for any singularity-free modification of gravity is that it reproduces the well-tested predictions of general relativity in the weak-field regime. In this section we demonstrate that the regular interior construction introduced above leaves all classical tests of gravity unchanged to observational accuracy. Deviations are parametrically suppressed and become relevant only in regimes far beyond current experimental reach.

Newtonian Limit

In the weak-field, low-velocity limit, the spacetime metric can be written as

where $\Phi(r)$ is the Newtonian gravitational potential. For the class of regular metrics considered here, the asymptotic behavior of the metric function yields

with $g$ denoting the characteristic interior regularization scale (this bracketed relative-correction form, matching the parallel expansions used elsewhere in this chapter, is equivalent to $\Phi(r)=-GM/r+GM\cdot\mathcal{O}(g^3/r^4)$; the correction is proportional to $M$, as required since the regularization itself is sourced by $M$). At distances $r \gg g$, the correction term is negligible, and the standard Newtonian potential is recovered. Consequently, the equations of motion for non-relativistic test particles reduce to Newton's law with high precision, ensuring consistency with laboratory, planetary, and astrophysical dynamics.

Light Deflection

The deflection of light by a massive body is determined by null geodesics in the exterior spacetime. For a light ray with impact parameter $b$, the leading-order deflection angle in general relativity is

In the present model, the exterior geometry coincides with the Schwarzschild solution up to corrections of order $(g/r)^3$. Expanding the null geodesic equation in the weak-field limit, we find

Since all observational tests of light bending involve impact parameters vastly larger than the regularization scale $g$, these corrections are many orders of magnitude below current experimental sensitivity. The model therefore reproduces the classical light-deflection results verified by solar-system experiments and gravitational lensing observations.

Perihelion Shift

The relativistic advance of the perihelion of bound orbits provides another stringent test of gravitational dynamics. For a test particle orbiting a central mass $M$ with semi-major axis $a$ and eccentricity $e$, general relativity predicts a perihelion shift per orbit of

In the regularized geometry, the effective potential governing orbital motion differs from the Schwarzschild case only by terms suppressed by powers of $g/r$. A perturbative analysis of the orbital equation yields

For planetary or stellar orbits, where $a \gg g$, the correction term is entirely negligible. As a result, the observed perihelion precession of Mercury and other systems is reproduced to the same accuracy as in classical general relativity.

Suppression of Post-Newtonian Corrections

More generally, deviations from general relativity in the weak-field regime appear only at higher post-Newtonian order. The regularization scale $g$ introduces corrections that scale as inverse powers of the radius,

ensuring rapid suppression at macroscopic distances. This guarantees that all parametrized post-Newtonian (PPN) parameters coincide with their general relativistic values up to corrections far below current bounds. Consequently, the no-singularity construction preserves the empirical success of classical gravity while modifying only the deep interior, where observational constraints are presently absent.