Unavoidable Singularities, Cosmic Censorship, and the Schwarzschild Metric

We revisit Penrose’s 1965 singularity theorem paper and note that its physical interpretation as-

sumes the exterior Schwarzschild metric and Einstein’s equations remain valid inside the horizon,

but the exterior Schwarzschild geometry becomes undefined there. We argue Einstein and Rosen

(ER) implicitly corrected a structural limitation of Schwarzschild metric in 1935 by their param-

eter substitution. The same structural limitation of the exterior Schwarzschild metric, remains in

Penrose’s argument. We show ER’s coordinate substitution which is a prarameterized full Newton–

Kepler balance equationa removed the central singularity. ER’s and our resulting modified metric,

implies a dual-curvature gravitational system, a discrete spacetime geometry, and a final equilib-

rium state. We show the interior Schwarzschild lapse function likewise can be separated into a

dual-curvature structure: consisting of a central (inner) curvature scale (Φinner := √1−

rs

rg ), and

an orbital (outer) (Keplerian) curvature scale contributions (Φouter(r) := √1−

r2 rs

r3

). Their nonlin-

g

ear combination, gtt =−

1

4 (3Φinner− Φouter)2c2

, reveals a discrete equilibrium condition Rnrn = 1.

The dual–curvature gravitational model developed, shows central singularities are avoidable with-

out the Penrose’s predicted catastrophic implications. We show not only the interior Schwarzschild

metric has dual–curvature, it is also equivalent to our modified exterior Schwarzschild metric, and

prevents trapped surfaces. This dual–curvature model generalizes the Einstein–Rosen construction

and replaces the classical interior with a discrete, nonsingular geometry.

A key observation is that the orbital curvature term (GM/r3

g )r2 is identical to the Keplerian

scaling (4π2r2/T 2). Kepler’s Third Law can be rewritten as T 2

2Vtorus

2

=

GM =

Gρ , showing that

the toroidal volume (Vtorus = 2π2r3) is already encoded in classical Keplerian orbital dynamics.

The mass density–orbital period relation (T ∝ 1/√ρ) provides a geometric mechanism for planet

formation, stellar collapse, and curvature evolution.