7. What Was Enforced, and What Was Chosen

now let’s step back and ask: which parts of this construction were forced by math, and which were choices?

We’ve shown how singularities can be resolved into finite absorbing cores. But along the way, not every step was unique. Some were enforced by math, while others were theoretical choices among alternatives.

7.1 Enforced by Math (No Wiggle Room)

1. Einstein’s equation

   \[ G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}. \]

   • Only local, second-order, conserved tensor equation consistent with GR principles.
   • Rock solid, both mathematically and observationally.

2. Spherical symmetry outside a mass → Schwarzschild geometry

   • Uniqueness theorem: the only vacuum, spherically symmetric solution is Schwarzschild.
   • So the exterior metric was not a choice.

3. Curvature blow-up at \(r=0\)

   • Invariants like the Kretschmann scalar diverge.
   • This is a true singularity, not a coordinate artifact.
   • Thus, a problem that must be addressed.

7.2 Theory Choices (Could Have Been Different)

1. Replacing the singularity with de Sitter

   • We chose a constant positive energy density interior (\(\rho_c\)).

   • Alternatives could include:

     • Quantum gravity inspired “bounces.”
     • Exotic matter cores.
     • Wormhole continuations.

   • We picked de Sitter because it’s the simplest, smoothest, and physically motivated (cosmology already gives us such a vacuum state).

2. Fixing \(\rho_c\) from cosmology

   • Anchoring \(\rho_c\) to the inflationary energy density is a choice.
   • One could imagine other scales (Planck density, QCD scale, etc.).
   • The inflationary anchor ties black holes to cosmology, which is elegant but not enforced.

3. Treatment of geodesics

   • We accept that geodesics end at \(r_c\).
   • Alternative: some models continue them through a bounce to another region (wormholes, baby universes).
   • We opted for a hard stop with finite curvature, which is minimal and conservative.

7.3 What Was Optional but Useful

1. Time-first formulation

   • Standard GR could also describe a de Sitter core (using the same Israel junction conditions).
   • But the lapse-first framework makes horizon regularity and flux laws far more transparent.
   • This was a technical choice that clarifies the math.

2. Absorbing boundary condition

   • We chose the condition \(\Phi'(r_c)=0\), which “freezes” the lapse gradient at the core.
   • This makes the core act like an absorber, not a reflector.
   • Could have chosen a reflective or bouncing condition, but that would alter dynamics drastically.

7.4 Why Our Choices Matter

• Simplicity: de Sitter interior requires no exotic matter or negative energies.
• Predictivity: one free parameter (\(\rho_c\)), fixed by cosmology, locks down the model.
• Consistency: exterior remains pure GR; no observable conflicts.
• Clarity: time-first variables expose which infinities are fake (horizons) and which are real (core).

In short: we picked the most conservative route to eliminate singularities, keep GR intact outside, and add the smallest consistent modification inside.

Glossary

We now see clearly what was dictated by math and what was our chosen resolution strategy.