8. Predictions and Testability

Now let’s ask the final natural question: If black holes really have finite absorbing cores instead of singularities, what predictions follow? Could we ever test this?

8.1 What the Model Predicts

1. Core radius scaling The core radius depends on mass as

   \[ r_c^3 = \frac{3 M c^2}{4 \pi \rho_c}. \]

   • Larger black holes have larger cores.
   • Scaling law is exact once \(\rho_c\) is fixed.

2. Finite curvature at the core

   • Instead of diverging, invariants like Ricci scalar and Kretschmann scalar remain finite.
   • Prediction: no singularity forms during collapse. The process halts at \(r_c\).

3. Absorbing boundary condition

   • Infalling matter and radiation disappear into the core, not a singularity.

   • The boundary grows/shrinks according to

     \[ \frac{dr_c}{dv} = \frac{\dot M c^2}{4\pi \rho_c r_c^2}. \]

4. Cosmology link

   • \(\rho_c\) is chosen to match the inflationary energy scale.
   • This links black hole interiors to the same physics that drove the early universe’s expansion.
   • Suggests a deep connection: black holes and the Big Bang may share the same “substrate.”

8.2 Observational Challenges

1. Tiny ratio of scales For a stellar-mass black hole:

   \[ \frac{r_c}{r_h} \sim 10^{-23}. \]

   • \(r_c\) is absurdly small compared to the horizon.
   • Practically impossible to probe directly with near-horizon observations.

2. Indistinguishable exterior

   • Outside the horizon, everything looks exactly like Schwarzschild.
   • All existing tests of GR are automatically satisfied.

8.3 Where Signals Might Appear

1. Cosmology

   • If \(\rho_c\) is anchored to the inflationary plateau \(V_*\), then measuring early-universe parameters constrains black hole interiors.
   • Conversely, black hole physics could constrain inflationary models.

2. Quantum gravity phenomenology

   • In principle, Hawking evaporation might terminate on a finite core rather than a complete disappearance.
   • Endpoints of evaporation could differ from traditional “Planck relic” scenarios.

3. Gravitational wave echoes?

   • Some models predict reflections near the core.
   • Our model imposes an absorbing boundary at \(r_c\), so no echoes are expected. That absence is itself a prediction.

8.4 What This Resolution Really Means

• The model does not make black holes observable in a new way today, the core is too small.

• Instead, it gives a conceptually clean resolution of singularities:

  • No infinities in curvature.
  • No violations of energy conditions.
  • A predictive link to cosmology.

It turns black holes from “mysteries hiding singular edges” into well-posed physical objects.

Glossary

The model resolves singularities while keeping exterior GR intact, with predictions tied to cosmology rather than astrophysical observations.