Now we finally tackle the real singularity at the center.
Up to now:
The fix is to replace the singular point with a finite, regular interior.
We hypothesize (this is a theory choice) that instead of letting spacetime run to infinite curvature, the innermost region is filled with a constant positive energy density \(\rho_c\).
Mathematically, that interior geometry is de Sitter spacetime:
\[ ds^2_{\text{int}} = -\left(1 - \frac{\Lambda r^2}{3}\right)c^2 d\tau^2 + \left(1 - \frac{\Lambda r^2}{3}\right)^{-1} dr^2 + r^2 d\Omega^2 , \]
with
\[ \Lambda = \frac{8 \pi G}{c^4} \, \rho_c. \]
So instead of an undefined blow-up, the interior is as “regular” as empty space with dark energy.
We glue this de Sitter core to the usual Schwarzschild exterior at some radius \(r=r_c\). The rules for gluing spacetimes are given by the Israel junction conditions (a math principle).
\[ \boxed{r_c^3 = \frac{3 M c^2}{4 \pi \rho_c}}. \]
This is the selector equation: the core size \(r_c\) is uniquely determined once you know the black hole mass \(M\) and the universal constant \(\rho_c\).
| Symbol / Term | Meaning | Value | Metaphor |
|---|---|---|---|
| \(\rho_c\) | Critical energy density of the core | Anchored to inflation scale | Like a “vacuum pressure” filling the core |
| \(\Lambda\) | Cosmological constant for the core | \(8\pi G\rho_c/c^4\) | A curvature “dial” set by \(\rho_c\) |
| de Sitter spacetime | Geometry of constant positive energy | Smooth interior metric | Like a perfectly stretched trampoline fabric |
| Israel junction conditions | Math rules for joining spacetimes | Continuity conditions | Sewing two fabrics seamlessly without tears |
| \(r_c\) | Core radius where match occurs | \((3Mc^2 / 4\pi\rho_c)^{1/3}\) | The “guardrail” that replaces the singular trapdoor |
We’ve replaced the singular center with a finite de Sitter-like core at \(r_c\), matched seamlessly to the Schwarzschild exterior.