2. The Spherical Solution

Schwarzschild Geometry

Given Einstein’s equation, the next natural question is: “What does spacetime look like outside a perfectly spherical, non-rotating mass (like an idealized star or planet)?”

Mathematically, this is enforced by symmetry:

• Spherical symmetry: the spacetime must look the same in all directions around the center.
• Static: no time dependence outside (the star just sits there).

Plugging these assumptions into Einstein’s equations yields the Schwarzschild solution:

\[ ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right)c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\Omega^2 . \]

where \(d\Omega^2 = d\theta^2 + \sin^2\theta\, d\phi^2\) is the angular part of a sphere.

What this solution tells us:

1. Outside the mass, this is the unique solution (no choices left once you demand spherical symmetry). → This is enforced by math.

2. Two “special” radii appear:

   • Schwarzschild radius

     \[ r_h = \frac{2GM}{c^2}. \]

     At this radius, the coefficient of \(dt^2\) goes to zero, and the coefficient of \(dr^2\) blows up.

   • Center \(r=0\). At this point, curvature invariants like the Kretschmann scalar

     \[ K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} \]

     diverge to infinity.

3. Interpretation:

   • The blow-up at \(r_h\) (the horizon) is a coordinate singularity: it can be removed by using different coordinates (like Eddington–Finkelstein or Painlevé–Gullstrand). It’s not physical. → This was a choice of coordinates.
   • The blow-up at \(r=0\) is a real singularity: curvature invariants (coordinate-independent) truly go infinite. That’s a problem of GR itself. → This is enforced by math, not removable by a trick.

Where we are now:

We now know that standard GR predicts unavoidable singularities at the center of a black hole. That’s the starting point for the “problem.”

Our paper’s contribution is: keep the exterior Schwarzschild intact, but replace the singular \(r=0\) region with a finite, physical core.

Glossary