Let’s start at the beginning. The most basic, undeniable principles. Then let's build carefully toward the singularity-free black hole solution.

1: Gravity’s Foundation. Einstein’s Equation

The unshakable bedrock is Einstein’s field equation of General Relativity (GR):

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

The left-hand side encodes the geometry of spacetime through the Einstein tensor \(G_{\mu\nu}\). It contains derivatives of the metric (the mathematical object that tells us how distances and times are measured).
The right-hand side encodes the matter and energy content through the stress–energy tensor \(T_{\mu\nu}\).
\(G\) is Newton’s gravitational constant, and \(c\) is the speed of light.

This equation is enforced by math + observation:

Mathematically, it’s the unique tensor equation that is local, second order in derivatives, and conserves energy–momentum (Noether’s theorem guarantees that the left side has zero divergence).
Observationally, it reduces to Newton’s law in the weak-field limit and has been tested in solar system dynamics, gravitational lensing, and gravitational waves.

So, Step 1: The geometry of spacetime is directly determined by matter and energy.

Symbol / Term	Meaning	Value	Metaphor
\(G_{\mu\nu}\)	Einstein tensor (curvature of spacetime)	Calculated from metric	Like a “curvature map” of spacetime’s bends
\(T_{\mu\nu}\)	Stress–energy tensor (matter + energy content)	Physical input	Like a detailed budget of energy, momentum, and pressure
\(G\)	Newton’s gravitational constant	\(6.67\times10^{-11}\,\text{N·m}^2/\text{kg}^2\)	Sets the strength of the “glue” of gravity
\(c\)	Speed of light	\(3\times 10^8\) m/s	Cosmic speed limit linking time and space
Metric \(g_{\mu\nu}\)	Rule for measuring intervals in spacetime	Variable solved by Einstein’s equation	A ruler + stopwatch woven into one object

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