Step 5. Dynamics of the Absorbing Core

Now that the singularity has been replaced by a finite core, we need to understand how that core evolves when the black hole is not static.

Black holes aren’t static in reality. They can gain mass (by swallowing stars, gas, radiation) or lose mass (through Hawking evaporation). So the question is: if the total mass \(M\) changes, what happens to the core radius \(r_c\)?

5.1 Recall the Selector Equation

From Step 4, the core radius is determined by

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

If \(M\) changes, then \(r_c\) must change accordingly. Differentiate both sides with respect to time (or advanced time coordinate \(v\)):

\[ 3 r_c^2 \frac{dr_c}{dv} = \frac{3 c^2}{4\pi \rho_c} \, \frac{dM}{dv}. \]

Simplify:

\[ \boxed{\frac{dr_c}{dv} = \frac{\dot M(v) \, c^2}{4\pi \rho_c \, r_c^2}}. \]

5.2 Physical Meaning

\(\dot M(v)\) is the rate of change of mass (positive for accretion, negative for evaporation).
This equation says the core grows or shrinks only in response to actual flux of matter/energy.
Nothing exotic needed: the same Einstein equations that gave us Schwarzschild now govern the core’s dynamics.

5.3 Relation to Vaidya Spacetime

In GR, the Vaidya solution describes a black hole that changes mass due to ingoing or outgoing radiation.

In standard form, Vaidya is written with a mass function \(M(v)\).
In the time-first language, this becomes a direct law for the core radius evolution.
Importantly, this equation is horizon-regular: it doesn’t blow up at \(r_h\). That’s another way the lapse-first formalism avoids fake singularities.

5.4 Big Picture

Now we have:

A static matching rule (Step 4) that sets the core size given the mass.
A dynamic law (this step) that evolves the core when the mass changes.

Together, these make the absorbing core a predictive feature: it responds smoothly to infalling matter or outgoing radiation, never diverges, and never needs arbitrary parameters.

Glossary

Symbol / Term	Meaning	Value	Metaphor
\(\dot M(v)\)	Time derivative of mass	In/out flux of energy	How fast the black hole is eating or evaporating
\(dr_c/dv\)	Time derivative of core radius	Given by evolution law	The “breathing” of the core as it absorbs/releases energy
\(v\)	Advanced time coordinate (regular at horizon)	Variable	A clock that smoothly follows infalling radiation
Vaidya spacetime	GR solution for a radiating/absorbing black hole	Matches with our flux law	A “dynamic Schwarzschild” tuned by energy flux
Horizon-regular	Behavior that stays smooth at \(r_h\)	Built-in via lapse-first	Like switching to a camera angle where nothing glitches

The absorbing core breathes with mass flow, governed by a clean evolution equation tied to energy flux.

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