3. Dynamics and the Role of the Lapse Field

So far, Schwarzschild showed us what spacetime looks like around a static spherical mass. But real black holes form from collapse and can change mass (by swallowing matter or radiating).

To describe that properly, we need coordinates that stay regular at the horizon and an evolution law that directly tracks energy flow.

3.1 The Lapse Function \(N = e^{\Phi}\)

In the time-first formulation, the central player is the lapse field \(N\), written as \(N = e^{\Phi}\).

The lapse tells you how proper time \(d\tau\) relates to coordinate time \(dt\):

\[ d\tau = N \, dt = e^{\Phi} dt. \]
In ordinary GR, the lapse is treated as a gauge multiplier. In time-first GR, it becomes the primary degree of freedom that governs temporal flow.

Why this helps:

Horizons appear when \(N \to 0\) (as in the Schwarzschild form), but that’s just a coordinate choice.
By switching to Eddington–Finkelstein (EF) or Painlevé–Gullstrand (PG) coordinates (which are natural in lapse-first form), the lapse remains regular across the horizon.
Thus, horizons are gauge artifacts, not singularities.

3.2 The Flux Law (Dynamic Evolution)

The mixed Einstein equation (the \(tr\) component) in spherical symmetry simplifies beautifully to:

\[ \partial_t \Phi = -4 \pi G \, r \, T_{tr}. \]

This says: the time field \(\Phi\) only evolves when energy flux crosses a spherical shell of radius \(r\).

If we integrate this at a fixed radius \(R\), we get:

\[ \partial_t \Phi = -\frac{G}{c^4} \, \frac{P(t)}{R}, \]

where \(P(t)\) is the total power (energy per unit time) crossing the sphere.

If nothing crosses, \(\Phi\) is frozen: the geometry is static.
If radiation or matter flows in, \(\Phi\) evolves accordingly.

This is essentially the Vaidya law (GR’s description of accreting/evaporating black holes), but here it’s written directly in terms of the lapse which makes regularity at the horizon manifest.

3.3 Why this step matters

We’ve clarified that horizons are not singular; they’re coordinate effects.
We now have a local, physical rule: spacetime evolves only when energy flux crosses a surface.
The true singularity problem remains at \(r=0\), but we’re set up to handle it: we can ask, “what if the lapse evolution naturally freezes before reaching a singular blow-up?”

Glossary

Symbol / Term	Meaning	Value	Metaphor
\(N = e^{\Phi}\)	Lapse function (time pacing)	Variable of the theory	A “gearbox” telling how fast proper time ticks compared to coordinate time
\(d\tau\)	Proper time (measured by a clock)	Depends on \(N\)	The ticking of a real wristwatch
EF / PG coordinates	Horizon-regular time slicings	Coordinate choice	A camera angle that doesn’t glitch at the horizon
\(\partial_t \Phi = -4 \pi G r T_{tr}\)	Flux law for lapse evolution	Einstein equation result	“Only real energy flow changes time’s gears”
\(P(t)\)	Power crossing a sphere	Energy flux input	Like the brightness of a flashlight passing through a surface

We now have a dynamic rule that cleanly separates fake singularities (horizons) from the real issue (the core).

← Chapter 2: Spherical Solution Chapter 4: Replacing Singularity →