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Gravity as Temporal Geometry (Φ‑first GR)

Big idea: Treat the lapse \(N=e^{\Phi}\) as primary and let space follow time via the constraints. In this view, only energy flux can change time curvature, cosmology becomes a one‑function story, and all classic GR results drop out with less machinery.

Why look at GR through time?

Einstein’s equations don’t privilege time or space, but our measurements do: clocks and light‑travel times are what we actually observe. The lapse‑first (time‑first) formulation leans into that: we write the metric so a single scalar \(\Phi\) controls the flow of proper time, and the radial piece of space is slaved to it. This flips the intuition—mass bends time; space adjusts to keep the constraints—while remaining exactly equivalent to GR.

The metric in one line

\[ ds^2 = -e^{2\Phi(t,r)} dt^2 + e^{-2\Phi(t,r)} dr^2 + r^2 d\Omega^2,\qquad A\equiv e^{2\Phi}. \]

With this diagonal, areal‑radius gauge, the Einstein equations simplify dramatically. Two consequences drive nearly everything that follows:

Vacuum enforces staticity (Birkhoff, in this gauge): the mixed equation gives \(G_{tr}\propto \partial_t\Phi\). In vacuum \(T_{tr}=0\Rightarrow\partial_t\Phi=0\). Schwarzschild follows from a single separable ODE.
With matter, flux drives time: the same equation becomes a flux law tying time evolution to radial energy flow.

Flux law (heart of the framework):

\[ \boxed{\;\partial_t\Phi = -4\pi G\,r\,T_{tr}\;} \quad \text{or, at radius }R:\quad \boxed{\;\partial_t\Phi(t,R) = -\dfrac{G}{c^4}\,\dfrac{P(t)}{R}\;} \]

Only actual energy flux can evolve the temporal potential. No flux, no evolution.

Schwarzschild in one step

Setting \(\partial_t\Phi=0\) (vacuum), the remaining components integrate to

\[ e^{2\Phi(r)} = 1 - \frac{2GM}{r},\qquad g_{rr} = \big(1-2GM/r\big)^{-1}. \]

All classic tests: redshift, light bending, Shapiro delay, precession, then follow from \(\Phi(r)\) alone; the spatial factor simply mirrors it via \(g_{rr}=e^{-2\Phi}\).

Vaidya, but lapse‑first

When null radiation falls in, the standard Eddington–Finkelstein/Vaidya picture tracks a mass function \(m(v)\). In lapse‑first variables we keep the metric diagonal and let \(A(t,r)=1-2GM(t,r)/r\). The flux law gives

\[ \partial_t A = -8\pi G\, r\,A\,T_{tr},\quad \text{so positive ingoing flux increases } M(t,r). \]

Changing to advanced time \(v\) reproduces Vaidya exactly; the physics is identical, the bookkeeping is cleaner.

“Space follows time” ≠ “no shift”

The diagonal gauge is for clarity, not dogma. A static re‑time gives the Painlevé–Gullstrand “flowing space” form, and advanced/retarded times give the EF/Vaidya form. The content is the same single function \(A=e^{2\Phi}\); you can move it between lapse and shift at will.

Rotation lives in the shift. In the weak‑field, quasi‑stationary limit the fields split like electromagnetism: \( \nabla^2\Phi=4\pi G\rho\) and \( \nabla^2\boldsymbol\omega=-16\pi G\mathbf J\). Frame dragging is the curl of the shift (gravitomagnetism), not a time‑derivative of \(\Phi\).

Cosmology from a single function

Homogeneous/isotropic spacetimes collapse to a one‑function story. Define proper (cosmic) time by \(d\tau=e^{\Phi}dt\) and the scale factor by \(a(\tau)=e^{-\Phi}\). Then

\[ H \equiv \frac{\dot a}{a} = -\dot\Phi,\qquad H^2+\frac{k}{a^2}=\frac{8\pi G}{3}\,\rho,\qquad \dot H=-4\pi G(\rho+p)+\frac{k}{a^2}. \]

This reproduces standard FRW exactly while making the ontology explicit: expansion is time diverging.

ADM in lapse‑first clothing (no extra graviton)

Promoting \(N=e^{\Phi}\) does not add a propagating scalar. In the 3+1 (ADM) split, \(N\) and the shift \(\omega_i\) are Lagrange multipliers enforcing the Hamiltonian/momentum constraints; after linearization only the two TT tensor modes propagate. Same physics, clearer ledger.

What’s unusual (and useful) in this lens

One‑field control: In spherical symmetry, a single scalar \(\Phi\) encodes time curvature; space follows as \(g_{rr}=e^{-2\Phi}\).
Flux‑first dynamics: \(\partial_t\Phi\) responds only to energy flux. This pins down which physical processes can really drive “time evolution.”
FRW as a definition: Setting \(a=e^{-\Phi}\) and \(d\tau=e^{\Phi}dt\) turns the Friedmann equations into identities for \(\Phi(\tau)\).
Gauge honesty: Horizons are gauge effects in lapse‑first variables; the only true pathology to cure is the central singularity (addressed in later papers).
Cleaner bridges: Vaidya (null flux), PG (flat slices), EF (horizon‑regular) all become simple re‑timings of the same \(A=e^{2\Phi}\).

Five‑minute derivation tour

Start from the ansatz \(ds^2=-A dt^2 + A^{-1}dr^2+r^2d\Omega^2\) with \(A=e^{2\Phi}\).
Compute \(G_{\mu\nu}\) and note the mixed piece: \(G_{tr}=-(2/r)\,\partial_t\Phi\).
Vacuum \(T_{tr}=0\Rightarrow\partial_t\Phi=0\); the radial equations separate to give Schwarzschild.
With flux you get the boxed law \(\partial_t\Phi=-4\pi G r T_{tr}\) and the time‑first Vaidya picture.
Cosmology with \(\Phi=\Phi(t)\) maps to FRW by \(d\tau=e^{\Phi}dt,\ a=e^{-\Phi}\).

Does anything change observationally?

No in the regimes we can test today. You recover all classic predictions of GR. What changes is the bookkeeping and (we think) the clarity: time carries the dynamical load, space follows from constraints. That clarity pays dividends in collapse, cosmology, and eventually quantum questions.

Bottom line: Same GR, different lens. Lapse‑first variables turn many multi‑equation derivations into one‑liners and make the causal driver, energy flux, explicit. If you like thinking with clocks, this math is for you.