TL;DR. We reorganize Einstein's gravity so time (the lapse N=eΦ) is the primary dial, and space follows from constraints. In spherical symmetry (zero shift), the spatial metric is fixed by grr=1/gtt; dynamics appear when there's actual energy flux. Cosmology maps cleanly via a=e-Φ and H=-Φ', reproducing standard FRW with different intuition.
When physicists hear about yet another "reformulation" of general relativity, skepticism is natural. Einstein's geometric theory works perfectly well. Why fix what isn't broken? Sometimes reorganizing familiar physics reveals hidden structures that change how we understand reality itself.
This framework recasts gravity not as curved spacetime, but as the geometry of time. Before dismissing this as mere mathematical shuffling, consider what the mathematics actually reveals.
The complete temporal geometry framework - read the papers, explore detailed explanations, and follow step-by-step derivations:
Scope (clear and testable). In spherical symmetry with zero shift, the entire spatial geometry is algebraically determined by a single temporal function Φ (the logarithm of the lapse): grr=e-2Φ=1/gtt. In that regime, space carries no independent degrees of freedom—the radial constraint enslaves it to Φ. Outside spherical symmetry or in time-dependent settings, we allow a nonzero shift (Painlevé–Gullstrand / Eddington–Finkelstein gauges) while keeping Φ primary.
This isn't just a coordinate trick valid only in one gauge. While the exact algebraic relation holds in the diagonal gauge, the broader principle extends everywhere: given the lapse Φ and matter content, the constraints determine spatial curvature. We can choose coordinates where time's flow is all in the lapse (static problems) or distributed into shift (flowing-space view) - the physics remains the same. The framework handles Painlevé-Gullstrand and Eddington-Finkelstein coordinates just as naturally.
Equivalence (classical): Given fields (Φ, ωi, γij) that satisfy the ADM constraints, the constructed metric solves Einstein's equations; conversely any GR solution can be written in this lapse-first form. No extra degrees of freedom are introduced.
Integrates to e2Φ = 1 − 2GM/r, with grr = 1/gtt.
Only real energy flux changes the temporal field Φ.
This mapping reproduces the standard FRW equations exactly.
Spherical evolution law. In the diagonal, spherical gauge the mixed Einstein equation gives a simple driver for time's geometry:
Radial energy flux directly modulates clock-rate drift. With no flux, ∂tΦ=0 in this gauge (Birkhoff in time-first clothing). For time-dependent inflow/outflow, this law is equivalent to the usual Vaidya picture after a coordinate change—same physics, different bookkeeping of lapse vs. shift.
Rotation and frame-dragging live in the gravitomagnetic shift ω, sourced by mass currents. In the linearized limit, ∇²ω = −16πG J and Bg = ∇×ω reproduces Lense–Thirring. The scalar Φ does not itself generate frame-dragging; currents do.
Every system with an internal frequency ω=E/ℏ is a clock. If Φ fluctuates quantum-mechanically, proper-time phases inherit those fluctuations. For an interferometer, the visibility reduces by a universal ω2 factor set by the two-point function of δΦ:
This universal ω² scaling means heavier objects lose quantum coherence faster because they're better clocks. Gravitationally-mediated entanglement would be direct evidence that Φ is a quantum field. Current experiments searching for mesoscopic superpositions are unknowingly probing temporal coherence.
Crucially, this doesn't add new particles to physics. The ADM analysis confirms that Φ and ω are Lagrange multipliers enforcing constraints - only the two standard gravitational wave polarizations propagate. We're not sneaking in a scalar graviton; this is exactly Einstein's theory, reorganized.
Degrees of freedom check. In ADM form, Φ (lapse) and ωi (shift) are non-dynamical multipliers enforcing constraints. Linearizing shows only the two tensor polarizations propagate—no extra scalar graviton is introduced.
The cosmological mapping a = e(-Φ) recasts the expanding universe in temporal terms. The universe doesn't expand "through" time - it expands as time itself diverges. Early universe conditions correspond to large positive Φ (rapid time flow), evolving toward negative Φ (slower time) in the future.
This reframing offers two interpretations of dark energy: either a constant "temporal vacuum pressure" or a potential V(Φ) that slowly rolls. Both reproduce ΛCDM-like results but suggest the acceleration stems from time's inherent dynamics. The Big Bang becomes temporal vacuum decay - the universe born from tunneling of Φ from a high-energy state. At the opposite extreme, Φ → -∞ represents a true end of evolution, avoiding eternal heat death.
Mathematically, inserting a=e-Φ and H=-Φ' into Einstein's equations reproduces the standard FRW pair exactly, so the background expansion history is identical once you specify the same stress–energy.
Define cosmic time by dτ = eΦ dt and set a = e−Φ. Then H = −Φ'(τ) and the standard Friedmann pair follows exactly.
At the Schwarzschild radius, the lapse function N = eΦ vanishes. This isn't where "space becomes infinitely curved" but where time stops flowing for external observers. The horizon is fundamentally a temporal phenomenon. A boundary in phase space rather than physical space.
This perspective reframes the information paradix: if information is encoded in temporal phase correlations, then understanding phase coherence near horizons might resolve the puzzle without exotic modifications.
In the static, spherically symmetric ansatz ds2 = -e2Φ(r)dt2 + e-2Φ(r)dr2 + r2 dΩ2, Einstein's equations collapse to a single separable ODE:
Integrating gives e2Φ(r) = 1 − 2GM/r. The spatial factor follows algebraically, grr = 1/gtt — space follows time.
The framework predicts distinctive quantum signatures that connect to ongoing quantum-gravity experiments:
These aren't abstract predictions. They're testable with current atomic interferometry and quantum sensing technology.
For an ingoing luminosity pulse L(t) at radius r (approximately null and radial):
So outside the source, clocks slow during the pulse by a predictable, sign-definite amount that scales like L(t)/r. This is the clean target for clock networks.
While mathematically equivalent to GR classically, this framework makes experimental predictions particularly transparent:
This framework suggests reality is primarily about causal relationships and temporal ordering rather than spatial embedding. Space emerges as consistency conditions on temporal flow. The universe isn't a 3D space evolving in time, but a self-consistent pattern of temporal phases that we perceive as spatial when we demand simultaneity.
Whether this temporal view represents gravity's "true nature" or simply a powerful alternative perspective remains to be determined. But by making gravity's effects directly measurable through humanity's most precise instruments, atomic clocks and interferometers, it transforms gravitational physics into experimental temporal metrology. The deepest questions about black holes, dark energy, and quantum gravity become questions about time itself.
Static A(r)=e2Φ(r) can be written in Painlevé–Gullstrand (flat slices, unit lapse) or Eddington–Finkelstein (null-adapted) coordinates; the mass-flux statements match Vaidya's dm/dv balance exactly after converting components.
Sometimes the most profound advances come not from new theories but from recognizing what was in front of us all along.
Recurring technical questions, answered tersely with pointers to the math used on this site.
We work lapse-first: choose N=eΦ as primary and allow shift when needed. In spherical symmetry with zero shift, the constraints give grr=1/gtt. For dynamics or rotation we use nonzero shift (Painlevé–Gullstrand / Eddington–Finkelstein) and recover the same physics.
No. In ADM, Φ and ωi are Lagrange multipliers enforcing the Hamiltonian and momentum constraints; linearization leaves only the two TT modes.
A local flux law: ∂tΦ = -4πG r Ttr. No radial energy flux ⇒ ∂tΦ=0 in this gauge (Birkhoff). With flux, a change of variables matches the Vaidya description (dm/dv).
In the gravitomagnetic shift ωi, sourced by mass currents Tti. The linearized solution reproduces Lense–Thirring and the Kerr gtφ term. Oscillating Φ alone doesn't produce frame dragging.
Use dτ=eΦdt, a=e-Φ, H=-Φ'. Plugging into Einstein's equations gives the usual FRW pair exactly; observational pipelines (distances, growth) carry over unchanged.
We frame the Big Bang as a boundary plus temporal vacuum decay: nucleation from very large +Φ (where a→0) to finite Φ, followed by roll. The Φ→−∞ limit is a degenerate "no-time" boundary, not a parent vacuum.
Yes—two tracks. Laboratory: clock-network tests of the flux law and universal ω2 dephasing in quantum-clock interferometry. Cosmology: the framework suggests concrete signatures (e.g., curvature floor and specific non-Gaussian relations) that next-gen surveys can probe; we detail these in Paper IV.2.
No. It's a re-parameterization. The constraint algebra (hypersurface deformation) is unchanged; we're choosing variables and gauges that make certain structures manifest.
They are different slices of the same physics. Ingoing null flux in EF (Vaidya) maps to the spherical flux law here via a simple tensor relation between Tvv and Ttr.
No. Rotation lives in the shift sector, but in GR radiation remains the usual tensor + and × modes; no additional vector/scalar modes are introduced by treating the lapse as primary.