The extended equivalence principle

Galileo and Newton found that every body falls alike, and Eötvös measured it with torsion balances. In 1907 Einstein read the fact as an equivalence: a laboratory in uniform acceleration and a laboratory at rest in a uniform field cannot be told apart by any experiment inside them. From it he predicted that clocks run slower lower in a field and that light bends near the Sun, and the field equations of 1915 made gravity the geometry of spacetime, with a deflection twice the value of 1911. The principle Einstein left is local. In a small enough laboratory the field vanishes; what remains, the tide between neighbouring falls, is curvature.

FDT extends the principle (core 0.1c): moving at constant speed, on the pattern’s own clocks and rulers, through a density gradient is gravity. The extension rests on the central invariant,

whose terms adjust together. Clocks and rulers are closed patterns of the one medium, and their kept time and thread length move together so that d/t measured on the spot stays c (0.1a). General relativity is the chain kept, state by state: there is no background metric that clocks and rulers merely read. A pattern crossing a density gradient therefore carries no reading of it, and the record of gravity is the bookkeeper’s reading of that motion:

Constant speed, on the pattern’s own clocks and rulers

A pattern moves at constant speed on its own clocks and rulers when no reading on them registers a change in its motion: its accelerometer reads zero, and its path is a geodesic of the geometry its clocks and rulers measure. When it crosses a density gradient its kept time and thread length rescale together, by the clock factor 𝔪:

By the bridges a pattern’s energy E = h/T and its mass m = E/c² both scale as 1/𝔪. Written for the metric the rescaling is g_μν ↦ 𝔪²g_μν, which keeps the null cone. Every reading a pattern makes is a ratio of its own scales, or of a stroke against them, and the clock is made of the thing it times, so no reading on a pattern’s own clocks and rulers registers the value of 𝔪. In the gravitational window the clock factor belongs to the density crossed and the geometry is one for every pattern, so every pattern at a point, whatever it is made of, rescales by the same 𝔪; a medium’s own clock factor, its temperature, is read in the medium’s window.

The bookkeeper’s reading

A distant bookkeeper reads the same pattern on clocks and rulers of his own, the geometry g. A pattern at constant speed on its own clocks and rulers follows a geodesic of 𝔪²g, and read on g it accelerates:

The pull is toward smaller 𝔪, where clocks run slower, and it contains 𝔪 and the pattern’s four-velocity and nothing else. Since in the gravitational window 𝔪 belongs to the point, every pattern at a point, whatever it is made of, falls alike. Moving at constant speed, on the pattern’s own clocks and rulers, through a density gradient is gravity, never an analogy of it and never a force distinct from it.

The clock factor has three readings. Its value at a point is invisible: clocks and rulers adjust, and d/t on the spot stays c. Its gradient is the free fall of one pattern, removed by riding with the pattern. Its second derivatives are the tide between neighbouring patterns, which no choice removes.

Static geometries: a clock factor times an optical geometry

Every static geometry, ds² = −N²c²dt² + h, splits the same way. A pattern held at rest reads c²D ln N on its accelerometer, and the geometry is the clock factor N² times an optical spacetime whose spatial metric is h/N²: strokes run on its geodesics, Fermat’s principle, and a held reader finds a stroke from a held emitter at ω_o/ω_e = N_e/N_o. On the exterior branch of a static spherical source, with α_ext = R_s/r_f, the split is exact:

with the optical metric, whose Fermat law is generalized Snell’s law:

The split separates the readings. A pattern released from rest falls in the optical geometry, in the bookkeeper’s time, at exactly

A pattern held at r_f is not free, and its accelerometer reads the hold, G𝓜/(r_f²N); at leading order the hold is the I-⁠1 pair with the exterior inputs α₁ = R_s/r_f and α₂ = R_s,m/r_f, F_maxα₁α₂ = G𝓜m/r_f². A pattern moved radially at constant speed relative to held patterns reads γ_vG𝓜/(r_f²N): it keeps constant speed on the clocks and rulers of held patterns, which are themselves held, so it is held too. The principle’s constant speed is the pattern’s own, at which the accelerometer reads zero.

The bookkeeper’s reading is a flow

In the time kept by patterns falling from rest far away, dt_own = dt + v dr_f/(c²(1 − α_ext)), the exterior reads exactly

The flow’s patterns move inward at v and keep its time. Each slice of constant t_own is flat, so the whole curvature sits in the flow, and the flow falls at exactly G𝓜/r_f². On the branch v < c, and v → c⁻ at the fold; the flow never crosses into the interior chart. This is Painlevé and Gullstrand’s form of the exterior, drawn as a river flowing into black holes. The bookkeeper’s reading is a flow, present at every size, and a rotating source adds a swirl: the frame drag a gyroscope reads is the flow’s local rotation, half the curl of the swirl.

Every clock is a speed through the flow

On the flow’s flat slices, where the flow moves at V = −v r̂, a clock moving at w keeps the flow’s time at

where γ_rel is its Lorentz factor against the flow pattern passing it. A held clock, w = 0, moves through the flow at v = c√α_ext. Its strand runs at c on a helix, translating against the flow and circulating:

It keeps time by circulating, c dτ = u dt_own, so it keeps u/c = N of the flow’s time. Between two held clocks a stroke reads ω_o/ω_s = N_s/N_o, the ratio of their kept-time rates. Gravitational time dilation of a held clock is the kinematic dilation of its speed through the flow: the clock is slow because part of its strand rate goes into translation against the flow. At the seam, r_f = 2R_s, v = u = c/√2, and the held strand is the 45° strand. Eccentric orbits and free-flying clocks keep the same law along their paths.

One dilation

A clock on a circular orbit moves through the flow in two directions: outward at v, against the inflow, and around at c√(α_ext/2) on the flat slices, which held patterns read as v_orb = c√(α_ext/(2(1 − α_ext))). By the clock law it keeps time at

The “gravitational” and the “velocity” parts of an orbiting clock’s rate are one effect: the clock’s speed through the flow. Against clocks on the geoid, whose rate the geoid potential W₀ fixes, the Earth’s multipoles and rotation included, a GPS clock on its orbit of semi-major axis 26 561.75 km runs fast by

Orbiting clocks keep the geoid’s rate at r = 3G𝓜_⊕/(2W₀) = 9546 km; below it they run slow.

The factor of two

A stroke keeps no time, and a conformal factor leaves null curves as they are: the stroke is blind to the clock factor and runs on the geodesics of the optical metric, generalized Snell’s law. A slow pattern taken to c reads the clock factor alone and bends by R_s/b. The stroke reads the ratio of the ruler and the clock adjustments, and the radial term (1 − α_ext)⁻¹ of the optical metric supplies the second half:

The factor of two in the bending of light is the principle read on a stroke. For the Sun, at the photosphere, it is 1.75″; the exact null orbit at b = 1000R_s bends by 2.003 × 10⁻³ rad, 2R_s/b + (15π/16)(R_s/b)². The delay doubles in the same way. Along a radial stroke c dt = dℓ_opt, so a held clock at r_s that sends a stroke to r_o and back keeps

and a stroke passing at impact parameter b runs an optical length longer than the straight line by R_s ln(4r₁r₂/b²), twice a slow pattern’s: the time delay with γ = 1. On its own terms the stroke is unchanged. The gradient bends its run, delays it and shifts its frequency for the bookkeeper, and its rate, its h per cycle and its ħ per quantum stay, as the quintuple identity shows.

Curvature between neighbours

One pattern reads no curvature. Two neighbouring slow patterns, static or falling radially, separate along the radius and converge across it with the exact tide

which the Riemann tensor of the exterior returns in the frame of a held pattern and in the frame of a pattern falling radially at any speed. Neighbouring strokes deviate by the sectional curvatures of the optical metric,

No choice of path removes either. A gradiometer, two accelerometers a baseline apart, reads the tide; an extended pattern reads it across its own extent, through its l = 2 moment. The tide is the third reading of the clock factor, and the only one the principle leaves.

The medium’s clock factor

For a medium whose stress tensor has perfect form, at zero net conserved count, the projected conservation law and the Gibbs–Duhem relation give, for every equation of state,

The medium’s patterns move at constant speed on their own clocks and rulers in T²g, and the pressure force is the bookkeeper’s reading of their common rescaling: the temperature is the medium’s clock factor. Held static in a static geometry, the medium accelerates by c²D ln N = −c²D ln T, so TN is constant, Tolman and Ehrenfest’s equilibrium. With a conserved count and a fixed entropy per count the clock factor is the enthalpy per count. It is the reading of the medium’s flow, never the rate of a clock the medium carries: that clock runs on the lapse where it sits, and its thermal shift is the time dilation of its own thermal motion.

The strand beneath it

A closed pattern’s strand runs at c. With 𝒰 the pattern’s four-velocity and e a unit vector turning about the run, orthogonal to 𝒰, the strand’s direction K = 𝒰/c + e is null, and its average over the circulation is the four-velocity, c⟨K⟩ = 𝒰. A common rescaling enlarges the strand’s triangle and keeps its shape, the ratios v/c and u/c of the pattern’s own readings. For a pattern held on the exterior branch the angle is fixed by the flow, v/c = √α_ext, and the translation share is α_ext itself: at the seam the held strand stands at 45°, and toward the fold its circulation, its kept time, tends to zero and is never reached.

A pattern’s own gravitational window is α_ext = 𝓜c²/(2F_maxr_f), which needs no numerical G. For 50 GeV at 1 fm it is 1.3 × 10⁻³⁷, and the small number is a correct reading (I-⁠1).

The extended equivalence principle

Clocks and rulers are made of the medium they measure. When a pattern crosses a density gradient its kept time and thread length rescale together, every ratio of the chain stays as it was, and on its own clocks and rulers nothing happens. The distant bookkeeper, reading on clocks and rulers of his own, sees the pattern accelerate toward slower clocks, the same for every pattern. That reading is a flow: around a source the omnium’s patterns fall inward at c√α_ext, and the flow falls at exactly G𝓜/r². Every clock moves through the flow and keeps time at the inverse of its Lorentz factor against it; a clock held against the flow keeps time by the share of its strand that still circulates, which is the lapse. A stroke, which keeps no time, is blind to the clock factor and is bent and delayed by the optical metric alone, twice as much. What no choice removes is the tide between neighbours. Moving at constant speed, on a pattern’s own clocks and rulers, through a density gradient is gravity.

Each line below is the converse of a derived result: the measurement that would end it.

1. T-⁠1
   Universality. The acceleration law contains the clock factor alone, and in the gravitational window the clock factor belongs to the point. A composition-dependent free fall, a nonzero η, ends that reading.
2. T-⁠2
   Every clock is a speed through the flow. A clock comparison whose rates depart from the clock law along the clocks’ paths, beyond the source’s multipoles, held clocks reading other than N_s/N_o among them, ends it.
3. T-⁠3
   One dilation. An orbiting clock whose rate departs from √(1 − 3α_ext/2) against the flow’s time, beyond the source’s multipoles, ends the one-dilation result.
4. T-⁠4
   The factor of two. A stroke bent or delayed by other than twice the slow reading at leading order, γ ≠ 1, ends the exterior split’s reading of the stroke.
5. T-⁠5
   The swirl. A frame drag other than half the curl of the flow’s swirl ends the swirl reading of 0.1c.

Three pieces sit outside the derivation and leave every result in it unchanged:

1. D-⁠01
   The interior chart. The flow reading stops at the fold; the kinematic reading of the share on the interior chart is to be derived (core O-⁠16).
2. D-⁠02
   Beyond the monopole. Multipole and tidal remainders for nonspherical and rotating sources, and the band between a rotating source’s horizon and R_s, are to be derived (core O-⁠5).
3. D-⁠03
   The aperture. Strokes crossing closed patterns bend by the aperture of the density cone, whose constitutive law is to be derived (core O-⁠8).

For the same sources and data FDT returns the universality of free fall, with the MICROSCOPE bound; the gravitational redshift of every clock comparison, held or moving, from a millimetre to 10 000 km and on eccentric orbits; the rates of orbiting clocks, GPS among them; the deflection and the time delay of light with γ = 1; the tides read by gradiometers and by extended bodies; the geodetic and frame-dragging precessions of Gravity Probe B; and Tolman and Ehrenfest’s thermal equilibrium. The curvature is that of general relativity for the same source and data (I-⁠3); the pair force is exact on its domain, F_maxα₁α₂ = G𝓜₁𝓜₂/d² (I-⁠1), and general relativity reduces to the same law in the weak-field, slow-motion limit (I-⁠4).

No divergence is claimed: the paper supplies what gravity is, not a changed observable.