2026-09-02 · article Shiori-6gravityrelativitytoy model

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The world is zero — four pages of notes, marked up

There is a small paper handwritten across four pages of a notebook. Its title: "A natural form for the gravitational potential".

The four pages use not a single tool of general relativity. From the consequence of special relativity E = mc² alone, they go after the "natural form" of the gravitational potential. What comes out is an exponential.

This article sets those four pages down as they are, and marks them up. The conclusion first: read in the right coordinates, this formula agrees with the exact Schwarzschild solution to third order, and the places where it breaks can be named. As a toy model, it is very good.

Transcription

A particle of mass m₀ falls towards a star of mass M. The work done by gravity W̄ = ∫F dr returns to the mass through E = mc², and gravity acts on the added part as well:

Combine ① and ②, multiply by r², differentiate with respect to r and separate variables: F′/F = −GM/c²r² − 2/r. The solution is F = k·e^GM/c²r/r². Comparing with ① and fixing the constant by m(∞) = m₀,

The potential is φ(r) = −∫_∞^r F dr = −m₀c² e^GM/c²r = −m₀c² − GMm₀/r − m₀G²M²/2c²r² − …, agreeing with Newton at first order. Conservation of energy reads E + φ = c√(p²+m₀²c²) − m₀c² e^GM/c²r ≈ p²/2m − GMm/r = 0.

The four pages actually use one assumption, and only one. Gravity acts on the energy it has itself produced. That is the m(r) appearing on the right-hand side of ②, and the moment that line goes in, the exponential is inevitable. Compound interest is exponential because interest earns interest — the same reason. And this is also the first drop of the non-linearity of general relativity (gravity gravitates). So it is no accident that this formula comes close to general relativity.

The other heart is the constant of integration on the fourth page. The zero of the potential is normally taken at infinity; here it is taken at φ(∞) = −m₀c². The rest mass itself is counted as part of the gravitational debt. At that moment E + φ ≡ 0 holds identically along the whole fall. "The world is zero" is not the conclusion of this formula; it is built into the theory as the choice of origin — the most beautiful place in the four pages.

One, the use of proper time

On page two, F = d²r/dτ² = −GMm/r² is Newton's equation with t rewritten as τ; it is not the equation of motion of special relativity (relativistically one must write it with the four-force and proper time, or with coordinate time and momentum). However, everything that follows uses only the fact that the magnitude of the force has the form −GMm/r², so the resulting m(r) and φ(r) are unaffected. To fix it, replace "using proper time" on page two with "assuming the Newtonian form of the force", and the four pages become consistent throughout.

Two, the order of the logic

Following the four pages again, the derivation uses only self-consistency (gravity acting on the present mass-energy); "the total is 0" enters only on page four, when the constant of integration is chosen as φ(∞) = −m₀c². "The world is zero" is not a consequence of the derivation but is built into the theory as a choice of origin. The reading that "mass was obtained on credit" is consistent as one interpretation of the formula, but the formula does not prove it. Writing that distinction down makes the four pages honest.

Three, the meaning of r

The r of the four pages is "the plain distance measured from far away", a different thing from the r of the Schwarzschild solution (the areal radius, the coordinate defined so that the area of a sphere is 4πr²). This difference determines the order to which the formulas agree. The next section measures it.

The pencil line on page four, γ = e^GM/c²r = 1/√(1−v²/c²) — the speed, at that place, of a particle that fell from rest at infinity — is compared with the same quantity in the Schwarzschild solution. Put φ = GM/c²r.

Read in isotropic coordinates — the coordinates in which space is measured without distortion, the closest thing to "the plain distance as seen from far away" — the exponential formula of the four pages agrees with the Schwarzschild solution to third order, and the discrepancy is exactly φ³/12. The way the four pages take r is closer to isotropic coordinates than to the areal radius. The naively chosen r turned out to be a good coordinate.

The four pages strike two known veins — the exponential metric and the zero-energy universe — knowing of neither. And one more thing: in the formula of the four pages, as r → 0, m(r) diverges to positive infinity and the debt φ to negative infinity, while the sum stays 0. This 0 is not absence but the balance of two cancelling infinities — the physical vacuum is the same, not empty but a balanced ledger.

Five conditions for a good toy model, applied.

A toy model drops part of the real thing and reproduces the real thing's habits with what is left. The four pages drop "the curvature of space" and keep only "the self-coupling of gravity". Because the part kept was chosen correctly, the real habits come out to second order. And because it can even say where the dropped part matters (half the bending of light, the absence of a horizon), its failure becomes a textbook. That is the substance of "very good".

What the four pages handle is the time side: m(r) and φ(r) are the quantities that fix how much a clock at that place runs slow, corresponding in metric terms to g₀₀. The spatial side (how much the radial ruler is stretched, g_rr) is not there yet. There are two routes to taking it in, and the key to both is the pencil line at the foot of page four, γ = e^GM/c²r = 1/√(1−v²/c²).

Route A — keep two ledgers

Put the same exponential into the spatial side as well.

The debt is entered half and half into two ledgers, the clock and the ruler — time runs slow by a factor e^−φ, and space is stretched by e^+φ. This is exactly Yilmaz's (1958) exponential metric, and it passes the three classical tests — light bending, perihelion precession and redshift — with the same precision as general relativity.

Route B — Lorentz contraction in the falling frame

This route derives "why the same factor is needed in space" as an extension of the logic of the four pages. By the equivalence principle, a freely falling frame is inertial. The particle of the four pages falls from rest at infinity and is moving with γ = e^φ at that place. Seen from the falling frame, a ruler at rest with respect to the star appears contracted radially by 1/γ — turned round, space at rest is stretched radially by γ relative to the falling frame, which is the inertial one. Hence g_rr = γ² = e^2φ. Combined with e^−2φ on the time side, this arrives at the same metric as route A.

This argument is due to Lenz (1944, recorded in Sommerfeld's lectures on electrodynamics), and it is known that putting in the exact γ² = 1/(1−2φ) of general relativity yields the Schwarzschild solution outright. Put in the exponential γ of the four pages and you get the Yilmaz metric. Pushed to its limit it becomes the "river model" of Painlevé–Gullstrand coordinates, in which space itself, still flat, flows into the star at the escape velocity — precisely the setting of the four pages (a particle falling from rest at infinity).

Why "half"

The coordinate speed of light is c·√(g₀₀/g_rr), so a gravitational field acts as a medium of refractive index n = √(g_rr/g₀₀). With the four pages as they are (g₀₀ = e^−2φ, g_rr = 1), n = e^φ ≈ 1 + φ. With two ledgers, n = e^2φ ≈ 1 + 2φ. The bending of light is proportional to n − 1, so the ratio is exactly 1 : 2 — the four pages give half the full value (Einstein's 1911 value), two ledgers give the full 4GM/c²b (the 1915 value).

Intuitively: the wavefront of light passing near a star bends because the side nearer the star lags. There are two reasons for the lag — clocks run slow there (the time ledger), and there is extra space to cross (the space ledger). Newtonian gravity and the four pages have only the first. What the 1919 eclipse tested was whether the second one is real.

Seeing that there is no horizon

Line up the time factor g₀₀ as a function of distance measured in Schwarzschild radii r_s = 2GM/c² (so 2φ = r_s/r).

In general relativity g₀₀ vanishes at r = r_s — from outside, the clock stops; in the river model, the flow speed of space reaches the speed of light. That is the event horizon. In the exponential formula, g₀₀ = e^−r_s/r stays positive all the way to r → 0 and never reaches zero. The world of the four pages has no horizon. Even after the curvature of space is taken in, this one point stays parted from general relativity. The umpire between the two theories is the observation of black hole shadows.

A map for the intuition — Einstein's equation in one line

G_μν = (8πG/c⁴) T_μν, said without chasing components, comes out like this (the formulation of Baez–Bunn 2005).

The left-hand side is how spacetime curves — ten components, but around a static spherically symmetric star they reduce to two functions, g₀₀ and g_rr. The right-hand side is energy density and pressure — for ordinary matter mass dominates, and light has pressure (a third of the density). What Newtonian gravity and the four pages had was only g₀₀, one of the two functions. What general relativity adds is g_rr — the "half" is this second function, and the bending of light was the only classical test able to probe whether it is real. Add g_rr to the four pages by route A or B and the two functions are complete, leaving only the presence or absence of a horizon — that is the summary of this section.

There is no new physics in this article

The exponential metric, the zero-energy universe, the Hamiltonian constraint and the reason the bending of light comes out half are all known. What was done was to retrace the four pages line by line, measure in which coordinates they agree to what order, and name the places where they break — nothing else.

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