Gravity / A point source

One unit on the lattice gives Newton’s 1/r in every direction.

Put one unit at one cell. In the point-source paper the static field equation of the lattice action, on the deformations that carry gravitational waves, is the lattice Poisson equation, and its exact response falls off as 1/(4πr) in every direction, with no constant adjusted. Move the unit to the next cell, leaving +1 and −1, and the 1/r field is gone. This kernel is that of the gravitational-wave amplitude; the field of a static mass lives in another component of the metric, which the paper leaves to the next step.

Newton’s kernel, one edge per tick, the growth: provedThe field of a mass, its strength and the clock: open

  • Along an axis
  • A face diagonal
  • A body diagonal
  • 4π|n|² P(n), along the axis
  • 4π|n| P(n), along the axis

Live · exact, Theorems 2 and 3

The plane through the source

Field times 4π|n|, against Newton’s value 1

What is computed. The field of one unit is the Green function W of the face Laplacian, L h(n) = 6h(n) − Σj(h(n + ej) + h(n − ej)), the operator that weighs a cell by 6 and each of its six face neighbours by −1. It is evaluated from the heat-kernel form of the paper’s Figure 5, W(n) = ∫0∞ Πj e−2t Inj(2t) dt, to about ten digits: the value at the source matches Watson’s 0.2527310…, the values at |n| ≈ 30 match the paper’s 1.00028, 0.99993 and 0.99981, and L W is one at the source and zero elsewhere to rounding. The moved unit’s field is P(n) = W(n) − W(n − e0). The colour scale is logarithmic for one unit and saturates at ±0.02 for the moved unit, as in the paper’s Figure 3.

Point-source paper, Sections 2 and 4, Theorems 2 and 3, Corollary 4 and Figures 3 and 5

With JavaScript on, this figure computes the field of one unit live. Its value at the source is W(0) = 0.2527310 (Watson’s integral); one cell away 4πW(e0) = 1.0815, 8.2% above Newton’s value 1; at |n| ≈ 30, 4π|n|W(n) is 1.00028 along an axis, 0.99993 on a face diagonal and 0.99981 on a body diagonal. A unit moved one cell leaves no 1/r field: 4π|n|P(n) tends to 0, and 4π|n|²P(n) to −1, the continuum dipole.

Reading one direction as time

One edge further each tick, and growing.

The same equation, with its fourth direction read as the ticks, computes each tick from the two before it. Started from rest with the unit at the source from tick 1 on, nothing reaches beyond t − 1 edges at tick t. With the sign the action gives, every non-uniform wave also has a solution that grows from tick to tick; with the wave equation’s sign the field oscillates instead, but only if the weight of space against time is below 1/3.

  • Positive
  • Negative
  • t − 1 edges from the source

Live · the tick rule (5), from rest

The plane through the source, log scale

What runs. The rule h(t + 2) = 2h(t + 1) − h(t) + μ L h(t + 1) − S(t + 1), the paper’s equation (5), with S the unit at the source from tick 1 on (its Figure 6b). The wave equation’s reading flips the sign of the neighbour weight, the face leapfrog of the paper’s Figure 6a. The field lives on a cube of 65 cells a side, wider than anything can reach in the 30 ticks shown, so its edge never enters; “Beyond reach” is checked over the whole cube at every tick. With the action’s sign, at μ = 1, the largest value climbs toward 7 + 4√3 ≈ 13.9 per tick, the rate of the shortest wave. At μ = 1 the wave equation’s sign grows too: in steps that reach only the six face neighbours, neither sign gives a stable evolution at the action’s weight.

Point-source paper, Section 6, Theorems 6 and 7, Proposition 8, and Section 7

With JavaScript on, this figure runs the tick rule live. Nothing ever lies more than t − 1 edges from the source at tick t (Theorem 7); with the action’s sign the largest value grows by up to 7 + 4√3 ≈ 13.9 per tick, and with the wave equation’s sign it stays bounded only when μ is below 1/3.

  • Growth per tick, the action’s sign
  • Frequency, the wave equation’s sign

Live · equation (7), at the weight you choose

Wave by wave, through the zone: Γ to X to M to R

What is drawn. The growth rate ω of the action’s reading solves sinh²(ω/2) = μ Σj sin²(kj/2); the wave equation’s frequency solves sin²(ω/2) = μ Σj sin²(kj/2), real only where the right side is at most 1 (shaded where it is not). Its frequencies are all real when 3μ ≤ 1, and bounded evolution needs 3μ < 1.

Point-source paper, Section 7, Theorem 10, equation (7) and Figure 7

With JavaScript on, this figure draws the rates wave by wave at the weight chosen above. At μ = 1 the action’s sign grows the alternating wave (π, 0, 0) by 3 + 2√2 ≈ 5.83 per tick and the shortest wave by 7 + 4√3 ≈ 13.9, and the wave equation’s sign has no real frequency beyond X; at μ = 1/3 its frequencies are real all the way to R.

The numbers

What one unit does.

Far away

1/(4πr)

The field of one unit, in every direction, though the lattice has preferred axes: Newton’s kernel, with no constant adjusted.

Point-source paper, Theorem 3

At the source

0.2527

Watson’s integral, finite where the continuum potential is infinite; within 0.3% of Newton’s kernel at ten cells and 0.03% at thirty.

Point-source paper, Sections 2 and 4

Reach

1 edge

per tick: read as ticks, the equation computes the present from the past, and nothing travels further than one lattice edge per tick.

Point-source paper, Theorem 7

Growth

5.83

per tick, 3 + 2√2, for the wave that alternates along one axis, at the action’s weight. Every non-uniform wave has a growing solution.

Point-source paper, Theorem 10

How it works

From one unit to the sign of time.

  1. 1 · The operator

    On the wave-carrying deformations the action is the face Laplacian.

    The lattice action is s times the Regge action of the Kuhn triangulation, which cuts each four-dimensional cube into 24 simplices. On the transverse-traceless deformations, the ones that carry gravitational waves, its second variation is the Einstein–Hilbert coefficient −¼ times s times the four-dimensional face symbol, exactly, at every wave number. Time and space enter it with the same sign: the action is Euclidean. A unit in the field’s amplitude profile is the transverse-traceless projection of a point source, not a point source of the metric itself.

    Point-source paper, Section 3 and Proposition 1

  2. 2 · Newton’s kernel

    The field of one unit falls off as 1/(4πr), in every direction.

    On a line of cells the field of one unit is a tent, −|x|/2 plus a straight line, and none of those solutions dies out far away: the field grows with distance. Gravity’s 1/r is a three-dimensional fact. On the cubic lattice the static equation for one unit is L W = δ0, and its solution, the lattice Green function, is finite at every cell. Far away, |n| W(n) tends to 1/(4π) along every sequence of cells, whatever the direction, so the anisotropy of the lattice survives only at short range: the shortfall falls off as |n|−2, with opposite signs on the axes and the diagonals.

    Point-source paper, Section 2 and Theorems 2 and 3

  3. 3 · A moved unit

    A balanced pair leaves no 1/r field.

    Move one unit to the next cell, +1 at one and −1 at the other, the paper’s test source for one transfer on the ledger, which it calls a posting. Its field W(n) − W(n − e0) has no 1/r part: only the net count is seen from far away, and the pair’s field approaches the continuum dipole, falling off as 1/r².

    Point-source paper, Corollary 4

  4. 4 · One edge per tick

    Read as ticks, the equation has one response from rest, and it spreads one edge per tick.

    Reading the fourth lattice direction as the ticks turns the equation into a rule that computes each tick from the two before it. From rest it has exactly one response, which first appears at tick 2 on the source itself, and a source within R edges of the origin gives a response within R + t edges at tick t + 1. A static field is a time-independent solution of the same rule, but it is not where the response from rest goes.

    Point-source paper, Theorems 6 and 7, Propositions 8 and 9

  5. 5 · The sign of time

    A field that evolves in ticks needs time to enter with the opposite sign.

    For every non-uniform spatial wave the rule has a solution that grows by a fixed factor each tick, up to 7 + 4√3 ≈ 13.9 for the shortest wave at the action’s weight, while wave by wave the equation’s own two-sided response decays toward past and future at that same rate. With the opposite sign, the wave equation’s, the waves oscillate, but in steps that reach only a cell’s six face neighbours that needs a weight of space below 1/3; at the action’s weight neither sign gives a stable evolution in face steps. A clock that runs continuously has real frequencies at every weight.

    Point-source paper, Section 7 and Theorems 10 and 11

  6. 6 · What remains

    Three steps stand between this kernel and the field of a mass.

    The kernel found here is that of the gravitational-wave amplitude. A static mass sources the time-time component of the metric, which the paper does not yet derive on the lattice; the strength of the coupling against Newton’s constant rests on the action scale s, which is open; and whether one tick of the field is a face step, a cube step or a continuous flow is still to be derived. For the field of a resting mass in the Lorentzian Regge lattice of the earlier paper, see Newton’s law from the lattice.

    Point-source paper, Section 8

On this page the paper’s own words differ from the other gravity pages: here space is a cubic lattice of cells, a tick is one step along the lattice’s fourth direction, and a posting is one unit moved to a neighbouring cell.

Sources: the point-source paper is What a Point Source Does to the Lattice: Newton’s Kernel, One Edge per Tick, and the Sign of Time, a preliminary preprint of 12 October 2026; the earlier paper is Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice.

Words on this page, in plain terms: face Laplacian · Green function · Kuhn lattice · Regge action.

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