Regge Calculus on the D4 Lattice through Second Order

Regge Calculus on the

Lattice through Second Order:

Exact Identities and the Post-Newtonian Static Field

Raghu Kulkarni

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA

raghu@idrive.com

Abstract

We study four-dimensional Euclidean Regge calculus on a closed periodic triangulation of the

root lattice through second order in the metric perturbation, and report three results. First,

the two-derivative quadratic and cubic terms of the action are ﬁnite lattice sums whose exact

coeﬃcient tensors, for arbitrary symmetric polarizations and momenta, equal the Fierz–Pauli and

Einstein–Hilbert terms coeﬃcient by coeﬃcient over

, and there is no one-derivative cubic term.

The identities are oﬀ-shell, so the lattice satisﬁes the ﬁrst-order gauge Ward identity of general

relativity about ﬂat space, and they hold for every axis-spine subdivision and every even torus

size

L ≥

4. Second, solving the linearized lattice equations on metric-sampled perturbations for

a static mass source gives Poisson’s equation with the Newtonian normalization and PPN

= 1,

with anisotropic corrections of relative order (

)

that we tabulate. Third, the projection of the

exact nonlinear Regge equations onto smooth metric variations, evaluated on the static exterior

ﬁeld with no expansion on the gravitational side, is consistent with the Schwarzschild exterior

and yields PPN β = 1.000 ± 0.002 and γ = 1.000 ± 0.002; the full lattice solution, including its

non-geometric edge amplitudes, is not constructed. The nonlinear test depends on two technical

points that apply to any test of this kind. Edge lengths must be geodesic to second order, for

which we give a closed form, because straight coordinate segments carry a non-metric error at

the order of the second-order ﬁeld. The ﬁeld equations must also be imposed in weak form. The

complex has ﬁfteen edges per vertex and a metric has ten components. Of the ﬁve non-geometric

edge directions, two are stiﬀ modes of the quadratic action, which also has three null directions

besides the gauge modes, lying predominantly in the non-geometric subspace. The per-edge

residual of an exact continuum solution lies almost entirely in these non-geometric directions

and does not vanish, while its projection onto smooth metric variations does. All results are

Euclidean and at leading order in the long-wavelength expansion; the second and third are static,

the third concerns the vacuum exterior, and the Regge action is assumed throughout.

1 Introduction

Regge calculus places the curvature of a piecewise-ﬂat manifold on codimension-two hinges and

takes the squared edge lengths as the dynamical variables [

]. That its action approaches the

Einstein–Hilbert action for suitable sequences of triangulations was established by Cheeger, M¨uller,

and Schrader [

], and the approach was analyzed for regular lattices by Feinberg, Friedberg, Lee,

and Ren [

]. Its perturbative expansion about ﬂat space goes back to Roˇcek and Williams, who

obtained the linearized graviton propagator on a periodic hypercubic triangulation [

]; diagrammatic

techniques were developed by Hamber and Liu [

] and more recently by Khatsymovsky [

]. A known

subtlety is that the lattice has an exact diﬀeomorphism symmetry only about ﬂat conﬁgurations and

breaks it elsewhere [

], and that the Regge equations evaluated on sampled smooth metrics need

not converge pointwise to Einstein’s equations [

], even though the simplicial solutions themselves

can converge [9]; this matters in numerical Regge calculus [10].

We carry the expansion on one lattice, the

root lattice, through second order in the metric

perturbation, and ask two questions that the convergence results do not settle. Do the lattice

quadratic and cubic terms reproduce Fierz–Pauli and Einstein–Hilbert exactly at leading derivative

order, as statements about ﬁnite lattice sums rather than numerical agreements? And is the static

exterior ﬁeld of general relativity consistent with the nonlinear lattice equations, and with which

post-Newtonian parameters?

Scope of the claims. All statements concern the leading term of the long-wavelength expansion,

|k|a ≪

1 or

a/r ≪

1 with

the lattice spacing, which is where a lattice can agree with a continuum

theory; subleading corrections are computed where we can and reported as such. The computation

is Euclidean and classical, and in Secs. 4 and 5 it is static. The Regge action is assumed, not derived.

Agreement of Regge calculus with general relativity at leading order is the expected outcome of the

convergence results above. The new content is an exact identity for the quadratic and cubic terms,

which replaces a sampled comparison with a statement about a ﬁnite sum; a nonlinear consistency

test that ﬁxes PPN

from the smooth-metric projection of the exact lattice equations; and two

technical ﬁndings, on the sampling of edge lengths at second order and on the non-geometric edge

modes of the complex, without which the nonlinear test appears to fail.

Relation to the selection–stitch model. The

lattice is the spacetime lattice of the selection–

stitch model, in which the constant-time slice is the face-centered cubic lattice [

]. Nothing

below depends on that framework. The linearized results of Ref. [

] are recomputed here on a

closed complex. At leading order they are not special to

: Roˇcek and Williams obtained the

continuum linearized theory on a hypercubic triangulation [

What is lattice-speciﬁc are the

subleading corrections, which we tabulate for the static ﬁeld, and the edge-mode structure of the

complex (Sec. 2.3).

Outline. Section 2 deﬁnes the complex, the action, its edge modes, and the assignment of edge

lengths to a continuum metric. Section 3 proves the quadratic and cubic identities. Section 4

solves the linearized static problem, and Sec. 5 performs the nonlinear consistency test. Section 6

discusses scope and limitations, and Sec. 7 concludes. Appendix A derives the geodesic length used

throughout, and Appendix B lists the reproduction scripts.

2 Complex, action, and edge lengths

2.1 The periodic D

complex

{x ∈ Z

even}

. We work in lattice coordinates, with unit coordinate spacing

= 1,

so the

bond length is

√

2 a

; momenta

and radii

are quoted in these units. The Delaunay

cells of

are 16-cells centered on the deep holes: integer points

with

odd, with vertices

c ±e

, and half-integer points

c ∈

(

)

, with the eight vertices

s ∈ {±

}

, that lie in

Each lattice point carries three cells. A 16-cell is triangulated by coning from one antipodal vertex

This corrects the statement in Ref. [

], Table 1, that

admits no ﬂat background. The standard Freudenthal

triangulation of

is ﬂat; on a periodic 3

torus we ﬁnd a maximum hinge deﬁcit of 9

−16

. The nonzero deﬁcits

reported there presumably arose from evaluating an open patch, whose boundary hinges are not fully surrounded;

compare Sec. 2.2.

spine

(a) coning from a spine (3D analog)

(lattice units)

|eigenvalue| per vertex

numerical zero: 4 gauge + 3 null

(b) 15 edge modes per vertex

6 metric modes

2 stiff non-geometric

∝

Figure 1: The complex and its edge modes. (a) A 16-cell is triangulated by coning from an antipodal

vertex pair, the spine, into eight four-simplices; drawn here is the three-dimensional analog, an

octahedron coned into four tetrahedra. Coning leaves the boundary faces whole, so neighboring

cells may choose spines independently. (b) Magnitudes of the eigenvalues of the linearized Regge

operator per vertex on the ﬁfteen edge classes of the periodic

complex, along a generic direction:

seven null directions (shaded; ﬁnite-diﬀerence noise below 10

−9

), six metric modes proportional to

, and two stiﬀ non-geometric modes of order one.

pair (the spine) into eight four-simplices (Fig. 1(a)); coning leaves the boundary tetrahedra whole,

so any per-cell choice of spine gives a consistent complex. The subdivision ambiguity is thereby

an explicit label, which we vary. In the axis-

assignment (

= 0

, . . . ,

3), cells centered on integer

points use the spine

c ± e

, and cells centered on half-integer points use the

-th antipodal pair

when their admissible sign vectors

are listed in lexicographic order with +1 before

−

1. These

four assignments are invariant under

translations. We also use a random assignment, with

the spine drawn independently in every cell. The edges are the lattice bonds, of squared length

2, and one cell diagonal of squared length 4 per cell. Each vertex has 24 bonds, and there are

three cells per vertex, so the complex has 12 + 3 = 15 edges per vertex for any spine assignment.

With an axis assignment every vertex has exactly 30 incident edges, and the edges fall into ﬁfteen

translation classes: twelve bond directions and three diagonal directions, which for axis-0 are

0), (1

1), and (1

, −

1). On an

torus (

= 4: 128 vertices, 3072 simplices, 1920

edges, 6400 hinges) every simplex has volume 1/12, every facet is shared by two simplices, and all

deﬁcits vanish to 10

−15

2.2 Action and the Schl¨aﬂi identity

The Euclidean action is

S = −

8πG

, δ

= 2π −

s⊃σ

σ,s

, (1)

normalized so that

→

√

g R

; Sec. 4 conﬁrms this normalization independently. The

Schl¨aﬂi identity

σ⊂s

dθ

σ,s

= 0 holds simplex by simplex. Summed over a complete hinge sum

it gives

dδ

= 0. On a complex with boundary this remains true once the boundary hinges

are included with their deﬁcits

π −

σ,s

, as in the Hartle–Sorkin boundary term [

]. It fails for

a truncated sum that keeps all simplices around a set of hinges but drops their other hinges, such

as the origin-hinge sum over a vertex star. With the identity, the ﬁeld equations take the form

≡

∂

∂ℓ

∂A

∂ℓ

(2)

for arbitrary edge lengths, and the second and third variations about ﬂat space, where

= 0,

reduce to

′

∂

A ∂

′

δ, S

′

′′



∂

′′

A ∂

′

δ + ∂

′

′′

A ∂

δ + ∂

′′

A ∂

′



, (3)

with

∂

≡ ∂/∂ℓ

and all derivatives evaluated at the ﬂat background. The second expression is

symmetric in (

e, e

′

, e

′′

) only on a complete hinge sum, which provides a check on any implementation.

2.3 Edge modes of the linearized operator

Fifteen edge lengths per vertex exceed the ten components of a metric. For perturbations uniform

within each edge class, the patterns

δℓ

·h·d

span a ten-dimensional metric subspace of

. We call its orthogonal complement, in the Euclidean inner product on class amplitudes, the

non-geometric subspace. Figure 1(b) shows the spectrum of

in Eq.

(3)

, per vertex, on Bloch

modes

δℓ

[e]

ik·x

, where [

] is the class of edge

. Along each of the three directions we tested,

a generic four-dimensional direction and the static directions [100] and [110], the spectrum splits as

7 + 6 + 2. (For static momenta,

= 0, we use spatial Miller indices.) Seven eigenvalues vanish: four

belong to the vertex translations, and three further null directions are not gauge. Six eigenvalues

are proportional to

; they carry the curvature of metric perturbations and include the graviton.

Two eigenvalues are of order one, and their eigenvectors lie entirely in the non-geometric subspace;

we call these the stiﬀ non-geometric modes.

The linearized deﬁcit map separates the subspaces diﬀerently. Metric directions produce deﬁcits

of order

and gauge directions none, so the ﬁve non-geometric directions are exactly those that

produce deﬁcits of order one as

k →

0. The three non-gauge null directions of

lie predominantly

(70% of their squared norm) in the non-geometric subspace and produce deﬁcits of the same size.

Their deﬁcit pattern is, however, orthogonal to the area gradients

∂A

/∂ℓ

to numerical precision,

so they cost no action at quadratic order.

2.4 From a metric to edge lengths

A continuum metric

is represented by assigning each edge a squared length. At ﬁrst order

the natural assignment integrates the metric along the edge,

δℓ

(

))(

, d

)

, with

the

edge vector. For a plane wave this is the midpoint value times

sinc

(

k·d

2), and it maps a linearized

diﬀeomorphism exactly onto a lattice vertex translation. At second order the straight coordinate

segment is not a geodesic. Its error is of order Γ

, and because it is quartic in

it is not of the

form

(

, d

): it is a non-metric edge pattern. Such patterns produce deﬁcits with no derivative

suppression (Sec. 2.3), so this error enters the ﬁeld equations at the same order as the second-order

metric itself. We therefore use the squared geodesic length to second order (Appendix A),

ℓ

g(∆, ∆)



(s)

ds −

(s) g

µν

K(s, s

′

) F

′

) ds ds

′

, F

= −2Γ

µ,αβ

∆

, (4)

where

(

) is the straight segment, ∆ =

µν

is evaluated at the edge midpoint, and

(

s, s

′

) =

min

(

s, s

′

) [1

− max

(

s, s

′

)] is the Green’s function of

−d

/ds

on [0

1]. The ﬁrst term is the aﬃne

energy of the segment, not its squared length; the diﬀerence between the two is itself a second-order

quantity.

3 Quadratic and cubic terms as ﬁnite lattice sums

Perturb the edge lengths by plane waves,

δℓ

⊗ d

·x

, with

= 0 for the

quadratic term and

= 0 for the cubic term. Let

denote the

(

) coeﬃcient of

the

term. The

(

) terms vanish identically, because a uniform strain is aﬃne and

an aﬃne map of a ﬂat complex is ﬂat. Here the metric is sampled at edge midpoints. Integrated

sampling (Sec. 2.4) multiplies each leg by 1 +

(

); since the

(

) terms vanish, this cannot

change the two-derivative coeﬃcients. With phases measured from a local origin on each hinge,

which is legitimate because the momenta sum to zero,

is a sum over hinge–simplex incidences of

products of the exact derivative tensors in Eq.

(3)

with the polarization weights

·ε

·d

and the

squared total phase



·x



. The

(

) terms are sums of the same form with the ﬁrst power

of the phase.

Arithmetic in

(

√

). At the ﬂat background the dihedral cosines satisfy

cos

θ ∈ {

}

and

the Heron discriminants take the values

{

}

, so every derivative lies in

(

√

). The local

geometry falls into seven congruence classes of simplex–hinge pairs and four hinge triangles. The

derivative tensors are computed symbolically once per class and cleared to integers over the common

denominator 288.

Reduction to local conﬁgurations. Every term in Eq.

(3)

factorizes into pieces attached to

individual slots. So, rather than evaluating

on sample kinematics, we accumulate its full coeﬃcient

tensor. For the cubic term, with k

= −k

− k

eliminated,

, κ = (k

, k

) ∈ R

, (5)

and analogously for the quadratic term with

∈ R

. The accumulation is carried out in

exact integer arithmetic, and

runs over the ten components of a general symmetric

, not only

transverse-traceless ones. A contribution depends only on the local data of its incidence: the

congruence class, the edge dyads, and the midpoint oﬀsets. Identical local data therefore give

identical contributions. The 30 720 incidences of the

= 4 torus reduce to 936 distinct local

conﬁgurations, and the 155 520 of L = 6 reduce to the same 936.

With the normalization of Eq.

(1)

, one vertex carries coordinate volume 2, so the per-vertex

Regge sum

is compared directly with the Lagrangian density below.

Theorem 1 (Exact equivalence through cubic order). Let the Regge action

(1)

be evaluated on the

closed periodic

triangulation of Sec. 2, with any of the four axis-spine assignments, on an

torus with even

L ≥

4, and let

√

g g

µν

(Γ

µβ

να

−

µν

αβ

) with

, which diﬀers from

√

g R

by a total derivative. For arbitrary symmetric perturbations

·x

with momenta

summing to zero, per vertex:

(a) the O(h

∂

) part of

equals the quadratic part of L, the Fierz–Pauli form;

(b) the O(h

∂) part vanishes; and

∂

) part equals the cubic part of L.

Equivalently, the coeﬃcient tensors of the lattice and continuum terms agree entry by entry over

Proof (computer-assisted, in exact arithmetic). (i) The ﬁrst and second derivatives of dihedral angles

and hinge areas at the ﬂat background are computed symbolically for each of the seven simplex–hinge

classes and four hinge classes, and lie in

(

√

) with common denominator 288. (ii) Because the

complex is invariant under

translations and, for

L ≥

4, the star of every hinge is isometric to its

image in

, each per-vertex tensor is a sum over the same 936 local conﬁgurations with the same

per-vertex multiplicities for every such

; this is conﬁrmed directly at

= 4 and

= 6. (iii) The

full tensors, over ten symmetric components per leg and all momenta, are accumulated in integer

arithmetic from Eq.

(3)

, and momentum conservation is then imposed. In each tensor the

√

component vanishes identically. The two orderings of the quadratic form, and the three eliminations

of a slot in the cubic form, give identical tensors, as required on a complete hinge sum. (iv) The

continuum tensors are obtained from the full expansion

det(1 + h)

exp

(

tr log

(1 +

)), with

no traceless assumption, by evaluation on a complete polarization basis. Every entry lies within

−9

of a rational with denominator dividing 64, which identiﬁes it unambiguously. (v) For (a), all

= 1 600 entries agree exactly, of which 132 are nonzero. For (b), all entries of the

(

)

cubic tensor vanish, for each of the three slot eliminations. For (c), all 10

= 64 000 entries

agree exactly, of which 3 744 are nonzero. Each statement holds for each of the four axis spines.

□

Statement (b) reﬂects the inversion symmetry

x 7→ −x

of the axis-spine complexes, which we veriﬁed

directly on the simplex sets; it rules out a one-derivative cubic term, a lattice artifact that would

be of lower order than the Einstein–Hilbert vertex. Two further checks lie outside the statement

of the theorem. The cubic tensor is unchanged for a spine drawn at random in every cell (3 740

distinct local conﬁgurations). And contracting it with transverse-traceless kinematics reproduces

values obtained by evaluating the third variation directly, for example −30, −40, and −8 on three

test conﬁgurations.

Gauge structure. Parts (a) and (c) hold oﬀ-shell, including pure-gauge and trace polarizations.

The lattice therefore has the same linearized gauge symmetry as general relativity about ﬂat space,

and that symmetry deforms at ﬁrst nonlinear order exactly as diﬀeomorphisms do: the ﬁrst-order

Ward identity relating the cubic and quadratic terms is satisﬁed, because it is satisﬁed by

. This

is a statement about gauge structure at ﬁrst nonlinear order, not about diﬀeomorphism invariance

away from ﬂat space, which Regge calculus is known to lack [7].

4 The linear static ﬁeld

We solve the linearized lattice equations with a static source. For a static wavevector,

= 0, we

build the 10

10 lattice quadratic form

(

) on metric-sampled edge perturbations, from Eq.

(3)

with integrated sampling. By Theorem 1(a), its leading term is the quadratic part of

, which

equals −q

, where

(ε, k) =

|ε|

− |k·ε|

+ (tr ε)(k·ε·k) −

(tr ε)

(6)

is the Fierz–Pauli form; with the sign of Eq.

(1)

, physical modes have positive Euclidean action.

Working in this ten-dimensional subspace excludes the non-geometric directions of Fig. 1(b). How a

matter source couples to them depends on how matter couples to individual edges, which we do not

specify. The source is a static mass density

, entering through

√

dτ →

ρ h

. The

direction [100] [110] [111]



Φk

/(−4πGρ) −1



/(ka)

0.0278 0.0697 0.0837

(γ − 1)/(ka)

0.0279 0.0070 < 10

−4

Table 1: Leading lattice corrections to the static ﬁeld from the linearized equations on metric-

sampled perturbations, with

the unit coordinate spacing. The values are unchanged across ﬁve

subdivisions.

(lattice units)

(

−

πGρ

)

−

(a) Newtonian normalization

[100]

[110]

[111]

∝

(lattice units)

−

(b) PPN

Figure 2: The linear static ﬁeld from the 10

10 lattice equations on metric-sampled perturbations.

(a) Departure of the potential from the Newtonian value

−

πGρ/k

; (b) departure of

from 1, with

markers as in (a); along [111],

|γ −

| <

−6

at all momenta shown, consistent with zero within

numerical precision, and is not plotted. Both departures vanish as

, with the direction-dependent

coeﬃcients of Table 1.

exact gauge directions

k ⊗ ξ

ξ ⊗ k

are projected out. They are exact lattice zero modes under

integrated sampling, and the source is orthogonal to them. We read oﬀ two quantities that are

gauge invariant because

= 0:

= 2Φ, and the transverse spatial trace Π

with Π = 1

−

In PPN form [14] h

= −2γΦ δ

, so γ = −Π

/(2h

Newtonian limit and γ. As |k| → 0,

Φ = −

4πGρ



1 + O(k

)



, γ = 1 + O(k

), (7)

with Φ

(

−

πGρ

) = 1

00007, 1

00017 and 1

00021 at

|k|

= 0

05 along [100], [110], and [111] (Fig. 2).

The normalization of Eq.

(1)

is thereby conﬁrmed with nothing ﬁtted, the potential is attractive,

and

= 1. The leading corrections are anisotropic (Table 1). They are identical, to the digits

quoted, for all four axis spines and the random spine. At this order they depend on conventions:

how the continuum metric is assigned to edges, which we ﬁx by integrated sampling, and how the

source couples, which we ﬁx by coupling it to the sampled component h

Conformal mode versus Newtonian potential. The static weak ﬁeld of general relativity is

not conformally ﬂat in four dimensions: in Euclidean isotropic gauge

= 2Φ while

−

2Φ

The four-dimensional conformal mode

= 2

1 has a kinetic term of the opposite sign. From

Eq.

(6)

with static

(1) =

−

, while

(

diag

(

−

, −

1)) = +

and transverse-traceless

modes give +

|ε|

. The lattice form

(

) reproduces all three with the overall sign

−q

as Theorem 1(a) requires. Solving within a restricted conformal ansatz returns

ϕ → −

3 as

k →

0, in every direction. A static computation restricted to that mode therefore computes the

conformal-factor response, not the Newtonian potential.

5 The nonlinear static ﬁeld

5.1 Test family and complex

Outside a mass,

is ﬁxed by the vacuum ﬁeld equations alone. By Birkhoﬀ’s theorem a static

isotropic metric with β = 1 is not a vacuum solution. We therefore take the family



1 − U/2

1 + U/2



+ 2(β − 1)U

, g



1 +



+ 2(γ − 1)U +

(ζ − 1)U

, (8)

with

M/r

, which is exact Euclidean Schwarzschild in isotropic coordinates at

= 1.

The parameter

frees the second-order spatial coeﬃcient. We sample the family onto the complex

with Eq.

(4)

and evaluate the exact residual

(2)

, with no expansion on the gravitational side. The

matter coupling enters only through

. This is a consistency test: we ask whether, and for which

parameters, the sampled family satisﬁes the lattice equations in the sense made precise in Sec. 5.3;

we do not solve for the lattice conﬁguration.

Evaluating at +

and

−M

separates odd from even orders exactly. The odd part

[

(

)

−

(

−M

)] is ﬁrst order up to

(

) and ﬁxes

. The even part

[

(

) +

(

−M

)] is second order

up to

(

) and ﬁxes

and

; ﬁrst-order lattice corrections, being odd, cancel from it identically.

The complex is built directly from the 16-cell tiling, with the spine rule of axis-0, in a slab through

the source, which sits at the generic spatial point (0

11). Its cells are those whose integer

part of the center lies in

|x|, |y| ≤

13,

|z| ≤

4, and

∈

[

−

4]: 551 120 simplices, 1 295 541 hinges,

and 435 667 edges. A hinge is complete exactly when its deﬁcit vanishes for ﬂat edge lengths, and

an edge is interior when all its hinges are complete; 263 578 edges qualify. The residual of ﬂat space

is 8 × 10

−16

5.2 Validation of the length assignment

We test Eq.

(4)

on ﬂat space in curved coordinates: the pullback of

under the radial displacement

x 7→ x

(

x − x

)

/|x − x

with

= 0

4, on a smaller slab (

|x|, |y| ≤

|z| ≤

3). For this metric

the exact lengths are known and the exact residual vanishes, to 10

−13

. The second-order (even)

residual on interior edges at 4

< r <

8 is 8

−5

with straight segments and 4

−6

with

Eq.

(4)

. Edge by edge, Eq.

(4)

reduces the length error by about a factor 30. Equation

(4)

is exact

at second order (Appendix A); the remaining error comes from the numerical quadrature of the

Green’s function

, whose kink on the diagonal makes

-point Gauss quadrature converge roughly

−2

. The largest per-edge error falls from 1

−7

= 8, the value used throughout, to

−9

= 64, where the remainder is of third order. With

= 32 the even ﬂat-space residual

falls to 4

−7

. Repeating the main ﬁt of Sec. 5.4 (

= 0

= 2

0) with

= 16 changes the

per-bump β by at most 4 × 10

−4

and β

∞

by 10

−4

5.3 Non-geometric modes and the weak form

Fitting Eq.

(8)

to the per-edge residual recovers

: the ﬁt gives

= 1

000

003 at

r ≥

5, and

the ﬁrst-order residual at

= 1 is 1–7% of the

= 0 signal, falling roughly as

−2

. It does not

recover

. Per radial shell the ﬁtted

drifts from 0

03 to 0

74 (Fig. 3a), and the even residual at

= 1 remains 81–90% of the

= 0 signal. That residual is exactly second order: its ratio between

= 0

4 and

= 0

2 is 4

01–4

07. It also falls as

−4

, like the

signal itself, and not like a lattice

correction.

The residual lies in the non-geometric directions of Sec. 2.3. Grouping the ﬁfteen edges at

each vertex and projecting onto the

k →

0 subspaces of Sec. 2.3, the even residual at

= 1

has 2–9% of its squared norm in the metric subspace, falling with

; 27–35% along the two stiﬀ

non-geometric modes; and 63–64% in the three-dimensional complement of the stiﬀ modes within

the non-geometric subspace, where the non-gauge null directions of

predominantly lie (10 938

vertices at 4

≤ r <

10). The

signal, by contrast, is 75% metric. The nonlinear terms of Eq.

(2)

therefore drive the non-geometric directions at the order of the signal, while the metric projection

of the residual is small. Along the stiﬀ modes the lattice solution can absorb this residual through a

length correction of relative size (

a/r

)

. Along the other three directions, which are null at quadratic

order, linear response cannot absorb it; their amplitudes in the lattice solution would be ﬁxed at

second order, as in degenerate perturbation theory. This is the

form of the known failure of

the Regge equations to converge pointwise on sampled solutions [

]. Brewin and Gentle showed,

for the Kasner cosmology, that simplicial solutions can converge while the residual of the Regge

equations evaluated on the continuum solution does not [

]; the same distinction underlies the weak

form below.

Variational meaning of the weak form. Which equation should a sampled continuum metric

satisfy? The ﬁrst variation of the lattice action is, exactly,

δS

δℓ

, by Eq.

(2)

. A continuum

metric variation

δg

induces edge variations through Eq.

(4)

. At linear order in

δg

these are

δℓ

·δg

(

))

·d

, since the aﬃne energy is linear in the metric, plus a variation of the

geodesic correction that is suppressed by

(

Λ) for

δg

varying on a scale Λ. Up to corrections of

relative order (a/Λ)

this is d

·δg(x

)·d

. For smooth δg, therefore,

δS



smooth δg

·δg(x

)·d

≡ P [δg]. (9)

The continuum ﬁeld equations,

δS

/δg

= 0, state that the action is stationary under metric

variations. Their lattice counterpart is

[

δg

] = 0 for all smooth

δg

: the ﬁrst variation must vanish

against metric-induced edge variations. It need not vanish separately along each of the ﬁfteen edge

directions per vertex. Ten of these are metric patterns; the other ﬁve are degrees of freedom of

the discretization, with no counterpart in the continuum metric. The full lattice equations E

= 0

impose those ﬁve conditions as well. They constrain the non-geometric amplitudes discussed above,

and whether satisfying them feeds back on the metric projection is the open question stated in

Sec. 6.

[

δg

] = 0 is thus the lattice form of the weak, or distributional, Einstein equations. This is

also the sense in which ﬁnite-element discretizations approximate their continuum equations, and

Regge calculus admits a ﬁnite-element interpretation [

]. It is not a projection chosen after the

fact: it is the equation whose continuum limit is Einstein’s, and the per-edge equations contain it

together with ﬁve further conditions that have no continuum counterpart.

We impose Eq.

(9)

with static test variations

δg

(

)

(

)

, twenty-seven in all. Here

one of

ˆrˆr

, 1

− ˆrˆr

, or

;

is a

cos

radial bump of half-width 2

5 centered at

∈ {

, . . . ,

}

;

and

is a

cos

window of half-width

Z ∈ {

}

, taken over one period in

. All test

supports lie entirely within the interior edges. In weak form the residual at

= 1 falls to 1

8% of

the β = 0 signal (M = 0.4, Z = 2.0).

M window Z weak residual / β=0 signal β

∞

0.4 2.0 0.018 0.9985 0.9984

0.4 1.5 0.021 0.9996 0.9992

0.2 2.0 0.021 1.0001 0.9991

0.2 1.5 0.023 1.0012 1.0000

Table 2: PPN parameters from the smooth-metric (weak-form) projection of the exact nonlinear

Regge equations, extrapolated in 1

over

= 5–9; on the test supports

M/r ≤

16. The

extended run of Sec. 5.4 (M = 0.4, Z = 1.5, r

= 5–12.5) gives β

∞

= 1.0000 and γ

∞

= 0.9995.

5.4 Results

For each bump we ﬁt

from the odd part and

from the even part with

= 1. A joint ﬁt of

and

over all bumps gives

= 0

986–1

002. The per-bump values rise monotonically toward 1 (Fig. 3b)

and ﬁt

(

) =

∞

χ/r

with rms residuals below 7

−4

. The extrapolated values are in

Table 2. The per-bump values at

= 0

2 and

= 0

4 agree to within 0

4% at every radius, so

the ﬁnite-

oﬀsets are lattice and test-window eﬀects rather than higher orders in

. With straight

segments in place of Eq.

(4)

, the ﬁnite-

bias of

doubles (0

956 at

= 5, against 0

980), although

the extrapolation still reaches 1.005.

Larger radii and the extrapolation form. Over

= 5–9 the data cannot distinguish a 1

from a 1

approach, and a 1

ﬁt would give

∞

= 1

010–1

017. We therefore extend the range

with a wider but thinner slab (

|x|, |y| ≤

17,

|z| ≤

∈

[

−

3]; 926 096 simplices) and 48 test

variations centered at

= 5–12

5, with

= 0

4 and

= 1

5. For

≤

9 it reproduces the earlier

per-bump values to four digits. Without any extrapolation,

= 0

9964 and

= 0

9973 at

= 12

A 1

ﬁt to

≤

9 predicts the new values at

= 9

5–12

5 to within 6

−4

; a 1

ﬁt misses

them by up to 2

−3

. A ﬁt with a free power prefers

= 1

8 for

and 1

65 for

. Over the

full range, the 1

, 1

+ 1

, and free-power forms give

∞

= 1

0000, 1

0005, and 1

0011, and

∞

= 0

9995, 1

0001, and 1

0008. The extrapolation is thus stable across the forms the data allow;

the 1

form is also the one expected from corrections of relative order (

a/r

)

from the lattice and

from the ﬁnite width of the test functions.

Across the four settings, the extended run, and the admissible extrapolation forms, the smooth-

metric projection of the exact nonlinear equations yields

= 1

000

002 and

= 1

000

002,

the ranges spanning all of these.

6 Discussion

Summary of results. On the closed

complex, the Regge action reproduces general relativity

at leading order in

a/r

through second order in the ﬁeld, in three ways. The quadratic and cubic

terms are Fierz–Pauli and Einstein–Hilbert as exact oﬀ-shell identities of ﬁnite lattice sums, with

no one-derivative cubic term. The linear static ﬁeld has the Newtonian normalization and

= 1.

And the smooth-metric projection of the exact nonlinear Regge equations is consistent with the

Schwarzschild exterior, with

= 1 to within 2

−3

. The third result is a nonlinear consistency

test: it probes the lattice ﬁeld equations at second order rather than a vertex about ﬂat space, but

it does not construct the full lattice solution.

4 6 8 10

(lattice units)

0.0

0.2

0.4

0.6

0.8

1.0

fitted

(a) strong vs. weak form,

= 0

per-edge (strong) fit

weak form

0.00 0.01 0.02 0.03 0.04

0.96

0.97

0.98

0.99

1.00

weak-form fit

(b) extrapolation

∞

χ/r

= 0

, straight segments

Figure 3: The nonlinear static ﬁeld. (a)

ﬁtted from the per-edge residual by radial shell drifts

far from 1; the weak form

(9)

recovers it. (b) Weak-form

(ﬁlled) and

(open) per test bump

against 1

for

= 0

4 and 0

2 with geodesic lengths (

= 2

0), for the extended run to

= 12

(diamonds, Z = 1.5), and β with straight segments (crosses); lines are ﬁts X

∞

+ χ/r

Relation to convergence theorems. Leading-order agreement is what the convergence theory

of Regge calculus predicts. Four results go beyond that prediction: the exactness of Theorem 1,

including its gauge sector; the subleading coeﬃcients in Table 1; the stiﬀ non-geometric modes and

the non-gauge null directions of Fig. 1(b); and the failure of the per-edge nonlinear equations at the

order of the signal, which occurs in those non-geometric directions.

Requirements for nonlinear tests. Two requirements apply to any nonlinear test of Regge

calculus by sampling a continuum solution. Edge lengths must be geodesic to second order, Eq.

(4)

since straight-segment lengths carry a non-metric error at the order of the second-order ﬁeld. And

the equations must be imposed in weak form, Eq.

(9)

, whenever the complex has more edges per

vertex than metric components. A related caution concerns reduced variational formulas: Eq.

(3)

requires the complete hinge sum of Sec. 2.2, and it is not valid for a truncated sum, such as the

origin-hinge sum over a vertex star, because the cancellation it relies on is then incomplete.

Limitations. The computation is Euclidean and, in Secs. 4–5, static; the Lorentzian continuation

and dynamical ﬁelds are not addressed. The Regge action is an assumption. The nonlinear test shows

that the weak-form residual of the Schwarzschild family is minimized at, and after extrapolation in

vanishes at,

= 1. It does not construct the lattice solution. In the conﬁgurations we test,

every edge length is ﬁxed by the sampled metric, so there is no independent non-geometric amplitude.

Whether the lattice solution carries such amplitudes, in particular along the three non-gauge null

directions of the quadratic action, and whether they feed back on the metric projection at second

order, is open. The extrapolation uses

≤

5 in slabs of half-thickness three to four cells; it

is stable across the ﬁt forms the data allow, but a larger complex would tighten it further. The

coupling of matter to the lattice geometry enters only through the exterior mass, and how lattice

defects source the ﬁeld is not addressed. The conventions of Sec. 4, integrated sampling and coupling

the source to

, ﬁx the subleading coeﬃcients of Table 1; a diﬀerent metric-to-edge assignment or

matter coupling would change them. Theorem 1 is proven for the four axis-spine subdivisions; for a

random subdivision it has been checked for one realization.

7 Conclusion

On a closed periodic triangulation of the

lattice, the two-derivative quadratic and cubic Regge

terms are Fierz–Pauli and Einstein–Hilbert as exact identities of their coeﬃcient tensors over

, and

there is no one-derivative cubic term. The linearized equations on metric-sampled perturbations,

with a static source, give the Newtonian potential with

= 1 and tabulated anisotropic corrections.

The projection of the exact nonlinear Regge equations onto smooth metric variations, evaluated

with geodesic edge lengths, is consistent with the Schwarzschild exterior and yields

= 1

000

002

and

= 1

000

002; the full lattice solution, including the amplitudes of the non-geometric edge

modes, is not constructed. The strong-form failure of the nonlinear test occurs in the non-geometric

edge directions of the complex, while the metric projection of the equations is satisﬁed.

A Geodesic squared length to second order

Let

(

) =

∆,

s ∈

1], and consider paths

with

(0) =

(1) = 0. The minimum over

paths of the energy E[x] =

g( ˙x, ˙x) ds is the squared geodesic length. Expanding,

E = E

ds +

˙η· ˙η ds + O(h ˙η ˙η, ∂h η ˙η, ∂

h ηη), E

g(∆, ∆) ds, (10)

with

. The linear term collects

∂

αβ

∆

from expanding

(

) and 2

µβ

∆

˙η

which after integration by parts (the boundary terms vanish) gives

= (

∂

αβ

−

∂

µβ

)∆

∆

−

2Γ

µ,αβ

∆

. Since

(

), the minimizing

(

), and every term omitted from the

expansion is of third order in

. Stationarity gives 2

¨η

, so

(

) =

−

(

s, s

′

)

(

′

)

′

with

as in Eq.

(4)

. The minimum is

F ·η

, which is Eq.

(4)

; there

µν

may be replaced by

µν

evaluated anywhere on the edge, the diﬀerence being of third order. Equation

(4)

is therefore exact

at second order in

, including the variation of

along the edge, which enters through

. The

third-order remainder is odd in

for the family of Sec. 5, so it cancels from the even combination.

For constant

the correction reduces to

−

αβ

µ,νλ

∆

, a form of the second-order

expansion of Synge’s world function [

]. Tangential components of

reparametrize the path, and

the use of E

rather than the squared straight-line length accounts for them.

B Reproducibility

Every result in this paper is reproduced by the self-contained archive

d4 regge second order.zip

at github.com/raghu91302/ssmtheory. It requires Python 3 with

numpy

scipy

sympy

, and

matplotlib

, and runs from a single directory; its

README.md

gives the commands. The complex

is built by

emergent periodic d4.py

and its Regge variations by

emergent periodic engine.py

The exact derivative tensors over

(

√

) are built by

emergent exact tensors.py

, and their output

is included.

Section 2 and Fig. 1(b):

edge spectrum.py

gives the Bloch spectrum on the ﬁfteen edge classes.

Section 3:

lattice tensor.py

and

lt constrain.py

build the exact cubic coeﬃcient tensor, in

about two minutes;

compare.py

compares it with the Einstein–Hilbert tensor of

eh general.py

and

reproduces the direct-contraction values;

check variant.py

repeats the construction for the other

axis spines, the random spine, and L = 6; and quadratic exact.py and linear k term.py prove

parts (a) and (b) of Theorem 1. Section 4:

gamma ppn.py

gives the static ﬁeld, the normalization, and

the conformal-ansatz ratio, and

gamma spines.py

gives Table 1 for all ﬁve subdivisions. Section 5:

regge static.py

builds the slab complex and evaluates the exact nonlinear residual, and

geo len.py

computes Eq.

(4)

flat test.py

performs the ﬂat-space validation and

flat q.py

its quadrature

check.

strong form fit.py

gives the per-shell ﬁts of Fig. 3(a), and

edge modes.py

decomposes

the per-edge residual into the metric, stiﬀ, and null subspaces.

beta weak.py

, run for all settings

run nonlinear.py

, gives the weak-form ﬁts of Table 2 in about two minutes per mass, with

the quadrature order set by the environment variable

beta weak big.py

performs the extended

run to

= 12

5 in about four minutes and 2

5 GB of memory, and

extrapolation forms.py

compares the extrapolation forms. Footnote 1:

z4flat.py

. Figures:

make fig complex.py

and

make figs paper.py

. The ﬁle

results reference.txt

records the outputs quoted in Sec. 5. The

helper modules

aniso.py

emergent eh vertex.py

, and

emergent star.py

are imported by the

scripts above.

Declarations

Funding. No funding was received for this study.

Competing interests. The author declares no competing interests.

Use of AI tools. During the preparation of this work the author used Anthropic’s Claude to assist

with the veriﬁcation and analysis scripts, the ﬁgures, the L

X preparation, and the analysis of

the numerical results. The author reviewed and veriﬁed all computational results and takes full

responsibility for the content of the publication.

Data availability. All code is openly available in the archive

d4 regge second order.zip

github.com/raghu91302/ssmtheory (Appendix B); no experimental data were used.

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