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1 \section{Geometric transport and local work}
2 \label{c:core:transport}
3 
4 The first gravitational question is local: how does a prescribed geometry
5 change transport and energy exchange? A second question is dynamical: which
6 equations determine that geometry? Keeping these questions distinct lets a
7 finite model establish an exact balance law before it is coupled to a
8 gravitational constraint system. Throughout this section the metric
9 coefficients are supplied functions. The scalar field evolves on them.
10 
11 \subsection{A finite scalar action on a diagonal metric}
12 
13 Take a periodic chain with $N\geq3$ sites, spacing $h>0$, and indices modulo
14 $N$. The continuum metric motivating its coefficients is
15 \[
16  ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,
17  \qquad N(t,x)>0,\quad a(t,x)>0.
18 \]
19 The repeated letter $N$ in this metric is the lapse, whereas the integer
20 $N$ counts lattice sites. In the finite formulas below only the coefficient
21 names $w$, $c$, and $v$ are needed:
22 \[
23  w_j=\frac{a_j}{N_j}>0,\qquad
24  c_{j+1/2}=\frac{N_{j+1/2}}{a_{j+1/2}}>0,\qquad
25  v_j=N_ja_jm^2\geq0.
26 \]
27 Thus $N/a$ is the coordinate characteristic speed of the continuum
28 massless equation. Assume these finite coefficient arrays are continuously
29 differentiable in time. For a real field $\varphi_j(t)$ set
30 $\Delta_j=\varphi_{j+1}-\varphi_j$ and define
31 \begin{equation}
32  L_h=\frac h2\sum_jw_j\dot\varphi_j^2
33  -\frac1{2h}\sum_jc_{j+1/2}\Delta_j^2
34  -\frac h2\sum_jv_j\varphi_j^2.
35  \label{c:core:scalar-action}
36 \end{equation}
37 Variations with compact time support give
38 \begin{equation}
39  w_j\ddot\varphi_j+\dot w_j\dot\varphi_j
40  =\frac{c_{j+1/2}\Delta_j-c_{j-1/2}\Delta_{j-1}}{h^2}
41  -v_j\varphi_j.
42  \label{c:core:scalar-evolution}
43 \end{equation}
44 The neighboring edge terms in the action produce the two terms in the
45 spatial numerator. The time integration by parts differentiates
46 $w_j\dot\varphi_j$, including the $\dot w_j\dot\varphi_j$ term.
47 
48 \Needspace{20\baselineskip}
49 \begin{proposition}[Local energy, flux, and geometric work]
50 \label{c:core:scalar-work}
51 For every twice differentiable solution of
52 \eqref{c:core:scalar-evolution}, define
53 \begin{align}
54  e_j&=\frac h2w_j\dot\varphi_j^2+\frac h2v_j\varphi_j^2
55  +\frac{c_{j+1/2}}{4h}\Delta_j^2
56  +\frac{c_{j-1/2}}{4h}\Delta_{j-1}^2,\nonumber\\
57  F_{j+1/2}&=-\frac{c_{j+1/2}}{2h}\Delta_j
58              (\dot\varphi_j+\dot\varphi_{j+1}),\nonumber\\
59  s_j&=-\frac h2\dot w_j\dot\varphi_j^2
60  +\frac h2\dot v_j\varphi_j^2
61  +\frac{\dot c_{j+1/2}}{4h}\Delta_j^2
62  +\frac{\dot c_{j-1/2}}{4h}\Delta_{j-1}^2.
63  \label{c:core:scalar-densities}
64 \end{align}
65 Then
66 \begin{equation}
67  \dot e_j+F_{j+1/2}-F_{j-1/2}=s_j.
68  \label{c:core:scalar-balance}
69 \end{equation}
70 In particular, $E_h=\sum_je_j$ satisfies
71 $\dot E_h=\sum_js_j=-\partial_tL_h$, where the partial derivative holds
72 $\varphi$ and $\dot\varphi$ fixed.
73 \end{proposition}
74 
75 \begin{proof}
76 Put $B_j=c_{j+1/2}\Delta_j$. Using the equation of motion, the derivative
77 of the kinetic and mass terms at site $j$ is
78 \[
79  \frac{\dot\varphi_j}{h}(B_j-B_{j-1})
80  -\frac h2\dot w_j\dot\varphi_j^2
81  +\frac h2\dot v_j\varphi_j^2.
82 \]
83 The right edge contribution differentiates to
84 \[
85  \frac{\dot c_{j+1/2}\Delta_j^2}{4h}
86  +\frac{B_j}{2h}(\dot\varphi_{j+1}-\dot\varphi_j),
87 \]
88 and the left edge has the analogous expression. Adding these terms
89 leaves
90 $B_j(\dot\varphi_j+\dot\varphi_{j+1})/(2h)$ on the right and its
91 negative counterpart on the left, plus $s_j$. These are exactly the flux
92 terms in \eqref{c:core:scalar-balance}. Periodic summation cancels every
93 interface flux. Each edge coefficient derivative appears twice with
94 weight $1/(4h)$, giving $-\partial_tL_h$.
95 \end{proof}
96 
97 Every $e_j$ is nonnegative under the stated coefficient assumptions. An
98 interval of cells $r,\ldots,s$ obeys the exact coarse balance
99 \[
100  \frac{d}{dt}\sum_{j=r}^se_j
101  +F_{s+1/2}-F_{r-1/2}=\sum_{j=r}^ss_j.
102 \]
103 Thus coarse energy is obtained by adding fine cell energies and retaining
104 only the exposed boundary fluxes. This is the same cancellation mechanism
105 as projector transfer, applied here to a classical field energy. Its
106 physical units and constitutive coefficients come from the action.
107 
108 \begin{example}[Time dependent expansion]
109 Let the lapse equal one, let $a(t)>0$ be spatially uniform, and let $m=0$.
110 Then $w=a$, $c=1/a$, and
111 \[
112  \dot E_h=-\frac h2\dot a\sum_j\dot\varphi_j^2
113  -\frac{\dot a}{2ha^2}\sum_j\Delta_j^2.
114 \]
115 An increasing $a$ transfers energy out of this coordinate-time field
116 Hamiltonian. The supplied background receives no evolution equation
117 from this calculation. A self-consistent matter--geometry model must
118 also evolve its gravitational variables.
119 \end{example}
120 
121 For a smooth field sampled at sites and a smooth coefficient sampled at
122 edge midpoints, the spatial expression is second-order consistent with
123 $\partial_x(c\partial_x\varphi)$. Indeed, expanding at $x_j$ gives
124 \[
125  c(x_j+h/2)[\varphi(x_j+h)-\varphi(x_j)]
126  -c(x_j-h/2)[\varphi(x_j)-\varphi(x_j-h)]
127  =h^2(c\varphi_x)_x(x_j)+O(h^4).
128 \]
129 Uniform bounds on the requisite fourth derivatives make the remainder
130 uniform on a compact spatial region. The exact finite balance holds
131 without any Taylor approximation.
132 
133 \gravityfigure{01_local_energy_work}{A six-cell example of prescribed
134 metric work. Each direct cell-energy derivative agrees with incoming
135 flux minus outgoing flux plus the coefficient work term. The curves
136 illustrate the exact finite identity for a prescribed background.}
137 \FloatBarrier
138 
139 
140 
141 \subsection{Conformal curvature as an exact boost holonomy}
142 
143 Consider a twice continuously differentiable real function $\phi$ on an
144 oriented $(t,x)$ rectangle, and the coframe
145 \[
146  e^0=e^\phi dt,\qquad e^1=e^\phi dx,
147  \qquad ds^2=e^{2\phi}(-dt^2+dx^2).
148 \]
149 Write
150 \[
151  K=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
152  \omega=(\phi_xdt+\phi_tdx)K.
153 \]
154 The coframe satisfies the torsion-free Cartan equations with this
155 connection. Since every connection value is a multiple of the same
156 matrix, $\omega\wedge\omega=0$. Therefore
157 \begin{equation}
158  \Omega=d\omega=(\phi_{tt}-\phi_{xx})K\,dt\wedge dx,
159  \qquad R=2e^{-2\phi}(\phi_{tt}-\phi_{xx}).
160  \label{c:core:conformal-curvature}
161 \end{equation}
162 This fixes the scalar-curvature sign convention used in this monograph.
163 
164 We use the archive's plus-sign edge convention: for each oriented
165 boundary edge define $U_e=\exp(+\int_e\omega)$. This specifies the
166 signed transport observable used below. Multiplying in boundary order
167 yields an exact identity
168 \begin{equation}
169  W_\square
170  =\exp\left(K\oint_{\partial\square}(\phi_xdt+\phi_tdx)\right)
171  =\exp\left(\frac K2\int_\square e^{2\phi}R\,dt\,dx\right).
172  \label{c:core:conformal-holonomy}
173 \end{equation}
174 To see the second equality directly, integrate $\phi_{tt}$ along the
175 two horizontal endpoints and $\phi_{xx}$ along the two vertical
176 endpoints, with the boundary orientation from $dt\wedge dx$. This is
177 the rectangular fundamental theorem of calculus. No path-order
178 correction survives because the edge exponents commute.
179 
180 If the integrated coefficient is $\chi$, then
181 \[
182  e^{\chi K}=\cosh\chi\,I+\sinh\chi\,K.
183 \]
184 This preserves the Lorentz form $\operatorname{diag}(-1,1)$ and has
185 positive eigenvalues $e^{\chi}$ and $e^{-\chi}$. Its real principal
186 matrix logarithm is $\chi K$. These are Lorentz boosts. The positive
187 Hilbert norm used for unitary color transport in the gauge monograph
188 is a different structure.
189 
190 \begin{example}[Constant curvature density]
191 For $\phi(t,x)=b(t^2-x^2)/2$, one has
192 \[
193  \phi_{tt}-\phi_{xx}=2b,
194  \qquad R=4b\,e^{-2\phi},
195  \qquad W_\square=e^{2b\,\operatorname{area}(\square)K}.
196 \]
197 The curvature density $e^{2\phi}R/2$ is constant although $R$ varies
198 with position. Adding oriented subrectangle integrals reconstructs
199 the coarse rapidity exactly.
200 \end{example}
201 
202 The plus-sign observable is assigned the endpoint action
203 \[
204  G(t,x)=e^{\chi(t,x)K},\qquad
205  U_{a\to b}'=G(b)U_{a\to b}G(a)^{-1}.
206 \]
207 It corresponds to the signed connection rule
208 $\omega_+'=\omega_++d\chi\,K$. Intermediate endpoint factors cancel,
209 so the loop observable is conjugated at its base point. For the
210 Cartan convention $D=d+\omega$, a coframe transformation $e\mapsto Ge$
211 instead gives $\omega'=\omega-d\chi\,K$ and parallel transport
212 $\exp(-\int_e\omega)$. Its closed-loop rapidity is the negative of the
213 rapidity of $W_\square$ defined here. The fixed-frame curvature identity
214 \eqref{c:core:conformal-holonomy} is unchanged. Declaring these signs
215 avoids mixing the archive's edge observable with a different Cartan
216 parallel-transport convention.
217 
218 
219 
220 \section{Weak gravity as a sign and consistency benchmark}
221 \label{c:core:weak-field}
222 
223 Weak gravity provides checks on attraction, variational signs, and
224 optical observables. These checks are performed within the weak-field
225 model. They do not furnish nonlinear gravitational evolution.
226 
227 \subsection{Radial Poisson equation and a finite variational problem}
228 
229 Let $G>0$, let $\rho(r)$ be a continuous spherical density, and impose
230 regularity $\lim_{r\downarrow0}r^2\Phi'(r)=0$. The radial Poisson
231 equation gives
232 \[
233  (r^2\Phi')'=4\pi G r^2\rho,
234  \qquad M_{\rm enc}(r)=4\pi\int_0^rs^2\rho(s)\,ds,
235  \qquad \Phi'(r)=\frac{GM_{\rm enc}(r)}{r^2}.
236 \]
237 The acceleration is $a_r=-\Phi'$. A positive density therefore gives
238 an inward radial acceleration.
239 
240 Putting $u(r)=r\Phi(r)$ removes the radial first derivative:
241 \[
242  r^2\Phi'=ru'-u,\qquad
243  (r^2\Phi')'=ru'',\qquad u''=4\pi G r\rho.
244 \]
245 For fixed endpoint values, this equation is the stationary condition of
246 \[
247  S[u]=\int_0^{R_0}\left[\frac{(u')^2}{2G}+4\pi r\rho u\right]dr.
248 \]
249 On $r_i=ih$, $i=0,\ldots,M$, with $Mh=R_0$, its finite counterpart is
250 \begin{equation}
251  S_h[u]=\sum_{i=0}^{M-1}\frac{(u_{i+1}-u_i)^2}{2Gh}
252  +4\pi h\sum_{i=1}^{M-1}r_i\rho_i u_i.
253  \label{c:core:poisson-action}
254 \end{equation}
255 
256 \begin{proposition}[Unique finite radial solution]
257 \label{c:core:poisson-minimum}
258 With $u_0,u_M$ fixed, $S_h$ has a unique minimizer. Its interior values
259 satisfy
260 \[
261  \frac{u_{i+1}-2u_i+u_{i-1}}{h^2}=4\pi G r_i\rho_i.
262 \]
263 \end{proposition}
264 
265 \begin{proof}
266 Differentiation with respect to $u_i$ gives
267 $(2u_i-u_{i-1}-u_{i+1})/(Gh)+4\pi h r_i\rho_i$. Setting it to zero
268 gives the equation. For a variation $v$ with $v_0=v_M=0$, the Hessian
269 quadratic form is
270 \[
271  \frac1{Gh}\sum_{i=0}^{M-1}(v_{i+1}-v_i)^2.
272 \]
273 Its vanishing forces $v$ to be constant and hence zero. The finite
274 Hessian is positive definite, so the quadratic action has one
275 stationary point and that point is its unique minimizer.
276 \end{proof}
277 
278 The boundary value $u_0=0$ allows a regular finite potential at the
279 origin. The quotient $u/r$ is used only for $r>0$, with its continuous
280 limit determined separately at $r=0$.
281 
282 For the normalized Gaussian density
283 \[
284  \rho(r)=\frac{M_0}{(2\pi)^{3/2}\sigma^3}e^{-r^2/(2\sigma^2)},
285  \qquad \sigma>0,
286 \]
287 direct differentiation verifies
288 \begin{align*}
289  M_{\rm enc}(r)&=M_0\left[
290  \operatorname{erf}\left(\frac r{\sqrt2\sigma}\right)
291  -\sqrt{\frac2\pi}\frac r\sigma e^{-r^2/(2\sigma^2)}\right],\\
292  \Phi(r)&=-\frac{GM_0}{r}
293  \operatorname{erf}\left(\frac r{\sqrt2\sigma}\right).
294 \end{align*}
295 The potential tends to zero at infinity and has the finite central
296 value $-\sqrt{2/\pi}\,GM_0/\sigma$. This removes the point-source
297 singularity while preserving the attractive sign.
298 
299 \subsection{Clocks and straight-ray optical integrals}
300 
301 In the stipulated first-order isotropic weak-field metric,
302 \[
303  ds^2=-\left(1+\frac{2\Phi}{c_0^2}\right)c_0^2dt^2
304        +\left(1-\frac{2\Phi}{c_0^2}\right)d\mathbf x^2,
305  \qquad \left|\frac{\Phi}{c_0^2}\right|\ll1,
306 \]
307 a static clock has lapse $\mathcal N=\sqrt{1+2\Phi/c_0^2}$.
308 The static-frequency ratio in this approximation is
309 \[
310  \frac{\nu_{\rm obs}}{\nu_{\rm emit}}
311  =\frac{\mathcal N_{\rm emit}}{\mathcal N_{\rm obs}}
312  =1+\frac{\Phi_{\rm emit}-\Phi_{\rm obs}}{c_0^2}
313  +O\left(\frac{\Phi^2}{c_0^4}\right).
314 \]
315 For $\Phi(x,b)=-GM_0/\sqrt{x^2+b^2}$, $b>0$, evaluating the
316 first-order straight-ray integrals over $[-L,L]$ gives
317 \begin{align*}
318  \Delta t_{\rm Shapiro}
319  &=-\frac2{c_0^3}\int_{-L}^L\Phi(x,b)\,dx
320  =\frac{4GM_0}{c_0^3}\operatorname{asinh}\left(\frac Lb\right),\\
321  \alpha(L,b)
322  &=\frac2{c_0^2}\left|\int_{-L}^L\partial_b\Phi(x,b)\,dx\right|
323  =\frac{4GM_0L}{c_0^2b\sqrt{L^2+b^2}}.
324 \end{align*}
325 Here $\alpha$ is the magnitude of the deflection toward the source.
326 The limit $L\to\infty$ gives $4GM_0/(c_0^2b)$. The displayed
327 equalities evaluate the weak-field integrals exactly. The approximation
328 to the physical ray already entered through the metric expansion and
329 the undeviated integration path.
330 
331 
332 
333 \section{Gowdy evolution and reconstruction of the remaining constraint}
334 \label{c:core:gowdy}
335 
336 The next step is an inhomogeneous nonlinear model with two gravitational
337 field channels. We take the archived Gowdy reduction as specified by its
338 action and constraint densities. The results below prove the internal
339 variational and reconstruction identities of that reduced system.
340 
341 Let $P,Q$ be real smooth fields on $I\times\T$, with
342 $\theta\in\R/(2\pi\Z)$ and $\gamma(\tau)=e^{-2\tau}$. Define
343 \begin{equation}
344  S_G=\frac12\int_I\int_0^{2\pi}
345  \left[P_\tau^2-\gamma P_\theta^2
346  +e^{2P}(Q_\tau^2-\gamma Q_\theta^2)\right]d\theta\,d\tau.
347  \label{c:core:gowdy-action}
348 \end{equation}
349 Time variations vanish at the endpoints of $I$ and spatial variations
350 are periodic. Integration by parts gives the two Euler--Lagrange
351 expressions
352 \begin{align}
353  \mathcal E_P&=P_{\tau\tau}-\gamma P_{\theta\theta}
354  -e^{2P}(Q_\tau^2-\gamma Q_\theta^2),\nonumber\\
355  \mathcal E_Q&=Q_{\tau\tau}-\gamma Q_{\theta\theta}
356  +2(P_\tau Q_\tau-\gamma P_\theta Q_\theta).
357  \label{c:core:gowdy-euler}
358 \end{align}
359 In particular, the $Q$ equation can first be written as
360 $\partial_\tau(e^{2P}Q_\tau)-\partial_\theta(e^{2P}\gamma Q_\theta)=0$.
361 The stationary equations are $\mathcal E_P=\mathcal E_Q=0$.
362 
363 Define the densities
364 \begin{align}
365  H_G&=P_\tau^2+\gamma P_\theta^2
366  +e^{2P}(Q_\tau^2+\gamma Q_\theta^2),\nonumber\\
367  M_G&=2(P_\tau P_\theta+e^{2P}Q_\tau Q_\theta).
368  \label{c:core:gowdy-densities}
369 \end{align}
370 The remaining reconstruction equations are
371 $\lambda_\tau=H_G$ and $\lambda_\theta=M_G$.
372 
373 \Needspace{14\baselineskip}
374 \begin{proposition}[Local integrability and global periodicity]
375 \label{c:core:gowdy-integrability}
376 For twice continuously differentiable $P,Q$,
377 \begin{equation}
378  \partial_\tau M_G-\partial_\theta H_G
379  =2P_\theta\mathcal E_P+2e^{2P}Q_\theta\mathcal E_Q.
380  \label{c:core:gowdy-identity}
381 \end{equation}
382 For a solution on $I\times\T$, a single-valued periodic $\lambda$ with
383 these two derivatives exists if and only if
384 $\int_0^{2\pi}M_G\,d\theta=0$ at one, and hence every, time. It is
385 unique up to an additive constant.
386 \end{proposition}
387 
388 \begin{proof}
389 Differentiate the two densities. The terms with $P_\tau
390 P_{\tau\theta}$ and $Q_\tau Q_{\tau\theta}$ cancel, leaving
391 \[
392 \begin{aligned}
393  \partial_\tau M_G-\partial_\theta H_G
394  ={}&2P_\theta(P_{\tau\tau}-\gamma P_{\theta\theta})\\
395  &+2e^{2P}\big[
396  Q_\theta(Q_{\tau\tau}-\gamma Q_{\theta\theta})
397  +2P_\tau Q_\tau Q_\theta
398  -P_\theta Q_\tau^2-\gamma P_\theta Q_\theta^2\big].
399 \end{aligned}
400 \]
401 Substitute \eqref{c:core:gowdy-euler} and combine the two occurrences of
402 $\gamma P_\theta Q_\theta^2$. This gives
403 \eqref{c:core:gowdy-identity}. On a solution the identity vanishes,
404 so periodicity gives $d\int M_G\,d\theta/d\tau=0$.
405 
406 If a periodic $\lambda$ exists, integrating $\lambda_\theta=M_G$
407 forces the zero-period condition. Conversely, fix $\tau_0$ and define
408 on the lifted strip
409 \[
410  \lambda(\tau,\theta)=\lambda_0
411  +\int_{\tau_0}^{\tau}H_G(s,0)\,ds
412  +\int_0^{\theta}M_G(\tau,\eta)\,d\eta.
413 \]
414 Its $\theta$ derivative is $M_G$. Its $\tau$ derivative is $H_G$ by
415 the integrability identity and the fundamental theorem of calculus.
416 The zero-period condition makes it periodic. The difference of two
417 such functions has both derivatives zero, so is constant.
418 \end{proof}
419 
420 This separates a local compatibility condition from a global topology
421 condition. Satisfying the evolution equations makes the reconstruction
422 one-form closed. A vanishing spatial period makes it the derivative of
423 a periodic potential.
424 
425 \subsection{An exact polarized mode}
426 
427 Set $Q=0$ and take an integer $n\geq1$. Define the entire function
428 \[
429  J_0(x)=\sum_{m=0}^{\infty}\frac{(-1)^m(x^2/4)^m}{(m!)^2}.
430 \]
431 The series and its differentiated series converge uniformly on compact
432 sets. Coefficient comparison gives
433 $x^2J_0''+xJ_0'+x^2J_0=0$. Thus
434 \begin{equation}
435  P(\tau,\theta)=A J_0(ne^{-\tau})\cos(n\theta)
436  \label{c:core:gowdy-mode}
437 \end{equation}
438 is an exact polarized solution. Indeed, for $x=ne^{-\tau}$,
439 $\partial_\tau^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x)$,
440 whereas $P_{\theta\theta}=-n^2P$. Also
441 $M_G=2P_\tau P_\theta$ has zero spatial integral, so periodic
442 $\lambda$ can be reconstructed.
443 
444 The Taylor channels can be separated without changing the physical
445 mode. Put $y=x^2/4$, $\mathcal B(y)=J_0(2\sqrt y)$, and for an integer
446 $r\geq2$ define
447 \[
448  \mathcal B_{r,j}(y)=\sum_{q=0}^{\infty}
449  \frac{(-1)^{rq+j}y^{rq+j}}{((rq+j)!)^2},
450  \qquad j=0,\ldots,r-1.
451 \]
452 Then $\mathcal B=\sum_j\mathcal B_{r,j}$. With
453 $\vartheta=y\,d/dy$, the coefficient identity
454 \[
455  \vartheta^2\mathcal B_{r,j}+y\mathcal B_{r,j-1}=0,
456  \qquad j\pmod r,
457 \]
458 follows from $m^2/(m!)^2=1/((m-1)!)^2$ for $m\geq1$.
459 Consequently the separated Taylor channels are coupled. Their sum
460 satisfies the Bessel equation. This decomposition resolves series
461 order and does not introduce additional independent gravitational
462 degrees of freedom.
463 
464 
465 
466 \section{An exact finite Gowdy constraint projection}
467 \label{c:core:gowdy-projection}
468 
469 Let $h=2\pi/N$ and use the periodic centered derivative
470 \[
471  (D_hf)_j=\frac{f_{j+1}-f_{j-1}}{2h},\qquad
472  (\Delta_hf)_j=\frac{f_{j+1}-2f_j+f_{j-1}}{h^2}.
473 \]
474 Here $D_h^{\mathsf T}=-D_h$. With $U=\dot P$, $V=\dot Q$, the raw
475 semidiscrete vector field $f_h(\tau,z)$ for $z=(P,Q,U,V)$ is
476 \begin{align}
477  \dot P&=U,&\dot Q&=V,\nonumber\\
478  \dot U&=\gamma\Delta_hP+e^{2P}[V^2-\gamma(D_hQ)^2],\nonumber\\
479  \dot V&=\gamma\Delta_hQ-2[UV-\gamma(D_hP)(D_hQ)].
480  \label{c:core:gowdy-raw}
481 \end{align}
482 Products and exponentials in these formulas are componentwise.
483 The finite global momentum is
484 \begin{equation}
485  C_h(z)=2h\sum_j[U_j(D_hP)_j+e^{2P_j}V_j(D_hQ)_j].
486  \label{c:core:gowdy-momentum}
487 \end{equation}
488 It is linear in the velocity variables when the configuration $P,Q$
489 is fixed.
490 
491 For the ordinary Euclidean gradients, skew-adjointness of $D_h$ gives
492 \begin{align*}
493  \nabla_PC_h&=-2hD_hU+4h e^{2P}V(D_hQ),\\
494  \nabla_QC_h&=-2hD_h(e^{2P}V),\\
495  g_U:=\nabla_UC_h&=2hD_hP,
496  &g_V:=\nabla_VC_h&=2h e^{2P}D_hQ.
497 \end{align*}
498 Put $G_h=(0,0,g_U,g_V)$ and $d_h=\norm{g_U}_2^2+\norm{g_V}_2^2$.
499 On the open region $d_h>0$, define
500 \[
501  \Lambda_h=\frac{\nabla C_h\cdot f_h}{d_h},\qquad
502  \widehat f_h=f_h-\Lambda_hG_h.
503 \]
504 The chain rule proves the exact tangency identity
505 \[
506  \nabla C_h\cdot\widehat f_h
507  =\nabla C_h\cdot f_h-\Lambda_hd_h=0.
508 \]
509 Every differentiable solution that remains in this region therefore
510 keeps $C_h$ constant.
511 
512 \begin{proposition}[Exact endpoint momentum]
513 \label{c:core:gowdy-endpoint}
514 Start from $z_n$ and let a provisional step produce
515 $\widetilde z=(\widetilde P,\widetilde Q,\widetilde U,\widetilde V)$.
516 Form $\widetilde g_U,\widetilde g_V$ at its configuration and assume
517 $\widetilde d_h>0$. The correction
518 \[
519  \mu=\frac{C_h(\widetilde z)-C_h(z_n)}{\widetilde d_h},\qquad
520  z_{n+1}=(\widetilde P,\widetilde Q,
521  \widetilde U-\mu\widetilde g_U,
522  \widetilde V-\mu\widetilde g_V)
523 \]
524 satisfies $C_h(z_{n+1})=C_h(z_n)$ exactly. It is the correction of
525 smallest Euclidean velocity norm at the fixed configuration.
526 \end{proposition}
527 
528 \begin{proof}
529 Linearity in the velocities gives
530 $C_h(z_{n+1})=C_h(\widetilde z)-\mu\widetilde d_h$.
531 For minimality, write $g=(\widetilde g_U,\widetilde g_V)$ and
532 $b=C_h(z_n)-C_h(\widetilde z)$. Every admissible velocity displacement
533 $v$ satisfies $g\cdot v=b$. Cauchy--Schwarz implies
534 $\norm v\geq|b|/\norm g$, with equality only at
535 $v=bg/\norm g^2$. This is the displayed correction.
536 \end{proof}
537 
538 \Needspace{11\baselineskip}
539 \begin{proposition}[Fixed-mesh second-order time accuracy]
540 \label{c:core:gowdy-time-order}
541 Fix $h$. Suppose a smooth exact solution of
542 $\dot z=\widehat f_h(\tau,z)$ and its numerical steps remain in a
543 compact regular neighborhood where $d_h\geq d_0>0$. Assume the mixed time and state
544 derivatives of $\widehat f_h$ through order three and the state
545 derivatives of $C_h$ through order four are bounded there. Apply the endpoint correction
546 of Proposition~\ref{c:core:gowdy-endpoint} after explicit midpoint.
547 Then its local error is $O(\Delta\tau^3)$ and its global error on a
548 fixed time interval is $O(\Delta\tau^2)$.
549 \end{proposition}
550 
551 \begin{proof}
552 The explicit midpoint expansion agrees with the Taylor expansion of
553 the exact flow through order two. Its provisional endpoint differs
554 from the exact endpoint by $O(\Delta\tau^3)$. Since the exact flow
555 keeps $C_h$ constant and $C_h$ has bounded first derivatives, the
556 provisional constraint defect is also $O(\Delta\tau^3)$. The lower
557 bound on the denominator and the upper bound on $g$ make the
558 correction $O(\Delta\tau^3)$. Thus the corrected local error has the
559 same order. Under the stated derivative bounds the corrected step
560 map has Lipschitz constant at most $1+L\Delta\tau$ for small steps.
561 If $e_n$ is its error against the exact solution, then
562 \[
563  e_{n+1}\leq(1+L\Delta\tau)e_n+M\Delta\tau^3.
564 \]
565 Iterating this inequality for $n\Delta\tau\leq T$ bounds the error by
566 $MT e^{LT}\Delta\tau^2$ when $e_0=0$.
567 \end{proof}
568 
569 These are regular-chart results. For a spatially homogeneous
570 configuration, $g_U=g_V=0$, and the displayed multiplier is
571 undefined even though $C_h=0$. Constants in the accuracy theorem
572 can depend on $h$. A claim about simultaneous spatial and temporal
573 refinement additionally needs bounds uniform in $h$.
574 
575 The local compatibility diagnostic is a separate quantity. Define
576 $H_{G,h},M_{G,h}$ by replacing spatial derivatives in
577 \eqref{c:core:gowdy-densities} with $D_h$. If
578 $\mathcal E_{P,h}=\dot U-\dot U_{\rm raw}$ and
579 $\mathcal E_{Q,h}=\dot V-\dot V_{\rm raw}$, then
580 \[
581  \mathcal I_h=\partial_\tau M_{G,h}-D_hH_{G,h}
582  -2(D_hP)\mathcal E_{P,h}
583  -2e^{2P}(D_hQ)\mathcal E_{Q,h}
584 \]
585 subtracts the explicit constraint-force contribution. Exact global
586 momentum preservation does not force $\mathcal I_h$ to vanish at each
587 site. Its refinement behavior tests whether the modified evolution
588 approaches the local continuum reconstruction identity. The archived
589 second-order numerical observations are experimental evidence for
590 that behavior, with their tested range recorded in the companion.
591 
592 
593 
594 \section{Finite linearized ADM gravity}
595 \label{c:core:adm}
596 
597 The symmetry-reduced model can now be compared with a finite
598 three-dimensional linearized constraint system. Its exact algebraic
599 closure rests on commuting linear derivatives. Nonlinear products
600 will introduce an additional requirement in the next section.
601 
602 \subsection{Finite calculus and the quadratic action}
603 
604 Let $V_h$ be a finite real inner-product space. Let
605 $\delta_1,\delta_2,\delta_3$ be commuting skew-adjoint operators on it:
606 \[
607  \delta_i\delta_j=\delta_j\delta_i,
608  \qquad\inner{\delta_iX}{Y}_h=-\inner{X}{\delta_iY}_h,
609  \qquad\delta^2=\sum_i\delta_i^2.
610 \]
611 The periodic centered differences in three coordinate directions
612 provide one example. All tensor inner products below sum over both
613 indices, using the Frobenius inner product on symmetric tensors.
614 
615 Let $h_{ij}=h_{ji}$ and $\pi_{ij}=\pi_{ji}$ take values in $V_h$.
616 Repeated spatial indices are summed. Put $h_{\rm tr}=h_{kk}$,
617 $\pi_{\rm tr}=\pi_{kk}$ and define
618 \begin{align}
619  C&=\delta_i\delta_jh_{ij}-\delta^2h_{\rm tr},
620  &C_i&=-2\delta_j\pi_{ij},\nonumber\\
621  \mathcal G_{ij}(h)&=\frac12\big[
622  \delta_k\delta_ih_{jk}+\delta_k\delta_jh_{ik}
623  -\delta^2h_{ij}-\delta_i\delta_jh_{\rm tr}
624  -\delta_{ij}C\big].
625  \label{c:core:adm-operators}
626 \end{align}
627 
628 \begin{lemma}[Linearized finite Bianchi and adjoint identities]
629 \label{c:core:adm-bianchi}
630 The spatial operator obeys $\delta_j\mathcal G_{ij}(h)=0$ and is
631 self-adjoint on symmetric tensors.
632 \end{lemma}
633 
634 \begin{proof}
635 Twice its divergence expands as
636 \begin{align*}
637  2\delta_j\mathcal G_{ij}
638  ={}&\delta_i\delta_j\delta_kh_{jk}
639  +\delta^2\delta_kh_{ik}-\delta^2\delta_jh_{ij}
640  -\delta_i\delta^2h_{\rm tr}\\
641  &-\delta_i\delta_k\delta_lh_{kl}
642  +\delta_i\delta^2h_{\rm tr}.
643 \end{align*}
644 The terms cancel in three pairs. For self-adjointness, write
645 $B_i(a)=\delta_ja_{ij}$ and $A(a)=\delta_i\delta_ja_{ij}$. Two
646 applications of the adjoint identity give
647 \begin{align*}
648  2\inner{a}{\mathcal G(b)}_h
649  ={}&-2\inner{B(a)}{B(b)}_h
650  +\sum_k\inner{\delta_ka}{\delta_kb}_h\\
651  &-\inner{A(a)}{b_{\rm tr}}_h
652  -\inner{a_{\rm tr}}{A(b)}_h
653  -\sum_k\inner{\delta_ka_{\rm tr}}{\delta_kb_{\rm tr}}_h.
654 \end{align*}
655 This expression is symmetric in $a,b$.
656 \end{proof}
657 
658 Set
659 \[
660  T_h=\inner{\pi}{\pi}_h-\tfrac12\inner{\pi_{\rm tr}}{\pi_{\rm tr}}_h,
661  \qquad U_h=\tfrac12\inner{h}{\mathcal G(h)}_h.
662 \]
663 With lapse perturbation $n$ and shift perturbation $n_i$, the action is
664 \begin{equation}
665  S_h^{(2)}=\int\left[
666  \inner{\pi}{\dot h}_h-T_h-U_h
667  +\inner{n}{C}_h-\inner{n_i}{C_i}_h\right]dt.
668  \label{c:core:adm-action}
669 \end{equation}
670 Variation of $n,n_i$ imposes $C=C_i=0$. The remaining equations are
671 \begin{align}
672  \dot h_{ij}&=2\pi_{ij}-\delta_{ij}\pi_{\rm tr}
673              +\delta_in_j+\delta_jn_i,\nonumber\\
674  \dot\pi_{ij}&=-\mathcal G_{ij}(h)
675               +\delta_i\delta_jn-\delta_{ij}\delta^2n.
676  \label{c:core:adm-evolution}
677 \end{align}
678 The kinetic gradient is $2\pi_{ij}-\delta_{ij}\pi_{\rm tr}$.
679 Lemma~\ref{c:core:adm-bianchi} makes the potential gradient
680 $\mathcal G(h)$. The lapse gradient follows by two spatial
681 integrations by parts. Integrating the shift term once gives
682 $-\langle\delta_in_j+\delta_jn_i,\pi_{ij}\rangle_h$ in the
683 Lagrangian, fixing the sign in the first equation.
684 
685 \Needspace{16\baselineskip}
686 \subsection{Constraint propagation and the canonical brackets}
687 
688 \Needspace{11\baselineskip}
689 \begin{theorem}[Exact finite linearized closure]
690 \label{c:core:adm-closure}
691 For every solution of \eqref{c:core:adm-evolution},
692 \[
693  \dot C=-\delta_iC_i,\qquad\dot C_i=0.
694 \]
695 Thus initially vanishing constraints remain zero. With the canonical
696 Frobenius bracket and smearings
697 $C[n]=\langle n,C\rangle_h$, $D[\xi]=\langle\xi_i,C_i\rangle_h$,
698 all three brackets $\{C[n],C[m]\}$, $\{D[\xi],D[\eta]\}$, and
699 $\{C[n],D[\xi]\}$ vanish identically.
700 \end{theorem}
701 
702 \begin{proof}
703 The trace of the momentum part of $\dot h$ is $-\pi_{\rm tr}$.
704 Its contribution to $\dot C$ is therefore
705 $2\delta_i\delta_j\pi_{ij}=-\delta_iC_i$. The shift contribution is
706 \[
707  \delta_i\delta_j(\delta_in_j+\delta_jn_i)
708  -\delta^2(2\delta_kn_k)=0.
709 \]
710 For $\dot C_i$, the divergence of $\mathcal G$ vanishes by
711 Lemma~\ref{c:core:adm-bianchi}, and
712 $\delta_j(\delta_i\delta_jn-\delta_{ij}\delta^2n)=0$.
713 This proves the propagation equations and their stated implication.
714 
715 For the bracket, use
716 $\{F,G\}=\langle\nabla_hF,\nabla_\pi G\rangle_h
717 -\langle\nabla_\pi F,\nabla_hG\rangle_h$.
718 Two scalar constraints depend only on $h$ and two momentum
719 constraints depend only on $\pi$, so their brackets vanish.
720 The gradients for the mixed bracket are
721 \[
722  (\nabla_hC[n])_{ij}=\delta_i\delta_jn-\delta_{ij}\delta^2n,
723  \qquad(\nabla_\pi D[\xi])_{ij}=\delta_i\xi_j+\delta_j\xi_i.
724 \]
725 Their contraction vanishes after one integration by parts because
726 the first tensor is divergence-free.
727 \end{proof}
728 
729 The momentum generator changes the metric perturbation by
730 $h_{ij}\mapsto h_{ij}+\delta_i\xi_j+\delta_j\xi_i$.
731 The corresponding change of the scalar constraint is
732 $2\delta^2\delta_k\xi_k-2\delta^2\delta_k\xi_k=0$.
733 No product rule is used in these cancellations.
734 
735 For a partition of fine spatial sites into coarse regions $A$, the
736 nonnegative diagnostic
737 \[
738  E_A^{\rm con}=\sum_{x\in A}h^3
739  \left(|C_x|^2+\sum_i|C_{i,x}|^2\right)
740 \]
741 adds exactly to the full fine diagnostic. This is an exact partition
742 identity. It does not assert that differentiating a coarse field
743 reconstructs its fine constraint values.
744 
745 
746 
747 \subsection{Tensor waves, positive reduced energy, and resolved dispersion}
748 
749 Choose $n=n_i=0$ and impose the transverse and traceless conditions
750 \[
751  \delta_jh_{ij}=0,\quad h_{\rm tr}=0,
752  \qquad\delta_j\pi_{ij}=0,\quad\pi_{\rm tr}=0.
753 \]
754 They are preserved by the evolution because
755 \[
756  \mathcal G_{ij}(h)=-\tfrac12\delta^2h_{ij},\qquad
757  \dot h=2\pi,\qquad\dot\pi=\tfrac12\delta^2h,
758  \qquad\ddot h=\delta^2h.
759 \]
760 On this reduced sector the energy is
761 \[
762  H_{\rm TT}=\norm{\pi}_h^2
763  +\frac14\sum_k\norm{\delta_kh}_h^2\geq0.
764 \]
765 Its derivative is zero: substituting the two first-order equations
766 reduces it to
767 $\langle\pi,\delta^2h\rangle_h
768 +\sum_k\langle\delta_kh,\delta_k\pi\rangle_h=0$.
769 
770 For a Fourier mode on the periodic centered lattice, let
771 $\widetilde k_i=\sin(k_ih)/h$. If $\widetilde{\mathbf k}\neq0$,
772 define
773 \[
774  \Pi_{ij}=\delta_{ij}
775  -\frac{\widetilde k_i\widetilde k_j}{|\widetilde{\mathbf k}|^2},
776  \qquad
777  \Lambda^{\rm TT}_{ij,kl}
778  =\tfrac12(\Pi_{ik}\Pi_{jl}+\Pi_{il}\Pi_{jk}-\Pi_{ij}\Pi_{kl}).
779 \]
780 For a symmetric tensor $X$, its image is
781 $\Pi X\Pi-\Pi\operatorname{tr}(\Pi X)/2$. It is transverse and
782 traceless because $\Pi^2=\Pi$ and $\operatorname{tr}\Pi=2$.
783 Applying the map again leaves it unchanged. On the transverse
784 two-plane, symmetric tensors have dimension three and the trace
785 condition removes one dimension. Thus the projector has rank two.
786 
787 The resulting frequency obeys
788 \begin{equation}
789  \omega_h^2=\sum_i\frac{\sin^2(k_ih)}{h^2}.
790  \label{c:core:adm-dispersion}
791 \end{equation}
792 For fixed $\mathbf k\neq0$ as $h\to0$,
793 \[
794  \omega_h^2=|\mathbf k|^2-\frac{h^2}{3}\sum_i k_i^4+O(h^4),
795  \qquad
796  \omega_h=|\mathbf k|
797  -\frac{h^2}{6|\mathbf k|}\sum_i k_i^4+O(h^4).
798 \]
799 These follow by expanding the sine and then the positive square root.
800 The regime keeps $\mathbf k$ fixed. It is not uniform up to the
801 Nyquist boundary.
802 
803 The spatial operator in \eqref{c:core:adm-dispersion} is the sum of
804 the squared centered first derivatives. It differs from the
805 nearest-neighbor second-difference operator used for the raw Gowdy
806 equations. For even site counts, an alternating sequence has centered
807 derivative zero. Such checkerboard modes, as well as constant modes,
808 belong to the derivative kernel. At a zero symbol the above TT
809 projector is undefined and the rank-two argument does not apply.
810 Exact constraint propagation consequently coexists with a finite
811 spatial kernel that a physical continuum comparison must address.
812 
813 
814 
815 
816 \gravityfigure{04_adm_dispersion}{The centered derivative has a Nyquist
817 zero, while a fixed resolved mode approaches its continuum frequency
818 with second-order error. The left panel is the one-dimensional symbol;
819 the right panel tests a fixed mode under refinement.}
820 
821 \section{The nonlinear pointwise obstruction and its algebraic alternative}
822 \label{c:core:nonlinear-obstruction}
823 
824 The linearized cancellations use commuting derivatives but never ask
825 them to differentiate a product. A nonlinear diffeomorphism generator
826 does. The following scalar model isolates that issue without relying
827 on the more involved gravitational tensor constraints.
828 
829 Let a periodic real lattice have canonical brackets
830 $\{q_a,p_b\}_h=\delta_{ab}/h$ and a skew matrix $D$. Define
831 \[
832  \mathcal D_h[\xi]=h\sum_a\xi_ap_a(Dq)_a,
833  \qquad
834  \mathfrak L_D(f,g)=D(fg)-(Df)g-f(Dg).
835 \]
836 All products on this lattice are pointwise.
837 
838 \begin{proposition}[Exact nonlinear defect identity]
839 \label{c:core:nonlinear-defect}
840 For arbitrary finite arrays $q,p,\xi,\eta$,
841 \begin{align}
842  \{\mathcal D_h[\xi],\mathcal D_h[\eta]\}_h
843  ={}&\mathcal D_h[\xi D\eta-\eta D\xi]\nonumber\\
844  &+h\sum_ap_a\big[
845  \xi_a\mathfrak L_D(\eta,Dq)_a
846  -\eta_a\mathfrak L_D(\xi,Dq)_a\big].
847  \label{c:core:nonlinear-defect-identity}
848 \end{align}
849 \end{proposition}
850 
851 \begin{proof}
852 Coordinate differentiation gives
853 $\partial_p\mathcal D_h[\xi]=h\xi Dq$ and
854 $\partial_q\mathcal D_h[\xi]=-hD(\xi p)$.
855 Insert these in the canonical bracket and use $D^{\mathsf T}=-D$:
856 \[
857  \{\mathcal D_h[\xi],\mathcal D_h[\eta]\}_h
858  =h\sum_ap_a\left[\xi_aD(\eta Dq)_a
859                   -\eta_aD(\xi Dq)_a\right].
860 \]
861 Expanding each derivative using the definition of $\mathfrak L_D$
862 cancels the two $\xi\eta D^2q$ terms. The remaining product-rule
863 terms give the first term of
864 \eqref{c:core:nonlinear-defect-identity}, and the two defect terms
865 give the second.
866 \end{proof}
867 
868 \begin{theorem}[Finite pointwise derivation obstruction]
869 \label{c:core:pointwise-no-go}
870 A linear derivation on the complete pointwise algebra $\C^N$ is zero.
871 Thus a nonzero finite pointwise derivative cannot obey the undeformed
872 Leibniz rule on every pair of arrays.
873 \end{theorem}
874 
875 \begin{proof}
876 Let $e_a$ be the coordinate idempotents. If $D$ is a derivation, then
877 $D(e_a)=D(e_a^2)=2e_aD(e_a)$. At coordinate $a$ this forces the
878 component to be twice itself, hence zero. At every other coordinate
879 the right side is zero. Therefore $D(e_a)=0$ for every $a$. Linearity
880 and the coordinate basis imply $D=0$.
881 \end{proof}
882 
883 The theorem rules out a universal ordinary derivation on the full
884 finite pointwise algebra. For a particular generator, one must still
885 inspect the contracted defect in
886 \eqref{c:core:nonlinear-defect-identity}. A nonzero Leibniz defect
887 does not by itself imply that every contraction is nonzero.
888 
889 \begin{example}[An exact bracket failure]
890 \label{c:core:bracket-negative-control}
891 Use $N=5$, $h=1$, and $(Df)_a=(f_{a+1}-f_{a-1})/2$.
892 Take $q=e_0$, $p=\xi=e_1$, and $\eta=e_2$. Then
893 $(Dq)_1=-1/2$, $(D\eta)_1=1/2$, and $\eta Dq=0$.
894 The direct bracket in the preceding proof is zero, while
895 \[
896  \mathcal D_h[\xi D\eta-\eta D\xi]=-\tfrac14.
897 \]
898 The defect is therefore exactly $+1/4$. This finite witness shows
899 that the naive centered scalar constraint does fail to close
900 universally in the proposed undeformed form.
901 \end{example}
902 
903 \subsection{Smooth resolved arrays and the matrix corner route}
904 
905 For samples of sufficiently smooth periodic functions, the centered
906 derivative obeys
907 \[
908  D_hf=f'+\frac{h^2}{6}f'''+O(h^4).
909 \]
910 If $f,g$ have uniformly bounded derivatives through order five, its
911 Leibniz defect consequently satisfies
912 \begin{equation}
913  \mathfrak L_{D_h}(f,g)
914  =\frac{h^2}{2}(f''g'+f'g'')+O(h^4).
915  \label{c:core:smooth-defect}
916 \end{equation}
917 For fixed smooth $p,\xi,\eta,q$, with sufficiently many additional
918 bounded derivatives of $q$ to control $D_hq$, the contraction in
919 \eqref{c:core:nonlinear-defect-identity} is $O(h^2)$ on a fixed periodic
920 interval. Each summand is uniformly $O(h^2)$ and $hN$ is the interval
921 length. The coordinate-idempotent witness above is outside this
922 uniformly smooth regime.
923 
924 A spectral derivative has an exact product rule on trigonometric
925 polynomials when every product remains in the represented band without
926 aliasing. Repeated products require further headroom. These restricted
927 domains differ from the complete pointwise algebra. The later resolved
928 constraint theorem includes explicit domain bounds.
929 
930 For the full matrix algebra, choose a Hermitian matrix $K$ and set
931 $\delta_KX=i[K,X]$. Direct multiplication gives
932 \begin{equation}
933  \delta_K(XY)=(\delta_KX)Y+X(\delta_KY).
934  \label{c:core:inner-derivation}
935 \end{equation}
936 It also obeys $\delta_K(X^*)=(\delta_KX)^*$,
937 $\operatorname{Tr}\delta_KX=0$, and is skew-adjoint for the
938 Hilbert--Schmidt inner product. The last identity follows by moving
939 $K$ through the trace. These properties supply an exact finite
940 product calculus on a noncommutative algebra.
941 
942 

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