# Question: 06-inertia-internal-energy

- **Subproject:** 04-ilang-time
- **Package:** 06-inertia-internal-energy
- **Equation tags:** (6.k)
- **Created:** 2026-10-10

## Goal

Determine whether, and how, the inertia of a body depends on its internal cost, here the cost of the clock it carries. Compare with the inverse inertia of 03-ilang-space/20-cost-and-mass. That inverse inertia is the response at $\lambda=0$, and for a body with internal labels it may differ from the response averaged over the internal oscillations. Both are determined here. This is criterion 5.

**Setting.** As in 03-ilang-space/20-cost-and-mass, items 2–4 (quoted under "Inputs"), with places that carry internal labels.
- **Objects.** A body $b$, the only instance of its type, and a medium $c$, one object of another type.
- **Places.** In item 1, the places of $b$ are pairs $h=(x,a)$; in items 2–4, triples $h=(x,s,a)$.
  - $x$ is a position label: from a finite set in item 1, and $x\in\mathbb Z$ in items 2–4.
  - $s\in\{1,2\}$ is a branch label (items 2–4).
  - $a\in\mathbb Z_N$, $N\ge2$, is a clock label.
- **Contract.** $C=C_b\otimes\mathbb 1_c+\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$, where $K_h$ depends only on the position label. Places with the same $x$ have the same $u_h$, so the edges between them lie inside one point of the background.
- **Blocks of $C_b$.** $C_b=\sum_{x,x'}\lvert x\rangle\langle x'\rvert\otimes T_{xx'}$, where $T_{xx'}$ acts on the internal labels and $T_{x'x}=T_{xx'}^\dagger$. The amplitudes of a start $\phi$ at position $x$ form the internal vector $\phi_x$.
- **Model D (items 2–4).** Places $(x,s,a)$ with $x\in\mathbb Z$, as the limit of long finite chains with the start far from both ends (A1). State how the limit is taken. Let
  $$C_b=A+B,\qquad A=\frac\tau2\sum_x\Bigl(\lvert x+1\rangle\langle x\rvert\otimes(-i\sigma_1)+\lvert x\rangle\langle x+1\rvert\otimes(i\sigma_1)\Bigr)\otimes\mathbb 1_N,\qquad B=\sum_x\lvert x\rangle\langle x\rvert\otimes\sigma_3\otimes(m\mathbb 1_N+D),$$
  with $\tau>0$ and $m>0$.
  - $\sigma_1,\sigma_3$ are the Pauli matrices on the branch label.
  - $D=\frac{2\pi}{N\lambda_0}\sum_{k=0}^{N-1}k\lvert f_k\rangle\langle f_k\rvert$ is the minimal ring clock of 02-clock on the clock label, with eigenvalues $d_k=\frac{2\pi k}{N\lambda_0}$.
  - The records are $K_x=\kappa\,x\,X$, with $\kappa>0$ and $X$ as in (17.15).
  - The **rest cost** of clock level $k$ is the positive eigenvalue of $C_b$ on the plane wave with label $p=0$ and clock level $k$.

1. **The inverse inertia of 20 for a body with internal labels.** Places $(x,a)$, general $C_b$, product start $\phi\otimes\chi$.
   - Express $L_{hh'}(0)$ and the inverse inertia $\mu$ of (20.10) through the blocks $T_{xx'}$, $x\neq x'$, and the internal vectors $\phi_x$. Determine whether edges between places with the same $x$ contribute.
   - For the decoupled body ($T_{xx'}=t_{xx'}\mathbb 1_N$ for $x\neq x'$, arbitrary $T_{xx}$) and a start $\phi_X\otimes\eta$, determine $\mu$. Determine whether it depends on $\eta$ or on the $T_{xx}$.

2. **Model D: response at $\lambda=0$.** Consider the start $\phi=\phi_X\otimes\lvert1\rangle\otimes\lvert f_k\rangle$, where $\lvert1\rangle$ is the branch label $s=1$ and $\phi_X$ is a normalized position state. Determine $L_{hh'}(0)$ on every edge and the inverse inertia $\mu$ of (20.10) for this start.

3. **Model D: averaged response.** Add a uniform cost gradient (in the sense of 20, by changing only the diagonal entries $t_{hh}$), with force $F$, and work to first order in $F$.
   - **Start.** Concentrated near the plane-wave label $p_0=0$, in the branch of positive eigenvalue, with clock level $k$.
   - **Regime.** Weak records: to leading order in $\kappa$ at fixed $\lambda$, the view of $b$ is that of $C_b$ plus the gradient term, and positions are measured in the background (17.15).
   - **Acceleration.** Determine $a(\lambda)=\mathrm d^2\bar x/\mathrm d\lambda^2$. Then determine its average over $\lambda$-intervals that are long compared with the inverse of the gap between the two branches and short enough for the first-order treatment. State both conditions.
   - **Inertia.** Define the averaged inverse inertia $\mu_{\rm av}(k)$ as the coefficient of $F$ in this average, and determine it.
   - **Dependence.** Determine how $\mu_{\rm av}(k)$ depends on $d_k$, and its relation to the rest cost of level $k$.
   - **Decoupled body.** Determine $\mu_{\rm av}$ for the decoupled body of item 1 on the chain, with nearest-neighbour hopping $t$ and any clock term, and whether it depends on the clock level.

4. **Comparison.** Compare $\mu$ of (20.10) with $\mu_{\rm av}$ for Model D and for the decoupled body. Determine for which of the two bodies they coincide, and why.

## Inputs

From 02-clock@v2. Notation fixed there: $S\lvert m\rangle=\lvert m+1\rangle$, $\lvert f_k\rangle=N^{-1/2}\sum_me^{2\pi ikm/N}\lvert m\rangle$; a uniform ring clock has $e^{-iC_K\lambda_0}\lvert m\rangle=e^{i\theta_m}\lvert m+1\rangle$ with tick $\lambda_0$, and $\Delta C_K$ is the spread of $C_K$.

Eq. (2.9), "where the $\alpha_m$ and $\varepsilon$ are arbitrary reals. The phases are then $\theta_m=\alpha_{m+1}-\alpha_m-\varepsilon\lambda_0$ (mod $2\pi$, with $\alpha_N:=\alpha_0$). If $\theta_m=0$ for all $m$, then $e^{-iC_K\lambda_0}=S$. The same argument applied to $S$, whose eigenvalues $e^{-2\pi ik/N}$ on $\lvert f_k\rangle$ are distinct, gives exactly $C_K=\sum_k\frac{2\pi\ell_k}{N\lambda_0}\lvert f_k\rangle\langle f_k\rvert$ with $\ell_k\equiv k$."

$$
C_K=\sum_{k\in\mathbb Z_N}E_k\lvert v_k\rangle\langle v_k\rvert,\quad
\lvert v_k\rangle=\frac{1}{\sqrt N}\sum_{m}e^{2\pi ikm/N}e^{i\alpha_m}\lvert m\rangle,\quad
E_k=\varepsilon+\frac{2\pi\ell_k}{N\lambda_0},\quad \ell_k\in\mathbb Z,\ \ell_k\equiv k\ (\mathrm{mod}\ N),
$$

Eq. (2.12), for the start $\lvert0\rangle$:

$$
\min\Delta C_K=\frac{2\pi}{\lambda_0}\sqrt{\frac{N^2-1}{12N^2}},\quad\text{attained iff } \{\ell_k\}=\{c,c+1,\dots,c+N-1\},\ c\in\mathbb Z .
$$

From 03-ilang-space/05-recording-contract@v1. These hold in the setting of 05: one recorded object, one medium object, the recording contract $\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$, and the start $\lvert\phi\rangle\otimes\lvert\chi\rangle$ with $\phi_h\neq0$ for all $h$. Notation fixed there: $g(v,w):=\operatorname{Re}\langle v\vert w\rangle$, and $h\sim_0h'$ iff $u_h=u_{h'}$.

Eq. (5.10):

$$
\lim_{\lambda\to0^+}\frac{\alpha(h,h';\lambda)}{\lambda}=d_0(h,h')=\bigl\lVert u_h-u_{h'}\bigr\rVert,
\qquad \lvert u_h\rangle=\bigl(K_h-\langle K_h\rangle\bigr)\lvert\chi\rangle .
$$

From 03-ilang-space/17-motion@v2. Setting there: $C=C_b\otimes\mathbb 1_c+\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$, $C_b=\sum_{h,h'}t_{hh'}\lvert h\rangle\langle h'\rvert$ with $t_{hh'}=t_{h'h}^*$; the hopping graph joins $h\neq h'$ iff $t_{hh'}\neq0$, and $E$ is its set of edges; the start is $\lvert\phi\rangle\otimes\lvert\chi\rangle$; $\langle X\rangle:=\langle\chi\vert X\vert\chi\rangle$. Definitions fixed there, with $p_h(\lambda)$ the place weights of the view of $b$:

$$\bar x(\lambda):=\sum_hp_h(\lambda)\,u_h,\qquad v(\lambda):=\frac{\mathrm d\bar x}{\mathrm d\lambda}.$$

Eq. (17.15), for the chain $h_m$, $m\in\mathbb Z$, with $K_{h_m}=m\,X$, $X$ self-adjoint on $\mathcal H_c$, $\sigma_X:=(\langle X^2\rangle-\langle X\rangle^2)^{1/2}>0$ and $e:=(X-\langle X\rangle)\chi$, so that $\lVert e\rVert=\sigma_X$ and $u_{h_m}=m\,e$:

$$
d_0(h_m,h_n)=\lvert m-n\rvert\,\sigma_X,\qquad \bar x=\bar m\,e,\quad \bar m:=\sum_mm\,p_{h_m},\qquad s^2=\sigma_X^2\sum_m(m-\bar m)^2p_{h_m} .
$$

From 03-ilang-space/20-cost-and-mass@v2. Setting: that of 17, items 2–4 of 20. Definitions fixed there:
- **Blocks.** $C_{hh'}:=(\langle h\rvert\otimes\mathbb 1)C(\lvert h'\rangle\otimes\mathbb 1)$; in this setting $C_{hh'}=t_{hh'}\mathbb 1_c$ for $h\neq h'$ and $C_{hh}=t_{hh}\mathbb 1_c+K_h$.
- **Costs.** The site cost is $L_h(\lambda):=\langle\psi_h\vert C_{hh}\vert\psi_h\rangle$ and the edge cost is $L_{hh'}(\lambda):=2\operatorname{Re}\langle\psi_h\vert C_{hh'}\vert\psi_{h'}\rangle$, with $\lvert\psi_h\rangle:=(\langle h\rvert\otimes\mathbb 1)\lvert\Psi\rangle$.
- **Site cost per unit weight.** $\Omega_h:=\langle\chi\vert C_{hh}\vert\chi\rangle=t_{hh}+\epsilon_h$, with $\epsilon_h:=\langle K_h\rangle$.
- **Acceleration.** $a:=\mathrm d^2\bar x/\mathrm d\lambda^2$.
- **Uniform cost gradient, force, inverse inertia.** "The site costs per unit weight have a uniform gradient if there is a vector $f\in\mathcal H_c$ with $\Omega_h-\Omega_{h'}=g\bigl(f,u_h-u_{h'}\bigr)$ for all $h,h'$. The **force** is $F:=-f$. Vary the description only through the diagonal entries $t_{hh}$; this changes the $\Omega_h$ and nothing else. Let $a_0$ be the value of $a(0)$ when all $\Omega_h$ are equal. The **inverse inertia** $\mu$ is the real-linear map on $(\mathcal H_c,g)$, if one exists, such that $a(0)=a_0+\mu F$ for every uniform gradient."

The derivation uses $\Delta_{hh'}:=u_h-u_{h'}$, and for the product start $L_{hh'}(0)=2\operatorname{Re}[t_{hh'}\phi_h^*\phi_{h'}]$. Eq. (20.10), "For a uniform gradient, $\Omega_h-\Omega_{h'}=g(f,\Delta_{hh'})=-g(F,\Delta_{hh'})$. Then (20.9) gives"

$$
a(0)=a_0+\mu F,\qquad \mu=-\sum_{\{h,h'\}\in E}L_{hh'}(0)\,\Delta_{hh'}\,g\bigl(\Delta_{hh'},\,\cdot\,\bigr) .
$$

## Assumptions

- The setting and the definitions under "Goal". All contracts are independent of $\lambda$ (A5), and $\lambda\ge0$.
- No object other than $b$ has the type of $b$, and $c$ is the only instance of its type.
- Item 3 uses the weak-record regime and first order in $F$; state every place where they are used, and the conditions on the averaging interval.

## Scope

- In scope:
  - the inverse inertia (20.10) for bodies with internal labels (item 1);
  - Model D at $\lambda=0$ (item 2) and averaged (item 3);
  - the decoupled body for comparison.
- Out of scope:
  - moving clocks and their rates (criterion 4, a separate package);
  - a common maximal speed for all types (criterion 6);
  - composite bodies of several objects;
  - branches of negative eigenvalue beyond what the averaging needs;
  - effects of the records beyond leading order in $\kappa$;
  - any physical meaning of $\lambda$ or of the costs beyond the definitions (M1, M3).

## Depth

- Item 1: short argument.
- Item 2: derive.
- Item 3: derive. Standard facts on plane waves, branches and their response to a uniform gradient may be used, stated.
- Item 4: short argument.

## Expected result

- Item 1: closed forms; a yes/no answer; a closed form and yes/no answers.
- Item 2: closed forms and a value.
- Item 3: a closed form for the average, the conditions on the interval, a closed form for $\mu_{\rm av}(k)$, a relation to the rest cost, and a closed form with a yes/no answer for the decoupled body.
- Item 4: a comparison and a reason.

Give every main result a tag $(6.k)$.

Consistency checks, at most three: for example $N$ arbitrary with $k=0$, $\tau\to0$, and the decoupled body against (20.10).

## Code

None.

Author: Ivan Honis · ivan.honis@ndot.io · License: CC BY 4.0 · Human–AI note