# Question: 09-proper-time

- **Subproject:** 04-ilang-time
- **Package:** 09-proper-time
- **Equation tags:** (9.k)
- **Created:** 2026-10-10

## Goal

Determine the time shown by a clock carried by a body whose motion changes, and compare two carried clocks between two meetings of their bodies. The motion is changed by a uniform cost gradient on one of the bodies. This is the part of criterion 4 that 05-moving-clock left open.

**Setting.** Model D of 05-moving-clock (items 2–4 there, quoted under "Inputs"), with two bodies.
- **Objects.** Two bodies, object $1$ of type $B_1$ and object $2$ of type $B_2$, and a medium $c$. The three types are pairwise different, and each object is the only instance of its type.
- **Places.** Each body has the places $(x,s,a)$ with $x\in\mathbb Z$ (position label), $s\in\{1,2\}$ (branch label) and $a\in\mathbb Z_N$, $N\ge2$ (clock label). $x\in\mathbb Z$ is the limit of long finite chains with the starts far from both ends (A1). State how the limit is taken.
- **Body contract.** On the places of one body,
  $$C_{\rm D}=A+B,\qquad A=\frac\tau2\sum_x\Bigl(\lvert x+1\rangle\langle x\rvert\otimes(-i\sigma_1)+\lvert x\rangle\langle x+1\rvert\otimes(i\sigma_1)\Bigr)\otimes\mathbb 1_N,\qquad B=\sum_x\lvert x\rangle\langle x\rvert\otimes\sigma_3\otimes(m\mathbb 1_N+D),$$
  with $\tau>0$, $m>0$, the Pauli matrices $\sigma_1,\sigma_3$ on the branch label, and $D=\frac{2\pi}{N\lambda_0}\sum_{k=0}^{N-1}k\lvert f_k\rangle\langle f_k\rvert$ on the clock label, where $\lvert f_k\rangle=N^{-1/2}\sum_ae^{2\pi ika/N}\lvert a\rangle$ and $\lambda_0>0$. Both bodies have the same $\tau$, $m$, $N$ and $\lambda_0$.
- **Gradient.** Let $P_x:=\lvert x\rangle\langle x\rvert\otimes\mathbb 1_2\otimes\mathbb 1_N$ on one body. Body $1$ has in addition the term $G:=-\mathcal F\sum_xx\,P_x$ with $\mathcal F>0$: a uniform gradient of its diagonal entries $t_{hh}$ along the position label. Body $2$ has no gradient.
- **Contract.**
  $$C=(C_{\rm D}+G)\otimes\mathbb 1\otimes\mathbb 1_c+\mathbb 1\otimes C_{\rm D}\otimes\mathbb 1_c+\sum_x\kappa\,x\,\bigl(P_x\otimes\mathbb 1+\mathbb 1\otimes P_x\bigr)\otimes X ,$$
  with $\kappa>0$ and $X$ self-adjoint on $\mathcal H_c$, $\sigma_X>0$ as in (17.15), and $\langle X\rangle=0$. There is no contract between the two bodies. Both bodies have the records $K_x=\kappa\,x\,X$, so both have the record vectors $u_x=\kappa\,x\,e$ of (17.15) with $X\to\kappa X$.
- **Start.** $\phi_1\otimes\phi_2\otimes\chi$. Each $\phi_i$ is a start of item 3 of 05-moving-clock: concentrated near a plane-wave label, in the branch of positive eigenvalue, with the clock label in the place $\lvert0\rangle$. The label is $p_0$ with $-\frac\pi2<p_0\le0$ for body $1$, and $0$ for body $2$. The two starts have the same mean position label.
- **Readings.** The clock reading on body $i$ has one class per clock label, with statistics $q_i(a;\lambda)$. The joint reading of both clocks has one class per pair of clock labels, with statistics $q_{12}(a_1,a_2;\lambda)$. Readings are applied at stated values of $\lambda$ (A8).
- **Positions.** For each body, $\bar x_i(\lambda):=\sum_hp^{(i)}_h(\lambda)\,u_h$ as in 17-motion, with the place weights $p^{(i)}_h$ of the view of body $i$. The **displacement** is $\Delta\bar x_i(\lambda):=\bar x_i(\lambda)-\bar x_i(0)$.
- **Meeting.** In the description above, equal position labels of the two bodies carry equal record vectors, so by (16.11) the matching of equal position labels is a correspondence with $t_\pi=0$. The bodies **meet** at $\lambda$ if $\bar x_2(\lambda)-\bar x_1(\lambda)=t_\pi$, here $\bar x_1(\lambda)=\bar x_2(\lambda)$, to the order to which the displacements are determined in item 1. By the choice of the starts they meet at $\lambda=0$.

1. **A carried clock under a uniform gradient.** Consider body $1$.
   - (a) Determine how the plane-wave label of the start changes with $\lambda$. State the condition on $\mathcal F$ under which the start stays in the branch of positive eigenvalue, and the size of what is neglected.
   - (b) Determine the displacement $\Delta\bar x_1(\lambda)$ to first order in $\epsilon$ relative to its leading term (the orders are defined under "Assumptions"). State what the oscillating part of the velocity (5.24) contributes to it.
   - (c) Determine $q_1(a;\lambda)$ to first order in $\epsilon$ and to leading order in the spread of the start. Determine whether it has the form of (5.21) with $\Gamma(p_0)\lambda/\lambda_0$ replaced by a function $\vartheta_1(\lambda)$. If it has, determine $\vartheta_1$, the **clock advance**; if not, determine what replaces it.
   - (d) Determine whether, to this order, $\mathcal F$ enters $\mathrm d\vartheta_1/\mathrm d\lambda$ otherwise than through the plane-wave label at $\lambda$.

2. **Rate along the path.** Let $V_1(\lambda):=\mathrm d\Delta\bar x_1/\mathrm d\lambda$, computed from the expression of item 1(b).
   - Determine the relation between $\lambda_0\,\mathrm d\vartheta_1/\mathrm d\lambda$ and $\lVert V_1(\lambda)\rVert/v_{\max}$, with $v_{\max}$ of (5.11), in two forms: without an expansion in the plane-wave label, to the orders in $\epsilon$ of items 1(b) and 1(c); and to leading order for small plane-wave labels.
   - Express $\vartheta_1(\lambda)$ as a functional of the path $\lambda'\mapsto V_1(\lambda')$, $0\le\lambda'\le\lambda$, to leading order for small plane-wave labels.

3. **Two clocks between two meetings.**
   - (a) Determine $q_{12}(a_1,a_2;\lambda)$ in terms of $q_1$ and $q_2$, and the clock advance $\vartheta_2(\lambda)$ of body $2$.
   - (b) Determine all $\lambda>0$ at which the bodies meet, and the first of them, $\lambda_R$.
   - (c) Determine $\vartheta_1(\lambda_R)$, $\vartheta_2(\lambda_R)$ and their ratio as functions of $p_0$, $\mathcal F\lambda_0$ and the parameters of the bodies.
   - (d) Determine which of the two clocks has advanced more at $\lambda_R$, for all admissible $p_0$ and $\mathcal F$, and when the advances are equal. Determine whether the same holds at every later meeting.
   - (e) Determine the deviation of the ratio of item (c) from $1$ to leading order for small $\lvert p_0\rvert$, and express it through the mean of $\lVert V_1\rVert^2/v_{\max}^2$ over $[0,\lambda_R]$.
   - (f) Determine the conditions under which both clock readings at $\lambda_R$ are certain to the order of item 1(c), and what the two readings then show.

## Inputs

From 05-moving-clock@v3 (this subproject). Setting there: one body with the places and the contract $C_b=C_{\rm D}$ above, the records $K_x=\kappa xX$, and the start $\lvert\phi\rangle\otimes\lvert\chi\rangle$. Notation fixed there:
- $d_k:=\frac{2\pi k}{N\lambda_0}$, $\epsilon:=\frac{2\pi}{N\lambda_0m}$, so $d_k=km\epsilon$;
- the plane waves $\lvert p\rangle=\sum_xe^{ipx}\lvert x\rangle$, $p\in(-\pi,\pi]$;
- $\hat e:=e/\sigma_X$, with $e$ of (17.15);
- $\gamma:=m/\omega_0(p_0)$;
- the label velocity $w_k(p):=\partial_p\omega_k(p)$ of the positive branch of level $k$;
- $\bar x_{\rm lab}:=\sum_xx\,P(x;\lambda)$, the mean position label, with $\bar x=\kappa\,\bar x_{\rm lab}\,e$.

Eq. (5.1), the reading distribution (2.13) of the minimal ring clock of 02-clock started at $\lvert0\rangle$, as a function of the advance $\vartheta$:

$$
P_N(a;\vartheta):=\Bigl\lvert\frac1N\sum_{k=0}^{N-1}e^{2\pi ik(a-\vartheta)/N}\Bigr\rvert^2=\frac{\sin^2(\pi\vartheta)}{N^2\sin^2\bigl(\pi(\vartheta-a)/N\bigr)},\qquad p(a;\lambda)=P_N(a;\lambda/\lambda_0)\ \text{in (2.13)}.
$$

Eq. (5.5):

$$
C_b\bigl(\lvert p\rangle\otimes\beta\otimes\lvert f_k\rangle\bigr)=\lvert p\rangle\otimes C_k(p)\beta\otimes\lvert f_k\rangle,\qquad C_k(p):=-\tau\sin p\,\sigma_1+\mu_k\sigma_3,\qquad\mu_k:=m+d_k .
$$

Eq. (5.6), the eigenvalues $\pm\omega_k(p)$ and the real eigenvectors $\lvert p\rangle\otimes\beta_{k,\pm}(p)\otimes\lvert f_k\rangle$ of $C_b$:

$$
\omega_k(p)=\sqrt{\mu_k^2+\tau^2\sin^2p},\quad
\beta_{k,+}=\begin{pmatrix}\cos\frac{\theta_k}2\\ \sin\frac{\theta_k}2\end{pmatrix},\quad
\beta_{k,-}=\begin{pmatrix}-\sin\frac{\theta_k}2\\ \cos\frac{\theta_k}2\end{pmatrix},\quad
\tan\theta_k(p)=-\frac{\tau\sin p}{\mu_k},\ \ \lvert\theta_k\rvert<\frac\pi2 .
$$

Eq. (5.11), the bound (17.7) for this $C_b$:

$$
v_{\max}=\tau\,\kappa\sigma_X=\max_h\sum_{h'}\lvert t_{hh'}\rvert\,d_0(h,h') .
$$

Eq. (5.20), with $\bar d:=\frac1N\sum_kd_k=\frac{N-1}2m\epsilon=\langle0\vert D\vert0\rangle$, $\bar\mu:=m+\bar d$ and $j:=k-\frac{N-1}2$:

$$
\omega_k=\bar\omega+\Gamma\,m\epsilon j+\tfrac12\bar c\,(m\epsilon j)^2+\bar\rho_k,\qquad
\bar\omega(p):=\sqrt{\bar\mu^2+\tau^2\sin^2p},\quad\Gamma(p):=\frac{\bar\mu}{\bar\omega(p)},\quad\bar c:=\frac{\tau^2\sin^2p}{\bar\omega^3}\le\frac1m,\quad\lvert\bar\rho_k\rvert\le\frac{(m\epsilon\lvert j\rvert)^3}{6m^2}.
$$

Eq. (5.21), for the start of item 3 there (concentrated near $p_0$, spinor of the positive branch of level $0$, clock label in $\lvert0\rangle$), in the weak-record regime. The remainder is $\mathcal R=O\bigl((N\epsilon)^2(1+\lambda/\lambda_0)^2\bigr)+O(\delta p^2)+O(\kappa)$, with $\delta p$ the spread of the start:

$$
q(a;\lambda)=P_N\Bigl(a;\Gamma(p_0)\frac{\lambda}{\lambda_0}\Bigr)+\mathcal R,\qquad
\Gamma(p_0)=\frac{\bar\mu}{\sqrt{\bar\mu^2+\tau^2\sin^2p_0}}=\gamma\Bigl[1+\frac{(N-1)\epsilon}2\bigl(1-\gamma^2\bigr)\Bigr]+O\bigl((N\epsilon)^2\bigr),\quad\gamma:=\frac m{\omega_0(p_0)} .
$$

Eq. (5.22), the tick of the first-order distribution (5.21) and its ratio to the tick at $p_0=0$, with $\omega:=\omega_0(p_0)$:

$$
\lambda_0(p_0)=\frac{\lambda_0}{\Gamma(p_0)},\qquad
R(p_0):=\frac{\lambda_0(p_0)}{\lambda_0(0)}=\frac{\bar\omega(p_0)}{\bar\mu}=\sqrt{1+\frac{\tau^2}{\bar\mu^2}\sin^2p_0}=\frac\omega m\Bigl[1-\frac{(N-1)\epsilon}2\,\frac{\tau^2\sin^2p_0}{\omega^2}\Bigr]+O\bigl((N\epsilon)^2\bigr)\ \ge1 .
$$

Eq. (5.24), the label velocity of the same start, exactly in the weak-record regime, with $\delta_k:=(\theta_0-\theta_k)/2$ and $g$ the plane-wave amplitude of the start:

$$
\frac{\mathrm d\bar x_{\rm lab}}{\mathrm d\lambda}=\int\frac{\mathrm dp}{2\pi}\lvert g\rvert^2\,\frac1N\sum_{k=0}^{N-1}\Bigl[\cos2\delta_k\,w_k-\sin2\delta_k\,\tau\cos p\,\cos\theta_k\,\cos(2\omega_k\lambda)\Bigr].
$$

Eq. (5.25), the mean of $v$ over $[0,\lambda]$ with $\lambda$ of the order of the tick, to first order in $\epsilon$:

$$
\bar v:=\frac{\bar x(\lambda)-\bar x(0)}{\lambda}=\kappa\sigma_X\,\bar w(p_0)\,\hat e+\mathcal R_v,\qquad\bar w(p):=w(p;\bar\mu)=\frac{\tau^2\sin p\cos p}{\sqrt{\bar\mu^2+\tau^2\sin^2p}} .
$$

Eq. (5.29), with $\lVert\bar v\rVert$ of (5.25) and $R$ of (5.22):

$$
R^2\Bigl(1-\frac{\lVert\bar v\rVert^2}{v_{\max}^2}\Bigr)=1+\frac{\tau^2}{\bar\mu^2}\sin^4p_0 .
$$

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 start is $\lvert\phi\rangle\otimes\lvert\chi\rangle$; $\langle X\rangle:=\langle\chi\vert X\vert\chi\rangle$, $\lvert u_h\rangle:=(K_h-\langle K_h\rangle)\lvert\chi\rangle$, $d_0(h,h')=\lVert u_h-u_{h'}\rVert$. 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. (17.7), with $W$ the witness weight of the view of $b$, $q_h:=\sqrt{p_h}$, and $a_{hh'}:=\lvert t_{hh'}\rvert d_0(h,h')$ with largest eigenvalue $\varrho(a)$; the bound holds for every start and every $\lambda$:

$$
\lVert v(\lambda)\rVert\le\sum_{\{h,h'\}\in E}2\lvert t_{hh'}\rvert d_0(h,h')\sqrt{W(h,h';\lambda)p_hp_{h'}}\le\sum_{h,h'}a_{hh'}q_hq_{h'}\le v_{\max}:=\varrho(a)\le\max_h\sum_{h'}\lvert t_{hh'}\rvert\,d_0(h,h') .
$$

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/16-common-space@v2. Setting there: objects $a$ (type $A$, places $h$), $b$ (type $B$, places $k$) and the medium $c$, of pairwise different types, each the only instance of its type; $C=\sum_h\lvert h\rangle\langle h\rvert_a\otimes\mathbb 1_b\otimes K^A_h+\sum_k\mathbb 1_a\otimes\lvert k\rangle\langle k\rvert_b\otimes K^B_k$; start $\lvert\phi\rangle_a\otimes\lvert\vartheta\rangle_b\otimes\lvert\chi\rangle_c$ with all $\phi_h\neq0$, $\vartheta_k\neq0$. Notation fixed there:
- $\lvert u^A_h\rangle:=(K^A_h-\langle K^A_h\rangle)\lvert\chi\rangle$, $\lvert u^B_k\rangle:=(K^B_k-\langle K^B_k\rangle)\lvert\chi\rangle$, $g(v,w):=\operatorname{Re}\langle v\vert w\rangle$, $\mathcal N:=\{v\in\mathcal H_c:\langle\chi\vert v\rangle=0\}$;
- $X_A$ and $X_B$ are the sets of leading-order points of the views of $\{a\}$ and $\{b\}$ (places with $d_0=0$ form one point), with record vectors $u^A_x$, $u^B_y$;
- a **correspondence** is a bijection $\pi:D_A\to D_B$ between subsets $D_A\subseteq X_A$, $D_B\subseteq X_B$ with $\lvert D_A\rvert\ge2$ such that, for all $x,x'\in D_A$, the joint places $(h,k')$ and $(h',k)$ have $d_0=0$ in the view of $\{a,b\}$, where $h\in x$, $h'\in x'$, $k\in\pi(x)$, $k'\in\pi(x')$.

Here body $1$ is the object $a$ and body $2$ is the object $b$.

Eq. (16.3), all replacements of the $K$'s that leave $C$ unchanged:

$$
K'^A_h=K^A_h+Z,\qquad K'^B_k=K^B_k-Z,\qquad Z=Z^\dagger\ \text{on }\mathcal H_c\ \text{arbitrary.}
$$

Eq. (16.4), with the possible shifts being exactly the vectors $z\in\mathcal N$:

$$
u^A_h\mapsto u^A_h+z,\qquad u^B_k\mapsto u^B_k-z,\qquad \lvert z\rangle=(Z-\langle Z\rangle)\lvert\chi\rangle .
$$

Eq. (16.11), "Hence $\pi$ is a correspondence iff"

$$
u^A_x-u^A_{x'}=u^B_{\pi(x)}-u^B_{\pi(x')}\ \ \forall x,x'\in D_A
\iff
\exists\,t_\pi\in\mathcal N:\ u^B_{\pi(x)}=u^A_x+t_\pi\ \ \forall x\in D_A .
$$

"Under (16.4), $t_\pi\mapsto t_\pi-2z$".

## Assumptions

- The setting and the definitions under "Goal". All contracts are independent of $\lambda$ (A5), and $\lambda\ge0$.
- **Weak records.** As in 05-moving-clock: work to leading order in $\kappa$ at fixed $\lambda$. The view of each body is then that of its own contract alone ($C_{\rm D}+G$ for body $1$, $C_{\rm D}$ for body $2$), and positions are measured in the background (17.15). State every place where this is used.
- **Starts.** As in 05-moving-clock: the weight of the start over the plane-wave label has mean $p_0$ (respectively $0$) and variance $\delta p^2\ll1$. "Leading order in the spread" means a remainder $O(\delta p^2)$.
- **Orders in $\epsilon$.** "To first order in $\epsilon$" means: at fixed $N$, $\tau/m$, $p_0$, $\lambda/\lambda_0$ and $\mathcal F\lambda_0$, with a remainder of relative order $\epsilon^2$. With $\mathcal F\lambda_0$ fixed, $\mathcal F/m=\frac{N\epsilon}{2\pi}\mathcal F\lambda_0$ is itself of first order.
- Standard facts on the motion of a plane-wave label under a uniform gradient, and on the adiabatic following of a branch separated from the other by a gap, may be used; state them with their conditions.

## Scope

- In scope:
  - one Model D body with a uniform gradient: label, displacement and clock reading;
  - the clock advance as a function of $\lambda$ and of the path;
  - two bodies with equal parameters, one with a gradient and one at rest, between two meetings.
- Out of scope:
  - two instances of the same type (Law 5), and bodies with different parameters (criterion 6, a separate package);
  - gradients that depend on $\lambda$, on the branch label or on the clock label;
  - transitions to the branch of negative eigenvalue, beyond the condition of item 1(a);
  - effects of the records beyond leading order in $\kappa$, and any effect of one body on the other through the medium;
  - any physical meaning of $\lambda$ beyond the definitions (M1); no relation of special relativity may be used as an input (M4).

## Depth

- Item 1: derive.
- Item 2: short argument.
- Item 3: derive; item 3(a): short argument.

## Expected result

- Item 1: a closed form for the label; a condition; a closed form for the displacement to the stated order; a distribution to the stated order with its remainder, and a yes/no answer with a function; a yes/no answer.
- Item 2: a relation in the two stated forms; a functional of the path.
- Item 3: a closed form; a set of values of $\lambda$; closed forms or one-dimensional integrals; an inequality with its case of equality, and a yes/no answer; a leading-order expression; conditions and two values.

Give every main result a tag $(9.k)$.

Consistency checks, at most three: for example $\mathcal F\to0$ at fixed $\lambda$, $p_0=0$, and $\tau\to0$.

## Code

None.

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