Cesàro average utilitarianism in relativistic spacetime

The good of any one individual is of no more importance, from the point of view of the Universe, than the good of any other.

Henry Sidgwick.

Abstract

Widely accepted theories in modern cosmology say that spacetime is probably infinite. This raises the question how to define a social welfare order (SWO) for an infinite population of people dispersed throughout time and space. Any such SWO should be Lorentz invariant: it should yield the same value independent of the position and velocity of the social observer. I define and axiomatically characterize spatiotemporal Cesàro average utilitarian SWOs as a solution to this problem.

Appendices

A Proofs

Proof of Lemma 1

Let $c\in {{\mathcal {S}}}$ and $\psi \in \Phi$, and suppose ${\varvec{{\mathcal {K}}}}$ is regular with respect c and $\psi$. Let $c'\in {{\mathcal {S}}}$ and $\psi '\in \Phi$; to show that ${\varvec{{\mathcal {K}}}}$ is also regular with respect $c'$ and $\psi '$, we shall check the three conditions of the definition.

• Mortality. Let $R_1:=d(c,c')$ and $R_2:=\delta (\psi ,\psi ')$ and let $R_*:=\max \{R_1,R_2\}$. Thus, for any $r\in {{\mathbb {R}}}_+$, we have ${{\mathcal {S}}}(c,r)\subseteq {{\mathcal {S}}}(c',r+R_*)$ and $\Phi (\psi ,r)\subseteq \Phi (\psi ',r+R_*)$, and thus, ${{\mathcal {O}}}(c,\psi ,r)\subseteq {{\mathcal {O}}}(c',\psi ',r+R_*)$. For any $n\in {{\mathcal {N}}}$, there exists $r\in {{\mathbb {R}}}_+$ such that $n\in {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)$, hence ${{\mathcal {K}}}_n\subseteq {{\mathcal {O}}}(c,\psi ,r)$, hence ${{\mathcal {K}}}_n\subseteq {{\mathcal {O}}}(c',\psi ',r+R_*)$, hence $n\in {{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r+R_*)$. Thus, $\displaystyle {{\mathcal {N}}}=\bigcup _{r=1}^{\infty }{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)$, as desired.

• Local finiteness. Let $R_*$ be as already defined above. For any $r\in {{\mathbb {R}}}_+$, we have ${{\mathcal {S}}}(c',r)\subseteq {{\mathcal {S}}}(c,r+R_*)$ and $\Phi (\psi ',r)\subseteq \Phi (\psi ,r+R_*)$, and thus, ${{\mathcal {O}}}(c',\psi ',r)\subseteq {{\mathcal {O}}}(c,\psi ,r+R_*)$. Thus, ${{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\subseteq {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R_*)$, so $\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\right|\leqslant \left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R_*)\right|$. Thus, $\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\right|<{\infty }$ because $\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R_*)\right|<{\infty }$. This holds for all $r\in {{\mathbb {R}}}_+$.

• Subexponential growth. As already noted, $\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\right|\leqslant \left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R_*)\right|$ for all $r\in {{\mathbb {R}}}_+$. By the same argument, $\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|\leqslant \left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r+R_*)\right|$ for all $r\in {{\mathbb {R}}}_+$. Thus, for any $r,R\in {{\mathbb {R}}}_+$, we have

  $$\begin{aligned} \frac{\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r+R)\right|}{\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\right|}\leqslant & {} \frac{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R+R_*)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r-R_*)\right|}.\\ \text{ Thus, }\quad \limsup _{r{\rightarrow }{\infty }} \frac{\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r+R)\right|}{\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\right|}\leqslant & {} \limsup _{r{\rightarrow }{\infty }} \frac{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R+R_*)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r-R_*)\right|} \\{} & {} \mathop {=}\limits _{\scriptscriptstyle {\mathrm {(*)}}} \limsup _{r'{\rightarrow }{\infty }} \frac{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r'+(R+2\,R_*))\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r')\right|} \quad \mathop {=}\limits _{\scriptscriptstyle {\mathrm {(\dagger )}}}\quad 1, \end{aligned}$$

  as desired. Here, $(*)$ is by the change of variables $r':=r-R_*$, and $(\dagger )$ is because ${\varvec{{\mathcal {K}}}}$ satisfies Subexponential growth relative to c and $\psi$ by hypothesis.

$\square$

Lemma A.1

Let ${\varvec{{\mathcal {K}}}}\in {{\mathfrak {K}}}$, let $c\in {{\mathcal {S}}}$, and let $\psi \in \Phi$.

• (a) For any $R>0$, we have $\displaystyle \lim _{r{\rightarrow }{\infty }}\frac{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R){\setminus } {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|} \ = \ 0$.

• (b) For any $c'\in {{\mathcal {S}}}$ and $\psi '\in \Phi$, we have $\displaystyle \lim _{r{\rightarrow }{\infty }}\frac{\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|} \ = \ 1$.

Proof

• (a) This follows from subexponential growth.

• (b) As in the proof of Lemma 1, let $R_*:=\max \{d(c,c'), \ \delta (\psi ,\psi ')\}$. Then for any $r>R_*$, we have ${{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r-R_*)\subseteq {{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\subseteq {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R_*)$, and thus

  $$\begin{aligned} \frac{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r-R_*)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|} \quad \leqslant \quad \frac{\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|} \quad \leqslant \quad \frac{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R_*)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|}. \end{aligned}$$

  Thus,

  $$\begin{aligned} 1 \ {}= & {} \ \liminf _{r{\rightarrow }{\infty }} \frac{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r-R_*)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|} \ \leqslant \ \lim _{r{\rightarrow }{\infty }}\frac{\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|} \ \\\leqslant & {} \limsup _{r{\rightarrow }{\infty }}\frac{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R_*)\right|}{\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|} \ = \ 1, \end{aligned}$$

  where the first and last equalities are by the subexponential growth of ${\varvec{{\mathcal {K}}}}$. The claim follows.

$\square$

Proof of Lemma 2

The limit (3) is a special case of the limit (4) when u is the indicator function of the set ${{\mathcal {B}}}$ (in which case $\{u(x_n)\}_{n\in {{\mathcal {N}}}}=\{0,1\}$ is obviously bounded). So it suffices to show that the existence and value of the limit (4) are independent of the choice of c and $\psi$.

So, let $u:{{\mathcal {X}}}{{\longrightarrow }}{{\mathbb {R}}}$, let ${{\textbf{x}}}\in \ell ^{\infty }({{\mathcal {N}}},{{\mathcal {X}}})$, and suppose that $M:= \sup \{|u(x_n)|$; $n\in {{\mathcal {N}}}\}$ is finite. Let $c\in {{\mathcal {S}}}$ and $\psi \in \Phi$, and suppose the limit (4) exists. Let $c'\in {{\mathcal {S}}}$ and $\psi '\in \Phi$; we must show that the limit

$$\begin{aligned} \mathop {\textrm{CesAve}}\limits _{c',\psi '} \ u({\varvec{{\mathcal {K}}}},{{\textbf{x}}}) \quad :=\quad \lim _{r{\rightarrow }{\infty }} \frac{1}{\left|{{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\right|} \sum _{n\in {{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)} u(x_n), \end{aligned}$$

(A1)

also exists, and has the same value as (4). For brevity, I shall use the following notation: for any $r>0$, ${{\mathcal {N}}}(r):= {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)$ and ${{\mathcal {N}}}\,'(r):= {{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)$. As in the proof of Lemma 1, let $R_*:=\max \{d(c,c'), \ \delta (\psi ,\psi ')\}$; then for all $r\in {{\mathbb {R}}}_+$, we have ${{\mathcal {N}}}\,'(r)\subseteq {{\mathcal {N}}}(r+R_*)$ and ${{\mathcal {N}}}(r)\subseteq {{\mathcal {N}}}\,'(r+R_*)$. Thus, for any $r\in {{\mathbb {R}}}_+$,

$$\begin{aligned}{} & {} \left|\sum _{n\in {{\mathcal {N}}}(r)} u(x_n) - \sum _{n\in {{\mathcal {N}}}\,'(r)} u(x_n) \right| \ \ = \ \ \left| \sum _{n\in {{\mathcal {N}}}(r)\setminus {{\mathcal {N}}}\,'(r) } u(x_n) \ - \sum _{n\in {{\mathcal {N}}}\,'(r)\setminus {{\mathcal {N}}}(r)} u(x_n) \right| \nonumber \\{} & {} \quad \leqslant \left| \sum _{n\in {{\mathcal {N}}}(r)\setminus {{\mathcal {N}}}\,'(r) } u(x_n)\right| +\left| \sum _{n\in {{\mathcal {N}}}\,'(r)\setminus {{\mathcal {N}}}(r)} u(x_n) \right| \ \ \nonumber \\{} & {} \quad \leqslant M\cdot \Big | {{\mathcal {N}}}(r)\setminus {{\mathcal {N}}}\,'(r) \Big | +M\cdot \Big | {{\mathcal {N}}}\,'(r)\setminus {{\mathcal {N}}}(r) \Big | \nonumber \\{} & {} \quad \mathop {{\displaystyle \leqslant }}\limits _{\scriptscriptstyle {\mathrm {(*)}}} M\cdot \Big | {{\mathcal {N}}}(r)\setminus {{\mathcal {N}}}(r-R_*) \Big | +M\cdot \Big | {{\mathcal {N}}}(r+R_*)\setminus {{\mathcal {N}}}(r) \Big |, \end{aligned}$$

(A2)

where $(*)$ is because ${{\mathcal {N}}}(r-R_*)\subseteq {{\mathcal {N}}}\,'(r)\subseteq {{\mathcal {N}}}(r+R_*)$. Thus,

$$\begin{aligned}{} & {} \left| \frac{1}{\left|{{\mathcal {N}}}(r)\right|} \sum _{n\in {{\mathcal {N}}}(r)} u(x_n) - \frac{1}{\left|{{\mathcal {N}}}\,'(r)\right|}\sum _{n\in {{\mathcal {N}}}\,'(r)} u(x_n) \right| \\{} & {} \quad \leqslant \left| \frac{1}{\left|{{\mathcal {N}}}(r)\right|} \sum _{n\in {{\mathcal {N}}}(r)} u(x_n) - \frac{1}{\left|{{\mathcal {N}}}(r)\right|}\sum _{n\in {{\mathcal {N}}}\,'(r)} u(x_n) \right|\\{} & {} \qquad + \ \left| \frac{1}{\left|{{\mathcal {N}}}(r)\right|} \sum _{n\in {{\mathcal {N}}}\,'(r)} u(x_n) - \frac{1}{\left|{{\mathcal {N}}}\,'(r)\right|}\sum _{n\in {{\mathcal {N}}}\,'(r)} u(x_n) \right| \\{} & {} \quad = \frac{1}{\left|{{\mathcal {N}}}(r)\right|} \left|\sum _{n\in {{\mathcal {N}}}(r)} u(x_n) \ - \ \sum _{n\in {{\mathcal {N}}}\,'(r)} u(x_n) \right| \ + \ \left| \frac{1}{\left|{{\mathcal {N}}}(r)\right|} - \frac{1}{\left|{{\mathcal {N}}}\,'(r)\right|}\right| \cdot \left|\sum _{n\in {{\mathcal {N}}}\,'(r)} u(x_n) \right| \\{} & {} \quad \mathop {{\displaystyle \leqslant }}\limits _{\scriptscriptstyle {\mathrm {(*)}}} \frac{M\cdot \big | {{\mathcal {N}}}(r)\setminus {{\mathcal {N}}}(r-R_*) \big | +M\cdot \big | {{\mathcal {N}}}(r+R_*)\setminus {{\mathcal {N}}}(r) \big |}{\left|{{\mathcal {N}}}(r)\right|} \\{} & {} \qquad + \ \left| \frac{1}{\left|{{\mathcal {N}}}(r)\right|} - \frac{1}{\left|{{\mathcal {N}}}\,'(r)\right|}\right| \cdot M\cdot \left|{{\mathcal {N}}}\,'(r) \right| \\{} & {} \quad = M \left(\frac{ \big |{{\mathcal {N}}}(r)\setminus {{\mathcal {N}}}(r-R_*) \big |}{\left|{{\mathcal {N}}}(r)\right|} + \frac{\big | {{\mathcal {N}}}(r+R_*)\setminus {{\mathcal {N}}}(r) \big |}{\left|{{\mathcal {N}}}(r)\right|} + \ \left| \frac{\left|{{\mathcal {N}}}\,'(r)\right|}{\left|{{\mathcal {N}}}(r)\right|} - 1\right|\right). \end{aligned}$$

where $(*)$ is by (A2). Thus,

$$\begin{aligned}{} & {} \lim _{r{\rightarrow }{\infty }} \left| \frac{1}{\left|{{\mathcal {N}}}(r)\right|} \sum _{n\in {{\mathcal {N}}}(r)} u(x_n) - \frac{1}{\left|{{\mathcal {N}}}\,'(r)\right|}\sum _{n\in {{\mathcal {N}}}\,'(r)} u(x_n) \right| \\{} & {} \quad \leqslant M \lim _{r{\rightarrow }{\infty }} \left(\frac{ \big | {{\mathcal {N}}}(r)\setminus {{\mathcal {N}}}(r-R_*) \big |}{\left|{{\mathcal {N}}}(r)\right|} + \frac{\big | {{\mathcal {N}}}(r+R_*)\setminus {{\mathcal {N}}}(r) \big |}{\left|{{\mathcal {N}}}(r)\right|} + \ \left| \frac{\left|{{\mathcal {N}}}\,'(r)\right|}{\left|{{\mathcal {N}}}(r)\right|} - 1\right|\right) \\{} & {} \quad \mathop {=}\limits _{\scriptscriptstyle {\mathrm {(\dagger )}}} M\cdot (0 + 0 + 0 ) \quad =\quad 0. \end{aligned}$$

where $(\dagger )$ is by Lemma A.1(a,b). It follows that the limit (A1) exists if and only if the limit (4) exists, and if they exist, then they have the same value.

Finally, if $u:{{\mathcal {X}}}{{\longrightarrow }}{{\mathbb {R}}}$ is continuous, and ${{\textbf{x}}}\in \ell ^{\infty }({{\mathcal {N}}},{{\mathcal {X}}})$, then $\{u(x_n)\}_{n\in {{\mathcal {N}}}}$ is a totally bounded subset of ${{\mathbb {R}}}$, hence bounded, so the previous analysis applies. $\square$

Proof of Lemma 3

Let $\gamma :=\eta '\circ \eta ^{-1}:{{\mathbb {N}}}{{\longrightarrow }}{{\mathbb {N}}}$. Then $\gamma$ is a bijection because $\eta$ and $\eta '$ are bijections. Clearly, $\eta '=\gamma \circ \eta$. We must show that $\gamma$ satisfies formula (8).

As in the proof of Lemmas 1 and 2, let $R_*:=\max \{d(c,c'), \ \delta (\psi ,\psi ')\}$; then for all $r\in {{\mathbb {R}}}_+$, we have ${{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\subseteq {{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r+R_*)$. Thus,

$$\begin{aligned} r_{c',\psi '}(n)\quad \leqslant \quad r_{c,\psi }(n) + R_*, \qquad \text{ for } \text{ all } n\in {{\mathcal {N}}}. \end{aligned}$$

(A3)

Next, for all $m\in {{\mathcal {N}}}$, formula (6) implies that

$$\begin{aligned} \eta '(m)\leqslant & {} \left| {{\mathcal {N}}}_{c',\psi '}\Big ({\varvec{{\mathcal {K}}}},r_{c',\psi '}(m)\Big )\right|. \end{aligned}$$

(A4)

Likewise, for all $m\in {{\mathcal {N}}}$ all $r<r_{c,\psi }(m)$, formula (6) implies that

$$\begin{aligned} \left| {{\mathcal {N}}}_{c,\psi }\Big ( {\varvec{{\mathcal {K}}}},r\Big )\right|< & {} \eta (m). \end{aligned}$$

(A5)

For any $N\in {{\mathbb {N}}}$, let $r_N:=r_{c,\psi }(n)$, where $n:=\eta ^{-1}(N)$. Finally, let $N':=\left|{{\mathcal {N}}}_{\psi ',c'}({\varvec{{\mathcal {K}}}},r_N+R_*)\right|$.

Claim 1

For all $N\in {{\mathbb {N}}}$, $\gamma [1\ldots N] \subseteq [1\ldots N']$.

Proof

For all $m\in \eta ^{-1}[1\ldots N]$, we have $r_{c,\psi }(m)\leqslant r_N$ by formula (6), and thus, $r_{c',\psi '}(m)\leqslant r_N+R_*$, by inequality (A3). Thus, $\eta '(m)\leqslant \left|{{\mathcal {N}}}_{\psi ',c'}({\varvec{{\mathcal {K}}}},r_N+R_*)\right|$ by inequality (A4). The claim follows. $\diamond\; [Claim 1]$

Meanwhile, for any $N\in {{\mathbb {N}}}$ and $\epsilon >0$,

$$\begin{aligned} N \quad > \quad \left| {{\mathcal {N}}}_{c,\psi }\Big ( {\varvec{{\mathcal {K}}}},r_N-\epsilon \Big )\right|, \end{aligned}$$

(A6)

by inequality (A5). Thus,

$$\begin{aligned}{} & {} \#{\left\{ n\in [1\ldots N] \; ; \; \gamma (n)>N \right\} } \nonumber \\{} & {} \quad \mathop {{\displaystyle \leqslant }}\limits _{\scriptscriptstyle {\mathrm {(\diamond )}}} N'-N \quad \mathop {{\displaystyle <}}\limits _{\scriptscriptstyle {\mathrm {(*)}}}\quad \Big | {{\mathcal {N}}}_{\psi ',c'}({\varvec{{\mathcal {K}}}},r_N+R_*)\Big |\nonumber \\{} & {} \qquad -\Big | {{\mathcal {N}}}_{\psi ,c}({\varvec{{\mathcal {K}}}},r_N-\epsilon )\Big | \nonumber \\{} & {} \quad \leqslant \Big | {{\mathcal {N}}}_{\psi ',c'}({\varvec{{\mathcal {K}}}},r_N+R_*)\setminus {{\mathcal {N}}}_{\psi ,c}({\varvec{{\mathcal {K}}}},r_N-\epsilon )\Big | \nonumber \\{} & {} \quad \mathop {{\displaystyle \leqslant }}\limits _{\scriptscriptstyle {\mathrm {(\dagger )}}} \Big | {{\mathcal {N}}}_{\psi ,c}({\varvec{{\mathcal {K}}}},r_N+2\,R_*)\setminus {{\mathcal {N}}}_{\psi ,c}({\varvec{{\mathcal {K}}}},r_N-\epsilon )\Big |, \end{aligned}$$

(A7)

where $(\diamond )$ is by Claim 1, $(*)$ is by inequality (A6) and $(\dagger )$ is because ${{\mathcal {N}}}_{c',\psi '}({\varvec{{\mathcal {K}}}},r)\subseteq {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r+R_*)$ for all $r\in {{\mathbb {R}}}_+$. Combining inequalities (A6) and (A7) yields

$$\begin{aligned}{} & {} \frac{ \#{\left\{ n\in [1\ldots N] \; ; \; \gamma (n)>N \right\} }}{N} \quad \nonumber \\{} & {} \quad \leqslant \quad \frac{ \Big |{{\mathcal {N}}}_{\psi ,c}({\varvec{{\mathcal {K}}}},r_N+2\,R_*)\setminus {{\mathcal {N}}}_{\psi ,c}({\varvec{{\mathcal {K}}}},r_N-\epsilon )\Big |}{ \left| {{\mathcal {N}}}_{c,\psi }\Big ( {\varvec{{\mathcal {K}}}},r_N-\epsilon \Big )\right|}. \end{aligned}$$

(A8)

But $r_N{\rightarrow }{\infty }$ as $N{\rightarrow }{\infty }$. Thus, Lemma A.1(a) implies that

$$\begin{aligned} \lim _{N{\rightarrow }{\infty }} \frac{ \Big | {{\mathcal {N}}}_{\psi ,c}({\varvec{{\mathcal {K}}}},r_N+2\,R_*)\setminus {{\mathcal {N}}}_{\psi ,c}({\varvec{{\mathcal {K}}}},r_N-\epsilon )\Big |}{ \left| {{\mathcal {N}}}_{c,\psi }\Big ( {\varvec{{\mathcal {K}}}},r_N-\epsilon \Big )\right|} \quad =\quad 0. \end{aligned}$$

(A9)

Now combine (A8) and (A9) to obtain (8). $\square\vspace{1em}$

The proof of the Main Theorem requires some preliminaries. For any subset ${{\mathcal {M}}}\subseteq {{\mathbb {N}}}$, recall that its density $\delta ({{\mathcal {M}}})$ is defined by formula (9), and ${{\mathfrak {D}}}$ is the collection of all subsets ${{\mathcal {D}}}\subset {{\mathbb {N}}}$ for which $\delta ({{\mathcal {D}}})$ is well-defined. This collection is closed under complementation (i.e. ${{\mathcal {D}}}\in {{\mathfrak {D}}}\iff {{\mathcal {D}}}^\complement \in {{\mathfrak {D}}}$) and under disjoint unions (i.e. if ${{\mathcal {A}}},{{\mathcal {B}}}\in {{\mathfrak {D}}}$ and ${{\mathcal {A}}}\cap {{\mathcal {B}}}=\varnothing$, then ${{\mathcal {A}}}\sqcup {{\mathcal {B}}}\in {{\mathfrak {D}}}$). But it is not a Boolean algebra. However, $\delta$ is “finitely additive”: for all disjoint ${{\mathcal {A}}},{{\mathcal {B}}}\in {{\mathfrak {D}}}$, we have $\delta ({{\mathcal {A}}}\sqcup {{\mathcal {B}}})=\delta ({{\mathcal {A}}})+\delta ({{\mathcal {B}}})$. In fact, $\delta$ can be extended (non-uniquely) to a finitely additive measure on $\wp ({{\mathbb {N}}})$. Such measures are called density measures (Letavaj et al. 2015; Sleziak and Ziman 2008).

Recall that ${{\mathfrak {B}}}({{\mathcal {X}}}):=\{{{\mathcal {O}}}\cap {{\mathcal {C}}}$; ${{\mathcal {O}}}\subseteq {{\mathcal {X}}}$ open and ${{\mathcal {C}}}\subseteq {{\mathcal {X}}}$ closed$\}$. A sequence ${{\textbf{x}}}\in {{\mathcal {X}}}^{{\mathbb {N}}}$ is regular if, for any ${{\mathcal {B}}}\in {{\mathfrak {B}}}({{\mathcal {X}}})$, the set $\{n\in {{\mathbb {N}}}$; $x_n\in {{\mathcal {B}}}\}$ is an element of ${{\mathfrak {D}}}$.¹⁷ Let $\textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$ denote the set of all regular sequences in $\ell ^{\infty }({{\mathbb {N}}},{{\mathcal {X}}})$. For any ${\varvec{{\mathcal {K}}}}\in {{\mathfrak {K}}}$ and any ${{\textbf{x}}}\in \ell ^{\infty }({{\mathcal {N}}},{{\mathcal {X}}})$ with enumeration ${\widetilde{{\textbf{x}}}}\in \ell ^{\infty }({{\mathbb {N}}},{{\mathcal {X}}})$, it is easy to see that ${{\textbf{x}}}\in \textrm{reg}({\varvec{{\mathcal {K}}}})$ if and only if ${\widetilde{{\textbf{x}}}}\in \textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$, by comparing formulae (3) and (9).

If ${{\textbf{x}}}\in {{\mathcal {X}}}^{{\mathbb {N}}}$ and $\gamma :{{\mathbb {N}}}{{\longrightarrow }}{{\mathbb {N}}}$ is a bijection (e.g. a Lévy permutation), then define $\gamma ({{\textbf{x}}}):=(y_n)_{n=1}^{\infty }\in {{\mathcal {X}}}^{{\mathbb {N}}}$ by setting $y_n:=x_{\gamma (n)}$ for all $n\in {{\mathbb {N}}}$. Clearly $\gamma ({{\textbf{x}}})\in \ell ^{\infty }({{\mathbb {N}}},{{\mathcal {X}}})$ if and only if ${{\textbf{x}}}\in \ell ^{\infty }({{\mathbb {N}}},{{\mathcal {X}}})$. Furthermore, if $\gamma \in \Gamma$, then $\gamma ({{\textbf{x}}})\in \textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$ if and only if ${{\textbf{x}}}\in \textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$ (Obata 1988, Prop. 3.1).

The Main Theorem is a consequence of a representation theorem of Pivato (2022) for preference orders on $\textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$. Let $({{\mathcal {X}}},d)$ be a connected metric space, and let $\,{\widetilde{\succcurlyeq }}\,$ be a preference order on $\textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$. We define $\succcurlyeq _*$ on ${{\mathcal {X}}}$ by restricting $\,{\widetilde{\succcurlyeq }}\,$ to constant sequences in the obvious way. We require $\,{\widetilde{\succcurlyeq }}\,$ to satisfy the following four axioms.

A1*.:

Let ${{\mathcal {M}}}\subseteq {{\mathbb {N}}}$ be non-null and let ${{\textbf{x}}},{{\textbf{y}}}\in \textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$. Suppose ${{\textbf{x}}}_{{{\mathbb {N}}}\setminus {{\mathcal {M}}}}={{\textbf{y}}}_{{{\mathbb {N}}}\setminus {{\mathcal {M}}}}$, and there exist $x,y\in {{\mathcal {X}}}$ such that $x_m=x$ and $y_m=y$ for all $m\in {{\mathcal {M}}}$. If $x\succ _* y$, then ${{\textbf{x}}}\,{\widetilde{\succ }}\,{{\textbf{y}}}$.

A2*.:

Let ${{\textbf{x}}}\in \textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$, and let $\{{{\textbf{x}}}^k\}_{k=1}^{\infty }$ be a sequence in $\textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$ such that $\displaystyle \lim _{k{\rightarrow }{\infty }} d_{\infty }({{\textbf{x}}}^k,{{\textbf{x}}})=0$. If ${{\textbf{x}}}^k\succcurlyeq {{\textbf{y}}}$ for all $k\in {{\mathbb {N}}}$, then ${{\textbf{x}}}\succcurlyeq {{\textbf{y}}}$. If ${{\textbf{x}}}^k\preccurlyeq {{\textbf{y}}}$ for all $k\in {{\mathbb {N}}}$, then ${{\textbf{x}}}\preccurlyeq {{\textbf{y}}}$.

A3*.:

Let ${{\textbf{x}}},{{\textbf{x}}}',{{\textbf{y}}},{{\textbf{y}}}'\in \textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$, and suppose there are subsets ${{\mathcal {J}}}\subseteq {{\mathbb {N}}}$, and ${{\mathcal {K}}}={{\mathbb {N}}}\setminus {{\mathcal {J}}}$ such that ${{\textbf{x}}}_{{\mathcal {J}}}= {{\textbf{x}}}'_{{\mathcal {J}}}$ and ${{\textbf{y}}}_{{\mathcal {J}}}= {{\textbf{y}}}'_{{\mathcal {J}}}$, while ${{\textbf{x}}}_{{\mathcal {K}}}= {{\textbf{y}}}_{{\mathcal {K}}}$ and ${{\textbf{x}}}'_{{\mathcal {K}}}= {{\textbf{y}}}'_{{\mathcal {K}}}$. Then ${{\textbf{x}}}\succcurlyeq {{\textbf{y}}}$ if and only if ${{\textbf{x}}}'\succcurlyeq {{\textbf{y}}}'$.

A4*.:

For all ${{\textbf{x}}}\in \textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$, and all $\gamma \in \Gamma$, ${{\textbf{x}}}\,{\widetilde{\approx }}\,\gamma ({{\textbf{x}}})$.

Axioms A1* - A4* are clearly analogous to Axioms A1 -  A4 from Sect. 3. We will also need $\,{\widetilde{\succcurlyeq }}\,$ to satisfy the following richness condition.

(B):

For any countable, totally bounded ${{\mathcal {Y}}}\subseteq {{\mathcal {X}}}$, there exist $x,z\in {{\mathcal {X}}}$ such that $x \preccurlyeq _* y \preccurlyeq _* z$ for all $y\in {{\mathcal {Y}}}$.

For any sequence ${{\textbf{r}}}=(r_1,r_2,r_3,\ldots )\in {{\mathbb {R}}}^{{\mathbb {N}}}$, the Cesàro average is defined

$$\begin{aligned} \mathop {\textrm{CesAve}}\limits _{n\in {{\mathbb {N}}}} \ r_n\quad :=\quad \lim _{N{\rightarrow }{\infty }}\frac{1}{N}\sum _{n=1}^N r_n, \end{aligned}$$

(A10)

when this limit exists; in this case, we say that ${{\textbf{r}}}$ is Cesàro summable. If ${{\textbf{r}}}\in \textrm{reg}({{\mathbb {N}}},{{\mathbb {R}}})$ then it is easily verified that the limit (A10) exists. Furthermore,

$$\begin{aligned} \text{ For } \text{ all } \gamma \in \Gamma ,\qquad \mathop {\textrm{CesAve}}\limits _{n\in {{\mathbb {N}}}} \ r_{\gamma (n)} \quad =\quad \mathop {\textrm{CesAve}}\limits _{n\in {{\mathbb {N}}}} \ r_n. \end{aligned}$$

(A11)

Proposition A.2

Let $\,{\widetilde{\succcurlyeq }}\,$ be a preference order on $\textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$ satisfying condition (B). Then $\,{\widetilde{\succcurlyeq }}\,$ satisfies Axioms A1* - A4* if and only if there is a continuous function $u:{{\mathcal {X}}}{{\longrightarrow }}{{\mathbb {R}}}$ such that, for all ${{\textbf{x}}},{{\textbf{y}}}\in \textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$, we have ${{\textbf{x}}}\,{\widetilde{\succcurlyeq }}\,{{\textbf{y}}}$ if and only if $\displaystyle \mathop {\textrm{CesAve}}\limits _{n\in {{\mathbb {N}}}} \ u(x_n) \geqslant \mathop {\textrm{CesAve}}\limits _{n\in {{\mathbb {N}}}} \ u(y_n)$.

Proof

See Pivato (2022).$\square$

Proof of the Main Theorem

Condition (R) and Axiom A5 imply that there is an order $\,{\widetilde{\succcurlyeq }}\,$ on $\textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$ such that, for all $({\varvec{{\mathcal {K}}}},{{\textbf{x}}})$ and $({\varvec{{\mathcal {L}}}},{{\textbf{y}}})$ in ${{\mathfrak {S}}}$, with enumerations ${\widetilde{{\textbf{x}}}}$ and ${\widetilde{{\textbf{y}}}}$,

$$\begin{aligned} \Big (({\varvec{{\mathcal {K}}}},{{\textbf{x}}})\succcurlyeq ({\varvec{{\mathcal {L}}}},{{\textbf{y}}})\Big ) \quad \iff \quad \Big ({\widetilde{{\textbf{x}}}}\ \,{\widetilde{\succcurlyeq }}\,\ {\widetilde{{\textbf{y}}}}\Big ). \end{aligned}$$

(A12)

Clearly, $\succcurlyeq$ satisfies A4 if and only if $\,{\widetilde{\succcurlyeq }}\,$ satisfies A4*; in this case, it does not matter which enumerations of $({\varvec{{\mathcal {K}}}},{{\textbf{x}}})$ and $({\varvec{{\mathcal {L}}}},{{\textbf{y}}})$ we use in statement (A12), by Lemma 3. Meanwhile, $\succcurlyeq$ satisfies  A1 - A3 if and only if $\,{\widetilde{\succcurlyeq }}\,$ satisfies A1* - A3*. Since the metric space $({{\mathcal {X}}},d)$ is complete, Axiom A2* implies that $\,{\widetilde{\succcurlyeq }}\,$ satisfies condition (B), by a proposition in Pivato (2022).¹⁸ Thus, $\,{\widetilde{\succcurlyeq }}\,$ has the representation described in Proposition A.2.

Now let $({\varvec{{\mathcal {K}}}},{{\textbf{x}}})$ be a scenario, and let ${\widetilde{{\textbf{x}}}}$ be an enumeration relative to some $c\in {{\mathcal {S}}}$ and $\psi \in \Phi$. Recall that ${{\textbf{x}}}\in \textrm{reg}({\varvec{{\mathcal {K}}}})$ if and only if ${\widetilde{{\textbf{x}}}}\in \textrm{reg}({{\mathbb {N}}},{{\mathcal {X}}})$, and in this case, it is easily verified that

$$\begin{aligned} \mathop {\textrm{CesAve}}\limits _{} \ u({\varvec{{\mathcal {K}}}},{{\textbf{x}}}) \quad =\quad \mathop {\textrm{CesAve}}\limits _{n\in {{\mathbb {N}}}} \ u({{\widetilde{x}}}_n). \end{aligned}$$

(A13)

(Recall that the left-hand side of (A13) is independent of the choice of c and $\psi$, by Lemma 2. Also, it does not matter which enumeration of $({\varvec{{\mathcal {K}}}},{{\textbf{x}}})$ we use on the right-hand side of (A13), by Lemma 3 and statement (A11).) Now let $({\varvec{{\mathcal {K}}}},{{\textbf{x}}})$ and $({\varvec{{\mathcal {L}}}},{{\textbf{y}}})$ be two regular scenarios with enumerations ${\widetilde{{\textbf{x}}}}$ and ${\widetilde{{\textbf{y}}}}$. Then

$$\begin{aligned} \Big (({\varvec{{\mathcal {K}}}},{{\textbf{x}}})\succcurlyeq ({\varvec{{\mathcal {L}}}},{{\textbf{y}}})\Big )&\iff \Big ({\widetilde{{\textbf{x}}}}\ \,{\widetilde{\succcurlyeq }}\,\ {\widetilde{{\textbf{y}}}}\Big ) \quad \mathop {\Longleftrightarrow }\limits _{\scriptscriptstyle {\mathrm {(*)}}}\quad \Big (\displaystyle \mathop {\textrm{CesAve}}\limits _{n\in {{\mathbb {N}}}} \ u({{\widetilde{x}}}_n)\geqslant \mathop {\textrm{CesAve}}\limits _{n\in {{\mathbb {N}}}} \ u({{\widetilde{y}}}_n)\Big ) \\ {}&\mathop {\Longleftrightarrow }\limits _{\scriptscriptstyle {\mathrm {(\dagger )}}} \Big (\displaystyle \mathop {\textrm{CesAve}}\limits _{} \ u({\varvec{{\mathcal {K}}}},{{\textbf{x}}}) \geqslant \mathop {\textrm{CesAve}}\limits _{} \ u({\varvec{{\mathcal {L}}}},{{\textbf{y}}}) \Big ), \end{aligned}$$

where $(*)$ is by Proposition A.2, and $(\dagger )$ is by equation (A13). $\square$

Fig. 3

The proof of Proposition 4.1

Proof of Proposition 4.1

If $\succcurlyeq$ is enumeration-based and satisfies Weak Pareto, then statement (11) implies that ${\widetilde{\succcurlyeq }}$ also satisfies Weak Pareto. In other words, for any ${\widetilde{{\textbf{x}}}},{\widetilde{{\textbf{y}}}}\in \ell ^{\infty }({{\mathbb {N}}},{{\mathcal {X}}})$, if ${{\widetilde{x}}}_n\succ _* {{\widetilde{y}}}_n$ for all $n\in {{\mathbb {N}}}$, then ${\widetilde{{\textbf{x}}}}\ {\widetilde{\succ }} \ {\widetilde{{\textbf{y}}}}$.

By hypothesis, there is a subspace ${{\mathcal {T}}}\subseteq {{\mathcal {S}}}$ and an isometry $\gamma :{{\mathbb {Z}}}{{\longrightarrow }}{{\mathcal {T}}}$. Let $\psi \in \Phi$ be a reference frame (a bijection $\psi :{{\mathcal {O}}}{{\longrightarrow }}{{\mathcal {S}}}$). Let ${{\mathcal {Z}}}:=\psi ^{-1}({{\mathcal {T}}})\subseteq {{\mathcal {O}}}$. It suffices to consider scenarios where people’s locations are all in ${{\mathcal {Z}}}$. Let $\varphi :=\gamma ^{-1}\circ \psi :{{\mathcal {Z}}}{{\longrightarrow }}{{\mathbb {Z}}}$ (a bijection). We shall identify each point in ${{\mathcal {Z}}}$ with its $\varphi$-image in ${{\mathbb {Z}}}$.

Recall that ${{\mathcal {N}}}$ is the indexing set. For notational convenience, suppose that ${{\mathcal {N}}}={{\mathbb {Z}}}$. Consider the scenarios $(\varvec{{\mathcal {J}}},{{\textbf{w}}})$, $({\varvec{{\mathcal {K}}}},{{\textbf{x}}})$ and $({\varvec{{\mathcal {L}}}},{{\textbf{y}}})$ shown in Fig. 3. In the somata $\varvec{{\mathcal {J}}}$, ${\varvec{{\mathcal {K}}}}$ and ${\varvec{{\mathcal {L}}}}$, each person’s spatiotemporal location is a single point in ${{\mathcal {Z}}}$. In other words, $\varvec{{\mathcal {J}}}=({{\mathcal {J}}}_n)_{n=-{\infty }}^{\infty }$, ${\varvec{{\mathcal {K}}}}=({{\mathcal {K}}}_n)_{n=-{\infty }}^{\infty }$ and ${\varvec{{\mathcal {L}}}}=({{\mathcal {L}}}_n)_{n=-{\infty }}^{\infty }$, where ${{\mathcal {J}}}_n=\{j_n\}$, ${{\mathcal {K}}}_n=\{k_n\}$ and ${{\mathcal {L}}}_n=\{\ell _n\}$, for all $n\in {{\mathcal {N}}}$, with $j_n$, $k_n$ and $\ell _n$ being points in ${{\mathcal {Z}}}$, which we identify with points in ${{\mathbb {Z}}}$ via $\varphi$. For all $n\in {{\mathcal {N}}}={{\mathbb {Z}}}$, let $\varphi (j_n):=9\,n$, $\varphi (k_n):=9\,n-3$ and $\varphi (\ell _n):=9\,n-6$, as shown in Fig. 3.

By hypothesis, there is a ${{\mathbb {Z}}}$-indexed sequence $(v_n)_{n=-{\infty }}^{\infty }$ in ${{\mathcal {X}}}$ such that $v_n \prec _* v_m$ whenever $n<m$. For notational simplicity, assume without loss of generality that ${{\mathbb {Z}}}\subseteq {{\mathcal {X}}}$ (with the natural ordering) and $v_n=n$ for all $n\in {{\mathbb {Z}}}$. For all $n\in {{\mathcal {N}}}={{\mathbb {Z}}}$, let $w_n:= 4\,n$, $x_n:=4\,n-1$ and $y_n:=4\,n-2$, as shown in Fig. 3.

Now, let $a:=\gamma (1)$. It is easily verified that $(\varvec{{\mathcal {J}}},{{\textbf{w}}})$ and $({\varvec{{\mathcal {K}}}},{{\textbf{x}}})$ have unique enumerations ${\widetilde{{\textbf{w}}}}$ and ${\widetilde{{\textbf{x}}}}$ with respect to a and $\psi$, namely ${\widetilde{{\textbf{w}}}}=(0$, 4, $-4$, 8, $-8$, 12, $-12$, $\ldots )$ and ${\widetilde{{\textbf{x}}}}=(-1$, 3, $-5$, 7, $-9$, 11, $-13$, $\ldots )$. Note that ${{\widetilde{x}}}_n={{\widetilde{w}}}_n-1$ for all $n\in {{\mathbb {N}}}$. Thus, ${\widetilde{{\textbf{w}}}}\ {\widetilde{\succ }} \ {\widetilde{{\textbf{x}}}}$ by Weak Pareto. Since $\succcurlyeq$ is enumeration-based and basepoint independent, statement (11) implies that $(\varvec{{\mathcal {J}}},{{\textbf{w}}})\succ ({\varvec{{\mathcal {K}}}},{{\textbf{x}}})$.

Next, let $b:=\gamma (-2)$. Again, $({\varvec{{\mathcal {K}}}},{{\textbf{x}}})$ and $({\varvec{{\mathcal {L}}}},{{\textbf{y}}})$ have unique enumerations ${\widetilde{{\textbf{x}}}}'$ and ${\widetilde{{\textbf{y}}}}$ with respect to b and $\psi$, namely ${\widetilde{{\textbf{x}}}}'= (-1$, 3, $-5$, 7, $-9$, 11, $-13$, $\ldots )$ and ${\widetilde{{\textbf{y}}}}=(-2$, 2, $-6$, 6, $-10$, 10, $-14$, $\ldots )$. Thus, ${\widetilde{{\textbf{x}}}}' \ {\widetilde{\succ }} \ {\widetilde{{\textbf{y}}}}$ by Weak Pareto. Thus, $({\varvec{{\mathcal {K}}}},{{\textbf{x}}})\succ ({\varvec{{\mathcal {L}}}},{{\textbf{y}}})$.

Finally, let $z_n:=4\,n+1$ for all $n\in {{\mathcal {N}}}={{\mathbb {Z}}}$, and consider the scenario $(\varvec{{\mathcal {J}}},{{\textbf{z}}})$. Let $c:=\gamma (-5)$. Then $({\varvec{{\mathcal {L}}}},{{\textbf{y}}})$ and $(\varvec{{\mathcal {J}}},{{\textbf{z}}})$ have unique enumerations ${\widetilde{{\textbf{y}}}}'$ and ${\widetilde{{\textbf{z}}}}$ with respect to c and $\psi$, namely ${\widetilde{{\textbf{y}}}}'=(-2$, 2, $-6$, 6, $-10$, 10, $-14$, $\ldots )$ and ${\widetilde{{\textbf{z}}}}=(-3$, 1, $-7$, 5, $-11$, 9, $-15$, $\ldots )$. Thus, ${\widetilde{{\textbf{y}}}}' \ {\widetilde{\succ }} \ {\widetilde{{\textbf{z}}}}$ by Weak Pareto. Thus, $({\varvec{{\mathcal {L}}}},{{\textbf{y}}})\succ (\varvec{{\mathcal {J}}},{{\textbf{z}}})$.

Combining the conclusions of the three previous paragraphs yields $(\varvec{{\mathcal {J}}},{{\textbf{w}}})\succ ({\varvec{{\mathcal {K}}}},{{\textbf{x}}})\succ ({\varvec{{\mathcal {L}}}},{{\textbf{y}}})\succ (\varvec{{\mathcal {J}}},{{\textbf{z}}})$, and hence $(\varvec{{\mathcal {J}}},{{\textbf{w}}})\succ (\varvec{{\mathcal {J}}},{{\textbf{z}}})$ by transitivity. But $z_n > w_n$ for all $n\in {{\mathcal {N}}}={{\mathbb {Z}}}$. Thus, Weak Pareto says that $(\varvec{{\mathcal {J}}},{{\textbf{z}}})\succ (\varvec{{\mathcal {J}}},{{\textbf{w}}})$, which is a contradiction. $\square$

B Examples of reference systems

Physics provides three important reference systems. In all three systems, ${{\mathcal {S}}}={{\mathbb {R}}}^4={{\mathbb {R}}}\times {{\mathbb {R}}}^3$, with the first dimension representing “time” and the last three dimensions representing “space”, as seen by some observer. Meanwhile, ${{\mathcal {O}}}={{\mathbb {R}}}^4$, and the various elements of $\Phi$ are different ways of assigning spacetime coordinates to ${{\mathcal {O}}}$.

Basic reference system. For any orthonormal basis ${{\mathcal {B}}}=\{{{\textbf{b}}}^1,{{\textbf{b}}}^2,{{\textbf{b}}}^3\}$ of ${{\mathbb {R}}}^3$, with ${{\textbf{b}}}^n=(b^n_1,b^n_2,b^n_3)$ for $n=1,2,3$, let $\rho _{{\mathcal {B}}}:{{\mathcal {O}}}={{\mathbb {R}}}^4{{\longrightarrow }}{{\mathbb {R}}}^4={{\mathcal {S}}}$ be the linear transformation with matrix

$$\begin{aligned} {\left(\begin{array}{cccc} 1 &{} 0 &{}0 &{}0 \\ 0 &{}b^1_1 &{} b^2_1 &{} b^3_1\\ 0 &{}b^1_2 &{} b^2_3 &{} b^3_2\\ 0 &{}b^1_3 &{} b^2_3 &{} b^3_3\end{array}\right)}. \end{aligned}$$

In other words, $\rho _{{\mathcal {B}}}$ leaves the time coordinate unchanged, while it applies to the three spatial coordinates a rotation and/or reflection that maps the standard basis of ${{\mathbb {R}}}^3$ to the basis ${{\mathcal {B}}}$. Meanwhile, for any vector ${{\textbf{v}}}\in {{\mathbb {R}}}^4$, define $\tau _{{\textbf{v}}}:{{\mathcal {O}}}={{\mathbb {R}}}^4{{\longrightarrow }}{{\mathbb {R}}}^4={{\mathcal {S}}}$ by $\tau _{{\textbf{v}}}({{\textbf{x}}}):={{\textbf{x}}}+{{\textbf{v}}}$ for all ${{\textbf{x}}}\in {{\mathbb {R}}}^4$; this is “spacetime translation” by the displacement vector ${{\textbf{v}}}$. Finally, let $\Psi _B:=\{\tau _{{\textbf{v}}}\circ \rho _{{\mathcal {B}}}$; ${{\textbf{v}}}\in {{\mathbb {R}}}^4$ and ${{\mathcal {B}}}$ any orthonormal basis of ${{\mathbb {R}}}^3\}$. Then $({{\mathcal {O}}},{{\mathcal {S}}},\Psi _B)$ is the reference system of Maxwellian electrodynamics. Note that this system has no way of representing motion.

Galilean reference system. For any vector ${{\textbf{v}}}\in {{\mathbb {R}}}^3$ (representing a velocity), define the Galilean boost $\gamma _{{\textbf{v}}}:{{\mathcal {O}}}={{\mathbb {R}}}^4{{\longrightarrow }}{{\mathbb {R}}}^4={{\mathcal {S}}}$ by setting $\gamma _{{\textbf{v}}}(t,{{\textbf{x}}}):=(t,{{\textbf{x}}}-t\,{{\textbf{v}}})$ for all ${{\textbf{x}}}\in {{\mathbb {R}}}^3$ and $t\in {{\mathbb {R}}}$. This represents a coordinate system assigned to four-dimensional spacetime by an observer traveling with velocity ${{\textbf{v}}}$. Let $\Phi _G=\{\psi \circ \gamma _{{\textbf{v}}}$; ${{\textbf{v}}}\in {{\mathbb {R}}}^3$ and $\psi \in \Psi _B\}$; then $({{\mathcal {O}}},{{\mathcal {S}}},\Phi )$ is the reference system of Newtonian mechanics.

Relativistic reference system. Let ${{\mathfrak {c}}}$ be the speed of light. For any vector ${{\textbf{v}}}=(v_1,v_2,v_3)\in {{\mathbb {R}}}^3$ (representing velocity), let ${\left\| {{\textbf{v}}} \right\| _{{}} } :=\sqrt{v_1^2+v_2^2+v_3^2}$ be its norm, and let ${\widehat{{\textbf{v}}}}=({{\widehat{v}}}_1,{{\widehat{v}}}_2,{{\widehat{v}}}_2):={{\textbf{v}}}/{\left\| {{\textbf{v}}} \right\| _{{}} }$ be the unit vector in the same direction as ${{\textbf{v}}}$. If ${\left\| {{\textbf{v}}} \right\| _{{}} } <{{\mathfrak {c}}}$, then let $\beta :={\left\| {{\textbf{v}}} \right\| _{{}} } /{{\mathfrak {c}}}$ where ${{\mathfrak {c}}}$ is the speed of light, and let $\alpha :=1/\sqrt{1-\beta ^2}$ be the “Lorentz factor”. We then define the Lorentz boost $\lambda _{{\textbf{v}}}:{{\mathcal {O}}}={{\mathbb {R}}}^4{{\longrightarrow }}{{\mathbb {R}}}^4={{\mathcal {S}}}$ to be the linear transformation with matrix

$$\begin{aligned} {\left(\begin{array}{cccc} \alpha &{} -\alpha \beta {{\widehat{v}}}_1 &{} -\alpha \beta {{\widehat{v}}}_2 &{} -\alpha \beta {{\widehat{v}}}_3 \\ -\alpha \beta {{\widehat{v}}}_1 &{} 1+(\alpha -1){{\widehat{v}}}_1^2 &{} (\alpha -1){{\widehat{v}}}_1{{\widehat{v}}}_2 &{} (\alpha -1){{\widehat{v}}}_1{{\widehat{v}}}_3\\ -\alpha \beta {{\widehat{v}}}_2 &{}(\alpha -1){{\widehat{v}}}_2{{\widehat{v}}}_1 &{}1+(\alpha -1){{\widehat{v}}}_2^2 &{} (\alpha -1){{\widehat{v}}}_2{{\widehat{v}}}_3\\ -\alpha \beta {{\widehat{v}}}_3 &{}(\alpha -1){{\widehat{v}}}_3{{\widehat{v}}}_1 &{} (\alpha -1){{\widehat{v}}}_3{{\widehat{v}}}_2 &{}1+(\alpha -1){{\widehat{v}}}_3^2 \end{array}\right).} \end{aligned}$$

This represents the relativistic coordinate system assigned to four-dimensional spacetime by an observer traveling with relative velocity ${{\textbf{v}}}$. Let $\Phi _R=\{\psi \circ \lambda _{{\textbf{v}}}$; $\psi \in \Psi _B$ and ${{\textbf{v}}}\in {{\mathbb {R}}}^3$ with ${\left\| {{\textbf{v}}} \right\| _{{}} } <{{\mathfrak {c}}}\}$; then $({{\mathcal {O}}},{{\mathcal {S}}},\Phi )$ is the reference system of relativistic physics.

Boost reference systems. For the purposes of the present paper, it is not necessary to incorporate rotations or reflections into the reference system $\Phi$. Open balls around a point in Euclidean space are already invariant under rotations and reflections around that point. So the sets ${{\mathcal {O}}}(c,\psi ,r)$ and ${{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)$ defined in formula (1) will be invariant under rotations and reflections whether or not they are specifically built into $\Phi$. Likewise, it is not necessary to encode spatial translations into $\Phi$, since applying a spatial translation is equivalent to changing the central location c in formula (1), and Lemmas 1 and 2 already guarantee that our key constructions are independent of the choice of c. Thus, instead of the basic reference system $\Psi _B$, we could use the trivial reference system $\Phi =\{\textrm{Id}\}$. Likewise, instead of the Galilean reference system $\Phi _G$, we could use the “Galilean boost” reference system $\Phi '_G=\{\gamma _{{\textbf{v}}}$; ${{\textbf{v}}}\in {{\mathbb {R}}}^3\}$, and instead of the relativistic reference system $\Phi _R$, we could use the “Lorentz boost” reference system $\Phi '_R=\{\lambda _{{\textbf{v}}}$; ${{\textbf{v}}}\in {{\mathbb {R}}}^3$ and ${\left\| {{\textbf{v}}} \right\| _{{}} } <{{\mathfrak {c}}}\}$.

Homogeneous reference systems. In all of the examples above, the reference system $\Phi$ has the form $\Phi = \{\gamma \circ \phi _0$; $\gamma \in {{\mathcal {G}}}\}$, where $\phi _0:{{\mathcal {O}}}{{\longrightarrow }}{{\mathcal {S}}}$ is one bijection, and ${{\mathcal {G}}}$ is a group (in fact, a Lie group) of transformations of ${{\mathcal {S}}}$; in this case, we say that $\Phi$ is a homogeneous reference system. In the abstract vector space example at the start of § 2.1, ${{\mathcal {G}}}$ was the general linear group ${{\mathbb {G}}}{{\mathbb {L}}}^N$. In the basic reference system $\Psi _B$, ${{\mathcal {G}}}$ was the group of all rigid motions of three-dimensional space, plus time translations. In the Galilean reference system $\Phi _G$, ${{\mathcal {G}}}$ was the group of Galilean transformations. In the relativistic reference system $\Phi _R$, ${{\mathcal {G}}}$ was the group of Poincaré transformations. In the two “boost” reference systems $\Phi '_G$ and $\Phi '_R$, ${{\mathcal {G}}}$ was a group of boosts. Such homogeneous reference systems are ubiquitous in modern physics: $\phi _0$ represents one (arbitrary) coordinate system or representation of the physical world, and ${{\mathcal {G}}}$ is the group of symmetries under which physical laws must remain invariant. However, the framework of this paper does not require the reference system to be homogeneous.

Natural metrics. The framework of Sect. 2 also posits a metric $\delta$ on $\Phi$. Clearly, if we want this framework to reflect “the point of view of the Universe”, then $\delta$ should be non-arbitrary and somehow canonical. In particular, its definition should not depend on a particular reference frame. Let us call this a natural metric.

In the two “boost” reference systems introduced above, there are natural metrics defined via kinetic energy. In both reference systems, reference frames are parameterized by velocities (which are themselves specified with respect to some fixed but arbitrary reference frame). We can thus define distance between two reference frames with velocity vectors ${{\textbf{v}}}$ and ${{\textbf{w}}}$ to be the square root of the kinetic energy of a particle of unit rest-mass travelling at velocity ${{\textbf{w}}}$, from the perspective of an observer at velocity ${{\textbf{v}}}$ (or vice versa).¹⁹ Equivalently, this is the square root of the minimum energy needed to accelerate a unit rest-mass particle from velocity ${{\textbf{v}}}$ to velocity ${{\textbf{w}}}$, from the perspective of an observer at velocity ${{\textbf{v}}}$ (or vice versa). It is easily verified that this defines a natural metric on the boost reference system. (In particular, it does not depend on the reference frame with respect to which the velocity vectors ${{\textbf{v}}}$ and ${{\textbf{w}}}$ were specified in the first place.)

In the Galilean boost reference system $\Phi '_G$, this kinetic energy metric reduces to $\delta (\gamma _{{\textbf{v}}},\gamma _{{\textbf{w}}})={\left\| {{\textbf{v}}}-{{\textbf{w}}} \right\| _{{}} } /\sqrt{2}$ for any velocity vectors ${{\textbf{v}}},{{\textbf{w}}}\in {{\mathbb {R}}}^3$, where ${\left\| \cdot \right\| _{{}} }$ is the ordinary Euclidean norm on ${{\mathbb {R}}}^3$. In the Lorentz boost reference system $\Phi '_R$, the exact formula is quite complicated, since it involves relativistic velocity addition. For simplicity, we will only consider the case of colinear velocities. Let ${{\mathfrak {c}}}$ be the speed of light, let $v,w\in (-{{\mathfrak {c}}},{{\mathfrak {c}}})$, and suppose that ${{\textbf{v}}}=(v,0,0)$ and ${{\textbf{w}}}=(w,0,0)$ (relative to some arbitrary reference frame). Then the relativistic speed of a ${{\textbf{w}}}$-velocity particle as seen from the ${{\textbf{v}}}$-reference frame is

$$\begin{aligned} u \ \ = \ \ \frac{w-v}{1-\frac{v\,w}{{{\mathfrak {c}}}^2}}. \end{aligned}$$

The relativistic kinetic energy of the ${{\textbf{w}}}$-particle from the ${{\textbf{v}}}$-reference frame is then

$$\begin{aligned} \frac{{{\mathfrak {c}}}^3 - {{\mathfrak {c}}}^2\sqrt{{{\mathfrak {c}}}^2-u^2}}{\sqrt{{{\mathfrak {c}}}^2-u^2}} \end{aligned}$$

(assuming a unit rest mass). Thus, in this special case of colinear motion,

$$\begin{aligned} \delta (\lambda _{{\textbf{v}}},\lambda _{{\textbf{w}}})= \sqrt{\frac{{{\mathfrak {c}}}^3 - {{\mathfrak {c}}}^2\sqrt{{{\mathfrak {c}}}^2-u^2}}{\sqrt{{{\mathfrak {c}}}^2-u^2}}}. \end{aligned}$$

(The general formula is more complicated.)

Setting aside kinetic energy, many homogeneous reference systems admit a natural metric. Let e be the identity element of ${{\mathcal {G}}}$, and let $d_{{\mathcal {G}}}$ be a right-invariant metric on ${{\mathcal {G}}}$ —i.e. a metric such that $d_{{\mathcal {G}}}(\gamma \circ \eta , \zeta \circ \eta ) =d_{{\mathcal {G}}}(\gamma , \zeta )$, for all $\gamma ,\zeta ,\eta \in {{\mathcal {G}}}$.²⁰ Then $d_{{\mathcal {G}}}$ induces a natural metric $\delta$ on $\Phi$ by stipulating that $\delta (\phi ,\phi '):=\inf \{d_{{\mathcal {G}}}(\gamma ,e)$; $\gamma \in {{\mathcal {G}}}$ and $\gamma \circ \phi =\phi '\}$.

C More on regular somata

This appendix contains some observations that provide a “recipe” for constructing a large class of regular somata. We shall suppose that ${{\mathcal {O}}}$ is a topological space, and for all $r>0$, suppose that ${{\mathcal {O}}}(c,\psi ,r)$ is an open subset of ${{\mathcal {O}}}$.

Lemma C.1

If ${{\mathcal {K}}}_n$ is compact in ${{\mathcal {O}}}$ for all $n\in {{\mathcal {N}}}$, then ${\varvec{{\mathcal {K}}}}$ satisfies Mortality.

Proof

${{\mathcal {O}}}(c,\psi ,r)\subseteq {{\mathcal {O}}}(c,\psi ,R)$ for any $r<R$. Also, $\bigcup \limits _{r=1}^{\infty }{{\mathcal {O}}}(c,\psi ,r)={{\mathcal {O}}}$ because $\bigcup \limits _{r=1}^{\infty }{{\mathcal {S}}}(c,r)={{\mathcal {S}}}$ and $\bigcup \limits _{r=1}^{\infty }\Phi (\psi ,r)=\Phi$. So $\{{{\mathcal {O}}}(c,\psi ,r)\}_{r=1}^{\infty }$ is an open cover of ${{\mathcal {O}}}$, hence of ${{\mathcal {K}}}_n$. There is a finite subcover of ${{\mathcal {K}}}_n$ by compactness; thus, there is some $r\in {{\mathbb {N}}}$ such that ${{\mathcal {K}}}_n\subseteq {{\mathcal {O}}}(c,\psi ,r)$, hence $n\in {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)$. This holds for all $n\in {{\mathcal {N}}}$. $\square$

Now suppose ${{\mathcal {O}}}$ is a metric space in which all open balls have compact closures (e.g. ${{\mathcal {O}}}={{\mathbb {R}}}^N$ with the Euclidean metric).

Lemma C.2

For all $r>0$, suppose ${{\mathcal {O}}}(c,\psi ,r)$ is contained in an open ball in ${{\mathcal {O}}}$. Suppose ${{\mathcal {K}}}_n\cap {{\mathcal {K}}}_m=\varnothing$ whenever $n\ne m$, and suppose there exists $\epsilon >0$ such that for all $n\in {{\mathbb {N}}}$, the set ${{\mathcal {K}}}_n$ contains an $\epsilon$-ball. Then ${\varvec{{\mathcal {K}}}}$ satisfies Local finiteness.

Proof

Any closed ball in ${{\mathcal {O}}}$ can be covered with a finite set of $\epsilon$-balls, because it is compact. Thus, it can only contain a finite number of disjoint $\epsilon$-balls. But ${{\mathcal {O}}}(c,\psi ,r)$ is contained in such a compact ball, so it also can contain only a finite number of disjoint $\epsilon$-balls. The elements of ${\varvec{{\mathcal {K}}}}$ are disjoint. Each contains an $\epsilon$-ball. So ${{\mathcal {O}}}(c,\psi ,r)$ can only contain a finite number of elements of ${\varvec{{\mathcal {K}}}}$. Thus, $\left|{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|$ is finite. $\square$

Finally, suppose that ${{\mathcal {O}}}$ is N-dimensional for some $N\in {{\mathbb {R}}}_+$, meaning that there is some $K>0$ such that for any $R>r>0$, a ball of radius $R\in {{\mathcal {O}}}$ can contain at most $K\cdot (R/r)^N$ disjoint balls of radius r. (For example, this is true if ${{\mathcal {O}}}={{\mathbb {R}}}^N$.) Furthermore, suppose there is some L and $M\in {{\mathbb {R}}}_+$ such that for all $r>0$, the set ${{\mathcal {O}}}(c,\psi ,r)$ is contained in a ball of radius $L\cdot r^M$.²¹ If every ${{\mathcal {K}}}_n$ contains a ball of radius $\epsilon$, then

$$\begin{aligned} \big |{{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\big |\quad \leqslant \quad K\cdot (L\cdot r^M/\epsilon )^N \quad =\quad C\, \cdot r^{M\,N}, \quad \text{ where } C:=K \, (L/\epsilon )^N. \end{aligned}$$

Thus, $\left| {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|$ grows at most polynomially in r, hence subexponentially. Now suppose that $\left| {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right|$ exhibits exactly polynomial asymptotic growth —in other words, there are constants $P\in {{\mathbb {R}}}_+$ and $H>h>0$ such that $h\, r^P \leqslant \left| {{\mathcal {N}}}_{c,\psi }({\varvec{{\mathcal {K}}}},r)\right| \leqslant H\,r^P$ for all sufficiently large $r\in {{\mathbb {R}}}_+$. Then it is easily verified that ${\varvec{{\mathcal {K}}}}$ satisfies Subexponential growth.