3.1 Definitions

Throughout this paper all geometries will be d-dimensional,Footnote ² for some fixed integer $d \geq 2$ , and defined over some field or ordered field $\mathfrak{F}$ with universe F. The points of coordinate space $F^d$ will be the points of our geometries.

When $\mathfrak{F}=\langle F,+,\cdot ,0,1\rangle $ is a field, the ternary relation ${\mathsf {Col}}$ of collinearity on $F^d$ is defined for all $\vec {p},\vec {q},\vec {r}\in F^d$ by

$$\begin{align*}{\mathsf{Col}}(\vec{p},\vec{q},\vec{r}\,)\ \stackrel{\textsf{ def}}{\iff}\ \vec{q}=\vec{p}+\lambda(\vec{r}-\vec{p}\,)\ \text{ for some }\ \lambda\in F,\ \text{ or }\ \vec{r}=\vec{p}. \end{align*}$$

We call model $\langle F^d,{\mathsf {Col}}\rangle $ the d-dimensional affine geometry over field $\mathfrak{F}$ . Our notion of affine geometry is a natural generalization of the affine Cartesian spaces of [Reference Szczerba and Tarski43, definition 1.5.] to fields which are not necessarily ordered.

Now suppose $\mathfrak{F}=\langle F,+,\cdot ,0,1,\leq \rangle $ is an ordered field. The ternary relation ${\mathsf {Bw}}$ of betweenness on $F^d$ is defined for every $\vec {p},\vec {q},\vec {r}\in F^d$ by

$$\begin{align*}{\mathsf{Bw}}(\vec{p},\vec{q},\vec{r}\,)\stackrel{\textsf{ def}}{\iff} \vec{q}=\vec{p}+\lambda(\vec{r}-\vec{p}\,)\ \text{ for some }\ \lambda\in [0,1] \subseteq F. \end{align*}$$

Of course, ${\mathsf {Col}}$ is also defined for ordered fields and is definable in terms of ${\mathsf {Bw}}$ by

$$\begin{align*}{\mathsf{Col}}(\vec{p},\vec{q},\vec{r}\,)\ \Longleftrightarrow\ {\mathsf{Bw}}(\vec{p},\vec{q},\vec{r}\,)\vee {\mathsf{Bw}}(\vec{q},\vec{p},\vec{r}\,)\vee {\mathsf{Bw}}(\vec{p},\vec{r},\vec{q}\,). \end{align*}$$

We call model $\langle F^d,{\mathsf {Bw}}\rangle $ the d-dimensional ordered affine geometry over ordered field $\mathfrak{F}$ . Our notion of ordered affine geometry coincides with the notion of affine Cartesian space given in [Reference Szczerba and Tarski43, definition 1.5.].

In much of what follows, we will need to re-express n-tuples over $M^d$ (n-tuples of (d-tuples over M)) as $(dn)$ -tuples over M. Given the potential confusion arising from treating the same collection of values and/or variables using tuples of different lengths, we will adopt the notational convention of denoting points in $M^d$ (d-tuples over M) using arrows and angle-brackets (e.g., $\vec {a} = \langle {a_1,\ldots ,a_d} \rangle $ ). Other tuples will be written out in the usual way using parentheses, and overlines (the context will always be clear). Given any d-tuple $\vec {p}$ , we denote its i’th component by $p_i$ , and note that $\vec {p} = \langle p_1, \dots , p_d \rangle $ .

Given any n-tuple $\bar p$ over $M^d$ , we write $\mathsf {flatten}(\bar {p})$ to denote the naturally corresponding ‘flattened’ $(dn)$ -tuple over M, i.e., given any n-tuple $\bar {p}=(\vec {p}_1, \dots , \vec {p}_n)$ of d-tuples $\vec {p}_1 = \langle { p_{11}, \ldots , p_{1d} } \rangle $ , …, $\vec {p}_n = \langle { p_{n1}, \ldots , p_{nd} } \rangle $ in $M^d$ , we define

$$\begin{align*}\mathsf{flatten}(\bar p) \ \stackrel{\textsf{ def}}{=} \ \left( p_{11},\ldots, p_{1d}, \dots, p_{n1},\ldots, p_{nd} \right). \end{align*}$$

For convenience, we will generally write $(\hat {p}_1, \dots , \hat {p}_n)$ for $\mathsf {flatten}(\bar {p})$ . For example, in the case that $d = 2$ , $\vec {p}_1 = \langle {1,2} \rangle $ , $\vec {p}_2 = \langle {3,4} \rangle $ and $\vec {p}_3 = \langle {5,6} \rangle $ , we have $(\hat {p}_1,\hat {p}_2) = (1,2,3,4)$ and $(\hat {p}_1, \dots , \hat {p}_3) = (1,2,3,4,5,6)$ . The “hatted” notation $\hat {p}$ always indicates that the underlying tuple is a d-tuple, namely, $\vec {p}$ . We use the analogous notation for sequences of variables.

Since $\mathsf {flatten}$ is a bijection from $(M^d)^n$ onto $M^{dn}$ it has an inverse (we call it $\mathsf {unflatten}$ ). If $\mathsf {R}$ is any n-ary relation on $M^d$ , we write $\widehat {\mathsf {R}}$ to denote the corresponding $(dn)$ -ary relation on M, i.e., for all $\bar p\in M^{dn}$ ,

$$\begin{align*}\bar p\in\widehat{\mathsf{R}} \quad\stackrel{\textsf{ def}}{\iff}\quad \mathsf{unflatten}(\bar p)\in\mathsf{R}. \end{align*}$$

Given any $\vec {p}_1, \dots , \vec {p}_n$ in $M^d$ ( $n \in \mathbb {N}^{+}$ ), the bijective nature of $\mathsf {flatten}$ and $\mathsf {unflatten}$ means this can also be written as

(3) $$ \begin{align} (\hat{p}_1, \dots, \hat{p}_n)\in\widehat{\mathsf{R}} \quad\iff\quad (\vec{p}_1,\dots, \vec{p}_n)\in\mathsf{R}. \end{align} $$

We say that n-ary relation $\mathsf {R}\subseteq \left (M^d\right )^n$ is definable over ${\mathfrak{M}}$ iff the corresponding relation $\widehat {\mathsf {R}}\subseteq M^{dn}$ is definable in ${\mathfrak{M}}$ . In particular, ${\mathsf {Col}}$ is definable over $\mathfrak{F}$ if $\mathfrak{F}$ is a field, and ${\mathsf {Bw}}$ is definable over $\mathfrak{F}$ if $\mathfrak{F}$ is an ordered field.

From now on, because constants and n-ary functions can be treated as unary and $(n + 1)$ -ary relations in the usual way, we assume without loss of generality that the languages of coordinate geometries contain only relation symbols.

3.2 Examples

Since relations ${\mathsf {Col}}$ and ${\mathsf {Bw}}$ are definable over the corresponding (ordered) field $\mathfrak{F}$ , affine and ordered affine geometries are FFD coordinate geometries for all $d\ge 2$ .

Euclidean geometry can be introduced as the following FFD coordinate geometry over any ordered field $\mathfrak{F}$ :

$$ \begin{align*} \mathcal{E}\!\mathit{ucl} \ \ \stackrel{\textsf{ def}}{=} \ \ \langle F^d,\cong,{\mathsf{Bw}}\rangle, \end{align*} $$

where the 4-nary relation $\cong $ of congruence is defined by

$$ \begin{align*} (\vec{p},\vec{q}\,)\,\boldsymbol{{ \cong}}_{}\,(\vec{r},\vec{s}\,) \stackrel{\textsf{ def}}{\iff} (p_1-q_1)^2+\dots +(p_d-q_d)^2=(r_1-s_1)^2+\dots +(r_d-s_d)^2. \end{align*} $$

Notable classical geometries such as Relativistic spacetime ( $\mathcal {R}\mathit {el}$ ), Minkowski spacetime ( $\mathcal {M}\hspace {-1pt}\mathit {ink}$ ), Galilean spacetime ( $\mathcal {G}\mathit {al}$ ) and Newtonian spacetime ( $\mathcal {N}\!\mathit {ewt}$ ), as examples for FFD coordinate geometries, will be introduced in a similar manner in §7.

The language of $\mathcal {E}\!\mathit {ucl}$ is exactly the language used by Tarski [Reference Tarski, Henkin, Suppes and Tarski45, Reference Tarski and Givant47] to axiomatize elementary Euclidean geometry in first-order logic. In Tarski’s work and in the literature following his axiomatization, $\mathcal {E}\!\mathit {ucl}$ is called a d-dimensional Cartesian space over an ordered field $\mathfrak{F}$ , and various representation theorems show that the models of suitable axiom systems are exactly those that are isomorphic to Cartesian spaces over, real-closed, Euclidean, ordered or pythagorean ordered fields (see, e.g., [Reference Gupta14, Reference Pambuccian33, Reference Pambuccian34, Reference Schwabhäuser, Szmielew and Tarski36, Reference Tarski, Henkin, Suppes and Tarski45]).

Our FFD coordinate geometries can be viewed as a natural generalization of Tarski-style Cartesian spaces. Instead of restricting attention to the relations of Euclidean geometry, we allow finitely many relations that are definable over a field or an ordered field. FFD coordinate geometries also admit an axiomatic treatment.

For cardinality reasons, it is clear that there must also be coordinate geometries which are not field-definable. A concrete example of such a geometry over the ordered field of reals is the geometry:

$$ \begin{align*} \langle \mathbb{R}^d, \mathbb{Z}^d,{\mathsf{Bw}}\rangle. \end{align*} $$

This is because the unary relation $\mathbb {Z}^d$ on points is not definable over the field of reals; this fact follows easily from Tarski’s theorem on quantifier elimination of real closed fields [Reference Tarski, Caviness and Johnson46].

4.1 Key theorems

In this section, we state our key theorems: they confirm that the concepts of a geometry are intimately associated with its affine automorphism group.

Definition 4.1 (Affine automorphisms)

Suppose F is the universe of some field or ordered field $\mathfrak{F}$ . A map $A\colon F^d\rightarrow F^d$ is called an affine transformation iff it is the composition of a linear bijection and a translation, i.e., $A=\tau \circ L$ for some translation $\tau $ and linear bijection L. We note that all affine transformations are bijections.

The set of affine transformations is denoted by $\mathsf {AffineTrf}$ .

Affine transformations that are automorphisms of a coordinate geometry $\mathcal {G}$ over $\mathfrak{F}$ are called affine automorphisms of $\mathcal {G}$ , and the set of these is denoted by $\mathsf {AffAut}\,\mathcal {G}$ . Thus,

(4) $$ \begin{align} \mathsf{AffAut}\,\mathcal{G} \ \stackrel{\textsf{ def}}{=} \ \mathsf{Aut}\; \mathcal{G}\cap\mathsf{AffineTrf}. \end{align} $$

If $\mathcal {G}$ is an affine or ordered affine geometry, we have $\mathsf {AffAut}\,\mathcal {G} = \mathsf {AffineTrf}$ .

Proving Theorem 4.2 requires that we first develop a number of supporting lemmas, and completing the proof is consequently relegated to §6.5 (p. 18, below). But first, we state and prove various corollaries of Theorem 4.2, showing that the subset-inclusion poset of the concept-sets associated with FFD coordinate geometries is dually isomorphic to the subgroup-inclusion poset of their respective automorphism groups.

We will apply these results in §7 to compare the sets of concepts of several historically significant spacetimes by determining their affine automorphism groups.

Remark 4.4. Let ${\mathfrak{M}}$ and ${\mathfrak{N}}$ be two models. Clearly,

$$\begin{align*}\mathsf{Conc}\,{\mathfrak{M}}\,=\mathsf{Conc}\,{\mathfrak{N}}\,\ \Longrightarrow\ \mathsf{Aut}\; {\mathfrak{M}}=\mathsf{Aut}\; {\mathfrak{N}}, \end{align*}$$

because definable relations must be closed under automorphisms. Furthermore, if the universes of ${\mathfrak{M}}$ and ${\mathfrak{N}}$ are the same, then

$$\begin{align*}\mathsf{Conc}\,{\mathfrak{M}}\,\subseteq\mathsf{Conc}\,{\mathfrak{N}}\,\ \Longrightarrow\ \mathsf{Aut}\; {\mathfrak{M}}\supseteq\mathsf{Aut}\; {\mathfrak{N}}. \end{align*}$$

The reverse implication ( $\Longleftarrow $ ) does not hold in general. For example, if ${\mathfrak{R}}$ is the field of reals and ${\mathfrak{R}}'$ is the expansion of ${\mathfrak{R}}$ with the unary relation N of being a natural number, then $\mathsf {Aut}\; {\mathfrak{R}} = \mathsf {Aut}\; {\mathfrak{R}}'$ because both ${\mathfrak{R}}$ and ${\mathfrak{R}}'$ have no nontrivial automorphisms, but $\mathsf {Conc}\,{\mathfrak{R}}'\,\not \subseteq \mathsf {Conc}\,{\mathfrak{R}}\,$ because the set of natural numbers is not definable in the field of real numbers. Nonetheless, this reverse implication does hold for FFD coordinate geometries—as we now show.

Observe that $\langle \mathsf {Aut}\; {\mathfrak{M}},\circ \rangle $ is a group for every model ${\mathfrak{M}}$ , and similarly $\langle \mathsf {AffAut}\,\mathcal {G},\circ \rangle $ is a group for every coordinate geometry $\mathcal {G}$ , where $\circ $ is the standard composition of functions. By (2), the following is a reformulation of Theorem 4.5 in terms of cylindric algebras.

As observed above, proving (and fully explaining) Theorem 4.2 requires that we first consider a number of preliminary results; the full proof concludes in §6.5.

6.1 Automorphisms and affine automorphisms

Suppose F is the universe of some field or ordered field $\mathfrak{F}$ . Given a function $f\colon F\rightarrow F$ , we write $\widetilde {f}\colon F^d\rightarrow F^d$ for the map induced on $F^d$ by f componentwise, i.e., $\widetilde {f}\bigl (\langle p_1,\ldots ,p_d\rangle \bigr )\ \stackrel {\textsf { def}}{=} \ \langle f(p_1),\ldots ,f(p_d)\rangle $ . We note that if f is a bijection, then so is $\widetilde {f}$ , and we define the induced automorphisms on $F^d$ to be the members of the set

$$\begin{align*}\widetilde{\mathsf{Aut}}\;\mathfrak{F}\ \stackrel{\textsf{ def}}{=} \ \{\widetilde{\alpha} : \alpha\in\mathsf{Aut}\; \mathfrak{F}\}. \end{align*}$$

Proposition 6.1. Assume that $\mathfrak{F}$ is an ordered field or a field that has more than two elements, and that $\mathcal {G}$ is a field-definable coordinate geometry over $\mathfrak{F}$ . Then

$$ \begin{align*} \mathsf{Aut}\; \mathcal{G}=\mathsf{AffAut}\,\mathcal{G}\;\circ\;\widetilde{\mathsf{Aut}}\;\mathfrak{F}. \end{align*} $$

Moreover, the decomposition of an automorphism of $\mathcal {G}$ into an affine automorphism following a map induced by an automorphism of $\mathfrak{F}$ is unique.

6.2 Definability of relations on d-tuples in arbitrary models

With geometries as our main application in mind, we now make general observations that are useful when investigating the definability of relations on d-tuples in arbitrary models.

Let ${\mathcal {L}}$ be an arbitrary first-order language and let $\mathit {Fm}_{{\mathcal {L}}}$ be the set of first-order formulas in the language $\mathcal L$ . Then we define the following set of relation symbols for $\mathcal L$ -definable relations of d-tuples:

$$ \begin{align*}\mathcal{R}\mathit{Symb}_{{\mathcal{L}}} \ \stackrel{\textsf{ def}}{=} \ \left\{\, \langle\varrho,n\rangle\in\mathit{Fm}_{{\mathcal{L}}}\times\mathbb{N}^{+}\: :\: \text{the free variables of } \varrho \text{ are in } \{{v}_1,\ldots,{v}_{dn}\}\, \right\}. \end{align*} $$

Let ${\mathfrak{M}}$ be an arbitrary model for language $\mathcal {L}$ with universe M. For each relation symbol $R=\langle \varrho ,n\rangle \in \mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ , we define a corresponding n-ary relation $R_{{\mathfrak{M}}d}$ on $M^d$ by

(10) $$ \begin{align} (\vec{p}_{1},\ldots, \vec{p}_{n})\in R_{{\mathfrak{M}}d} \quad\stackrel{\textsf{ def}}{\iff}\quad {\mathfrak{M}}\models \varrho [ \hat{{p}}_{1},\dots, \hat{{p}}_{n} ]. \end{align} $$

For every $\Psi \subseteq \mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ , let ${\mathcal {L}}_{\Psi }$ denote the first-order language with relation symbols in $\Psi $ such that the arity of each $R=\langle \varrho ,n\rangle \in \Psi $ is n.

In particular, we define the following two formulas $\gamma $ and $\beta $ , the first in the languages of fields and the second in that of ordered fields, and each having $3d$ free variables, $v_1,\ldots ,v_{3d}$ . Given $\vec {x}=\langle v_1,\ldots , v_d\rangle $ , $\vec {y}=\langle v_{d+1},\ldots ,v_{2d}\rangle $ , $\vec {z}=\langle v_{2d+1},\ldots , v_{3d}\rangle $ and $v=v_{3d+1}$ , we define

$$ \begin{align*} \gamma(\hat{x},\hat{y},\hat{z}) & := \exists v\, \left(\vec{y}+ v\vec{x}=\vec{x}+v\vec{z}\ \vee\ \vec{z}=\vec{x}\,\right) \text{ and}\\ \beta(\hat{x},\hat{y},\hat{z}) & := \exists v\, \left(\vec{y}+v\vec{x}=\vec{x}+ v\vec{z}\ \wedge\ 0\leq v \leq 1\right). \end{align*} $$

Throughout the remainder of this paper, we consider two cases in parallel: the case where $\mathfrak{F}$ is a field, ${\mathcal {L}}$ is the language of fields, $\gamma $ is the relevant formula and ${\mathsf {Col}}$ the relevant relation; and the case where $\mathfrak{F}$ is an ordered field, ${\mathcal {L}}$ is the language of ordered fields, $\beta $ is the relevant formula and ${\mathsf {Bw}}$ the relevant relation. To simplify the presentation of results below, we will write $\mathsf {\kappa }$ (‘key formula’) in place of the formulas $\gamma $ and $\beta $ , i.e.,

$$\begin{align*}\mathsf{\kappa} \quad = \begin{cases} \gamma & \text{ if the language of } \mathfrak{F} \text{ is the language of fields; and } \\ \beta & \text{ if the language of } \mathfrak{F} \text{ is the language of ordered fields}. \end{cases} \end{align*}$$

We likewise define the ‘key relation’ to be $\mathsf {K} = \langle \mathsf {\kappa }, 3 \rangle _{\mathfrak{F} d}$ , and note that $\mathsf {K}$ coincides either with ${\mathsf {Col}}$ or ${\mathsf {Bw}}$ depending on the language of $\mathfrak{F}$ (cf. Remark 6.3). In both cases, we continue to write F for the universe of $\mathfrak{F}$ and note that the language of $\mathfrak{F}$ is usually clear from context.

For every $\Psi \subseteq \mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ , we define a model $\mathcal {G}_\Psi ({\mathfrak{M}})$ for language ${\mathcal {L}}_{\Psi }$ to be the model with universe $M^d$ in which the interpretation of every $R\in \Psi $ is $R_{{\mathfrak{M}}d}$ , i.e.,

(11) $$ \begin{align} \mathcal{G}_\Psi({\mathfrak{M}})=\langle M^d,R_{{\mathfrak{M}}d}\rangle_{R\in\Psi}. \end{align} $$

To express that two models ${\mathfrak{M}}$ and ${\mathfrak{N}}$ for possibly different languages are the same, we introduce the following notation: we write ${\mathfrak{M}}\doteq {\mathfrak{N}}$ iff their universes are the same and there is a one-to-one correspondence between their languages such that the corresponding relation symbols and function symbols have the same interpretations in the two models.

Remark 6.6. For arbitrary models ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ , and (a possibly infinite) subset $\Psi $ of $\mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ , we have that $\mathcal {G}_{\Psi }({\mathfrak{M}}_1)$ and $\mathcal {G}_{\Psi }({\mathfrak{M}}_2)$ are elementarily equivalent whenever ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are elementarily equivalent. This is because, using the formula parts of the symbols of $\Psi $ , one can define a translation $\mathsf {Tr}$ from the common language of $\mathcal {G}_{\Psi }({\mathfrak{M}}_1)$ and $\mathcal {G}_{\Psi }({\mathfrak{M}}_2)$ to the common language of ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ , and it can be shown by formula induction that

$$ \begin{align*} \mathcal{G}_{\Psi}({\mathfrak{M}})\models \sigma \iff {\mathfrak{M}} \models \mathsf{Tr}(\sigma) \end{align*} $$

holds for every model ${\mathfrak{M}}$ and sentence $\sigma $ of common language of ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ .

6.3 Extending automorphisms from one geometry to another

In this section, we establish the conditions under which affine automorphisms of certain coordinate geometries are also automorphisms of their finite extensions. In particular, if $\mathcal {G}_{\Psi }(\mathfrak{F})$ is a coordinate geometry for some finite $\Psi \subseteq \mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ , and $R\in \mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ , Lemma 6.11 establishes conditions under which every affine automorphism of $\mathcal {G}_{\Psi }(\mathfrak{F})$ must also be an affine automorphism of $\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F})$ .

The following simple proposition is well-known from linear algebra. We restate it here to fix notation, and then re-express the concepts embedded within it in a form suitable for our purposes.

We now introduce various formulas, in the language ${\mathcal {L}}$ of $\mathfrak{F}$ , needed to capture the concepts expressed in Proposition 6.8.Footnote ⁹ These are:

• ι: Formula $\iota \in \mathit {Fm}_{{\mathcal {L}}}$ has $d^2+d$ free variables and expresses the idea that vectors $(\vec {{x}}_{1}-\vec {{x}}_{0}),\ldots ,(\vec {{x}}_{d}-\vec {{x}}_{0})$ are linearly independent:

  $$\begin{align*}\begin {aligned} \iota (\hat {x}_0,\hat {x}_{1}, & \ldots ,\hat {x}_{d}) \ \stackrel {\textsf { def}}{=} \ \\ & \left [ \forall \lambda _1\ldots \forall \lambda _d \left ( \textstyle \sum _{i=1}^d\,\lambda _i(\vec {{x}}_{i}-\vec {{x}}_{0})=0\ \rightarrow \ \textstyle \bigwedge _{i=1}^d \lambda _i=0 \right ) \right ]. \end {aligned}\end{align*}$$

• θ: Taken together with $\iota $ , formula $\theta \in \mathit {Fm}_{{\mathcal {L}}}$ (with $d^2+3d$ free variables) expresses the idea that affine map $A_{\vec {e}_0:\vec {\boldsymbol {e}}}$ takes vector $\vec {x}=\langle x_1,\ldots ,x_d\rangle $ to vector $\vec {y}=\langle y_1,\ldots ,y_d\rangle $ :

  $$\begin{align*}\theta (\hat {e}_{0},\hat {e}_{1},\ldots ,\hat {e}_{d},\hat {x},\hat {y}) \ \stackrel {\textsf { def}}{=} \ \left [ \vec {y}=\vec {{e}}_{0}+\textstyle \sum _{i=1}^d\, x_i\cdot (\vec {{e}}_{i}-\vec {{e}}_{0})\right ].\end{align*}$$

• θ _R : For each $R=\langle \varrho ,n\rangle \in \mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ , we define a formula $\theta _{R}$ which expressesFootnote ¹⁰ in the language of $\mathfrak{F}$ that $A_{\vec {e}_0:\vec {\boldsymbol {e}}}$ exists and respects R:

  $$ \begin{align*} \begin{aligned} \theta_{R} & (\hat{e}_{0},\hat{e}_{1},\ldots,\hat{e}_{d}) \ \ \stackrel{\textsf{ def}}{=} \ \ \iota(\hat{e}_0,\hat{e}_1, \ldots, \hat{e}_{d}) \\ & \land \ \left[ \ \forall \hat{x}_{1}\ldots\forall \hat{x}_{n}\forall\hat{y}_{1}\ldots\forall\hat{y}_{n} \right. \\ & \quad \left. \left( \left( \textstyle\bigwedge_{i=1}^{n}\, \theta(\hat{e}_{0}, \hat{e}_{1}, \ldots, \hat{e}_{d}, \hat{x}_{i}, \hat{y}_{i}) \right) \rightarrow\ (\varrho(\hat{x}_{1}, \ldots, \hat{x}_{n})\ \leftrightarrow\ \varrho(\hat{y}_{1}, \ldots, \hat{y}_{n}))\right) \ \right]. \end{aligned} \end{align*} $$

• θ _Ψ: Finally, if $\Psi $ is a finite subset of $\mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ , we define

  $$ \begin{align*} \theta_\Psi(\hat{e}_{0},\hat{e}_{1},\ldots,\hat{e}_{d}) \ \ \stackrel{\textsf{ def}}{=} \ \ \bigwedge_{R\in\Psi}\, \theta_{R}(\hat{e}_{0},\hat{e}_{1},\ldots,\hat{e}_{d}). \end{align*} $$

6.5 Proof of Theorem 4.2

We are now in a position to complete the proof of Theorem 4.2.

Proof. (i) $\Rightarrow $ (ii): Since $\mathsf {R}$ is definable in $\mathcal {G}$ it must be closed under automorphisms of $\mathcal {G}$ . So to prove (ii), it is enough to show that $\mathsf {R}$ is definable over $\mathfrak{F}$ . To do this, we note that it’s possible to define a translation $\mathsf {Tr}$ from the language of $\mathcal {G}$ to the language of $\mathfrak{F}$ and prove (see [Reference Madarász, Stannett and Székely25] for details) that, for every formula $\varphi ({\boldsymbol {\mathit {v}}}_1,\ldots ,{\boldsymbol {\mathit {v}}}_n)$ in the language of $\mathcal {G}$ and points $\vec {p}_1, \dots , \vec {p}_n$ , we have

(14) $$ \begin{align} \mathcal{G}\models \varphi [\vec{p}_1,\dots,\vec{p}_n] \iff \mathfrak{F} \models \mathsf{Tr}(\varphi)[\hat{p}_1,\dots,\hat{p}_n]. \end{align} $$

Now let $\varrho $ be the formula defining $\mathsf {R}$ in $\mathcal {G}$ . Then, by (14), $\mathcal {G}\models \varrho [\vec {p}_1,\dots ,\vec {p}_n]$ iff $\mathfrak{F}\models \mathsf {Tr}(\varrho )[\hat {p}_1,\dots ,\hat {p}_n]$ , i.e., $\mathsf {Tr}(\varrho )$ defines $\widehat {\mathsf {R}}$ in $\mathfrak{F}$ , and hence $\mathsf {R}$ is definable over $\mathfrak{F}$ as stated.

(ii) $\Rightarrow $ (iii): Trivial as affine automorphisms are also automorphisms.

(iii) $\Rightarrow $ (i): Assume that $\mathsf {R}$ is definable over $\mathfrak{F}$ and closed under affine automorphisms of $\mathcal {G}$ . We have to prove that $\mathsf {R}$ is definable in $\mathcal {G}$ . As affine automorphisms of $\mathcal {G}$ form a group under composition, they respect $\mathsf {R}$ .

Let ${\mathcal {L}}$ be the language of $\mathfrak{F}$ , let $R\in \mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ be the relation symbol for which $\mathsf {R}=R_{\mathfrak{F}d}$ , and let $\Psi $ be a finite subset of $\mathcal {R}\mathit {Symb}_{{\mathcal {L}}}$ containing $\langle \mathsf {\kappa }, 3 \rangle $ , such that $\langle \mathcal {G}, \mathsf {K} \rangle \doteq \mathcal {G}_{\Psi }(\mathfrak{F})$ . Such R and $\Psi $ exist by Remarks 6.2, 6.3 and 6.5. Since $\Psi $ contains $\langle \kappa ,3\rangle $ , and $\langle \kappa ,3\rangle _{{\mathfrak{F}}{d}}$ coincides with $\mathsf {K}$ , $\mathcal {G}_{\Psi }(\mathfrak{F})$ is a coordinate geometry as well, and it follows easily that the definable relations, automorphisms and affine automorphisms of $\mathcal {G}$ and $\mathcal {G}_{\Psi }(\mathfrak{F})$ coincide. Hence affine automorphisms of $\mathcal {G}_{\Psi }(\mathfrak{F})$ respect $\mathsf {R}=R_{\mathfrak{F}d}$ , and to prove that $\mathsf {R}=R_{\mathfrak{F}d}$ is definable in $\mathcal {G}$ , it is enough to prove that $R_{\mathfrak{F}d}$ is definable in $\mathcal {G}_{\Psi }(\mathfrak{F})$ . Clearly, the expansion of $\mathcal {G}_{\Psi }(\mathfrak{F})$ with $R_{\mathfrak{F}d}$ to the language ${\mathcal {L}}_{\Psi \cup \{R\}}$ is $\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F})$ and every affine automorphism of $\mathcal {G}_{\Psi }(\mathfrak{F})$ is an affine automorphism of $\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F})$ , so by Lemma 6.11, $\mathfrak{F}\models (\theta _{\Psi }\rightarrow \theta _{R})$ . We will apply Theorem 6.12 to obtain a proof that $R_{\mathfrak{F}d}$ is definable in $\mathcal {G}_{\Psi }(\mathfrak{F})$ , namely, we will prove that for any ultrafilter $\mathcal {U}$ , every automorphism of $(\mathcal {G}_{\Psi }(\mathfrak{F}))^{\mathcal {U}}$ is an automorphism of $(\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F}))^{\mathcal {U}}$ . To prove this, let $\mathcal {U}$ be any ultrafilter (on any index set). Applying Lemma 6.15 (twice), we observe that the function $\vec {\boldsymbol{\pi }}$ is both an isomorphism from $(\mathcal {G}_{\Psi }(\mathfrak{F}))^{\mathcal {U}}$ to $\mathcal {G}_{\Psi }(\mathfrak{F}^{\mathcal {U}})$ , and also an isomorphism from $(\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F}))^{\mathcal {U}}$ to $\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F}^{\mathcal {U}})$ .

It is therefore enough to prove that every automorphism of $\mathcal {G}_{\Psi }(\mathfrak{F}^{\mathcal {U}})$ is an automorphism of $\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F}^{\,\mathcal {U}})$ . To this end, we observe that the model $\mathcal {G}_{\Psi }(\mathfrak{F}^{\,\mathcal {U}})$ is a coordinate geometry over $\mathfrak{F}^{\,\mathcal {U}}$ because $\Psi $ contains $\langle \kappa ,3\rangle $ and relation $\langle \mathsf {\kappa },3\rangle _{{\mathfrak{F}^{\mathcal {U}}} d}$ of $\mathcal {G}_{\Psi }(\mathfrak{F}^{\,\mathcal {U}})$ coincides with the relation $\mathsf {K}$ of the affine geometry over $\mathfrak{F}^{\mathcal {U}}$ (cf. Remark 6.3).

Moreover, we know that $\mathfrak{F}^{\mathcal {U}}\models (\theta _{\Psi }\rightarrow \theta _{R})$ because $\mathfrak{F} \models (\theta _{\Psi }\rightarrow \theta _{R}) $ . By Lemma 6.11, this implies that every affine automorphism of $\mathcal {G}_{\Psi }(\mathfrak{F}^{\mathcal {U}})$ is an affine automorphism of $\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F}^{\mathcal {U}})$ as well.

Assume therefore that $\alpha $ is an arbitrary automorphism of $\mathcal {G}_{\Psi }(\mathfrak{F}^{\mathcal {U}})$ . By Proposition 6.1, we can write $\alpha = A\circ \widetilde {\varphi }$ for some affine automorphism A of $\mathcal {G}_{\Psi }(\mathfrak{F}^{\,\mathcal {U}})$ and some automorphism $\varphi $ of $\mathfrak{F}^{\,\mathcal {U}}$ . Since A is also an affine automorphism of $\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F}^{\,\mathcal {U}})$ , applying Proposition 6.1 once again shows that $\alpha = A\circ \widetilde {\varphi }$ is an automorphism of $\mathcal {G}_{\Psi \cup \{R\}}(\mathfrak{F}^{\,\mathcal {U}})$ as required.

7.1 Assumptions

Throughout, we restrict attention to d-dimensional coordinate geometries over an ordered field $\mathfrak{F}=\langle F,+,\cdot ,0,1,\leq \rangle $ where, as always, we assume that $d \geq 2$ .

In particular, the Euclidean scalar product $\boldsymbol {\cdot }$ of vectors $\vec {p}, \vec {q} \in F^d$ is defined, as usual, by

$$\begin{align*}\vec{p}\boldsymbol{\cdot}\vec{q} \ \stackrel{\textsf{ def}}{=} \ p_1q_1+\cdots+p_dq_d, \end{align*}$$

and their Minkowski product ( $\unicode{x27D0} $ ) by

$$\begin{align*}\vec{p}\unicode{x27D0} \vec{q} \ \stackrel{\textsf{ def}}{=} \ p_1q_1-p_2q_2-\cdots-p_dq_d. \end{align*}$$

The squared distance functions corresponding to the Euclidean and Minkowski products will be denoted by $\mathsf {d}^2$ and $\mathsf {d}_\mu ^2$ , respectively.

7.2 Unit-free spacetimes

Thinking of $F^d$ as the set of spacetime points, we regard the first component, $p_1$ , of any vector $\vec {p} = \langle p_1,p_2,\dots ,p_d \rangle $ as its time component, and the rest, $(p_2,\dots ,p_d)$ as its spatial component. This is reflected in the following terminology.

On the set of points $F^d$ , the binary relations ${\mathsf {S}}$ of absolute simultaneity, $\mathsf {Rest}$ of being at rest and $\boldsymbol {\unicode{x3bb} }$ of lightlike relatedness are defined for $\vec {p}, \vec {q} \in F^d$ by

$$ \begin{align*} \vec{p}\,{\mathsf{S}}\,\vec{q} &\ \stackrel{\textsf{ def}}{\iff}\ p_1=q_1,\\ \vec{p}\,\mathsf{Rest}\,\vec{q} &\ \stackrel{\textsf{ def}}{\iff}\ ( p_2,\ldots,p_d) = (q_2,\ldots,q_d), \\ \vec{p}\,\boldsymbol{\unicode{x3bb}}\,\vec{q} & \ \stackrel{\textsf{ def}}{\iff}\ (p_1-q_1)^2=(p_2-q_2)^2+\cdots+(p_d-q_d)^2. \end{align*} $$

We note that

(15) $$ \begin{align} \vec{p}\,\boldsymbol{\unicode{x3bb}}\,\vec{q} \quad\Longleftrightarrow\quad \mathsf{d}_{\mu}^2\left(\vec{p},\vec{q}\,\right)=0. \end{align} $$

On the set of points $F^d$ , the $4$ -ary relations $\boldsymbol {{ \cong }}_{}$ of (Euclidean) congruence, $\boldsymbol {{ \cong }}_{\mu }$ of Minkowski congruence, and $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ of congruence on simultaneity are defined for $\vec {p},\vec {q},\vec {r},\vec {s}\in F^d$ by

$$ \begin{align*} (\vec{p},\vec{q}\,)\,\boldsymbol{{ \cong}}_{}\,(\vec{r},\vec{s}\,) & \stackrel{\textsf{ def}}{\iff} \mathsf{d}^2(\vec{p},\vec{q}\,)=\mathsf{d}^2(\vec{r},\vec{s}\,), \\ (\vec{p},\vec{q}\,)\,\boldsymbol{{ \cong}}_{\mu}\,(\vec{r},\vec{s}\,) & \stackrel{\textsf{ def}}{\iff} \mathsf{d}_{\mu}^2\left(\vec{p},\vec{q}\,\right)=\mathsf{d}_{\mu}^2\left(\vec{r},\vec{s}\,\right), \\ (\vec{p},\vec{q}\,)\,\boldsymbol{{ \cong}}_{{\mathsf{S}}}\,(\vec{r},\vec{s}\,) & \stackrel{\textsf{ def}}{\iff} (\vec{p},\vec{q}\,)\,\boldsymbol{{ \cong}}_{}\,(\vec{r},\vec{s}\,)\ \land\ \vec{p}\,{\mathsf{S}}\,\vec{q}\ \land\ \vec{r}\,{\mathsf{S}}\,\vec{s.} \end{align*} $$

When $(\vec {p},\vec {q}\,)\,\boldsymbol {{ \cong }}_{}\,(\vec {r},\vec {s}\,)$ holds, we say that $(\vec {p},\vec {q}\,)$ and $(\vec {r},\vec {s}\,)$ are congruent segments, and when $(\vec {p},\vec {q}\,)\,\boldsymbol {{ \cong }}_{{\mathsf {S}}}\,(\vec {r},\vec {s}\,)$ holds, we say that they are spatially congruent.

Note that the relations $\boldsymbol {\unicode{x3bb} }$ , ${\mathsf {S}}$ , $\mathsf {Rest}$ , $\boldsymbol {{ \cong }}_{}$ , $\boldsymbol {{ \cong }}_{\mu }$ and $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ are all definable over $\mathfrak{F}$ .

We also define a ternary relation $\boldsymbol {\unicode{x3b4} }$ on $F^d$ by

$$\begin{align*}\boldsymbol{\unicode{x3b4}}(\vec{p},\vec{q},\vec{r}\,)\ \stackrel{\textsf{ def}}{\iff}\ \vec{p}\,{\mathsf{S}}\,\vec{q}\ \land\ \vec{p}\,\boldsymbol{\unicode{x3bb}}\,\vec{r}. \end{align*}$$

Notice that $\langle F^d,{\mathsf {S}},\boldsymbol {\unicode{x3bb} }\rangle $ and $\langle F^d,\boldsymbol {\unicode{x3b4} }\rangle $ are definitionally equivalent, i.e.,

(16) $$ \begin{align} \mathsf{Conc}\,\langle F^d,{\mathsf{S}},\boldsymbol{\unicode{x3bb}}\rangle\, = \mathsf{Conc}\,\langle F^d,\boldsymbol{\unicode{x3b4}}\rangle\, \end{align} $$

because $\boldsymbol {\unicode{x3b4} }$ is defined in terms of ${\mathsf {S}}$ and $\boldsymbol {\unicode{x3bb} }$ , and it is not difficult to prove that both ${\mathsf {S}}$ and $\boldsymbol {\unicode{x3bb} }$ can be defined in terms of $\boldsymbol {\unicode{x3b4} }$ by $ \vec {p}\,{\mathsf {S}}\,\vec {q}\; \Leftrightarrow \; \exists \vec {r}\, \boldsymbol {\unicode{x3b4} }(\vec {p},\vec {q},\vec {r}\,)$ and $\vec {p}\,\boldsymbol {\unicode{x3bb} }\,\vec {r}\; \Leftrightarrow \; \exists \vec {q}\, \boldsymbol {\unicode{x3b4} }(\vec {p},\vec {q},\vec {r}\,)$ .

All spacetimes and geometries considered in this section are unit-free.Footnote ¹² We now recall the definition of (d-dimensional) ordered affine geometry, and introduce Euclidean geometry and, respectively, Relativistic, Minkowski, Galilean, Newtonian and Late ClassicalFootnote ¹³ spacetimes over $\mathfrak{F}$ :

$$ \begin{align*} \mathcal{OA}\mathit{ff} &\ \ \stackrel{\textsf{ def}}{=} \ \ \langle F^d, {\mathsf{Bw}} \rangle, \\ \mathcal{E}\!\mathit{ucl} &\ \ \stackrel{\textsf{ def}}{=} \ \ \langle F^d,\boldsymbol{{ \cong}}_{} ,{\mathsf{Bw}}\rangle,\\ \mathcal{R}\mathit{el} &\ \ \stackrel{\textsf{ def}}{=} \ \ \langle F^d,\boldsymbol{\unicode{x3bb}},{\mathsf{Bw}} \rangle,\\ \mathcal{M}\hspace{-1pt}\mathit{ink} &\ \ \stackrel{\textsf{ def}}{=} \ \ \langle F^d,\boldsymbol{{ \cong}}_{\mu},{\mathsf{Bw}}\rangle,\\ \mathcal{G}\mathit{al} &\ \ \stackrel{\textsf{ def}}{=} \ \ \langle F^d,\boldsymbol{{ \cong}}_{{\mathsf{S}}},{\mathsf{Bw}}\rangle,\\ \mathcal{N}\!\mathit{ewt} &\ \ \stackrel{\textsf{ def}}{=} \ \ \langle \mathcal{G}\mathit{al},\mathsf{Rest} \rangle,\\ \mathcal{LC}\mathit{lass} &\ \ \stackrel{\textsf{ def}}{=} \ \ \langle \mathcal{G}\mathit{al}, \boldsymbol{\unicode{x3bb}} \rangle. \end{align*} $$

Note that these are all FFD coordinate geometries over $\mathfrak{F}$ .

Given the close historical relationship between Relativistic spacetime and Minkowski spacetime, one would naturally expect to find, as shown in Figure 4, that $\mathcal {M}\hspace {-1pt}\mathit {ink}$ and $\mathcal {R}\mathit {el}$ are conceptually identical. Both this relationship and the other subset relationships shown in Figure 4 are summarised in the following theorem.

Theorem 7.3. The subset relations between $\mathsf {Conc}\,\mathcal {R}\mathit {el}\,$ , $\mathsf {Conc}\,\mathcal {M}\hspace {-1pt}\mathit {ink}\,$ , $\mathsf {Conc}\,\mathcal {E}\!\mathit {ucl}\,$ , $\mathsf {Conc}\,\mathcal {G}\mathit {al}\,$ , $\mathsf {Conc}\,\mathcal {N}\!\mathit {ewt}\,$ , $\mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ and $\mathsf {Conc}\,\mathcal {OA}\mathit {ff}\,$ are as shown in Figure 4. In particular, the concepts ${\mathsf {S}}$ , $\mathsf {Rest}$ , $\boldsymbol {\unicode{x3bb} }$ , $\boldsymbol {{ \cong }}_{}$ , $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ , $\boldsymbol {{ \cong }}_{\mu }$ and $\boldsymbol {\unicode{x3b4} }$ allow us to distinguish between different geometries, as shown explicitly in Table 1. In more detail:

1. (i) $\mathsf {Conc}\,\mathcal {R}\mathit {el}\,=\mathsf {Conc}\,\mathcal {M}\hspace {-1pt}\mathit {ink}\,$ , i.e., $\mathcal {R}\mathit {el}$ and $\mathcal {M}\hspace {-1pt}\mathit {ink}$ are definitionally equivalent.

2. (ii) $\mathsf {Conc}\,\mathcal {G}\mathit {al}\,\subseteq \mathsf {Conc}\,\mathcal {N}\!\mathit {ewt}\,$ .

3. (iii) $\mathsf {Conc}\,\mathcal {OA}\mathit {ff}\,\subseteq \left (\mathsf {Conc}\,\mathcal {E}\!\mathit {ucl}\,\cap \mathsf {Conc}\,\mathcal {R}\mathit {el}\,\cap \mathsf {Conc}\,\mathcal {G}\mathit {al}\,\right )$ .

4. (iv) $\left (\mathsf {Conc}\,\mathcal {E}\!\mathit {ucl}\,\cup \mathsf {Conc}\,\mathcal {R}\mathit {el}\,\cup \mathsf {Conc}\,\mathcal {N}\!\mathit {ewt}\,\right )\subseteq \mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ .

5. (v) Table 1 correctly identifies which of the cited geometries do and do not contain the concepts ${\mathsf {S}}$ , $\mathsf {Rest}$ , $\boldsymbol {\unicode{x3bb} }$ , $\boldsymbol {{ \cong }}_{}$ , $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ , $\boldsymbol {{ \cong }}_{\mu }$ and $\boldsymbol {\unicode{x3b4} }$ .

Table 1

The geometries shown in Figure 4 which do and do not contain the concepts ${\mathsf {S}}$ , $\mathsf {Rest}$ , $\boldsymbol {\unicode{x3bb} }$ , $\boldsymbol {{ \cong }}_{}$ , $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ , $\boldsymbol {{ \cong }}_{\mu }$ and $\boldsymbol {\unicode{x3b4} }$

Table 1 Long description

The table consists of 6 columns and 8 rows.

Column headers from left to right are:

* Empty header for row labels

* Conc E u c l

* Conc R e l

* Conc G a l

* Conc N e w t

* Conc L C l a s s

Row labels from top to bottom are:

* Congruence symbol

* Congruence sub mu symbol

* Bold lambda symbol

* Congruence sub S symbol

* S symbol

* Rest symbol

* Bold delta symbol

Data mapping:

* Congruence symbol: In Conc E u c l and Conc L C l a s s. Not in others.

* Congruence sub mu symbol: In Conc R e l and Conc L C l a s s. Not in others.

* Bold lambda symbol: In Conc R e l and Conc L C l a s s. Not in others.

* Congruence sub S symbol: In Conc G a l, Conc N e w t, and Conc L C l a s s. Not in others.

* S symbol: In Conc G a l, Conc N e w t, and Conc L C l a s s. Not in others.

* Rest symbol: In Conc N e w t and Conc L C l a s s. Not in others.

* Bold delta symbol: In Conc L C l a s s only. Not in others.

Cells containing the ‘In’ symbol are shaded green, while cells containing the ‘Not in’ symbol are shaded red.

Note: This information confirms which geometries are definitionally distinct from one another.

The next theorem tells us that, with just a few exceptions, if we expand any of the geometries we have been studying with any of the concepts from $\{\boldsymbol {{ \cong }}_{},\boldsymbol {{ \cong }}_{\mu },\boldsymbol {\unicode{x3bb} }, \boldsymbol {{ \cong }}_{{\mathsf {S}}}, {\mathsf {S}}, \mathsf {Rest}, \boldsymbol {\unicode{x3b4} }\}$ that are not concepts of the geometry in question, the result is definitionally equivalent to $\mathcal {LC}\mathit {lass}$ .

7.3 Affine automorphisms and similarities

In this section, we show that the affine automorphisms of various geometries can be characterised as similarity transformations. We begin by defining the various types of similarities we will be using.

Definition 7.5 ( $\otimes $ -similarities)

Suppose $A\colon F^d\to F^d$ is a function and $\otimes $ is a product on $F^d$ . As a natural generalization of the notion to arbitrary ordered fields $\mathfrak{F}$ , we call A a $\otimes $ -similarity iff it is an affine transformation that respects squared $\otimes $ -distances up to scaling, i.e., there is $a\in F$ such that

(17) $$ \begin{align} \mathsf{d}_{\otimes}^2\left(A(\vec{p}\,),A(\vec{q}\,)\,\right)=a\cdot \mathsf{d}_{\otimes}^2\left(\vec{p},\vec{q}\,\right) \quad\text{for every } \vec{p},\vec{q}\in F^d. \end{align} $$

The value a is called the squared scale factor (or square-factor).

In particular, if $\otimes $ is the Euclidean scalar product, we call A a Euclidean similarity, and if $\otimes $ is the Minkowski product, we call A a Poincaré similarity. When the square-factor a is $1$ , these similarities are traditionally called Euclidean and Poincaré transformations.

If $\mathfrak{F}$ is a Euclidean field and A a Euclidean similarity with square-factor a, then A is the composition of a Euclidean transformation with the uniform scaling $\vec {p}\mapsto \sqrt {a}\vec {p}$ with scale factor $\sqrt {a}$ . For non-Euclidean fields this need not hold, e.g., the map $L(t,x)=(t+x,t-x)$ is a Euclidean similarity with square-factor 2 which cannot be decomposed in this way over the field of rationals. We also note that, for any ordered field, the composition of a Euclidean transformation with a uniform scaling of factor $\lambda \neq 0$ is a Euclidean similarity with square-factor $\lambda ^2$ . Analogous statements hold for Poincaré transformations and similarities when $d>2$ .

The sets of Euclidean, Poincaré and trivial Euclidean similarities are denoted by $\mathsf {EuclSim}$ , $\mathsf {PoiSim}$ and $\mathsf {TrivEuclSim}$ , respectively.

The following key theorem shows that these similarities are precisely the affine automorphisms of their associated spacetimes. It is this result which makes it easy for us to check for definitional equivalence between spacetimes.Footnote ¹⁶

Because $\mathsf {Aut}\; {\mathfrak{M}}$ is always a group under composition for any model ${\mathfrak{M}}$ , it immediately follows that all of these similarity sets are groups.

The results collected in the following lemma can all be proven using standard linear algebraic methods; full details are provided in [Reference Madarász, Stannett and Székely26].

7.4 Proof sketches for Theorem 7.8

Theorem 7.8 identifies which similarity groups act as automorphism groups for which geometries. We provide here sketch versions of the proofs, and note that full details are provided in [Reference Madarász, Stannett and Székely26].

Proof of (ii) $(\mathsf {AffAut}\,\mathcal {R}\mathit {el}=\mathsf {AffAut}\,\mathcal {M}\hspace {-1pt}\mathit {ink}=\mathsf {PoiSim})$

It follows easily from the definitions that $\mathsf {PoiSim}\subseteq \mathsf {AffAut}\,\mathcal {M}\hspace {-1pt}\mathit {ink}$ , and we know that $\mathsf {AffAut}\,\mathcal {M}\hspace {-1pt}\mathit {ink}\subseteq \mathsf {AffAut}\,\mathcal {R}\mathit {el}$ because $\boldsymbol {\unicode{x3bb} }$ is definable in terms of $\boldsymbol {{ \cong }}_{\mu }$ (as $\vec {p}\,\boldsymbol {\unicode{x3bb} }\,\vec {q} \Longleftrightarrow (\vec {p},\vec {q}\,)\boldsymbol {{ \cong }}_{\mu }(\vec {p},\vec {p}\,)$ ). This shows that

$$\begin{align*}\mathsf{PoiSim} \subseteq \mathsf{AffAut}\,\mathcal{M}\hspace{-1pt}\mathit{ink} \subseteq \mathsf{AffAut}\,\mathcal{R}\mathit{el}, \end{align*}$$

so we only need to show the reverse inclusion, $\mathsf {AffAut}\,\mathcal {R}\mathit {el}\subseteq \mathsf {PoiSim}$ . For the case $d>2$ , this follows easily from Lester [Reference Lester24]. In [Reference Madarász, Stannett and Székely26] we provide a self-contained proof which works for the case when $d=2$ as well.

Proof of (iii) $\mathsf {AffAut}\,\mathcal {LC}\mathit {lass}=\mathsf {TrivEuclSim}$

By definition, $\mathcal {LC}\mathit {lass} \doteq \langle \mathcal {R}\mathit {el}, \boldsymbol {{ \cong }}_{{\mathsf {S}}} \rangle $ , and we know that ${\mathsf {S}}$ is definable from $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ (as $\vec p \,{\mathsf {S}} \vec q \Longleftrightarrow (\vec p,\vec q\,) \boldsymbol {{ \cong }}_{{\mathsf {S}}} (\vec p, \vec q\,)$ ). It follows from this, from Lemma 7.10(ii) and Theorem 7.8(ii) that

$$\begin{align*}\mathsf{AffAut}\,\mathcal{LC}\mathit{lass} = \mathsf{AffAut}\,\mathcal{R}\mathit{el} \cap \mathsf{Aut}\; \langle F^d, \boldsymbol{{ \cong}}_{{\mathsf{S}}} \rangle \subseteq \mathsf{PoiSim} \cap \langle F^d, {\mathsf{S}} \rangle = \mathsf{TrivEuclSim}. \end{align*}$$

On the other hand, by Lemma 7.10(i) and item (i) we have

$$\begin{align*}\mathsf{TrivEuclSim} = \mathsf{EuclSim} \cap \mathsf{Aut}\; \langle F^d, {\mathsf{S}}\rangle = \mathsf{AffAut}\,\mathcal{E}\!\mathit{ucl} \cap \mathsf{Aut}\; \langle F^d, {\mathsf{S}} \rangle, \end{align*}$$

whence elements of $\mathsf {TrivEuclSim}$ respect both $\boldsymbol {{ \cong }}_{}$ and ${\mathsf {S}}$ , and hence also $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ (which is defined in terms of them). So $\mathsf {TrivEuclSim} \subseteq \mathsf {Aut}\; \langle F^d, \boldsymbol {{ \cong }}_{{\mathsf {S}}} \rangle $ . By Lemma 7.10(ii), $\mathsf {TrivEuclSim}\subseteq \mathsf {PoiSim}$ . It therefore follows, as required, that

$$\begin{align*}\mathsf{TrivEuclSim} \subseteq \mathsf{PoiSim} \cap \mathsf{Aut}\; \langle F^d, \boldsymbol{{ \cong}}_{{\mathsf{S}}} \rangle = \mathsf{AffAut}\,\mathcal{R}\mathit{el} \cap \mathsf{Aut}\; \langle F^d, \boldsymbol{{ \cong}}_{{\mathsf{S}}} \rangle = \mathsf{AffAut}\,\mathcal{LC}\mathit{lass}. \end{align*}$$

7.5 Proof sketches for Theorems 7.3 and 7.4

In the proofs of Theorems 7.3 and 7.4, we use Corollary 4.6(ii), according to which, for any FFD coordinate geometries $\mathcal {G}$ and $\mathcal {G}'$ over $\mathfrak{F}$ ,

$$\begin{align*}\mathsf{Conc}\,\mathcal{G}\,=\mathsf{Conc}\,\mathcal{G}'\,\ \Longleftrightarrow\ \mathsf{AffAut}\,\mathcal{G}=\mathsf{AffAut}\,\mathcal{G}', \end{align*}$$

together with Theorem 4.2 which implies that for any FFD coordinate geometry $\mathcal {G}$ and relation $\mathsf {R}$ on $F^d$ which is definable over $\mathfrak{F}$ ,

$$\begin{align*}\mathsf{R}\in\mathsf{Conc}\,\mathcal{G}\,\ \Longleftrightarrow\ \mathsf{R} \text{ is closed under the elements of } \mathsf{AffAut}\,\mathcal{G}. \end{align*}$$

The proofs as to which concept sets are contained within one another are generally straightforward (full details are provided in [Reference Madarász, Stannett and Székely26]). For example:

Demonstrating that Table 1—which shows which specific concepts are included in the various geometries considered above—is correct is only slightly more involved. We have already shown (Theorem 7.3(i)) that

$$\begin{align*}\mathsf{Conc}\,\langle F^d,\boldsymbol{\unicode{x3bb}},{\mathsf{Bw}}\rangle\, \ \stackrel{\textsf{ def}}{=} \ \mathsf{Conc}\,\mathcal{R}\mathit{el}\, = \mathsf{Conc}\,\mathcal{M}\hspace{-1pt}\mathit{ink}\, \ \stackrel{\textsf{ def}}{=} \ \mathsf{Conc}\,\langle F^d,\boldsymbol{{ \cong}}_{\mu},{\mathsf{Bw}}\rangle\,. \end{align*}$$

Therefore, for any coordinate geometry $\mathcal {G}$ over $\mathfrak{F}$ , we have that

(18) $$ \begin{align} \boldsymbol{\unicode{x3bb}}\in\mathsf{Conc}\,\mathcal{G}\,\quad\Longleftrightarrow\quad \boldsymbol{{ \cong}}_{\mu}\in\mathsf{Conc}\,\mathcal{G}\,. \end{align} $$

Similarly, by (16), we have

(19) $$ \begin{align} \boldsymbol{\unicode{x3b4}}\in\mathsf{Conc}\,\mathcal{G}\,\quad \Longleftrightarrow\quad \{ \boldsymbol{\unicode{x3bb}},{\mathsf{S}} \} \subseteq \mathsf{Conc}\,\mathcal{G}\,. \end{align} $$

Furthermore, since ${\mathsf {S}}$ is definable in terms of $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ (as $\vec {p}\,{\mathsf {S}}\, \vec {q}\; \Leftrightarrow \; (\vec {p},\vec {q}\,)\boldsymbol {{ \cong }}_{{\mathsf {S}}}(\vec {p},\vec {q}\,)$ ), we have that

(20) $$ \begin{align} \left(\boldsymbol{{ \cong}}_{{\mathsf{S}}}\in\mathsf{Conc}\,\mathcal{G}\,\ \Longrightarrow\ {\mathsf{S}}\in \mathsf{Conc}\,\mathcal{G}\,\right)\ \text{ and }\ \left({\mathsf{S}}\not\in\mathsf{Conc}\,\mathcal{G}\,\ \Longrightarrow\ \boldsymbol{{ \cong}}_{{\mathsf{S}}}\not\in\mathsf{Conc}\,\mathcal{G}\,\right). \end{align} $$

We use these observations to prove the correctness of the table, one column at a time. For example, consider Table 1, column 1, which concerns $\mathcal {E}\!\mathit {ucl} \ \stackrel {\textsf { def}}{=} \ \langle F^d,\boldsymbol {{ \cong }}_{},{\mathsf {Bw}}\rangle $ . We need to prove that $\boldsymbol {{ \cong }}_{}$ is a concept—and that $\boldsymbol {{ \cong }}_{\mu }$ , $\boldsymbol {\unicode{x3bb} }$ , $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ , ${\mathsf {S}}$ , $\mathsf {Rest}$ and $\boldsymbol {\unicode{x3b4} }$ are not concepts—of $\mathcal {E}\!\mathit {ucl}$ . Clearly, $\boldsymbol {{ \cong }}_{}\in \mathsf {Conc}\,\mathcal {E}\!\mathit {ucl}\,$ . To prove that the other relations are not concepts, it is enough to find an automorphism of $\mathcal {E}\!\mathit {ucl}$ that does not respect them. Let $E\colon F^d\rightarrow F^d$ be the linear transformation that takes the unit vectors and , respectively, to and , and keeps the rest of the unit vectors fixed. It can be proven that E is a Euclidean similarity, hence by Theorem 7.8(i), E is an automorphism of $\mathcal {E}\!\mathit {ucl}$ . Transformation E respects neither $\mathsf {Rest}$ , ${\mathsf {S}}$ nor $\boldsymbol {\unicode{x3bb} }$ because , and all hold, but none of , , do. Therefore, $\mathsf {Rest},{\mathsf {S}},\boldsymbol {\unicode{x3bb} }\not \in \mathsf {Conc}\,\mathcal {E}\!\mathit {ucl}\,$ . By these and (18)–(20), we have that $\boldsymbol {{ \cong }}_{\mu }, \boldsymbol {{ \cong }}_{{\mathsf {S}}},\boldsymbol {\unicode{x3b4} } \not \in \mathsf {Conc}\,\mathcal {E}\!\mathit {ucl}\,$ . $\Box $

7.5.1 Proof sketch of Theorem 7.4

Since the proofs of parts (i) and (ii) overlap to some extent, we will prove them together, considering each possible choice of $\mathcal {G}$ in turn.

Proof (i) and (ii): Here we consider only those cases where $\mathsf {R} \in \{ \mathsf {Rest}, {\mathsf {S}}, \boldsymbol {\unicode{x3bb} }, \boldsymbol {{ \cong }}_{} \}$ because the claims follow for the other relations easily by (16), Theorem 7.3(i) and the fact that ${\mathsf {S}}$ is definable in terms of $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ . By Theorem 7.8(iii) (which says that $\mathsf {AffAut}\,\mathcal {LC}\mathit {lass}=\mathsf {TrivEuclSim}$ ) and by Corollary 4.6(ii), we have, for any FFD coordinate geometry $\mathcal {G}$ over $\mathfrak{F}$ and any relation $\mathsf {R}$ definable over $\mathfrak{F}$ , that

(21) $$ \begin{align} \mathsf{Conc}\,\langle \mathcal{G},\mathsf{R}\rangle\,=\mathsf{Conc}\,\mathcal{LC}\mathit{lass}\,\ \ \Longleftrightarrow\ \mathsf{AffAut}\,\langle \mathcal{G}, \mathsf{R}\rangle=\mathsf{TrivEuclSim}. \end{align} $$

Case $\mathcal {G} = \mathsf {Eucl}$ . We will prove that $\langle \mathcal {E}\!\mathit {ucl}, {\mathsf {S}}\rangle $ , $\langle \mathcal {E}\!\mathit {ucl}, \mathsf {Rest}\rangle $ and (if $d>2$ ) $\langle \mathcal {E}\!\mathit {ucl},\boldsymbol {\unicode{x3bb} }\rangle $ are all definitionally equivalent to $\mathcal {LC}\mathit {lass}$ . By Theorem 7.8(i), $\mathsf {AffAut}\,\mathcal {E}\!\mathit {ucl}=\mathsf {EuclSim}$ . By this and Lemma 7.10(i), we have that

(22) $$ \begin{align} \begin{aligned} \mathsf{AffAut}\,\langle \mathcal{E}\!\mathit{ucl},{\mathsf{S}}\rangle & = \mathsf{AffAut}\,\mathcal{E}\!\mathit{ucl}\cap \mathsf{Aut}\; \langle F^d,{\mathsf{S}}\rangle \\ & = \mathsf{EuclSim}\cap \mathsf{Aut}\; \langle F^d,{\mathsf{S}}\rangle = \mathsf{TrivEuclSim} \end{aligned} \end{align} $$

and likewise

(23) $$ \begin{align} \mathsf{AffAut}\,\langle \mathcal{E}\!\mathit{ucl},\mathsf{Rest}\rangle=\mathsf{EuclSim}\cap\mathsf{Aut}\; \langle F^d,\mathsf{Rest}\rangle=\mathsf{TrivEuclSim}. \end{align} $$

Similarly, by Lemma 7.10(iii), by $\mathcal {R}\mathit {el} \ \stackrel {\textsf { def}}{=} \ \langle F^d,\boldsymbol {\unicode{x3bb} },{\mathsf {Bw}}\rangle $ and by $\mathsf {AffAut}\,\mathcal {R}\mathit {el}= \mathsf {PoiSim}$ (Theorem 7.8(ii)), we have, when $d> 2$ ,

(24) $$ \begin{align} \begin{aligned} \mathsf{AffAut}\,\langle \mathcal{E}\!\mathit{ucl},\boldsymbol{\unicode{x3bb}}\rangle &= \mathsf{AffAut}\,\mathcal{E}\!\mathit{ucl}\cap\mathsf{AffAut}\,\mathcal{R}\mathit{el} \\ & = \mathsf{EuclSim}\cap\mathsf{PoiSim} = \mathsf{TrivEuclSim}. \end{aligned} \end{align} $$

Therefore, by (21)–(23) we have that $\mathsf {Conc}\,\langle \mathcal {E}\!\mathit {ucl},\mathsf {Rest}\rangle \, = \mathsf {Conc}\,\langle \mathcal {E}\!\mathit {ucl}, {\mathsf {S}}\rangle \, = \mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ . And by (21) and (24) we also have $ \mathsf {Conc}\,\langle \mathcal {E}\!\mathit {ucl},\boldsymbol {\unicode{x3bb} }\rangle \, = \mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ when $d>2$ .

The rest follow from these, e.g., $ \mathsf {Conc}\,\langle \mathcal {E}\!\mathit {ucl}, \boldsymbol {{ \cong }}_{{\mathsf {S}}}\rangle \, = \mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ because $\boldsymbol {{ \cong }}_{{\mathsf {S}}} \in \mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ , ${\mathsf {S}}$ is definable in terms of $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ and $\mathsf {Conc}\,\langle \mathcal {E}\!\mathit {ucl},{\mathsf {S}}\rangle \, = \mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ .

Case $\mathcal {G} =\mathcal {R}\mathit {el}$ : We need to show that $\langle \mathcal {R}\mathit {el},{\mathsf {S}}\rangle $ , $\langle \mathcal {R}\mathit {el},\mathsf {Rest}\rangle $ and (if $d>2$ ) $\langle \mathcal {R}\mathit {el}, \boldsymbol {{ \cong }}_{}\rangle $ are all definitionally equivalent to $\mathcal {LC}\mathit {lass}$ . By Theorem 7.8(ii), we have that $\mathsf {AffAut}\,\mathcal {R}\mathit {el}=\mathsf {PoiSim}$ . By this and Lemma 7.10(ii),

(25) $$ \begin{align} \mathsf{AffAut}\,\langle\mathcal{R}\mathit{el},{\mathsf{S}}\rangle & = \mathsf{PoiSim}\cap\mathsf{Aut}\; \langle F^d,{\mathsf{S}}\rangle = \mathsf{TrivEuclSim},\ \text{ and } \end{align} $$

(26) $$ \begin{align} \mathsf{AffAut}\,\langle\mathcal{R}\mathit{el},\mathsf{Rest}\rangle & = \mathsf{PoiSim}\cap\mathsf{Aut}\; \langle F^d,\mathsf{Rest}\rangle &~= \mathsf{TrivEuclSim}. \end{align} $$

The proof is analogous to that just given. By (21), (25) and (26), we have that $\mathsf {Conc}\,\langle \mathcal {R}\mathit {el},\mathsf {Rest}\rangle \,= \mathsf {Conc}\,\langle \mathcal {R}\mathit {el},{\mathsf {S}}\rangle \, = \mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ .

From the definitions of $\mathcal {E}\!\mathit {ucl}$ and $\mathcal {R}\mathit {el}$ , we have that $\langle \mathcal {R}\mathit {el},\boldsymbol {{ \cong }}_{}\rangle \doteq \langle \mathcal {E}\!\mathit {ucl},\boldsymbol {\unicode{x3bb} }\rangle $ . We have already seen that $\mathsf {Conc}\,\langle \mathcal {E}\!\mathit {ucl},\boldsymbol {\unicode{x3bb} }\rangle \,=\mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ if $d>2$ . Thus $\mathsf {Conc}\,\langle \mathcal {R}\mathit {el},\boldsymbol {{ \cong }}_{}\rangle \,=\mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ if $d>2$ .

Case $\mathcal {G} = \mathcal {G}\mathit {al}$ . We need to show that $\langle \mathcal {G}\mathit {al},\boldsymbol {{ \cong }}_{}\rangle $ and $\langle \mathcal {G}\mathit {al},\boldsymbol {\unicode{x3bb} }\rangle $ are definitionally equivalent to $\mathcal {LC}\mathit {lass}$ . This is straightforward because $\mathcal {LC}\mathit {lass} \ \stackrel {\textsf { def}}{=} \ \langle \mathcal {G}\mathit {al},\boldsymbol {\unicode{x3bb} }\rangle $ , $\mathcal {G}\mathit {al} \ \stackrel {\textsf { def}}{=} \ \langle F^d,\boldsymbol {{ \cong }}_{{\mathsf {S}}},{\mathsf {Bw}}\rangle $ and $\mathcal {E}\!\mathit {ucl} \ \stackrel {\textsf { def}}{=} \ \langle F^d,\boldsymbol {{ \cong }}_{},{\mathsf {Bw}}\rangle $ ; whence it follows immediately that $\langle \mathcal {G}\mathit {al},\boldsymbol {{ \cong }}_{}\rangle \doteq \langle \mathcal {E}\!\mathit {ucl},\boldsymbol {{ \cong }}_{{\mathsf {S}}}\rangle $ . As we have already seen that $\mathsf {Conc}\,\langle \mathcal {E}\!\mathit {ucl},\boldsymbol {{ \cong }}_{{\mathsf {S}}}\rangle \,= \mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ , the claim follows.

Proof of (iii)

We have to show that $\langle \mathcal {R}\mathit {el}, \boldsymbol {{ \cong }}_{} \rangle $ and $\langle \mathcal {E}\!\mathit {ucl}, \boldsymbol {\unicode{x3bb} } \rangle $ are definitionally equivalent to one another; and that they are not definitionally equivalent to $\mathcal {LC}\mathit {lass}$ when $d=2$ . Note first that $\langle \mathcal {E}\!\mathit {ucl},\boldsymbol {\unicode{x3bb} }\rangle $ and $\langle \mathcal {R}\mathit {el},\boldsymbol {{ \cong }}_{}\rangle $ are clearly definitionally equivalent, because $\mathcal {E}\!\mathit {ucl} \ \stackrel {\textsf { def}}{=} \ \langle F^d,\boldsymbol {{ \cong }}_{},{\mathsf {Bw}}\rangle $ and $\mathcal {R}\mathit {el} \ \stackrel {\textsf { def}}{=} \ \langle F^d,\boldsymbol {\unicode{x3bb} },{\mathsf {Bw}}\rangle $ . However, geometries $\langle \mathcal {E}\!\mathit {ucl},\boldsymbol {\unicode{x3bb} }\rangle $ and $\mathcal {LC}\mathit {lass}$ are not definitionally equivalent if $d=2$ , because the linear transformation that interchanges the unit vectors and is an automorphism of $\langle \mathcal {E}\!\mathit {ucl},\boldsymbol {\unicode{x3bb} }\rangle $ , but it is not an automorphism of $\mathcal {LC}\mathit {lass}$ as it does not respect $\boldsymbol {{ \cong }}_{{\mathsf {S}}}$ .

Proof of (iv)

We have to show that $\langle \mathcal {OA}\mathit {ff},\boldsymbol {\unicode{x3b4} }\rangle $ is definitionally equal to $\mathcal {LC}\mathit {lass}$ . Recall that $\mathcal {OA}\mathit {ff} \ \stackrel {\textsf { def}}{=} \ \langle F^d,{\mathsf {Bw}}\rangle $ and $\mathcal {R}\mathit {el} \ \stackrel {\textsf { def}}{=} \ \langle F^d,\boldsymbol {\unicode{x3bb} },{\mathsf {Bw}}\rangle $ , and that $\mathsf {Conc}\,\langle F^d,\boldsymbol {\unicode{x3b4} }\rangle \,= \mathsf {Conc}\,\langle F^d,\boldsymbol {\unicode{x3bb} },{\mathsf {S}}\rangle \,$ by (16). We have already seen that $\mathsf {Conc}\,\langle \mathcal {R}\mathit {el},{\mathsf {S}}\rangle \,= \mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ . Therefore, $\mathsf {Conc}\,\langle \mathcal {OA}\mathit {ff},\boldsymbol {\unicode{x3b4} }\rangle \,= \mathsf {Conc}\,\langle \mathcal {OA}\mathit {ff},\boldsymbol {\unicode{x3bb} },{\mathsf {S}}\rangle \,= \mathsf {Conc}\,\langle \mathcal {R}\mathit {el},{\mathsf {S}}\rangle \,=\mathsf {Conc}\,\mathcal {LC}\mathit {lass}\,$ .

9.1 Open questions