International Journal of Theoretical Physics

, Volume 52, Issue 5, pp 1706–1718

Special Relativity over the Field of Rational Numbers

Authors

  • Judit X. Madarász
    • Alfréd Rényi Institute of MathematicsHungarian Academy of Sciences
    • Alfréd Rényi Institute of MathematicsHungarian Academy of Sciences
Article

DOI: 10.1007/s10773-013-1492-8

Cite this article as:
Madarász, J.X. & Székely, G. Int J Theor Phys (2013) 52: 1706. doi:10.1007/s10773-013-1492-8

Abstract

We investigate the question: what structures of numbers (as physical quantities) are suitable to be used in special relativity? The answer to this question depends strongly on the auxiliary assumptions we add to the basic assumptions of special relativity. We show that there is a natural axiom system of special relativity which can be modeled even over the field of rational numbers.

Keywords

Relativity theorySpecial relativityRational numbersAxiomatic theoriesFirst-order logic

1 Introduction

In this paper, we investigate, within an axiomatic framework, the question: what structures of numbers (as physical quantities) are suitable to be used in special relativity? There are several reasons to investigate this kind of questions in the case of any theory of physics. First of all, we cannot experimentally verify whether the structure of quantities is isomorphic to the field of real numbers. Moreover, the fact that the outcome of every measurement is a finite decimal suggests that rational numbers (or even integers) should be enough to model physical quantities. Another reason is that these investigations lead to a deeper understanding of the connection of the mathematical assumptions about the quantities and the other (physical) assumptions of the theory. Hence these investigations lead to a deeper understanding of any theory of physics, which may come handy if we have to change some of the basic assumptions for some reason. For a more general perspective of this research direction, see [4].

So in general we would like to investigate the question

What structure can numbers have in a certain physical theory?

To introduce the central concept of our investigation, let Th be a theory of physics that contains the concept of numbers (as physical quantities) together with some algebraic operations on them (or at least these concepts are definable in Th). In this case, we can introduce notation Num(Th) for the class of the possible quantity structures of theory Th:
$$ \mathit{Num}( \mathsf{Th} )=\{\mathfrak{Q}:\mathfrak{Q} \text{ is a structure of quantities over which $\mathsf{Th}$ has a model.}\} $$
(1)

In this paper, we investigate our question only in the case of special relativity. However, this question can be investigated in any other physical theory in the same way.

We will see an axiom on observers implying that positive numbers have square roots. Therefore, we recall that Euclidean fields, which got their names after their role in Tarski’s first-order logic axiomatization of Euclidean geometry [18], are ordered fields in which positive numbers have square roots.

Our axiom system for d-dimensional special relativity (\(\mathsf{SpecRel_{d}} \), see p. 5) captures the kinematics of special relativity perfectly if d≥3, see Theorem 1. Without any extra assumptions \(\mathsf{SpecRel_{d}} \) has a model over every ordered field, i.e.,
$$ \mathit{Num}( \mathsf{SpecRel_d} )=\{\mathfrak {Q}:\mathfrak{Q} \text{ is an ordered field} \}, $$
see Remark 1. Therefore, SpecRel has a model over ℚ (the field of rational numbers), too. However, if we assume that inertial observes can move with arbitrary speed less than that of light, see AxThExp on p. 6, then every positive number has to have a square root if d≥3 by Theorem 2, i.e.,
$$ \mathit{Num}( \mathsf{SpecRel_d} + \mathsf {AxThExp} )=\{\mathfrak{Q}:\mathfrak{Q} \text{ is a Euclidean field} \} \quad\text{if\ } d\ge3, $$
see [4]. In particular, the number structure cannot be the field of rational numbers if AxThExp is assumed and d≥3, i.e.,
$$ \mathbb{Q}\not\in\mathit{Num}( \mathsf{SpecRel_d} + \mathsf{AxThExp} )\quad\mathrm{if\ } d\ge3. $$
Theorem 3, the main result of this paper, shows that our axiom system SpecRel has a model over ℚ (in any dimension) if we assume axiom AxThExp only approximately, i.e.,
$$ \mathbb{Q}\in\mathit{Num}\bigl( \mathsf{SpecRel_d} + \mathsf{AxThExp^-} \bigr)\quad\mathrm{if\ } d\ge2, $$
see the precise formulation of \(\mathsf{AxThExp^{-}} \) on p. 6. Assuming \(\mathsf{AxThExp^{-}} \) instead of AxThExp is reasonable because we cannot be sure of anything perfectly accurately in physics. Theorem 3 implies that \(\mathsf{SpecRel} + \mathsf{AxThExp^{-}} \) can be modeled over every subfield of the field of real numbers (ℝ), see Corollary 1; and we conjecture that this axiom system has a model over every ordered field, see Conjecture 1.

Physical theories over ordered fields different from the field of real numbers are interesting not only within theoretical physics, e.g., [10] is a biology paper developing its results over the field of rational numbers.

An interesting and related approach of Mike Stannett introduces two structures, one for the measurable numbers and one for the theoretical numbers and assumes that the set of measurable numbers is dense in the set of theoretical numbers, see [16].

We chose first-order predicate logic to formulate our axioms because experience (e.g., in geometry and set theory) shows that this logic is the best logic for providing an axiomatic foundation for a theory. A further reason for choosing first-order logic is that it is a well defined fragment of natural language with an unambiguous syntax and semantics, which do not depend on set theory. For further reasons, see, e.g., [3, Sect. Why FOL?], [5], [17, Sect. 11], [19, 20].

2 The Language of Our Theories

To our investigation, we need an axiomatic theory of special relativity. Therefore, we will recall our axiom system \(\mathsf{SpecRel_{d}} \) in Sect. 3. To write up any axiom system, we have to choose the set of basic symbols of its language, i.e., what objects and relations between them will be used as basic concepts.

Here we will use the following two-sorted1 language of first-order logic parametrized by a natural number d≥2 representing the dimension of spacetime:
$$ \{ \mathit{B} , \mathit{Q} ; \mathsf{IOb} , \mathsf{Ph} ,+,\,\cdot,\le, \mathsf{W} \}, $$
(2)
where B (bodies) and Q (quantities) are the two sorts, IOb (inertial observers) and Ph (light signals) are one-place relation symbols of sort B, + and ⋅ are two-place function symbols of sort Q, ≤ is a two-place relation symbol of sort Q, and W (the worldview relation) is a d+2-place relation symbol the first two arguments of which are of sort B and the rest are of sort Q.

Relations IOb(m) and Ph(p) are translated as “mis an inertial observer,” and “pis a light signal,” respectively. To speak about coordinatization of observers, we translate relation W(k,b,x1,x2,…,xd) as “bodykcoordinatizes bodybat space-time locationx1,x2,…,xd〉,” (i.e., at space location 〈x2,…,xd〉 and instant x1).

Quantity terms are the variables of sort Q and what can be built from them by using the two-place operations + and ⋅, body terms are only the variables of sort B. IOb(m), Ph(p), W(m,b,x1,…,xd), x=y, and xy where m, p, b, x, y, x1, …, xd are arbitrary terms of the respective sorts are so-called atomic formulas of our first-order logic language. The formulas are built up from these atomic formulas by using the logical connectives not (¬), and (∧), or (∨), implies (→), if-and-only-if (↔) and the quantifiers exists (∃) and for all (∀).

To make them easier to read, we omit the outermost universal quantifiers from the formalizations of our axioms, i.e., all the free variables are universally quantified.

We use the notation Qn for the set of all n-tuples of elements of Q. If \(\bar{\mathbf{x}}\in \mathit{Q} ^{n}\), we assume that \(\bar{\mathbf{x}}=\langle x_{1},\ldots,x_{n}\rangle\), i.e., xi denotes the i-th component of the n-tuple \(\bar{\mathbf{x}}\). Specially, we write \(\mathsf{W} (m,b,\bar{\mathbf{x}})\) in place of W(m,b,x1,…,xd), and we write \(\forall\bar{\mathbf{x}}\) in place of ∀x1…∀xd, etc.

We use first-order logic set theory as a meta theory to speak about model theoretical terms, such as models, validity, etc. The models of this language are of the form
$$ {\mathfrak{M}} = \langle \mathit{B} , \mathit{Q} ; \mathsf{IOb} _\mathfrak{M}, \mathsf{Ph} _\mathfrak {M},+_\mathfrak{M},\cdot_\mathfrak{M},\le_\mathfrak{M}, \mathsf{W} _\mathfrak{M}\rangle, $$
(3)
where B and Q are nonempty sets, \(\mathsf{IOb} _{\mathfrak{M}}\) and \(\mathsf{Ph} _{\mathfrak{M}}\) are subsets of B, \(+_{\mathfrak{M}}\) and \(\cdot_{\mathfrak{M}}\) are binary functions and \(\le_{\mathfrak{M}}\) is a binary relation on Q, and \(\mathsf{W} _{\mathfrak{M}}\) is a subset of B×B×Qd. Formulas are interpreted in \(\mathfrak{M}\) in the usual way. For the precise definition of the syntax and semantics of first-order logic, see, e.g., [7, Sect. 1.3], [8, Sects. 2.1, 2.2].

3 Axioms for Special Relativity

Now having our language fixed, we can recall axiom system \(\mathsf{SpecRel_{d}} \), as well as two theorems on \(\mathsf{SpecRel_{d}} \) related to our investigation.

The key axiom of special relativity states that the speed of light is the same in every direction for every inertial observers.
AxPh
For any inertial observer, the speed of light is the same everywhere and in every direction (and it is finite). Furthermore, it is possible to send out a light signal in any direction (existing according to the coordinate system) everywhere:2
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-013-1492-8/MediaObjects/10773_2013_1492_Eque_HTML.gif
where \(\mathsf{space}^{2}(\bar{\mathbf{x}},\bar{\mathbf {y}}):=(x_{2}-y_{2})^{2}+\cdots+(x_{d}-y_{d})^{2}\) and \(\mathsf{time}(\bar{\mathbf{y}},\bar{\mathbf{y}}):=x_{1}-y_{1}\).

To get back the intended meaning of axiom AxPh (or even to be able to define subtraction from addition), we have to assume some properties of numbers.

In our next axiom, we state some basic properties of addition, multiplication and ordering true for real numbers.
AxOField
The quantity part 〈Q,+, ⋅,≤〉 is an ordered field, i.e.,
  • Q,+,⋅〉 is a field in the sense of abstract algebra; and

  • the relation ≤ is a linear ordering on Q such that
    1. (i)

      xyx+zy+z and

       
    2. (ii)

      0≤x∧0≤y→0≤xy holds.

       

Using axiom AxOFiled instead of assuming that the structure of quantities is the field of real numbers not just makes our theory more flexible, but also makes it possible to meaningfully investigate our main question. Another reason for using AxOField instead of ℝ is that we cannot experimentally verify whether the structure of physical quantities are isomorphic to ℝ. Hence the assumption that the structure of quantities is ℝ cannot be empirically supported. The two properties of real numbers which are the most difficult to defend from empirical point of view are the Archimedean property, see [11], [12, Sect. 3.1], [13, 14], and the supremum property.3

We also have to support AxPh with the assumption that all observers coordinatize the same “external” reality (the same set of events). By the event occurring for observer m at point \(\bar{\mathbf{x}}\), we mean the set of bodies m coordinatizes at \(\bar{\mathbf{x}}\):
$$ \mathsf{ev} _m(\bar{\mathbf{x}}):=\bigl\{ b : \mathsf{W} (m,b,\bar{\mathbf{x}})\bigr\}. $$
(4)
AxEv
All inertial observers coordinatize the same set of events:
$$ \mathsf{IOb} (m)\land \mathsf {IOb} (k)\rightarrow\exists\bar{\mathbf{y}}\,\forall b \big[ \mathsf{W} (m,b,\bar{\mathbf{x}})\leftrightarrow \mathsf{W} (k,b,\bar{\mathbf{y}})\big]. $$
From now on, the subformula \(\forall b [ \mathsf{W} (m,b,\bar{\mathbf {x}})\leftrightarrow \mathsf{W} (k,b,\bar{\mathbf {y}})]\) of AxEv will be abbreviated to \(\mathsf{ev} _{m}(\bar{\mathbf{x}})= \mathsf{ev} _{k}(\bar {\mathbf{y}})\).
These three axioms are enough to capture the essence of special relativity. However, let us assume two more simplifying axioms.
AxSelf
Any inertial observer is stationary relative to himself:
$$ \mathsf{IOb} (m)\rightarrow\forall\bar{\mathbf {x}}\big[ \mathsf{W} (m,m,\bar{\mathbf{x}}) \leftrightarrow x_2=\cdots=x_d=0\big]. $$
Our last axiom on inertial observers is a symmetry axiom saying that they use the same units of measurement.
AxSymD
Any two inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them; furthermore, the speed of light is 1 for all observers:
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-013-1492-8/MediaObjects/10773_2013_1492_Equh_HTML.gif
Let us introduce an axiom system for special relativity as the collection of the five simple axioms above:
$$ \mathsf{SpecRel_d} := \mathsf {AxPh} + \mathsf{AxOField} + \mathsf {AxEv} + \mathsf{AxSelf} + \mathsf{AxSymD} . $$
To show that the five simple axioms of \(\mathsf {SpecRel_{d}} \) capture special relativity well, let us introduce the concept of worldview transformation between observers m and k (in symbols, wmk) as the binary relation on Qd connecting the coordinate points where m and k coordinatize the same events:
$$ \mathsf{w} _{mk} \bigl(\bar{\mathbf{x}},\bar{\mathbf {x}}'\bigr)\quad \stackrel{\mathrm{def}}{\Longleftrightarrow} \quad \forall b\big[ \mathsf{W} (m,b,\bar{\mathbf{x}})\leftrightarrow \mathsf{W} \bigl(k,b,\bar{\mathbf{x}}'\bigr)\big]. $$
(5)
Map P:QdQd is called a Poincaré transformation iff it is an affine bijection having the following property
$$ \mathsf{time}(\bar{\mathbf{x}},\bar{\mathbf{y}})^2-\mathsf {space}^2(\bar{\mathbf{x}},\bar{\mathbf{y}})=\mathsf{time}\bigl (\bar{\mathbf{x}}',\bar{\mathbf{y}} '\bigr)^2-\mathsf{space}^2\bigl(\bar{\mathbf{x}}',\bar{\mathbf {y}}'\bigr) $$
(6)
for all \(\bar{\mathbf{x}},\bar{\mathbf{y}},\bar{\mathbf{x}}',\bar {\mathbf{y}}'\in \mathit{Q} ^{d}\) for which \(P(\bar {\mathbf{x}})=\bar{\mathbf{x}}'\) and \(P(\bar{\mathbf{y}})=\bar{\mathbf{y}}'\).

Theorem 1 shows that our axiom system \(\mathsf{SpecRel_{d}} \) captures the kinematics of special relativity since it implies that the worldview transformations between inertial observers are Poincaré transformations.

Theorem 1

Letd≥3. Assume\(\mathsf{SpecRel_{d}}\). Thenwmkis a Poincaré transformation ifmandkare inertial observers.

For the proof of Theorem 1, see [4]. For a similar result over Euclidean fields, see, e.g., [1, Theorems 1.4 and 1.2], [2, Theorem 11.10], [17, Theorem 3.1.4].

Let us now introduce a further auxiliary axiom about the possibility of motion of inertial observers.
AxThExp
Inertial observers can move along any straight line with any speed less than the speed of light:
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-013-1492-8/MediaObjects/10773_2013_1492_Equj_HTML.gif

Theorem 2 below shows that axiom AxThExp implies that positive numbers have square roots if \(\mathsf{SpecRel_{d}} \) is assumed.

Theorem 2

Ifd≥3, then
$$ \mathit{Num}( \mathsf{SpecRel_d} + \mathsf {AxThExp} )=\{\mathfrak{Q}:\mathfrak{Q} \text{ is a Euclidean field}\}. $$

Remark 1

Axiom AxThExp cannot be omitted from Theorem 2 since \(\mathsf{SpecRel_{d}}\) has a model over every ordered field, i.e., for all d≥2,
$$ \mathit{Num}( \mathsf{SpecRel_d} )=\{\mathfrak {Q}:\mathfrak{Q} \text{ is an ordered field}\} $$
for all d≥2. Moreover, \(\mathsf{SpecRel_{d}}\) also has non trivial models in which there are several observers moving relative to each other. We conjecture that there is a model of \(\mathsf {SpecRel_{d}}\) over every ordered field such that the possible speeds of observers are dense in interval [0,1], see Conjecture 1 on p. 7.
Since our measurements have only finite accuracy, it is natural to assume AxThExp only approximately.
AxThExp
Inertial observers can move roughly with any speed less than the speed of light roughly in any direction, see Fig. 1:4
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-013-1492-8/MediaObjects/10773_2013_1492_Equm_HTML.gif
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-013-1492-8/MediaObjects/10773_2013_1492_Fig1_HTML.gif
Fig. 1

Illustration for axiom \(\mathsf {AxThExp^{-}} \)

By Theorem 3, a model of \(\mathsf {SpecRel_{d}} + \mathsf{AxThExp^{-}} \) has a model over the field of rational numbers in any dimension. We use the notation \(\mathfrak{Q}\in\mathit{Num}( \mathsf {Th} )\) for algebraic structure \(\mathfrak{Q}\) the same way as the model theoretic notation \(\mathfrak{Q}\in Mod( \mathsf{AxField} )\), e.g., ℚ∈Num(Th) means that ℚ, the field of rational numbers, can be the structure of quantities in theory Th.

Theorem 3

For alld≥2,
$$ \mathbb{Q}\in\mathit{Num}\bigl( \mathsf{SpecRel_d} + \mathsf{AxThExp^-} \bigr). $$

For the proof of Theorem 3, see Sect. 4.

An ordered field is called Archimedean field iff for all a, there is a natural number n such that
$$ a<\underbrace{1+\cdots+1}_n $$
(7)
holds. By Hölder’s theorem, every Archimedean field is isomorphic to a subfield of the field of real numbers, see, e.g., [9, Sect. IV]. Consequently, the field of rational numbers is dense in any Archimedean field since it is dense in the field of real numbers. Therefore, the following is a corollary of Theorem 3.

Corollary 1

For alld≥2,
$$ \{\mathfrak{Q}:\mathfrak{Q}\ \text{\textit{is an Archimedean field}}\} \subsetneqq\mathit{Num}\bigl( \mathsf{SpecRel_d} + \mathsf{AxThExp^-} \bigr). $$

The question “exactly which ordered fields can be the quantity structures of theory \(\mathsf{SpecRel_{d}} + \mathsf{AxThExp^{-}} \)?” is open. By Lövenheim–Skolem Theorem it is clear that \(\mathit{Num}( \mathsf{SpecRel_{d}} + \mathsf{AxThExp^{-}} )\) cannot be the class of Archimedean fields since it has elements of arbitrarily large cardinality while an Archimedean field has at most the cardinality of continuum since Archimedean fields are subsets of the field of real numbers by Hölder’s Theorem. We conjecture that there is a model of \(\mathsf {SpecRel_{d}} + \mathsf{AxThExp^{-}} \) over every ordered field in any dimension, i.e.:

Conjecture 1

For alld≥2,
$$ \mathit{Num}\bigl( \mathsf{SpecRel_d} + \mathsf{AxThExp^-} \bigr)=\{\mathfrak {Q}:\mathfrak{Q}\ \text{\textit{is an ordered field}}\}. $$

4 Proof of Theorem 3

In this section, we are going to prove our main result. To do so, let us recall some concepts and theorems from the literature. The following theorem is well-known, see, e.g., [15, Theorem 2.1].

Theorem 4

The unit sphere ofnhas a dense set of points with rational coordinates.

The Euclidean length of \(\bar{\mathbf{x}}\in \mathit{Q} ^{n}\) if n≥1 is defined as:
$$ |\bar{\mathbf{x}}|:=\sqrt{x_1^2+\cdots+x_n^2}. $$
(8)
Let us recall that the norm of linear map A:ℝd→ℝd, in symbols ∥A∥, is defined as follows:
$$ \|A\|:=\max\bigl\{ |A\bar{\mathbf{x}}|: \bar{\mathbf{x}}\in \mathbb{R}^d \text{ and } |\bar{\mathbf{x}} |=1\bigr\}. $$
(9)
Linear bijection A is called orthogonal transformation if it preserves the Euclidean distance.

Theorem 4 implies Theorem 5, see [15, Theorem 3.1].

Theorem 5

For all orthogonal transformationT:ℝn→ℝnand anyε>0, there is an orthogonal transformationA:ℚn→ℚnsuch thatTA∥<ε.

Using Theorem 5, let us prove that its statement also holds for Poincaré transformations.

Theorem 6

For every Poincaré transformationL:ℝd→ℝdand positive real number ε, there is a Poincaré transformationL:ℚd→ℚdsuch thatLL∥<ε.

We are going to prove Theorem 6 by using the fact that every Poincaré transformation is a composition of a Lorentz boost and two orthogonal transformations. Lorentz boost corresponding to velocity v∈[0,1), in symbols Bv, is defined as the following linear map:
$$ B_v\bar{\mathbf{x}}=\biggl\langle \frac{x_1-vx_2}{\sqrt{1-v^2}},\frac{x_2-vx_1}{\sqrt {1-v^2}},x_3,\ldots,x_d\biggr\rangle\quad \text{for all } \bar{\mathbf{x}}\in \mathit{Q} ^d. $$
(10)

Lemma 1

For any Lorentz boostBv:ℝd→ℝdand positive numberε, there is a Lorentz boostBw:ℚd→ℚdsuch thatBvBw∥<ε.

Proof

Since, by Theorem 4, the set of rational points are dense in the unit circle, we have that, for all δ>0 and v∈[0,1), there is a w∈ℚ∪[0,1) such that |vw|<δ and \(\sqrt{1-w^{2}}\in\mathbb{Q}\), i.e., Bw takes rational point to rational ones. So we have to show that ∥BvBw∥<ε if δ is small enough. Since in a finite-dimensional vector space all norms are equivalent, see [6, Sect. 8.5], it is enough to show that the norm of BvBw can be less than any positive real number according to the Euclidean norm, which is
$$ \sqrt{2\biggl\vert\frac{1}{\sqrt{1-v^2}} -\frac{1}{\sqrt {1-w^2}}\biggr \vert^2 +2\biggl\vert\frac{v}{\sqrt{1-v^2}} -\frac{w}{\sqrt {1-w^2}}\biggr\vert^2}. $$
(11)
By the continuity of functions \(v\mapsto(1-v^{2})^{-\frac{1}{2}}\) and \(v\mapsto v(1-v^{2})^{-\frac{1}{2}}\), the Euclidean norm of BvBw is less than any fixed positive real number if |vw| is small enough. Therefore, there is a Lorentz boost Bw such that Bw maps rational points to rational ones and ∥BwBv∥<ε. □

Lemma 2

LetAandBbe linear bijections of d. LetAandBlinear maps such thatAA′∥<ε1andBB′∥<ε2. ThenBABA′∥≤ε1B∥+ε1ε2+ε2A∥.

Proof

First let us note that
$$ \bigl\|A'\bigr\|=\bigl\|A'-A+A\bigr\|\le\bigl\|A'-A\bigr\|+\|A\| =\varepsilon_1 + \|A\| $$
(12)
by the triangle inequality. Let \(\bar{\mathbf{x}}\in\mathbb{R}^{d}\) such that \(|\bar{\mathbf {x}}|=1\). We have to show that
$$ \bigl|BA\bar{\mathbf{x}}-B'A'\bar{\mathbf{x}}\bigr|\le \varepsilon_1\|B\| +\varepsilon_1\varepsilon_2 +\varepsilon_2\|A\| $$
(13)
By the triangle inequality and the fact that \(|M\bar{\mathbf{y}}|\le\|M\|\cdot|\bar{\mathbf{y}}|\), we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-013-1492-8/MediaObjects/10773_2013_1492_Equ14_HTML.gif
(14)
and this is what we wanted to prove. □

Proof of Theorem 6

Every Poincaré transformation is a composition of a translation, a Lorentz-boost Bv and an orthogonal transformation. Therefore, Lemmas 1 and 2, together with Theorem 5 imply our statement. □

Now we are going to prove Theorem 3. Let Id be the identity map of ℚd. We denote the origin of Qn by \(\bar{\mathbf{o}}\), i.e.,
$$ \bar{\mathbf{o}}:=\langle0,\ldots,0\rangle. $$
(15)
Let the time-axis be defined as the following subset of Qd:
$$ \mathsf{t\text{-}axis}:=\{\bar{\mathbf{x}}:x_2=\cdots=x_d=0\}. $$
(16)
Let H be a subset of Qd and let f:QdQd be a map. The f-image of set H is defined as:
$$ f[H]:=\bigl\{f(\bar{\mathbf{x}}):\bar{\mathbf{x}}\in H\bigr\}. $$
(17)
The so-called worldline of body b according to observer m is defined as follows:
$$ \mathsf{wl} _m(b):=\bigl\{ \bar{\mathbf{x}}: \mathsf{W} (m,b,\bar{\mathbf{x}})\bigr\}. $$
(18)

Proof of Theorem 3

We are going to construct a model of \(\mathsf{SpecRel_{d}} + \mathsf{AxThExp^{-}} \) over ℚ. So let 〈Q,+, ⋅ ,≤〉 be the ordered field of rational numbers. Let
$$ \mathsf{Ph} :=\{l : l \text{ is a line of slope 1}\}, $$
(19)
$$ \mathsf{IOb} :=\bigl\{m: m \text{ is a Poincar\'{e} transformation from $\mathbb{Q} ^{d}$ to $\mathbb{Q}^{d}$}\bigr\}, $$
(20)
and let B=IObPh. First we are going to give the worldview of observer Id. Let
$$ \mathsf{W} ( \mathsf{Id} , \mathsf{Id} ,\bar{\mathbf{x}})\quad\stackrel{\mathrm {def}}{\Longleftrightarrow} \quad x_2=\cdots=x_d=0; $$
(21)
for any other inertial observer m, let
$$ \mathsf{W} ( \mathsf{Id} ,m,\bar{\mathbf {x}}) \quad\stackrel{\mathrm{def}}{\Longleftrightarrow} \quad\bar {\mathbf{x}}\in m\ [{\mathsf{t\text{-}axis}}]; $$
(22)
and for any light signal pPh, let
$$ \mathsf{W} ( \mathsf{Id} ,p,\bar{\mathbf {x}})\quad\stackrel{\mathrm{def}}{\Longleftrightarrow} \quad\bar {\mathbf{x}}\in p. $$
(23)
Now the worldview of observer Id is given. From the worldview of Id, we construct the worldview of another inertial observer m as follows:
$$ \mathsf{W} (m,b,\bar{\mathbf{x}})\quad\stackrel {\mathrm{def}}{\Longleftrightarrow} \quad \mathsf {W} \big( \mathsf{Id} ,b,m(\bar{\mathbf{x}})\big) $$
(24)
for all body bB, see Fig. 2.
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-013-1492-8/MediaObjects/10773_2013_1492_Fig2_HTML.gif
Fig. 2

Illustration for the proof of Theorem 3

Now we have given the model. Let us see why the axioms of \(\mathsf{SpecRel_{d}} \) and \(\mathsf {AxThExp^{-}} \) are valid in it.

By the above definition of W, if m and k are inertial observers, then
$$ \mathsf{W} (m,k,\bar{\mathbf{x}}) \quad\text{holds iff } m(\bar{\mathbf{x}})\in k\ [\mathsf{t\text{-}axis}], $$
(25)
and if mIOb and pPh, then
$$ \mathsf{W} (m,p,\bar{\mathbf{x}})\quad\text{holds iff } m(\bar{\mathbf{x}})\in p. $$
(26)
The worldview transformations between inertial observers m and Id is m, i.e., wmId=m by equation (24). Therefore, the worldview transformation between inertial observers m and k is k−1m, i.e.,
$$ \mathsf{w} _{mk}=k^{-1} \circ m $$
(27)
since wmk=wIdkwmId and wIdk=(wkId)−1 by the definition of the worldview transformation (5). In particular, the worldview transformations between inertial observers are Poincaré transformations in these models (as Theorem 1 requires it). Hence
$$ \mathsf{w} _{mk} \text{ is a bijection for all inertial observers $m$ and $k$.} $$
(28)

Axiom AxPh is valid for observer Id by the definition of Ph and that of his worldview. It is also clear that the speed of light is 1 for observer Id. Axiom AxPh is valid for the other observers since Poincaré transformations take lines of slope one to lines of slope one. This also show that the speed of light is 1 according to every inertial observer, which is the second half of AxSymD.

Axiom AxOField is valid in this model since ℚ is an ordered field. Axiom AxEv is valid in this model since Poincaré transformations are bijections. Axiom AxSelf is valid in this model since
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-013-1492-8/MediaObjects/10773_2013_1492_Equ29_HTML.gif
(29)

Any Poincaré transformation P preserves the spatial distance of points \(\bar{\mathbf{x}}\), \(\bar{\mathbf{y}}\) for which x1=y1 and \(P(\bar{\mathbf{x}})_{1}=P(\bar{\mathbf{y}})_{1}\). Therefore, inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them. We have already shown that the speed of light is 1 for each inertial observers in this model. Hence axiom AxSymD is also valid in this model.

Now we are going to show that \(\mathsf{AxThExp^{-}} \) is valid in this model. The ∃hIOb(h) part of axiom AxThExp is valid, since there are Poincaré transformations (e.g., Id is one). To show that the rest of axiom \(\mathsf{AxThExp^{-}} \) is valid, let m be an inertial observer and let us fix an ε>0 and a \(\bar{\mathbf {v}}\in\mathbb{Q}^{d}\) for which \(v_{2}^{2}+\cdots+ v_{d}^{2}<1\) and v1=1. Let \(\bar{\mathbf{1}}\) be vector 〈1,0,…,0〉. Let L be a Lorentz transformation (i.e., linear Poincaré transformation) for which
$$ \bar{\mathbf{v}}=\frac{L(\bar{\mathbf{1}})}{L(\bar{\mathbf{1}})_1}. $$
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Let 0<δ<1 be such that
$$ \delta< \frac{\varepsilon L(\bar{\mathbf{1}})_1}{2} $$
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and
$$ \biggl|\frac{1}{L(\bar{\mathbf{1}})_1}-\frac{1}{x}\biggr|<\frac {\varepsilon }{2(\|L\|+1)} $$
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for any x for which \(|L(\bar{\mathbf{1}})_{1}-x|<\delta\). By Theorem 6, there is a Lorentz transformation L which takes rational points to rational ones and ∥LL∥<δ. Then
$$ \bigl|L(\bar{\mathbf{1}})-L^*(\bar{\mathbf{1}})\bigr|<\delta $$
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since \(|\bar{\mathbf{1}}|=1\). We have \(|L(\bar{\mathbf {1}})_{1}-L^{*}(\bar{\mathbf{1}})_{1}|<\delta\) since \(|x_{1}|<|\bar{\mathbf{x}}|\) for all \(\bar{\mathbf{x}}\in\mathbb {Q}^{d}\). By triangle inequality, we also have
$$ \bigl|L^*(\bar{\mathbf{1}})\bigr|\le\bigl|L^*(\bar{\mathbf {1}})-L(\bar{\mathbf{1}})\bigr|+\bigl |L(\bar{\mathbf{1}})\bigr|\le \delta+\|L\|\le 1+\|L\|. $$
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Let
$$ \bar{\mathbf{w}}:=\frac{L^*(\bar{\mathbf{1}})}{L^*(\bar{\mathbf{1}})_1} $$
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By triangle inequality, we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-013-1492-8/MediaObjects/10773_2013_1492_Equ36_HTML.gif
(36)
Let \(\bar{\mathbf{x}},\bar{\mathbf{y}}\in\mathbb{Q}^{d}\) such that there is a λ∈ℚ such that \(\bar{\mathbf{y}}-\bar{\mathbf{x}}=\lambda\bar{\mathbf {w}}\). To finish the proof of \(\mathsf{AxThExp^{-}} \), we have to show that there is an inertial observer k such that \(\mathsf{W} (m,k,\bar{\mathbf{x}})\) and \(\mathsf{W} (m,k,\bar{\mathbf{y}})\), i.e., \(\bar{\mathbf{x}},\bar {\mathbf{y}}\in \mathsf{wl} _{m}(k)\). Let \(P^{*}=L^{*}+\bar{\mathbf{x}}\). P is a Poincaré transformation taking rational points to rational ones. Therefore, there is an inertial observer k such that wkm=P. Since \(\mathsf{wl} _{m}(k)= \mathsf{w} _{km}\ [\mathsf{t\text{-}axis}]\), we have that \(\mathsf{w} _{km}(\bar{\mathbf{o}})=\bar {\mathbf{x}}\in \mathsf{wl} _{m}(k)\) and that \(\bar{\mathbf{y}}=a L^{*}(\bar{\mathbf{1}})+\bar{\mathbf{x}}= \mathsf {w} _{km}(a\bar{\mathbf{1}})\in \mathsf{wl} _{m}(k)\), where \(a=\lambda/L^{*}(\bar{\mathbf{1}})_{1}\). This shows that \(\mathsf{AxThExp^{-}} \) is also valid in our model. □
Footnotes
1

That our theory is two-sorted means only that there are two types of basic objects (bodies and quantities) as opposed to, e.g., Zermelo–Fraenkel set theory where there is only one type of basic objects (sets).

 
2

That is, if m is an inertial observer, there is a is a positive quantity cm such that for all coordinate points \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\) there is a light signal p coordinatized at \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\) by observer m if and only if equation \(\mathsf{space}^{2}(\bar{\mathbf{x}},\bar{\mathbf{y}})= c_{m}^{2}\cdot\mathsf{time}(\bar{\mathbf{x}},\bar{\mathbf{y}})^{2}\) holds.

 
3

The supremum property (i.e., that every nonempty and bounded subset of the numbers has a least upper bound) implies the Archimedean property. So if we want to get ourselves free from the Archimedean property, we have to leave this one, too.

 
4

That is, for every vector \(\bar{\mathbf{v}}=\langle v_{1},\ldots,v_{d}\rangle\) determining a spatial direction and a slower than light speed, there is another vector \(\bar{\mathbf{w}}\) in the ε-neighborhood of \(\bar {\mathbf{v}}\), such that, if points \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\) are on a line parallel to \(\bar{\mathbf{w}}\) (i.e., on a line corresponding to a uniform motion with the speed and in the direction determined by \(\bar{\mathbf{w}}\)), then there is an inertial observer k moving through \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\), see Fig. 1.

 

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© Springer Science+Business Media New York 2013