Axiomatizing relativistic dynamics using formal thought experiments

Abstract

Thought experiments are widely used in informal explanations of Relativity Theories; however, they are not present explicitly in formalized versions of Relativity Theory. In this paper, we present an axiom system of Special Relativity which is able to express thought experiments formally and explicitly. Moreover, using these thought experiments, we can provide an explicit definition of relativistic mass based merely on kinematical concepts and thought experiments on collisions. Using this definition, we can geometrically prove the Mass Increase Formula $m_0 = m \mathrel {\cdot }\sqrt{1-v^2}$ in a natural way, without postulates of conservation of mass and momentum.

Appendix

1.1 Poincaré Transformation Theorem

Here we are going to prove Theorem 1 sating that:

$$\begin{aligned} {\mathsf{MSpecRel}}\vdash (\forall k,h \in {\mathrm {IOb}})\ ``{\mathrm {w}}_{{k}{h}}\, \mathrm{is\, a\, Poincar}\acute{\mathrm{e}} \,\mathrm{transformation.}\hbox \mathrm{''} \end{aligned}$$

To prove that ${\mathrm {w}}_{{k}{h}}$ is a Poincaré transformation, it is enough to show that it takes lines of slope 1 to lines of slope 1, since there is an Alexandrov–Zeeman type theorem which works only with these premises, see Madarász and Székely (2013). To prove this lemma, let us introduce the following notation for the speed corresponding to coordinate points $\bar{x}$ and $\bar{y}$:

$$\begin{aligned} {\mathrm {v}}(\bar{x},\bar{y}) \mathop {=}\limits ^{{ def. }}\frac{{\mathrm {Space}}(\bar{x} , \bar{y})}{{\mathrm {Time}}(\bar{x} , \bar{y})}. \end{aligned}$$

Lemma 1

(Light-line) Assume MSpecRel. Then every worldview transformation is a bijection taking lines of slope 1 to lines of slope 1.

Proof

Worldview transformations are bijections by Proposition 2.

Now we prove that worldview transformations take lines of slope 1 to lines of slope 1. By AxEField, ${\mathrm {v}}(\bar{x} ,\bar{y})={\mathrm {v}}(\bar{y} ,\bar{z})={\mathrm {v}}(\bar{z} ,\bar{x})=1$ implies that $\bar{x}$, $\bar{y}$ and $\bar{z}$ are collinear. Therefore, to finish our proof, it is enough to derive the following formula:

$$\begin{aligned} (\forall k,h \in {\mathrm {IOb}}) (\forall \bar{x},\bar{y}) [{\mathrm {v}}(\bar{x} ,\bar{y})=1 \rightarrow {\mathrm {v}}({\mathrm {w}}_{{k}{h}}({\bar{x}}),{\mathrm {w}}_{{k}{h}}({\bar{y}}))=1]. \end{aligned}$$

(14)

Let $k$ and $h$ be arbitrary observers in a world $w$, and let $\bar{x}$ and $\bar{y}$ be coordinate points such that ${\mathrm {v}}(\bar{x} ,\bar{y})=1$. By ${{\mathsf{AxPhExp}}}$, in every $w$ world, there is an accessible world $w'$ such that $wRw'$ and in $w'$ there is a light signal $p\in {\mathrm {ev}}_{k}({\bar{x}})\cap {\mathrm {ev}}_{k}({\bar{y}})$ in $w'$. By ${\mathsf{AxMFrame}}$, $k$ still exists as an observer in $w'$. So $p\in {\mathrm {ev}}_{k}({{\mathrm {w}}_{{k}{h}}({\bar{x}})}) \cap {\mathrm {ev}}_{k}({{\mathrm {w}}_{{k}{h}}({\bar{y}})})$ by ${\mathsf{AxMEv}}$. Consequently, by ${\mathsf{AxPhObs}}$:

$$\begin{aligned} {\mathrm {v}}({\mathrm {w}}_{{k}{h}}({\bar{x}}),{\mathrm {w}}_{{k}{h}}({\bar{y}}))=1; \end{aligned}$$

and this is what we wanted to prove.$\square$

1.2 Proof of Symmetric Collision Theorem

Here we are going to prove Theorem 2 stating that:

Fig. 10

The symmetric collision theorem

Proof

Let $k$ and $h$ be observers and let $b$ and $c$ be ordinary bodies in a world $w$ such that ${\mathrm {v}}_{k}(b)=0$, ${\mathrm {v}}_{h}(c)=0$ and ${\mathrm {SymColl}}({b},{c})$ holds for them. Since ${\mathrm {SymColl}}({b},{c})$ holds, there is an observer $m$ (the so-called median observer) in $w$ such that $\bar{{\mathrm {v}}}_{m}(b) + \bar{{\mathrm {v}}}_{m}(c)=\bar{0}$ and $({b}\mathop {:}{c})_{m} = 1$. See Fig. 10.

The time dilation effect, i.e.,

$$\begin{aligned} (\forall k,h\! \in \!{\mathrm {IOb}}) (\forall \bar{x},\bar{y}\! \in \!{\mathrm {wline}}_{k}({h}))\, \, {\mathrm {Time}}(\bar{x},\bar{y}) = \frac{{\mathrm {Time}}({\mathrm {w}}_{{k}{h}}({\bar{x}}),{\mathrm {w}}_{{k}{h}}({\bar{y}}))}{\sqrt{1- {\mathrm {v}}_{k}(h)^2}}, \end{aligned}$$

(15)

is a consequence of Theorem 1, see (Andréka et al. (2007), Theorem 2.4, (2)):

We know from (15) that if the clocks of $k$ and $h$ show $0$ at $A$, and the clock of $h$ shows $-1$ at $C$, then the clock of $k$ shows $-\sqrt{1-v^2}$ where $v$ is $\mathrm {v}_h(k)$.

We are interested in $({c}\mathop {:}{b})_{h}$, which is now:

$$\begin{aligned} ({c}\mathop {:}{b})_{h} = \frac{BD}{DC}. \end{aligned}$$

(16)

Since worldview transformation are affine transformations, this ratio is the same in the worldview of the median observer $m$, i.e.,

$$\begin{aligned} \frac{BD}{DC} = \frac{B'D'}{D'C'}. \end{aligned}$$

(17)

Since the worldline of $d$ is an angle bisector of the triangle $A'B'C'$ in the worldview of $m$, by the angle bisector theorem,

$$\begin{aligned} \frac{B'D'}{D'C'} = \frac{B'A'}{A'C'}. \end{aligned}$$

(18)

Since the clocks of $k$ and $h$ slow down with the same rate for the median observer $m$, we know that

$$\begin{aligned} \frac{B'A'}{A'C'} = \sqrt{1-v^2}. \end{aligned}$$

(19)

Since the collision was symmetric, $({b}\mathop {:}{c})_{k} = ({c}\mathop {:}{b})_{h}$. Therefore, from (16), (17), (18) and (19) we have

$$\begin{aligned} ({c}\mathop {:}{b})_{k} = \sqrt{1-v^2}, \end{aligned}$$

and this is what we wanted to prove. $\square$

1.3 Consistency and independence

To prove the consistency of ${\mathsf{MSpecRelDyn}}$, it is enough to construct a model for it. It is easy to construct a model of ${\mathsf{MSpecRelDyn}}$—if we choose ${\mathrm {IOb}}^\mathfrak M$ and ${\mathrm {W}}^\mathfrak M$ to be empty, then almost all axioms will be satisfied, since all of them has the form “$(\forall k \in {\mathrm {IOb}})\dots$”. The only exception is ${\mathsf{AxMFrame}}$, since this requires the reflexivity of the alternative relation, the rigidity of mathematics, etc., but it is easy to satisfy these statements by choosing the set of possible worlds to be a singleton $\{w\}$ such that $wRw$.

However, we would not only like to prove that that ${\mathsf{MSpecRelDyn}}$ is consistent but also that it has some complex models, models where there are several observers moving relative to one another and measuring ordinary inertial bodies, e.g., where we see something similar to what we had in mind during the proofs of Sect. 4. This type of model construction is too complex to be included in detail in this paper. However, in this section, we give a sketch of such a construction.

The main difficulty in the construction of such a model is the following: If two ordinary bodies, say $b$ and $c$ collide, then every observer has to measure the two bodies and the resulting body $b {\small \oplus } c$ directly, pseudo-directly or indirectly. There are plenty of other measurements we have to include because, for example, in the successor world, where the pseudo-direct measurement of $b$ is done by ${\mathrm {e}}_{k}$, there will be an ordinary body $b\oplus {\mathrm {e}}_{k}$, which again, has to be measured directly, pseudo-directly, indirectly, and so on and so forth. So building any model $\mathfrak M$ in which ${\mathsf{MSpecRelDyn}}$ is true (i.e., true in every world of $\mathfrak M$), and in which $\exists {\mathsf{2IOb}}$ and $(\exists b ) {\mathrm {OIB}}(b)$ is satisfied (i.e., true in some world of $\mathfrak M$) involves an infinite process of measuring.

Another difficulty is that by $\exists {\mathsf{2IOb}}$ two observers sooner or later will compare their equivalents by ${\mathsf{AxPDirComp}}$, which postulates the existence of a median-observer as well. So the cardinality of observers cannot be 2 or any finite number.

From now on, to describe a complex model $\mathfrak M$, we take the perspective of an observer $M$. To guarantee the truth of $\exists {\mathsf{2IOb}}$ and the existence of medians (needed for ${\mathsf{AxPDirComp}}$), we include infinitely many²³ observers to meet with $M$ in its origin in all worlds (where $M$ exists) such that for every possible velocity in the interval $[-0.5,0.5]$, there will be an observer in the $tx$-plane having that velocity. There will be a set of possible worlds, $S_0$, where these observers compare their mass-standard-equivalents and measure the other’s equivalent directly in the origin at time $0$ according to $M$. Every such comparison and direct measurement will result a body, which again has to be measured; these measurements will happen 1 second later according to $M$, and these worlds will constitute the set $S_1^{\oplus }$. Note that mass-standard-equivalents of $S_1^{\oplus }$ have different worldlines than they had in $S_0$ since their worldlines end at the time $1$ instead of $0$. So there will be a set $S_1^{et}$ very similar to $S_0$, where the observers compare their mass-standard-equivalents 1 second later. The construction continues in the same way with $S_2^{\oplus }$, $S_2^{et}$, $S_3^{\oplus }$, $S_3^{et}$ ...   into infinity. A part of that model, measurements concerning two equivalents $a$ and $b$ of observers $A$, $B$ having opposite velocity according $M$, is illustrated on Fig. 11. This figure shows also how are the indirect measurements fulfilled in that model: $a\oplus b$ is measured in the central world of the figure indirectly through worlds having the resulting body $a\oplus a\oplus b$ and $a\oplus e$ (the latter is from $S_1^{et}$).

Fig. 11

A part of the model: measurements concerning two equivalents $a$ and $b$ of observers $A$, $B$ having opposite velocity according to $M$

This construction will satisfy all the axioms of ${\mathsf{MSpecRelDyn}}$ except the light signal-sending axiom ${\mathsf{AxPhExp}}$. To make this axiom true in every world of our model, it is enough to extend this construction with only one world, say $ph$, which is an alternative of every world of the model, and which realizes every possible photon needed by any world.²⁴ This means that $ph$ will be a classical model of SpecRel. For such a construction, see (Andréka et al. (2007), Corollary 11.12, especially pp. 643–644).

This construction can be carried out even in a way such that the conservation of mass and linear moment fails in the resulting model. This construction shows that the Mass Increase Theorem can be proved without the usual conservation postulates, which is a result similar to that of Andréka et al. (2008).²⁵ However, ${\mathsf{MSpecRelDyn}}$ has models (different from the above construction) where even the key axiom of Andréka et al. (2008), $\mathsf{AxCenter}$, the conservation of centerline of mass, is refutable.²⁶

Independence This model $\mathfrak M$ can easily be modified to show the independence of several axioms of ${\mathsf{MSpecRelDyn}}$. For example, if we remove a pair of worlds $\langle w,w'\rangle$ from the alternative relation where $w'$ is a world where the mass-standard exists, then we can falsify ${\mathsf{AxDir}}$ while the other axioms remain true, i.e., ${\mathsf{AxDir}}$ is independent from the rest axioms of ${\mathsf{MSpecRelDyn}}$. If we do the same with a $w'$ where instead of the mass-standard, one of its equivalents exists, we have the independence of ${\mathsf{AxPDirComp}}$. To show that ${\mathsf{AxCollRel}}$ is independent, it is enough to copy an arbitrary world of $\mathfrak M$ which sees the same alternatives, but which is seen by no one. If, in this copied world, we alter the speed of one body, we can violate ${\mathsf{AxCollRel}}$ while the other axioms remain true.

The independence of ${\mathsf{AxIndir}}$ from the rest of the axioms of ${\mathsf{MSpecRelDyn}}$ can be showed by a one-world model where there is only one observer and only one resting ordinary inertial body resting to that observer. However, in the light of our motivation of $\mathfrak M$, it is worth to examine the independence of ${\mathsf{AxIndir}}$ in the axiom system ${\mathsf{MSpecRelDyn}}\cup \{\exists {\mathsf{2IOb}}\}$ as well. The existence of indirect measurements is a consequence of ${\mathsf{MSpecRelDyn}}\cup \{\exists {\mathsf{2IOb}}\}$, the idea is used in the proof of the Mass Increase Theorem (Thm. 3). However, the independence of the uniqueness of the results of indirect measurements is a question for further research. Our conjecture is that it is independent.

It is also a question for further research whether the last axiom of MSpecRelDyn, ${\mathsf{AxEqSym}}$ is independent from the rest of ${\mathsf{MSpecRelDyn}}$ or not; our conjecture is that it is. However, a model which would be capable of showing this must be entirely different from the above outlined $\mathfrak M$.²⁷