1 Introduction↩︎

The topic of superluminal particles, which are nowadays called tachyons (term coined by Feinberg [1]), is still controversial among physicists for two reasons. The first one is that the existence of tachyons carrying information is commonly believed to lead to violations of causality [2]. The second one is that no experimental evidence has been found for their existence. On the theoretical side, different attempts of constructing a legitimate theory were made. However, none of them was accepted by the whole community. On the experimental side, some physicists claimed that they registered tachyons, but the result of their experiment was either not repeatable or it was later shown that a mistake was made in the measurement process (see [3] for review of such efforts).

The recent publication of Dragan and Ekert has once again renewed interest in tachyon-related physics [4]. In their work, they proposed that the theory of relativity and quantum mechanics are deeply connected. They argued that the violations of causality associated with tachyons explain the probabilistic nature of quantum effects. According to them, the presence of tachyons in the theory does not lead to causal paradoxes as in [2], but rather changes our notion of causality. Their novel ideas triggered a heated debate in the scientific community [5]–[11].

In traditional relativity, where only subluminal reference frames are considered, a tachyon is a particle that is never at rest. Thereby, various authors considered the concept of superluminal reference frames. Such studies date back at least to the work of Parker [12], where the Lorentz transformation has the following form \[\begin{align} &|u|<1: \; t' = \gamma(u)(t-u x), \quad x'=\gamma(u)(x-u t),\\ &|u|>1: \;t' = -\mathop{\mathrm{\text{sgn}}}(u)\gamma(u)(t-u x), \quad x'=-\mathop{\mathrm{\text{sgn}}}(u)\gamma(u)(x-ut) \tag{1} \end{align} \tag{2}\] and the \(-\mathop{\mathrm{\text{sgn}}}(u)\to\mathop{\mathrm{\text{sgn}}}(u)\) version of (2 ) is said to lead to the same physics (\(\mathop{\mathrm{\text{sgn}}}(x)=0,\pm1\) is the sign function). According to these equations, the primed reference frame moves with the dimensionless velocity \(u\) relative to the unprimed one (the velocity is measured in the units of the speed of light) and \(\gamma(u)=1/|1-u^2|^{1/2}\). We say that the velocity \(u\) is subluminal for \(|u|<1\) and superluminal for \(|u|>1\).

Besides (2 ), there are other versions of the Lorentz transformation present in the literature such as \[u\in\mathbb{R}\setminus\{1,-1\}: \; t' = \gamma(u)(t-u x), \quad x'=\gamma(u)(x-u t) \label{Itr}\tag{3}\] or its \(\gamma(u)\to\mathop{\mathrm{\text{sgn}}}(1-u^2)\gamma(u)\) version (see e.g. [13], references cited in [14], and [15]). However, it can be shown that none of them is actually acceptable [14], which illustrates the fact that the extension of the standard Lorentz transformation to the superluminal regime involves some subtle issues.

In this work, we will introduce the \(2\)-velocity of a reference frame and the clockwork postulate in Sec. 2, which will allow us to present superluminal Lorentz transformations from a different angle. We will examine in Secs. 3 and 4 the new formalism in the context of basic special relativity effects. The discussion of (2 ) and (3 ) from the perspective of our formalism will be presented in Sec. 5. The summary of our work will be given in Sec. 6.

2 \(2\)-velocity of reference frame↩︎

When one thinks about the Lorentz transformation in \(1+1\) dimensional spacetime, one considers two reference frames moving relative to each other with some velocity, say \(u\). Then, one uses such a velocity to parametrize the coefficients in the transformation relating the spacetime coordinates in the two reference frames. Such coefficients take the form \[\pm \gamma(u), \;\pm \gamma(u)|u|, \label{ab}\tag{4}\] where the signs in both expressions are independently chosen. While there is in principle nothing wrong with such a procedure, we find it hardly satisfactory for the following reason. Namely, the special theory of relativity is about physics happening in spacetime, where the time-like and space-like features appear on equal footing in different contexts. Thereby, it is our opinion that it would be more natural to use the \(2\)-velocity to characterize the relation between the spacetime coordinates in the two reference frames. Such an observation also naturally follows from (4 ), which suggests the consideration of the \(2\)-vector \[U=(\pm \gamma(u),\pm \gamma(u)|u|), \label{Upm}\tag{5}\] which is reminiscent of the \(2\)-velocity of a relativistic particle. The basic property of (5 ) is that \[U\cdot U \equiv \left(U^0\right)^2 - \left(U^1\right)^2= \mathop{\mathrm{\text{sgn}}}(1-u^2)= \left\{ \begin{array}{l} +1 \;{ \ \text{for} \ } u \;\text{subluminal}\\ -1 \;{ \ \text{for} \ } u \;\text{superluminal} \end{array} \right. \label{UUs}\tag{6}\] regardless of the choice of signs in (5 ) (all dot products are defined in this work with the metric tensor \(\text{diag}(1,-1)\); the signs in (5 ) are independently chosen so that there are four expressions encoded in such an equation).

We propose the following parametrization of the Lorentz transformation \[t' = U\cdot U(U^0 t- U^1 x), \quad x'=U\cdot U(U^0 x- U^1 t), \label{tprimuu}\tag{7}\] which leads to the following inverse transformation after the employment of (6 ) \[t = U^0 t'+ U^1 x', \quad x=U^0 x'+ U^1 t'. \label{tuu}\tag{8}\]

To give physical meaning to \(U\), we remark that the orientation of the \(t'\) axis on the spacetime diagram \((t,x)\) is given by \(U\). Such an observation follows from the \(x'=0\) version of (8 ). The four superluminal options resulting from different sign choices in (5 ) are illustrated in Fig. 1.

Clockwork postulate. In order to add meaning to the four possible orientations of the \(t'\) axis, we introduce the clockwork postulate according to which the proper time of an inertial observer always increases. In other words, an observer always becomes older (never younger) in his own rest frame, which nicely fits the basic assumption of special relativity that all inertial observers and/or reference frames are equivalent. This postulate implies that the stationary observer in the primed reference frame always moves in the positive direction of the \(t'\) axis. The above-stated clockwork postulate should not be confused with the clock postulate (also known as the clock hypothesis), which is based on the assumption that the rate of operation of the ideal clock in motion depends only on its instantaneous velocity (see [16] for a recent critical discussion of the clock postulate).

The remark formulated below (8 ), along with the clockwork postulate, leads to the conclusion that \(U\) is the relativistic \(2\)-velocity of the primed reference frame relative to the unprimed one. Thereby, the four options displayed in Fig. 1 come from the fact that the primed reference frame can be moving either forwards (\(U^0>0\)) or backwards (\(U^0<0\)) in time \(t\) and either in the positive (\(U^1>0\)) or in the negative (\(U^1<0\)) direction of the \(x\) axis. In other words, all spacetime options for the motion of the reference frame are encoded in \(U\). Finally, we note that reference frames moving either forwards or backwards in time and in different directions in space were considered in \(1+1\) (\(1+3\)) dimensional spacetime in the paper of Viera [17] (Sutherland and Shepanski [18]). However, the above introduced \(U\)-parametrization of these transformations was not explored in [17], [18].

3 Properties of \(U\)-parametrized transformation↩︎

We begin the discussion here from the derivation of the addition law for \(2\)-velocities. We consider three inertial reference frames \(O\), \(O'\) and \(O''\). We assume that \(O'\) is moving with the \(2\)-velocity \(V\) relative to \(O\), \(O''\) is moving with the \(2\)-velocity \(U\) relative to \(O\) and \(U'\) relative to \(O'\). To determine \(U'\), we note that such a \(2\)-velocity is defined via \[\binom{t'}{x'}= \binom{U'^0 \quad U'^1}{U'^1 \quad U'^0} \binom{t''}{x''},\] which can be compared to \[\binom{t'}{x'}= V\cdot V\binom{V^0U^0-V^1U^1 \quad V^0U^1-V^1U^0}{V^0U^1-V^1U^0 \quad V^0U^0-V^1U^1} \binom{t''}{x''}\] that is obtained via the transformation \(O'\to O\) followed by the \(O\to O''\) transformation.¹ This leads to the identification \[\label{Uti} U'=V\cdot V(V\cdot U, \varepsilon_{\mu\nu} V^\mu U^\nu), \;\varepsilon_{\mu\nu}=-\varepsilon_{\nu\mu}, \; \varepsilon_{01}=+1.\tag{9}\] Alternatively, one may obtain (9 ) by the Lorentz transformation of the \(2\)-velocity \(U\). This is achieved by the following replacements imposed on (7 ): \(U\to V\) followed by \((t,x)\to (U^0,U^1)\), and \((t',x')\to (U'^0,U'^1)\). Moving on, we note that (9 ) results in \[\begin{align} U'\cdot U' =(V\cdot V)(U\cdot U)= \left\{ \begin{array}{l} +1 \; { \ \text{for} \ } U { \ \text{and} \ } V \;\text{subluminal}\\ +1 \; { \ \text{for} \ } U { \ \text{and} \ } V \;\text{superluminal}\\ -1 \; \;\text{otherwise} \end{array} \right.. \label{tU2} \end{align}\tag{10}\] Thereby, the relative velocity of two reference frames is always superluminal when, relative to some other reference frame, one of them is superluminal while the other one is subluminal. Otherwise, the discussed relative velocity is subluminal, which we find interesting (obvious) when both reference frames are superluminal (subluminal) with respect to some other reference frame.

Equation (9 ) can be cast into the following more familiar form. With every \(2\)-vector \(U\), we associate the velocity \(u\) via \[u=\frac{U^1}{U^0}. \label{1vel}\tag{11}\] Such a definition follows from the \(x'=0\) version of (8 ) and it leads to the following \(U\)-parametrization \(U=\left(U^0,U^0u\right)=\pm\left(\gamma(u),\gamma(u)u\right)\). Combining (9 ) and (11 ), we arrive at \[u'=\frac{V^0 U^1-V^1 U^0}{V^0U^0-V^1U^1}, \label{util}\tag{12}\] which is antisymmetric with respect to the \(U\leftrightarrow V\) transformation in accordance with standard expectations. We find it curious that (9 ) does not exhibit such a symmetry. We note that whenever \(V\sim(1,v)\) and \(U\sim(1,u)\), (12 ) leads to the standard formula (see e.g. [19], [20]) \[u'=\frac{u-v}{1-u v}. \label{add}\tag{13}\]

Keeping in mind that \(V^0=\mathop{\mathrm{\text{sgn}}}(V^0)\gamma(v)\), \(V^1=\mathop{\mathrm{\text{sgn}}}(V^0)\gamma(v)v\), etc., we arrive at another representation of (9 ) \[\begin{align} \tag{14} &V\cdot U\neq0: \;U'=V\cdot V\mathop{\mathrm{\text{sgn}}}(V\cdot U)\left(\gamma(u'),\gamma(u')u'\right),\\ &V\cdot U=0: \;U'=\left(0,\mathop{\mathrm{\text{sgn}}}(V^0 U^1)\right). \tag{15} \end{align}\] Two remarks are in order now.

Firstly, we note that \(V\cdot U=0\) is satisfied by \(V=\pm(U^1,U^0)\) or equivalently \(U=\pm(V^1,V^0)\), which can be easily visualized on spacetime diagrams such as the ones depicted in Fig. 1. In the traditional nomenclature, \(V\cdot U=0\) amounts to the well-known condition \(uv=1\) for \(|u'|=\infty\) (see e.g. [19], [20]), where \(u\) and \(v\) are defined as in (11 ). We would like to stress that the condition \(V\cdot U=0\), unlike \(uv=1\), has clear geometrical meaning in Minkowski spacetime. We also note that the right-hand side of (15 ) implies \(|u'|=\infty\).

Secondly, (14 ) can be used to argue that reference frames propagating backwards in time inevitably appear in our formalism when superluminal velocities are considered, which is counterintuitive. Indeed, taking \(V=(\gamma(v),\gamma(v)v)\) and \(U=(\gamma(u),\gamma(u)u)\), where \(V\cdot U\neq0\) and both \(2\)-velocities describe reference frames moving forwards in time \(t\), we see from (14 ) that \(U'\) describes propagation backwards in time \(t'\) when² \[\mathop{\mathrm{\text{sgn}}}(1-v^2)\mathop{\mathrm{\text{sgn}}}(1-uv)=-1, \label{minus1}\tag{16}\] which can be satisfied when at least one of the velocities is superluminal.

Finally, we mention that the transformation \[(t,x) \;\leftrightarrow \;(t',x') \label{tyghj}\tag{17}\] is induced by \[\left(U^0,U^1\right) \;\rightarrow \;U\cdot U \left(U^0,-U^1\right), \label{Urec}\tag{18}\] which can be seen as velocity reciprocity in our formalism (see [21] for the recent comprehensive discussion of velocity reciprocity in relativity theories). For subluminal \(U\), (18 ) reduces to \(\left(U^0,U^1\right) \rightarrow \left(U^0,-U^1\right)\): the spatial component of the \(2\)-velocity gets flipped. For superluminal \(U\), (18 ) reads \(\left(U^0,U^1\right) \rightarrow \left(-U^0,U^1\right)\): the temporal component of the \(2\)-velocity gets flipped. We mention in passing that (17 ) is enforced by the flip of the velocity, i.e. \(u\to-u\), in subluminal and superluminal transformations (2 ). This is not the case in our formalism in the superluminal case.

4 Length “contraction” and time “dilation”↩︎

We discuss here further properties of transformation (7 ) and its inverse (8 ), i.e. we again assume that the primed reference frame moves with the \(2\)-velocity \(U\) with respect to the unprimed reference frame.

Length “contraction”. Measuring the length of a rod requires that an observer, in his own reference frame, simultaneously determines the spatial positions of the rod’s endpoints. We consider the rod whose endpoints are located at \(A=(t'_A,x'_A)\) and \(B=(t'_B,x'_B)\) in the primed reference frame and at \(C=(t_C,x_C)\) and \(D=(t_D,x_D)\) in the unprimed reference frame, respectively. We assume that the rod is at rest in the primed reference frame and so its proper length \(\ell'\) is given by \(|x'_A-x'_B|\) even when the events \(A\) and \(B\) are not simultaneous in the primed reference frame. The rod’s length in the unprimed reference frame is \(\ell=|x_C-x_D|\) as long as \(t_C=t_D\). It follows from (7 ) that the spatial coordinates of the events \(C\) and \(D\), as observed in the primed reference frame, are \[x'_C=U\cdot U (U^0 x_C - U^1 t_C), \;x'_D=U\cdot U (U^0 x_D - U^1 t_D).\] Subtracting these two equations from each other, and keeping in mind that \(x'_C=x'_A\), \(x'_D=x'_B\), and \(t_C=t_D\), we obtain \[\ell=\frac{\ell'}{|(U\cdot U)U^0|}=\frac{\ell'}{\gamma(u)}\] via (5 ). These considerations are illustrated in Fig. 2, where we set \(t'_A=t'_B\) and consider the primed reference frame moving either forwards or backwards in time with a superluminal velocity.

Time “dilation”. We consider a clock at rest in the primed reference frame. This clock measures the time interval \(\Delta t'_{AB}=t'_B-t'_A\) between events \(A=(t'_A,x'_A)\) and \(B=(t'_B,x'_B)\) occurring at \(x'_A=x'_B\). As measured by the clock resting in the unprimed reference frame, the time interval separating these events is \[\Delta t_{AB}=t_B-t_A=U^0\Delta t'_{AB}=\mathop{\mathrm{\text{sgn}}}(U^0)\gamma(u)\Delta t'_{AB} \label{DilT}\tag{19}\] according to (8 ) and (5 ). It is clear from (19 ) that the sequence of events \(A\) and \(B\) is swapped when \(\mathop{\mathrm{\text{sgn}}}(U^0)=-1\), i.e. when the primed reference frame is moving backwards in time \(t\). Such a conclusion can be easily visualized with the help of the diagrams from Fig. 1 (one may assume for simplicity that the events in the primed reference frame take place on the \(t'\) axis: \(x'_A=x'_B=0\)).

Finally, we note that for \(|u|>\sqrt{2}\): \(\ell>\ell'\) (contraction is not always seen) and \(|\Delta t_{AB}|<|\Delta t'_{AB}|\) (dilation is not always seen). These remarks explain our use of quotation marks around the words contraction and dilation. We remark that the special role of \(u=\sqrt{2}\) in the context of length “contraction” and time “dilation” was noted in [22].

5 Restricted transformations↩︎

We discuss here transformations restricted to only two (out of four possible) spacetime orientations of the \(2\)-velocity \(U\). On the one hand, this will give us another opportunity to illustrate how our \(U\)-parametrization works in practice. On the other hand, this will allow us to discuss from a different angle superluminal Lorentz transformations that were studied in the literature.

We begin the discussion of such transformations from (3 ), which looks like a natural extension of the standard Lorentz transformation to superluminal velocities. In our \(U\)-parametrization, (3 ) reads \[u\in\mathbb{R}\setminus\{1,-1\}: \;U = \mathop{\mathrm{\text{sgn}}}(1-u^2)\left(\gamma(u),\gamma(u)u\right). \label{ItrU}\tag{20}\] Transformations having such a structure, however, do not form a group, which can be shown with the help of (14 ). Namely, we choose \(u\) and \(v\) such that \(u v\neq1\), which allows us to put (20 ) and \(V=\mathop{\mathrm{\text{sgn}}}(1-v^2)\left(\gamma(v),\gamma(v)v\right)\) into (14 ). This leads to \[U'= \mathop{\mathrm{\text{sgn}}}(1-u^2)\mathop{\mathrm{\text{sgn}}}(1-uv)\left(\gamma(u'),\gamma(u')u'\right)=\mathop{\mathrm{\text{sgn}}}(1-v^2)\mathop{\mathrm{\text{sgn}}}(1-u v)\mathop{\mathrm{\text{sgn}}}(1-u'^2)\left(\gamma(u'),\gamma(u')u'\right), \label{sdwdewd}\tag{21}\] where the last equality follows from \[\mathop{\mathrm{\text{sgn}}}(1-u'^2)=\mathop{\mathrm{\text{sgn}}}(1-v^2)\mathop{\mathrm{\text{sgn}}}(1-u^2)\] obtained from (13 ). Result (21 ) does not agree with the \(u\to u'\) version of (20 ) when (16 ) holds. If both \(u\) and \(v\) are subluminal, then (16 ) cannot be satisfied, which is expected because we deal in such a case with the standard Lorentz transformation. However, when at least one of these velocities is superluminal, then the other can always be chosen so as to satisfy (16 ). The very same problem appears when one uses the \(\gamma(u)\to\mathop{\mathrm{\text{sgn}}}(1-u^2)\gamma(u)\) version of (3 ), where \[u\in\mathbb{R}\setminus\{1,-1\}: \; U =\left(\gamma(u),\gamma(u)u\right). \label{ItrUU}\tag{22}\] Thereby, (3 ) and its \(\gamma(u)\to\mathop{\mathrm{\text{sgn}}}(1-u^2)\gamma(u)\) version break the group property that any Lorentz transformation should satisfy, which was noted in [14]. In our formalism, the clockwork principle leads to the conclusion that superluminal (20 ) [(22 )] describes reference frames moving only backwards [forwards] in time \(t\) and either in the positive or negative direction of the \(x\) axis (this is evident from the fact that superluminal \(U\) representing (20 ) [(22 )] are depicted in Figs. 1c and 1d [Figs. 1a and 1b]). This observation shows that the restriction to reference frames, which according to our formalism propagate in a fixed direction in time, is insufficient when superluminal velocities are considered.

Then, we take a close look at transformation (2 ) employed in [4], [12], [14], [22]. Transformations having such a structure form a group (see e.g. [4] for a recent insight into this issue as well as [14]). This is seen in our formalism as follows. The \(U\)-parametrization of (2 ) is \[\begin{align} &|u|<1: \;U=\left(\gamma(u),\gamma(u)u\right),\\ &|u|>1: \;U=\left(\mathop{\mathrm{\text{sgn}}}(u)\gamma(u),\gamma(u)|u|\right), \tag{23}\end{align} \tag{24}\] which can be equivalently written as \[u\in\mathbb{R}\setminus\{1,-1\}: \; U=\mathop{\mathrm{\text{sgn}}}(1+u)\left(\gamma(u),\gamma(u)u\right) \label{UIV}\tag{25}\] thanks to the compact representation of (2 ) proposed in [6]. We choose \(u\) and \(v\) such that \(u v\neq1\) and substitute (25 ) and \(V=\mathop{\mathrm{\text{sgn}}}(1+v)\left(\gamma(v),\gamma(v)v\right)\) into (14 ) getting \[U'=\mathop{\mathrm{\text{sgn}}}(1-v)\mathop{\mathrm{\text{sgn}}}(1+u)\mathop{\mathrm{\text{sgn}}}(1- uv)\left(\gamma(u'),\gamma(u')u'\right)=\mathop{\mathrm{\text{sgn}}}(1+u')\left(\gamma(u'),\gamma(u')u'\right), \label{hjkop}\tag{26}\] where the last equality is obtained from (13 ). Result (26 ) proves that \(U'\) is given by the \(u\to u'\) version of (25 ) when \(u v\neq1\). For \(uv=1\), one may easily arrive at the same conclusion by using (15 ) instead of (14 ). Moreover, by the same token one may show that the \(-\mathop{\mathrm{\text{sgn}}}(u)\to\mathop{\mathrm{\text{sgn}}}(u)\) version of (2 ), which is \(U\)-parametrized as \[u\in\mathbb{R}\setminus\{1,-1\}: \; U=\mathop{\mathrm{\text{sgn}}}(1-u)\left(\gamma(u),\gamma(u)u\right) =\left\{ \begin{array}{l} |u|<1: \;U=\left(\gamma(u),\gamma(u)u\right) \\ |u|>1: \;U=-\left(\mathop{\mathrm{\text{sgn}}}(u)\gamma(u),\gamma(u)|u|\right) \end{array} \right., \label{UV}\tag{27}\] also satisfies the group property. Proceeding as above, we note that the clockwork principle leads to the observation that superluminal (25 ) [(27 )] describes reference frames moving in the positive [negative] direction of the \(x\) axis and either forwards or backwards in time \(t\): \(U\) corresponding to superluminal (25 ) [(27 )] are shown in Figs. 1a and 1d [Figs. 1b and 1c].

To fix such a problematic one-directional movement in space, one is forced to use the reinterpretation principle, which states that motion backwards in time \(t\) in the positive (negative) direction of the \(x\) axis represents motion forwards in time \(t\) in the negative (positive) direction of the \(x\) axis [23]. Therefore, the reinterpretation principle allows one to work with only two orientations of spacetime axes as in Figs. 1a and 1d or Figs. 1b and 1c (instead of four as in Figs. 1a–1d). However, one should be aware of the fact that by choosing (2 ) as in [4], [12], [14], [22], one deals with the situation, where the superluminal observer moving in the positive (negative) direction of the \(x\) axis uses a clock with a normal (inverted) mechanism (the hands of the clock are moving clockwise (counterclockwise) in these two cases, so to speak). Similarly, by using (2 ) subjected to the \(-\mathop{\mathrm{\text{sgn}}}(u)\to\mathop{\mathrm{\text{sgn}}}(u)\) replacement, one assumes that the superluminal observer uses a clock with a normal (inverted) mechanism when it moves in the negative (positive) direction of the \(x\) axis. We see such dependence of the clock’s mechanism on the direction of motion as a counterintuitive feature.

6 Summary↩︎

In the spirit of the theory of relativity, we have considered reference frames moving in all possible directions in space and time. In particular, this implies that we have taken into account reference frames moving both forwards and backwards in time. While the exploration of both options does not seem to be necessary in the subluminal context, the situation is far less clear in the superluminal context, where one is bound to encounter counterintuitive features and lack of experimental data leaves various possibilities open. We hope that our work will stimulate the discussion of this issue.

In our formalism, we have parameterized Lorentz transformations via the \(2\)-velocity of a reference frame and introduced the clockwork principle to give physical meaning to this quantity. We have studied then the group property of such transformations and re-examined basic effects such as the length “contraction” and the time “dilation”. The new formalism has been then compared to the standard approach discussed in [4], [12], [14], [22]. In the course of these studies, we have identified counterintuitive features both in our formalism and in the above-mentioned standard approach (see the discussion around (16 ) and by the end of Sec. 5). In fact, we believe that any approach to superluminal systems is going to encounter some conceptual difficulties needing detailed discussion. Such a remark partly motivates research pursuits discussed in this work, which we hope gives a non-standard perspective on superluminal physics. Finally, we note that it is of interest to extend the formalism presented in this work to higher dimensional spacetimes, particularly in light of recent developments presented in [24].

References↩︎