Introduction

The theory of thermodynamic relativity describes velocity (kinematics) and entropy (thermodynamics) under a common physical framework, also leading to an anisotropic adaptation of Einstein’s special relativity 1. Here, we show the kinematic derivation of the addition rule of one-way velocities that leads to exactly the same description of anisotropic special relativity.

The derivation of thermodynamic relativity emerged from the investigation of a longstanding problem in regard to the energy distributions of space plasma particle populations, called kappa distributions 2,3,4,5,6,7,8,9,10,11,12,13,14. These distributions deviate from the classical Maxwellian behavior 15, and thus cannot be described by classical thermodynamics 2,5.

The theoretical foundation of kappa distributions, and their associated thermodynamics, has its origin in the entropy defect 2,3,16,17,18,19,20. This is the concept that describes the ways that the entropy of a system can be shared among its constituents and provides the most generalized addition rule for entropies 2,3,4,5. The concept of entropy defect and the addition rule of entropies is based on three postulates 1. Surprisingly, these are identical to the postulates of special relativity, and thus both velocities and entropies are expressed by a common addition rule. For instance, in classical physics, the addition rule of velocities is restricted to a simple summation (Galilean kinematics 21), while the addition rule of entropies is also restricted to a simple summation (Boltzmann-Gibbs thermodynamics 22,23).

In thermodynamic relativity 1, the derived addition rule of velocities can lead to Einstein’s special relativity, which is an isotropic theory. Moreover, it can naturally lead to a generalized adaptation of special relativity, which also corresponds to linear spacetime (Lorentz) transformations 24. This is an anisotropic theory, which brings up the problem of synchronization. This paper sheds light on the synchronization principle, and how this can lead to an anisotropic adaptation of special relativity, identical to that, which we found through the theory of thermodynamic relativity, and therefore further support the new theory of thermodynamic relativity.

The speed of light is generally assumed to be isotropic, or independent of direction. This speed can be readily measured over a two-way (round trip) path, but this only provides the average speed over the two directions. In contrast, the one-way speed along one or the other direction is unknown and even worse, potentially undefined. This problem arises from the fact that we cannot have both the knowledge of synchronization of clocks (time) and the one-way speed, but one of these must be predetermined. Indeed, synchronized clocks at different locations are needed for measuring the one-way speeds, but knowledge of the one-way speeds is necessary for the synchronization. As Anderson et al. 25 summarized, “one cannot hope even to test the isotropy of the speed of light without, in the course of the same experiment, deriving a one-way numerical value at least in principle, which then would contradict the conventionality of synchrony.”

The way that synchronization is defined has a direct effect on the spacetime measurements, as well as on their transformations among various stationary or nonstationary observers. As said, once the synchronization principle is given, then, the one-way speed can be determined and measured. Einstein’s convention provides such a synchronization 26, which is simple and consistent with an isotropic relativity. However, an arbitrary synchronization should include anisotropy as a free parameter, which can be either a universal constant, or more generally, a spacetime scalar function. We note that the respective Lorentz transformation is linear only when the anisotropy parameter is constant.

In Einstein’s synchronization convention, one assumes zero anisotropy; namely, distant clocks are synchronized so that the one-way speed of light becomes equal to the two-way speed of light. This is formulated as follows. Let the two observers A and B be stationary to each other in distance L. Then, A sends a signal at time \(t={t_0}\), arriving at B at time \(t={t_0}+{\tau _{{\text{B,A}}}}\), which returning to A at time \(t={t_0}+{\tau _{{\text{B,A}}}}+{\tau _{{\text{A,B}}}}\), where t stands for the time measured by the clock of A.

Einstein’s isotropic synchronization requires equal time periods for both of the one-way trips,

$${\tau _{{\text{B,A}}}} - {\tau _{{\text{A,B}}}}=0,$$
(1a)

and thus, B receives the signal at exactly the half of the period of the two-way trip \({\tau _{{\text{2-way}}}}\), i.e.,

$${\tau _{{\text{B,A}}}}=\tfrac{1}{2} \cdot {\tau _{{\text{2-way}}}}, {\tau _{{\text{2-way}}}}={\tau _{{\text{B,A}}}}+{\tau _{{\text{A,B}}}}.$$
(1b)

There is only one free parameter that is subject to generalization in Einstein’s convention, and this is the proportion of the time periods of each one-way trip. If this proportion is not \({\tau _{{\text{B,A}}}}:{\tau _{{\text{A,B}}}}=1:1\), then, there is a nonzero difference between the time periods, \({\tau _{{\text{B,A}}}} - {\tau _{{\text{A,B}}}} \ne 0\), which can be normalized to the two-way period, and define the anisotropy through this ratio:

$$r \equiv ({\tau _{{\text{B,A}}}} - {\tau _{{\text{A,B}}}})/({\tau _{{\text{B,A}}}}+{\tau _{{\text{A,B}}}}) \ne 0,$$
(2a)

thus, B receives the signal at some portion of the period of the two-way trip, i.e.,

$${\tau _{{\text{B,A}}}}=\tfrac{{r+1}}{2} \cdot {\tau _{{\text{2-way}}}}, {\tau _{{\text{2-way}}}}={\tau _{{\text{B,A}}}}+{\tau _{{\text{A,B}}}},$$
(2b)

where Eqs. (2a, 2b) recover the respective Eqs. (1a, 2b) in the isotropic case (i.e., when r= 0). This generalized synchronization scheme was adopted by multiple authors over the previous decades 1,25,27,28,29,30,31, but is still sometimes overlooked by the broader community.

Each of the one-way speeds has a unique upper limit (speed of light), whose value is finite but, in principle, could be anywhere between zero and infinity; the only requirement is that the one-way speeds are consistent with their two-way average, as this is subject to observations and measurements. If c+ and c− denote the one-way speeds of light in the departing and approaching directions with respect to an observer, respectively, then, for the previous formalism for the stationary observers A and B, we have the time periods \({\tau _{{\text{B,A}}}}=L/{c_+}\) and \({\tau _{{\text{A,B}}}}=L/{c_ - }\) (with respect to A), and thus, the anisotropy r is derived from Eq. (2a) as

$$r=\left( {{c_+}^{{ - 1}} - {c_ - }^{{ - 1}}} \right) \cdot {c_{\rm O}},$$
(3a)

while the observable speed of light is given by the harmonic average

$${c_{\rm O}}^{{ - 1}} \equiv \tfrac{1}{2}\left( {{c_+}^{{ - 1}}+{c_ - }^{{ - 1}}} \right).$$
(3b)

The measurable value of the upper limit is given by the average of the one-way limits given by Eq. (3b). This defies Einstein’s synchronization, which assumed isotropy. However, all of the experimental tests that isotropic relativity has passed are independent of this convention (e.g., 1,25), and thus, do not actually demonstrate isotropy. Given any nonzero anisotropy r, and the measured speed of light, \({c_{\rm O}}\), the two one-way speeds of light are given by:

and

$${c_+}={c_{\rm O}}/(1+\tfrac{1}{2}r)\quad and\quad {c_ - }={c_{\rm O}}/(1 - \tfrac{1}{2}r),$$
(4a)

which stand as the upper limit of any object’s speed V, in the departing (+) and approaching (-) directions, respectively, i.e., \({c_ - }<V<{c_+}\).

We call the departing and approaching directions (with respect to an observer) as positive (+) and negative (-) directions, respectively; (corresponding to positive and negative radial velocity, respectively). Importantly, we note that this does not generally refer to the sign of the velocity, but to the effect of the anisotropy on the one-way speeds. For instance, let a particle P with a constant velocity with respect to another distant observer A, approach, reach, and then, pass an observer B; while the particle will be always moving in the positive direction with respect to A, it will be moving in the negative direction and then at the positive direction with respect to B.

The one-way transmission of information is realized for any finite speed V, and not just for their upper limits \({c_ \pm }\). Indeed, the framework of anisotropic relativity relies on the basis that the anisotropy not only applies to the speed of light, but in addition, and in a similar way, to the speed of any object. If \({V_ \pm }\) are the one-way speeds, consistent with the measurable two-way speed \({V_{\text{O}}}\), then, their connection involves the anisotropy r, as follows:

and

$$r=\left( {{V_+}^{{ - 1}} - {V_ - }^{{ - 1}}} \right) \cdot {c_{\rm O}} {\quad}and {\quad}{V_{\rm O}}^{{ - 1}} \equiv \tfrac{1}{2}\left( {{V_+}^{{ - 1}}+{V_ - }^{{ - 1}}} \right).$$
(5)

Thus, given the anisotropy r, and the measured speed \({V_{\rm O}}\), the two one-way speeds \({V_ \pm }\) are

 

$${V_+}={V_{\rm O}}/(1+\tfrac{1}{2}r \cdot {V_{\rm O}}/{c_{\rm O}}) {\quad}and{\quad}{V_-}={V_{\rm O}}/(1+\tfrac{1}{2}r \cdot {V_{\rm O}}/{c_{\rm O}}).$$
(6)

The connection between speeds in the positive and negative directions is.

 

$${V_+}={V_ - }/(1+r \cdot {V_ - }/{c_{\rm O}}) {\quad}and{\quad} {V_ - }={V_+}/(1 - r \cdot {V_+}/{c_{\rm O}}).$$
(7a)

Note that our approach here is restricted to one-dimensional velocities, and in the above relations V stands for speeds. We chose to work with speeds to have smooth transition from the expression of anisotropy in terms of light speed (Eq. (3a)) to its expression in terms of any other finite speed (Eq. (5)). Then, we can easily convert a speed description into a velocity one, simply by considering the respective signs, e.g., in Eq. (7a), and switching \({V_ - } \to - {V_ - }\), leading to:

 

$${V_+}= - {V_ - }/(1 - r \cdot {V_ - }/{c_{\rm O}})\quad and \quad {V_ - }= - \,\,{V_+}/(1 - r \cdot {V_+}/{c_{\rm O}}).$$
(7b)

We emphasize the fact that the relations in Eqs. (7b) are derived directly from Eq. (5), namely, on the basis that the anisotropy applies to any object’s speed.

Furthermore, the rule of addition for one-way velocities, \({V_{{\text{B,O}}}}={V_{{\text{A,O}}}} \oplus {V_{{\text{B,A}}}}\), is given by:

$${V_{{\text{B,O}}}}=\frac{{{V_{{\text{B,A}}}}+{V_{{\text{A,O}}}} - (\tfrac{1}{{{c_+}}} - \tfrac{1}{{{c_ - }}}) \cdot {V_{{\text{B,A}}}}{V_{{\text{A,O}}}}}}{{1+\tfrac{1}{{{c_+}{c_ - }}} \cdot {V_{{\text{B,A}}}}{V_{{\text{A,O}}}}}}$$
(8a)

recovering Einstein’s isotropic relativity for \({c_+}={c_ - }=c\),

$${V_{{\text{B,O}}}}=\frac{{{V_{{\text{B,A}}}}+{V_{{\text{A,O}}}}}}{{1+\tfrac{1}{{{c^2}}} \cdot {V_{{\text{B,A}}}}{V_{{\text{A,O}}}}}}$$
(8b)

Therefore, the Lorentz transformation has been generalized to apply to any one-way speeds \({V_ \pm }\) 1,25,31. The transformation is represented by an asymmetric matrix, from which, the respective generalized Minkowsky nondiagonal metric was also derived by 1, having direct consequences for the energy-momentum-velocity relationships.

In1 we started from a very different basis than that of modifying the Lorentz transformation. Instead, we first developed the rule of addition for one-way velocities, and then showed that it characterizes an anisotropic adaptation of relativity. In particular, we found the existence of two upper limits, each bounding the one-way speed of one of the directions, while the speeds were subject to an algebra that differentiates the speeds along the two opposite directions, as given by Eq. (7b) (see also 16). The theoretical framework that provided a natural derivation of the anisotropic addition rule of velocities came from what we call “thermodynamic relativity,” a unified theoretical framework for describing both entropies and velocities, and their respective physical disciplines of thermodynamics and kinematics. Here, we start from the other end and show a kinematic derivation of the addition rule of the one-way velocities, and thus, to the framework of anisotropic relativity (Fig. 1).

In the next section, we show how the thermodynamic relativity leads to the anisotropic addition rule of velocities. Then, we describe three thought experiments, which apply the anisotropic synchronization principle expressed in Eqs. (3–5), and progressively lead to the same addition rule.

Fig. 1
figure 1

Thermodynamic relativity provides a natural derivation of addition of velocities and the anisotropic adaptation of Einstein’s relativity. This paper shows a kinematic derivation of the addition of velocities.

Full size image

Thermodynamic relativity as the origin of anisotropic relativity

In thermodynamic relativity, entropy and velocity are characterized by three identical postulates, interpreting potential physical laws that provide the basis of a broader framework of relativity. (Note that physical postulates are fundamental assumptions that are taken for granted in the development of physical theories; once experimentally tested and validated, they become physical laws 32.) The postulates lead to a unique form of addition for entropies and for velocities, called kappa-addition, and to a systematic method of constructing a nonlinear theory of relativity, fully consistent with both thermodynamics and kinematics.

These postulates are stated in a unified framework for both entropies and velocities (taken verbatim from 1):

  1. 1.

    There is no absolute reference frame in which entropy or velocity are zero; both entropy and velocity are always relative quantities, connected with the properties of symmetry and transitivity, broadly defined under a general addition rule that forms a mathematical group on the sets of their allowable values.

  2. 2.

    There exists a nonzero, finite value of entropy and of velocity, which remains fixed (i.e., constant for all times) and invariant (i.e., constant for all observers), thus, it has the same value in all reference frames, and constitutes the upper limit of any entropy and velocity, respectively.

  3. 3.

    If a system is stationary in thermodynamics (i.e., constant entropy - zeroth law of thermodynamics) and/or kinematics (i.e., constant velocity – characterization of inertia) for a reference frame O, it will be stationary for all reference frames that are stationary for O.

Below we show how these postulates lead to a unique rule of addition, common for both entropies S and velocities V. The kappa-addition of xA and xB, noted with \({x_{\text{A}}} \oplus {x_{\text{B}}}\), is interpreted as the x-value of the combined system A and B, noted with \({x_{{\text{A}} \oplus {\text{B}}}}\), and is given by:

$$H({x_{{\text{A}} \oplus {\text{B}}}}) \equiv H({x_{\text{A}}} \oplus {x_{\text{B}}})=H({x_{\text{A}}})+H({x_{\text{B}}}) - \tfrac{1}{\kappa }H({x_{\text{A}}})H({x_{\text{B}}}) ,$$
(9a)

that is, writing it for entropy

$${S_{{\text{A}} \oplus {\text{B}}}} \equiv {S_{\text{A}}} \oplus {S_{\text{B}}}={H^{ - 1}}\left[ {H({S_{\text{A}}})+H({S_{\text{B}}}) - \tfrac{1}{\kappa }H({S_{\text{A}}})H({S_{\text{B}}})} \right] ,$$
(9b)

and velocity

$${V_{{\text{A}} \oplus {\text{B}}}} \equiv {V_{\text{A}}} \oplus {V_{\text{B}}}={H^{ - 1}}\left[ {H({V_{\text{A}}})+H({V_{\text{B}}}) - \tfrac{1}{\kappa }H({V_{\text{A}}})H({V_{\text{B}}})} \right] ,$$
(9c)

where the involved function H is called the partitioning function as it defines the way that a quantity \({x_{{\text{A}} \oplus {\text{B}}}}\) of the combined system is shared among its parts, i.e., A and B subsystems, \({x_{\text{A}}}\) and \({x_{\text{B}}}\), respectively.

In thermodynamic relativity, the parameter kappa, κ, defines and determines the “interdependence” between any thermodynamically connected systems or the reference systems of the observers. It provides the upper limit of the described variables (entropies or speeds), and this property of having an upper limit is sufficient for connecting the variables such that they deviate from their classical sum; when the upper limit tends to infinity (namely, it doesn’t exist), then, the kappa addition of entropies or velocities recovers their classical sums of Boltzmann-Gibbs entropies 22 or Galilean velocities 21. In particular, we have for each:

  1. (a)

    In the context of thermodynamics, the kappa addition provides the generalized addition rule of entropies. The interdependence is interpreted as the correlations among the velocities/energies of particles: Correlations induce order in the involved systems and reduce their entropy, emerging thus as the entropy defect. 1/κ provides the magnitude of the entropy defect 1,2,16 and the correlations of particle kinetic energy per degrees of freedom 10,17,38. The kappa provides the upper limit of entropies 2. For κ⟶∞, correlations tend to zero, the system’s constituents become independent, and the entropy defect vanishes leaving the addition rule of the classical summing of entropies 20.

  2. (b)

    In the context of kinematics, the kappa addition provides the generalized addition rule of velocities. The interdependence allows for anisotropy among the connected systems, so that the anisotropy vanishes for κ⟶∞. The kappa provides the upper limit of speeds (that is, the speed of light in vacuum); in particular, any speed is bounded by V < 2κ. For κ⟶∞, interdependence tends to zero, and the addition rule of speed reduces to the classical Galilean summing of velocities 1.

The partitioning function is not uniquely determined function in \({\mathbb{R}} \to {\mathbb{R}}\), but it does have several thermodynamic properties that lead to the following mathematical constraints (see 1,16  for more details): (i) H ≥ 0, i.e., positive entropies should remain positive, while the zero holds at S = 0; the non-negativity comes from the requirement of H being equal S in the classical limit (see property (v) below); (ii) H(0) = 0, because adding zero entropy must have zero effect; indeed, setting \({S_{\text{B}}}=0\) in Eq. (9b), requiring that \(H({S_{{\text{A}} \oplus {\text{B}}}})=H({S_{\text{A}}})\), we obtain \(H(0) \cdot [1 - \tfrac{1}{\kappa }H({S_{\text{A}}})]=0\) for any \({S_{\text{A}}}\), thus H(0) = 0; (iii) \(H^{\prime}(0)=1\), where \(H^{\prime}(0)\) only appears in a ratio with kappa, thus its value is arbitrary and can be absorbed into the value of kappa; we show this for small entropies, where the kappa addition expanded to square terms gives \({S_{{\text{A}} \oplus {\text{B}}}} \cong {S_{\text{A}}}+{S_{\text{B}}} - H^{\prime}(0) \cdot \tfrac{1}{\kappa }{S_{\text{A}}}{S_{\text{B}}}\), which should coincide with the entropy defect for H = 1 2,5, \({S_{{\text{A}} \oplus {\text{B}}}}={S_{\text{A}}}+{S_{\text{B}}} - \tfrac{1}{\kappa } \cdot {S_{\text{A}}} \cdot {S_{\text{B}}}\), thus \(H^{\prime}(0)=1\); (iv) H monotonically increases, because its inverse H−1(S) must be defined, and it is surely positive at one point, \(H^{\prime}(0)=1\); (v) if H ≠ 1, then it is kappa dependent, but at the classical limit where κ→∞, it must reduce to the identity function H(S) = S 5,2,20; and (vi) the entropy defect must be positive, i.e., \({S_{\text{D}}} \equiv {S_{\text{A}}}+{S_{\text{B}}} - {S_{{\text{A}} \oplus {\text{B}}}}>0\), or \({S_{\text{A}}}+{S_{\text{B}}} - {H^{ - 1}}\left[ {H({S_{\text{A}}})+H({S_{\text{B}}}) - \tfrac{1}{\kappa }H({S_{\text{A}}})H({S_{\text{B}}})} \right]>0\).

The kappa-addition has the properties of symmetry and transitivity, which are crucial for characterizing the zeroth law of thermodynamic generalized equilibrium 16 (that is, the postulate of stationarity). To properly describe these properties, we consider three reference frames O, A and B. They are all thermodynamically connected with each other, through interactions that allow for the observations and exchange of information, forming also a single combined system. Hence, O, A and B constitute thermodynamically connected reference frames or observers, each behave as a correlated subsystem of the combined system. Notations \({x_{{\text{A,O}}}}\), \({x_{{\text{B,O}}}}\), and \({x_{{\text{A,B}}}}\), provide the entropy or velocity (respectively for the contexts of thermodynamics or kinematics) of the three connections, i.e., A as measured by O, B as measured by O, and A as measured by B, respectively.

Equation (9a) provides how the x value of B measured by O, \({x_{{\text{B,O}}}}\), and the x value of A measured by B, \({x_{{\text{A,B}}}}\), are combined to provide the x value of A measured by O, \({x_{{\text{A,O}}}}\); this comprises the transitivity property. On the other hand, the existence of the negative value of any value of x comprises the symmetry property. Then, the transitivity and symmetry properties are formulated by:

1. Transitivity, \({x_{{\text{A,O}}}}={x_{{\text{A,B}}}} \oplus {x_{{\text{B,O}}}}\)

$$H({x_{{\text{A,O}}}})=H({x_{{\text{A,B}}}} \oplus {x_{{\text{B,O}}}})=H({x_{{\text{A,B}}}})+H({x_{{\text{B,O}}}}) - \tfrac{1}{\kappa }H({x_{{\text{A,B}}}})H({x_{{\text{B,O}}}})$$
(10a)

2. Symmetry, \(0={x_{{\text{A,B}}}} \oplus {x_{{\text{B,A}}}}\)

$$0=H({x_{{\text{A,B}}}} \oplus {x_{{\text{B,A}}}})=H({x_{{\text{A,B}}}})+H({x_{{\text{B,A}}}}) - \tfrac{1}{\kappa }H({x_{{\text{A,B}}}})H({x_{{\text{B,A}}}})$$
(10b)

Furthermore, there is a specific partitioning function, corresponding to linear adaptations of special relativity. Using the thermodynamic relativity for velocities, we construct the basic formulations of the linear anisotropic special relativity, using the specific partitioning function, expressed in terms of an arbitrary parameter a,

$${H_a}(x)=x/(1+\tfrac{a}{\kappa }x)$$
(11)

which leads to the kappa-addition.

$${x_{{\text{A}} \oplus {\text{B}}}} \equiv {x_{\text{A}}} \oplus {x_{\text{B}}}=\frac{{{x_{\text{A}}}+{x_{\text{B}}} - \tfrac{{1 - 2a}}{\kappa }{x_{\text{A}}}{x_{\text{B}}}}}{{1+\tfrac{{a(1 - a)}}{{{\kappa ^2}}}{x_{\text{A}}}{x_{\text{B}}}}}, \quad {for} \,x: S\, or\, V$$
(12)

The generalized partitioning function Ha recovers the known extreme cases of the isotropic relativity of Einstein for a = 1/2 and of the framework of Tsallis nonextensive statistical mechanics 6,33,34,35,36,37 for a = 0:

$${x_{{\text{A}} \oplus {\text{B}}}}=\frac{{{x_{\text{A}}}+{x_{\text{B}}}}}{{1+\tfrac{1}{{{{(2\kappa )}^2}}}{x_{\text{A}}}{x_{\text{B}}}}}, \,for \, a = 1/2,$$
(13a)
$${x_{{\text{A}} \oplus {\text{B}}}}={x_{\text{A}}}+{x_{\text{B}}} - \tfrac{1}{\kappa }{x_{\text{A}}}{x_{\text{B}}}, \, for\,a=0,$$
(13b)

while intermediate cases correspond to anisotropic relativity.

Anisotropic relativity makes distinctions about the directionality of the observed system as compared to the reference frame of the observer. The characterization of the directions in the case of velocities is trivial: the direction of an observed system is set to be positive when it is departing from the observer, or otherwise, when the velocity is positive, as measured in the frame of the observer. Similarly, the direction of an observed system is set to be negative, when it is approaching the observer, or otherwise, when the velocity is negative, again, in the frame of the observer. Following the same pattern, the positive direction is defined for positive values of entropy of a system as measured by an observer, while the negative direction is defined for negative values of entropy of a system as measured by an observer. Note that our approach is one-dimensional, which is expected in the case of entropies, but is simplified for the case of velocities; (in the 3-D scenario, we would refer to the radial velocity component).

Anisotropy affects both the velocities and their limits. There are two upper limits of entropy and velocity, one at the positive and one at the negative direction (as said, corresponding to positive and negative values, respectively), and we can think of them as one-way values of entropies or velocities.

For both entropy and velocity, i.e., x = S or V, we specify these upper limits with κ+ and κ−, i.e., for the positive, \(0<x\): \(x<{\kappa _+}\), and for the negative direction, \(\bar {x}<0\): \(\left| {\bar {x}} \right|<{\kappa _ - }\). These limits are:

$$x<{\kappa _+} \equiv \tfrac{1}{{1 - a}}\kappa \quad and \quad \left| {\bar {x}} \right|<{\kappa _ - } \equiv \tfrac{1}{a}\kappa.$$
(14)

The physical meaning of kappa (κ) in thermodynamics is connected with the entropy defect (thermodynamic definition) and the phase-space correlations (kinetic definition)17,18 (see also 14,38:); in both cases, the kappa appears in its reciprocal 1/κ 2,17, so its physical representation is retained through its reciprocal. Therefore, the anisotropy produces a nonzero difference between these two upper limits, and is determined by.

$$\frac{1}{{{\kappa _+}}} - \frac{1}{{{\kappa _ - }}}=\frac{{1 - 2a}}{\kappa } \equiv \frac{1}{{{\kappa _{\rm O}}}} \cdot r,\, or\, r \equiv \left( {{\kappa _+}^{{ - 1}} - {\kappa _ - }^{{ - 1}}} \right) \cdot {\kappa _{\rm O}}.$$
(15a)

while its harmonic average is given by.

$$\frac{1}{{{\kappa _{\rm O}}}} \equiv \tfrac{1}{2}\left( {\frac{1}{{{\kappa _+}}}+\frac{1}{{{\kappa _ - }}}} \right)=\frac{1}{{2\kappa }}, \, or\, {\kappa _{\rm O}}=2\kappa.$$
(15b)

Then, the one-way limits are given by.

and

$${\kappa _+}={\kappa _{\rm O}}/(1+\tfrac{1}{2}r) \, and \, {\kappa _ - }={\kappa _{\rm O}}/(1 - \tfrac{1}{2}r).$$
(16)

The entropies or velocities may be expressed as normalized to their average limit, \(\beta =S/{\kappa _{\rm O}}\) or \(\beta =V/{c_{\rm O}}\). If a system B is observed with respect to the system A with entropy SB, A and velocity VB, A, and A is observed with respect to an observer O with entropy SA, O and velocity VA, O, then, B is observed with entropy SB, O and velocity VB, O with respect to O, both unified given by the kappa-addition, as

$${\beta _{{\text{B,O}}}} \equiv {\beta _{{\text{B,A}}}} \oplus {\beta _{{\text{A,O}}}}=\frac{{{\beta _{{\text{B,A}}}}+{\beta _{{\text{A,O}}}} - r \cdot {\beta _{{\text{B,A}}}}{\beta _{{\text{A,O}}}}}}{{1+[1 - {{(\tfrac{1}{2}r)}^2}] \cdot \,{\beta _{{\text{B,A}}}}{\beta _{{\text{A,O}}}}}}$$
(17)

.

Hereafter, we focus on the anisotropic relativity for velocities. The addition rule of velocities \({V_{{\text{B,O}}}}={V_{{\text{B,A}}}} \oplus {V_{{\text{A,O}}}}\), becomes

$${V_{{\text{B,O}}}}={V_{{\text{B,A}}}} \oplus {V_{{\text{A,O}}}}=\frac{{{V_{{\text{B,A}}}}+{V_{{\text{A,O}}}} - (\tfrac{1}{{{c_+}}} - \tfrac{1}{{{c_ - }}}) \cdot {V_{{\text{B,A}}}}{V_{{\text{A,O}}}}}}{{1+\tfrac{1}{{{c_+}{c_ - }}} \cdot {V_{{\text{B,A}}}}{V_{{\text{A,O}}}}}}$$
(18)

.

The velocity in the negative direction, noted by \(\bar {V}\), is expressed in terms of V as.

$${V_{{\text{A,B}}}}={\bar {V}_{{\text{B,A}}}}=\frac{{ - {V_{{\text{B,A}}}}}}{{1 - (\tfrac{1}{{{c_+}}} - \tfrac{1}{{{c_ - }}}){V_{{\text{B,A}}}}}}\, or \, {\beta _{{\text{A,B}}}}={\bar {\beta }_{{\text{B,A}}}}=\frac{{ - {\beta _{{\text{B,A}}}}}}{{1 - r{\beta _{{\text{B,A}}}}}},$$
(19)

where it recovers the classical algebraic \(\bar {V}= - V\) when \(r \to 0\).

The addition rule of velocities can be used as a starting point to derive the anisotropic adaptations of the (asymmetric) Lorentz transformation and the (nondiagonal) Minkowsky metric 39. Let the addition of speeds \(V^{\prime}=V \oplus u\). In 1 we showed how one can derive the Lorentz transformation by setting \(V=dx/dt\) and \(V^{\prime}=dx^{\prime}/dt^{\prime}\), leading to,

$$dX^{\prime}={L_r}(\beta ) \cdot dX, dX \equiv \left( {\begin{array}{*{20}{c}} {{c_{{{\rm O}}}}dt} \\ {dx} \end{array}} \right),{L_r}(\beta ) \equiv {\gamma _r} \cdot \left( {\begin{array}{*{20}{c}} 1&{(1 - \tfrac{1}{4}{r^2})\beta } \\ \beta &{1 - r\beta } \end{array}} \right),$$
(20a)

with the involved γ-factor, given by:

 

$${\gamma _r}=1/\sqrt {(1 - u/{c_1}) \cdot (1+u/{c_2})}\, or\, {\gamma _r}=1/\sqrt {{{(1 - \tfrac{1}{2}r\beta )}^2} - {\beta ^2}}.$$
(20b)

The metric ηr can be derived from the Lorentz transformation defining property that keeps the spacetime length ds2 invariant, that is, \((ds)^2=d{X^t} \cdot {\eta _r} {\cdot} dX=dX^{{\prime}{t}} {\cdot} {\eta _r}{\cdot} dX^{\prime}=d{X^t} {\cdot} \left[ {{L_r}^{t}(\beta ) {\cdot} {\eta _r} {\cdot} {L_r}(\beta )} \right] {\cdot} dX\), for any 4-vector dX, thus, \({L_r}^{t}(\beta ) \cdot {\eta _r} \cdot {L_r}(\beta )={\eta _r}\), from where we have derived the form of the metric 1:

$${\eta _r}=\left( {\begin{array}{*{20}{c}} { - 1}&{\tfrac{1}{2}r} \\ {\tfrac{1}{2}r}&{1 - \tfrac{1}{4}{r^2}} \end{array}} \right)$$
(21)

This recovers the isotropic Minkowsky metric, η0 = diag(−1,1), for zero anisotropy, r = 0.

Then, the invariant spacetime length, \({(ds)^2}=d{X^t} \cdot {\eta _r} \cdot dX\), is given by:

$$d{s^2}= - {({c_{{{ O}}}}dt)^2}+r\,{c_{{{\rm O}}}}dtdx+[1 - {(\tfrac{1}{2}r)^2}]\,d{x^2} ,$$
(22)

ds2 is invariant, thus equals the infinitesimal time period when there is no displacement, dx = 0, that is, the time ticking of a clock at its own frame, \(d{s^2}= - {({c_{{{\rm O}}}}d{t_0})^2}\); combining the two expressions of ds2, we obtain

$${({c_{{{\rm O}}}}dt)^2} - 2(\tfrac{1}{2}r){c_{{{\rm O}}}}dtdx - \left\{ {[1 - {{(\tfrac{1}{2}r)}^2}]\,d{x^2}+{{({c_{{{\rm O}}}}d{t_0})}^2}} \right\}=0 .$$
(23)

Note that by dividing the whole expression in Eq. (23) with \({({c_{{{\rm O}}}}dt)^2}\), and setting \(\beta =V/{c_{{{\rm O}}}}\) and \(V=dx/dt\), we conclude in the time-dilation expression 1,39,

$$dt=\frac{{d{t_0}}}{{\sqrt {1 - r\beta - [1 - {{(\tfrac{1}{2}r)}^2}]\,{\beta ^2}} }}={\gamma _r} \cdot d{t_0}$$
(24)

Next, solving Eq. (23) for the (absolute) value of time period, we find

$${\left| {{c_{{{\rm O}}}}dt} \right|_ \pm }=\sqrt {d{x^2}+{{({c_{{{\rm O}}}}d{t_0})}^2}} \pm \tfrac{1}{2}r \cdot dx$$
(25)

The motions in the one-way direction have respective time periods that differ by:

$$\tfrac{1}{2}\left( {{{\left| {{c_{{{\rm O}}}}dt} \right|}_+} - {{\left| {{c_{{{\rm O}}}}dt} \right|}_ - }} \right)=\tfrac{1}{2}r \cdot dx$$
(26a)

while their average is

$${\left| {{c_{{{\rm O}}}}dt} \right|_{{O}}} \equiv \tfrac{1}{2} \cdot ({\left| {{c_{{{\rm O}}}}dt} \right|_+}+{\left| {{c_{{{\rm O}}}}dt} \right|_ - })=\sqrt {d{x^2}+{{({c_{{{\rm O}}}}d{t_0})}^2}}$$
(26b)

Then, Eq. (25) is written as \({\left| {{c_{{{\rm O}}}}dt} \right|_ \pm }={\left| {{c_{{{\rm O}}}}dt} \right|_{\text{O}}} \pm \tfrac{1}{2}r \cdot dx\), or

$${\left| {dt} \right|_ \pm }={\left| {dt} \right|_{\text{O}}} \pm \tfrac{1}{2}r \cdot dx/{c_{{{\rm O}}}}$$
(27)

Dividing by the displacement dx, we get \({\left| {dt} \right|_ \pm }/dx={\left| {dt} \right|_{\text{O}}}/dx \pm \tfrac{1}{2}r \cdot /{c_{{{\rm O}}}}\); then, setting \(\left| {{V_ \pm }} \right|=dx/{\left| {dt} \right|_ \pm }\) and \({V_{\text{O}}}=dx/{\left| {dt} \right|_{\text{O}}}\), we obtain the relationship between the speeds in the one-way directions,

$$1/\left| {{V_ \pm }} \right|=1/{V_{\text{O}}} \pm \tfrac{1}{2}r \cdot /{c_{{{\rm O}}}}$$
(28)

or, using \({V_ \pm }= \pm \left| {{V_ \pm }} \right|\),

$${V_ \pm }=\frac{{ \pm {V_{\text{O}}}}}{{1 \pm \tfrac{1}{2}r \cdot {V_{\text{O}}}/{c_{{{\rm O}}}}}}$$
(29)

which coincides with the kinetically derived relations in Eq. (7b).

The whole framework of anisotropic relativity can be built from this velocity addition. Thermodynamic relativity develops this addition rule from the three above postulates, and is further used to construct the mathematical formulation of the (asymmetric) Lorentz transformation and the (nondiagonal) Minkowsky metric. With this, we were able to derive the relationship between the speeds in the opposite one-way directions (Eq. (29)), exactly as they are known from the asynchronous kinematics (Eq. (7b)). Therefore, the reasonable next step is to derive the addition rule from asynchronous kinematics.

Asynchronous kinematics

Einstein’s famous effort to provide a simpler description of special relativity 26, was carried out through thought experiments that made clear that the premise of synchronization must be given first, and thus assumed the principles of an isotropic relativistic framework. In the case of anisotropic relativity, a more inclusive set of thought experiments is needed. In this section we develop the kinematic derivation of the addition of speeds for anisotropic relativity, by performing three different thought experiments 40,41: (1) Stationary observers that exchange a signal of particles and light beams in the positive and negative direction, respectively; (2) Nonstationary observers that exchange light signals; and (3) Nonstationary observers that exchange a signal of particle and light beams.

These three thought experiments will progressively lead to the anisotropic addition of speeds. Through them we show: (1) the relationship of velocities in positive and negative directions; (2) the distinction between speed with negative sign and speed in negative direction (with respect to anisotropy); and (3), the addition rule of velocities, where the results of the previous thought experiments are exploited.

Experiment 1. Stationary observers exchange particle and light signals

Let two observers A and B be set in stationary coordinates at a distance L from each other. Observer A sends a signal to B with a particle beam, while B decides to reply back with a light signal (Fig. 2). For observer A, the particle signal is sent in the positive direction with speed V, while the light signal is received in the negative direction with speed c−. For the observer B, the particle signal is received in the negative direction with speed \(\left| {\bar {V}} \right|\), while the light signal is sent in the positive direction with speed c+. The process of A and B exchanging these signals can be repeated periodically. Then, the time period τ between any two signals is constant and the same for all stationary observers. Indeed, according to the first relativity postulate, the perspective of B from A is equivalent to the perspective of A from B. Therefore, if τ is a time scale characterizing the connection of B with respect to A, then the same scale τ is characterizing the connection of A with respect to B.

Fig. 2
figure 2

Thought Experiment 1: Two stationary observers exchange information through a signal made of a particle beam sent from A to B, and a light beam sent from B to A, once the particle beam is received. Kinematics is examined in the perspectives of observer’s A frame (a) and of observer’s B frame (b).

Full size image

Hence, we have for the two reference frames:

  • Reference frame of A:

$$\tau =L/V+L/{c_ - }$$
(30a)

.

  • Reference frame of B:

$$\tau =L/\left| {\bar {V}} \right|+L/{c_+}$$
(30b)

For simplicity, in the respective mathematical formalism, we use speeds instead of one-dimensional velocities, showing with absolute values when the motion is in the negative direction. Connecting the two equations, we obtain:

$$\tau =L/V+L/{c_ - }=L/\left| {\bar {V}} \right|+L/{c_+}$$
(31)

leading to \(1/V - 1/\left| {\bar {V}} \right|=1/{c_+} - 1/{c_ - }=r/{c_{\text{O}}}\), or \(\left| {\bar {V}} \right|=V/(1 - r \cdot V/{c_{\text{O}}})\), and including the sign,

$$\bar {V}= - \frac{V}{{1 - r \cdot V/{c_{\text{O}}}}}$$
(32)

This is the negative (or inverse) element of the kappa addition, given by Eq. (19). Indeed, if we set the kappa addition of two elements to be zero, that is, \(0=V \oplus \bar {V}=V+\bar {V} - (r/{c_{\text{O}}}) \cdot V\,\,\bar {V}\), then, we end up with Eq. (32). Moreover, if we use the property that the negative of the sum of two elements, equals the sum of their negatives, i.e., \({\bar {V}_1} \oplus {\bar {V}_2}=\overline {{{V_1} \oplus {V_2}}}\), then, we can show that Eq. (32) leads to the kappa addition formulation. Nevertheless, we prefer here to show the kappa addition through the same kinematical intuition as we followed to derive the formulation of the negative element.

Experiment 2. Nonstationary observers exchange light signals

We next consider two observers A and B in relative motion, but exchanging information only with light signals. In the frame of A, observer B moves with speed \({V_{{\text{B,A}}}}\), while in the frame of B, observer A moves with the same speed \(\left| {{V_{{\text{A,B}}}}} \right|\) (but with opposite velocity sign). The result of anisotropy is the same in both cases as they all move in the same direction (both in positive or in negative directions); here we set it to be on the positive direction, i.e., B departs away from A in the frame of A, and A departs away from B in the frame of B. In this section, we will show the validity of this concept, namely \({V_{{\text{B,A}}}}=\left| {{V_{{\text{A,B}}}}} \right|\).

We have exchange of a light signal between A and B; the light signal is sent from A to B and then reemitted from B, back to A (Fig. 3). In the frame of A, the time period needed for the light to travel to B and return to A is \(\tau ={L_{\operatorname{Re} }}/{c_+}+{L_{\operatorname{Re} }}/{c_ - }=2{L_{\operatorname{Re} }}/{c_{\text{O}}}\), with \({L_{\operatorname{Re} }}\) noting the position of B (from A) at the moment in time when the light is received by B and reemitted to A; the respective initial position of B from A is \({L_0}\) (at the moment of the initial light emission from A). Hence, we have \(({L_{\operatorname{Re} }} - {L_0})/{V_{{\text{B,A}}}}={L_{\operatorname{Re} }}/{c_+}\), because light travels from A towards B in the positive direction in the frame of A, covering distance \({L_{\operatorname{Re} }}\), in the same time that B covers distance \({L_{\operatorname{Re} }} - {L_0}\) with speed \({V_{{\text{B,A}}}}\); thus, we find that \({L_{\operatorname{Re} }}={L_0}/(1 - {V_{{\text{B,A}}}}/{c_+})\). The light is then reemitted back to A, received after time \({L_{\operatorname{Re} }}/{c_ - }\), during which B has moved \(({L_{\operatorname{Re} }}/{c_ - }) \cdot {V_{{\text{B,A}}}}\). Therefore, the total distance between A and B, increases from \({L_0}\) (initial, i.e., at the time A emits light signal) to \(L={L_{\operatorname{Re} }} \cdot (1+{V_{{\text{B,A}}}}/{c_ - })\) (final, i.e., at the time A receives back the light signal); substituting \({L_{\operatorname{Re} }}\) with \({L_0}\), we find: \(L={L_0} \cdot (1+{V_{{\text{B,A}}}}/{c_ - })/(1 - {V_{{\text{B,A}}}}/{c_+})\). The respective time period is \(\tau =2{L_{\operatorname{Re} }}/{c_{\text{O}}}\); substituting \({L_{\operatorname{Re} }}\) with \({L_0}\), we find \(\tau =2({L_0}/{c_{\text{O}}})/(1 - {V_{{\text{B,A}}}}/{c_+})\).

On the other hand, in the frame of B, observer A moves with speed \(\left| {{V_{{\text{A,B}}}}} \right|\) in the positive direction with respect to B. The time needed for the light to travel to B is \({L_0}/{c_ - }\), because light travels from A towards B, in the negative direction with respect to B; then, light travels back to A, in the positive direction, covering distance L in time \(L/{c_+}\). Light travels in total time period \(\tau ={L_0}/{c_ - }+L/{c_+}\), during which A has moved by \(L - {L_0}=\left| {{V_{{\text{A,B}}}}} \right| \cdot \tau\); thus, we find \(\tau ={L_0}/{c_ - }+L/{c_+}=(L - {L_0})/\left| {{V_{{\text{A,B}}}}} \right|\), solve in terms of the total distance \(L={L_0} \cdot (1+\left| {{V_{{\text{A,B}}}}} \right|/{c_ - })/(1 - \left| {{V_{{\text{A,B}}}}} \right|/{c_+})\), and express the time period as \(\tau =(2{L_0}/{c_{\text{O}}})/(1 - \left| {{V_{{\text{A,B}}}}} \right|/{c_+})\).

Fig. 3
figure 3

Thought Experiment 2: Two nonstationary observers exchange information through a signal made of a light beam sent from A to B, and then, from B back to A. Similar to Fig. 2, kinematics is examined from the perspectives of observer’s A frame (a) and of observer’s B frame (b). Distance between A and B: L0 at the time of emission from A, LRe at the time B receives the signal, and L when A receives the signal from B; L0 and LRe not shown.

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Overall, we have:

  • Reference frame of A:

 

$$\tau =\frac{{2{L_0}/{c_{\text{O}}}}}{{1 - {V_{{\text{B,A}}}}/{c_+}}}\, and \, L={L_0} \cdot \frac{{1+{V_{{\text{B,A}}}}/{c_ - }}}{{1 - {V_{{\text{B,A}}}}/{c_+}}}.$$
(33a)

  • Reference frame of B:

 

$$\tau =\frac{{2{L_0}/{c_{\text{O}}}}}{{1 - \left| {{V_{{\text{A,B}}}}} \right|/{c_+}}}\, and \, L={L_0} \cdot \frac{{1+\left| {{V_{{\text{A,B}}}}} \right|/{c_ - }}}{{1 - \left| {{V_{{\text{A,B}}}}} \right|/{c_+}}}.$$
(33b)

The total time travel period of the light signal τ and the involved total length L covered between A and B in this time τ is independent of the frames of A or B. Then, comparing the respective relationships for the two frames, we find

$$\left| {{V_{{\text{A,B}}}}} \right|={V_{{\text{B,A}}}}$$
(34)

Of course, the anisotropy and velocity vector directions may be the same or opposite. We conclude that independent of the vector direction, when the velocities are in the same anisotropy direction, Eq. (34) holds, while when they are in opposite anisotropy directions, Eq. (32) holds.

Experiment 3. Nonstationary observers exchange particle and light signals

In this final thought experiment, we consider the previous two observers A and B in relative motion, but the exchange of information occurs through a combination of a particle and light signals, as in the first example (Fig. 4). The particle (noted with P) has speed \({V_{{\text{P,A}}}}\) in the frame of observer A (positive direction) and \(\left| {{{\bar {V}}_{{\text{P,B}}}}} \right|\) in the frame of observer B (negative direction). The formulation is similar to the previous example, where the particle speed \({V_{{\text{P,A}}}}\) is substituted for the light speed \({c_+}\) in the frame of A, while the particle speed \(\left| {{{\bar {V}}_{{\text{P,B}}}}} \right|\) substitutes the light speed \({c_ - }\) in the frame of B.

Fig. 4
figure 4

Thought Experiment 3: Two nonstationary observers exchange information through a signal made of a particle beam sent from A to B, and then, a light beam sent from B to A. Similar to Figs. 2 and 3, kinematics is examined in the perspectives of observer’s A frame (a) and of observer’s B frame (b). Distances are defined in Fig. 3.

Full size image

We derive the following total time travel period τ, summing the travel of the particle beam and light signal, and the respective total length L covered between A and B in this time τ. Overall, these are:

  • Reference frame of A (as in Eq. (33a) after substituting \({c_+} \to {V_{{\text{P,A}}}}\)):

$$\tau ={L_0} \cdot \frac{{1/{V_{{\text{P,A}}}}+1/{c_ - }}}{{1 - {V_{{\text{B,A}}}}/{V_{{\text{P,A}}}}}}\, and \, L={L_0} \cdot \frac{{1+{V_{{\text{B,A}}}}/{c_ - }}}{{1 - {V_{{\text{B,A}}}}/{V_{{\text{P,A}}}}}}.$$
(35a)

  • Reference frame of B (as in Eq. (33b) after substituting \({c_ - } \to \left| {{{\bar {V}}_{{\text{P,B}}}}} \right|\)):

 

$$\tau ={L_0} \cdot \frac{{1/{c_+}+1/\left| {{{\bar {V}}_{{\text{P,B}}}}} \right|}}{{1 - \left| {{V_{{\text{A,B}}}}} \right|/{c_+}}}\, and \, L={L_0} \cdot \frac{{1+\left| {{V_{{\text{A,B}}}}} \right|/\left| {{{\bar {V}}_{{\text{P,B}}}}} \right|}}{{1 - \left| {{V_{{\text{A,B}}}}} \right|/{c_+}}}.$$
(35b)

As in the previous thought experiments, the total time travel period τ and length L are independent of the frames of A or B. Combining the respective equations from each frame, and using Eq. (34), we obtain:

 

$$\tau /{L_0}=\frac{{1/{V_{{\text{P,A}}}}+1/{c_ - }}}{{1 - {V_{{\text{B,A}}}}/{V_{{\text{P,A}}}}}}=\frac{{1/{c_+}+1/\left| {{{\bar {V}}_{{\text{P,B}}}}} \right|}}{{1 - {V_{{\text{B,A}}}}/{c_+}}}\, and \, L/{L_0}=\frac{{1+{V_{{\text{B,A}}}}/{c_ - }}}{{1 - {V_{{\text{B,A}}}}/{V_{{\text{P,A}}}}}}=\frac{{1+{V_{{\text{B,A}}}}/\left| {{{\bar {V}}_{{\text{P,B}}}}} \right|}}{{1 - {V_{{\text{B,A}}}}/{c_+}}}.$$
(36)

We see that the two relationships are equivalent: the left-hand side one can be transformed to the right-hand side one, through

$${V_{{\text{B,A}}}}(\tau /{L_0})+1=\frac{{1+{V_{{\text{B,A}}}}/{c_ - }}}{{1 - {V_{{\text{B,A}}}}/{V_{{\text{P,A}}}}}}=\frac{{1+{V_{{\text{B,A}}}}/\left| {{{\bar {V}}_{{\text{P,B}}}}} \right|}}{{1 - {V_{{\text{B,A}}}}/{c_+}}}=L/{L_0}$$
(37)

In the speed relationship of Eq. (37), the particle speed with respect to B is transformed from negative to positive direction, \(\left| {{{\bar {V}}_{{\text{P,B}}}}} \right|={V_{{\text{P,B}}}}/[1 - (1/{c_+} - 1/{c_ - }) \cdot {V_{{\text{P,B}}}}]\) (following Eq. (32)), leading to

$$\frac{{1+{V_{{\text{B,A}}}}/{c_ - }}}{{1 - {V_{{\text{B,A}}}}/{V_{{\text{P,A}}}}}}=\frac{{1+{V_{{\text{B,A}}}}/{V_{{\text{P,B}}}} - (1/{c_+} - 1/{c_ - }){V_{{\text{B,A}}}}}}{{1 - {V_{{\text{B,A}}}}/{c_+}}}$$
(38)

Then, solving in terms of \({V_{{\text{P,A}}}}\), we find

$${V_{{\text{P,A}}}}=\frac{{{V_{{\text{P,B}}}}+{V_{{\text{B,A}}}} - (\tfrac{1}{{{c_+}}} - \tfrac{1}{{{c_ - }}}) \cdot {V_{{\text{P,B}}}}{V_{{\text{B,A}}}}}}{{1+\tfrac{1}{{{c_+}{c_ - }}} \cdot {V_{{\text{P,B}}}}{V_{{\text{B,A}}}}}}$$
(39)

which is the exact addition rule of velocities, shown in Eq. (18), that is, \({V_{{\text{P,A}}}}={V_{{\text{P,B}}}} \oplus {V_{{\text{B,A}}}}\).

Discussion and Conclusions

We showed that there are two independent and equivalent paths for deriving the framework of anisotropic relativity. The first path is through the theory of thermodynamic relativity, which was developed in1. The second path is through the asynchronous kinematics, as shown and developed in this paper.

The theory of thermodynamic relativity is a unified framework for kinematics and thermodynamics, where the entropy and velocity are characterized by three identical postulates: (1) no privileged reference frame with zero value; (2) existence of an invariant and fixed value for all reference frames; and (3) existence of stationarity. The postulates lead to the most generalized adaptation of addition rule, common for entropies and velocities. In1 we derived the most generalized adaptation of this rule, whose velocities are related with linear Lorentz transformations. Therefore, thermodynamic relativity provides both a natural derivation of addition of velocities and the anisotropic adaptation of Einstein’s special relativity.

Asynchronous kinematics provides another, different path to the framework of anisotropic relativity. We have shown this path of derivation via three progressive thought experiments: (1) Stationary observers that exchange a signal of particle and light beams in the positive and negative direction, respectively; (2) Nonstationary observers that exchange light signals; and (3) Nonstationary observers that exchange a signal of particles and light. Through these experiments, we have shown, respectively: (1) The relationship of the one-way velocities in positive and negative directions is the same as derived from thermodynamic relativity; (2) the distinction between speed with negative sign and speed in negative direction (with respect to anisotropy); and (3) the addition rule of the one-way velocities, which is the same as derived from thermodynamic relativity.

Therefore, the broadest formalism of special relativity, the one derived from the generality of thermodynamic relativity and still associated with linear Lorentz transformations, is connected with asynchronous kinematics and describes anisotropic relativity. Now that we have anisotropic relativity, it is straightforward to seek the next steps and paths, towards: (i) a better understanding the nature of uncertainty that accompanies the one-way motion; (ii) a nonlinear theory, where the arbitrary H-partitioning would correspond to nonlinear transformations; (iii) a generalized anisotropic relativity through Finsler geometry 42,43, which allows for directional distinction of the motion.

Relativistic kinematics and nonextensive thermodynamics are unified into the theory of thermodynamic relativity. Einstein’s special relativity is the limit of fully isotropic, while nonextensive thermodynamics is the opposite limit of fully anisotropic. The adaptation of anisotropic relativity naturally originates from both thermodynamic relativity 1, and in parallel through kinematics as shown in this paper. Some experimental tests of the kinematic theory have been performed 31, but so far none that are indisputable. On the other hand, space plasmas provide an observational ground truth in the development of anisotropic theory of relativity. These provide a natural laboratory for directly observing and characterizing the thermodynamics of plasma particle distributions that reside in stationary states with correlations among the particles. Good examples are provided by the thermodynamics of the outer boundaries of our heliosphere via existing measurements from the Interstellar Boundary Explorer mission 44,45, and even more precise observations from the Interstellar Mapping and Acceleration Probe mission 46.

Returning back to where we started with the problem of synchronization, we recall that one cannot have both the knowledge of synchronization of clocks and the one-way speed, unless one of the two is predetermined. On one hand, an assumption on the synchronization of clocks acting as the fundamental principle to lead to the characterization of one-way speeds, is perhaps metaphysical. On the other hand, the existence of speed anisotropy inserts a lack of knowledge of all observers as regards their description of a nonstationary system, preventing them from having a complete picture of its kinetic state.

Various conceptual or real experiments of anisotropic relativity can be setup and explored, all interwoven with famous physical consequences of relativity, such as, (i) matter-antimatter baryonic asymmetry, (ii) time dilation, and (iii) Doppler effect. Per our previous comments, anisotropic relativity leads to an incomplete description of all observers that cannot be surpassed within kinetic experiments 1. We conjecture, though, that this could possibly be achieved when many different observers are examined through a statistical and thermodynamic analysis (e.g., see, as an example, the matter-antimatter baryonic asymmetry in the early universe, and the decay of muons in their passage through the upper atmosphere, also examined in 1). Despite the complexity and difficulty of performing experimental tests of the kinematic implementation, the thermodynamic portion is clearly and easily testable; such results support both aspects of this unified theory of thermodynamic relativity.