Relativistic Relative Velocities and Relativistic Acceleration

ACTA PHYSICA POLONICA A No. 4 Vol. 139 (2021)

Special issue, XLVI Extraordinary Congress of Polish Physicists, Warsaw, Poland, October 16–18, 2020

G.M. Koczan∗

Warsaw University of Life Sciences (WULS/SGGW), Warsaw, Poland

Doi: 10.12693/APhysPolA.139.401 ∗e-mail: [email protected]

It turns out that the standard application of the four-vector SR formalism does not include the concept of relative velocity. Only the absolute velocity is described by the four-vector, and even the Lorentz transformation parameters are described by the three-dimensional velocity. This gap in the development of the SR formalism reflects the lack of some significant velocity subtraction operations. The differential application of these operations leads to a relativistic acceleration. topics: velocity subtraction, 3D and 4D relative velocity, differential of velocity, relativistic acceleration

1. Introduction It is also worth calculating the velocity differential in relation to the velocity subject to the boost Relative velocity is the defined difference of two (compare [7]): velocities. In the theory of relativity, such a differ- ∂u0 du du0 := x du = x = γ2 du , ence is most easily realized using the Lorentz trans- x x 2 2 v x ∂ux ux=v 1 − v /c formation [1]: 0 0 (5) ∂uy ∂uy 0 0 v du0 := du + du = γ du . x = γv(x − vt), t = γv t − x , y x y v y c2 ∂ux uy =0 ∂uy ux=v (6) 0 0 y = y, z = z, (1) In vector terms, this equation takes the form [8]: p 2 2 where γv = 1/ 1 − v /c . The Lorentz transfor- 0 0 0 ∂u mation of velocity is some way of subtracting veloc- du := (du )v := dui = ∂u u=v ities i dx0 u − v γ3 v(v du) u0 = = x =: u v, (2) v x 0 2 x γv du + 2 . (7) dt 1 − uxv/c γv + 1 c It turns out that the differential calculated with re- 0 p 2 2 0 dy 1 − v /c spect to the second variable is basically only differ- uy = 0 = uy 2 =: uy ux v. (3) dt 1 − uxv/c ent in sign The component parallel to the boost velocity is 0 0 ∂u subject to the subtraction, and the perpendic- (du )u := dvi = ∂vi v=u ular components are subject to the ux operation, 3 which is parameter dependent and is more like mul- γu u(udv) −γu dv − 2 . (8) tiplication than subtraction. If we apply appropri- γu + 1 c ate operations for all components at the same time At high speeds, it is natural that the dux dif- ( , ux , ux ) then we will get the full vector sub- ferential, which is also the differential of u value, traction 0 of the Einstein velocity [2]: scales as in (5) with the appropriate power of the u − v + γv (u × v) × v/c2 gamma factor [9]. Such scaling is necessary so that u0 = γv +1 =: u v. 1 − uv/c2 0 the transformed speed value does not exceed the (4) speed of light. On the other hand, other scaling Einstein, like many behind him (but not all [3, 4]), of the perpendicular speed differential (6) results considered addition, not subtraction. However, from time dilation between transformed systems. the addition here is less natural and leads to an In [10] an attempt was made to reconcile the scal- unequally ambiguous sequence u ⊕ v = u 0 (−v) ing of both differentials. However, the scaling of vs. u ⊕ v = v 0 (−u) or v ⊕ u = u 0 (−v). the perpendicular differential does not follow the The latter convention is most often used [5, 6], idea of an essentially directional (axial) change in which changes the right-hand character of the op- the velocity vector — an analogous discrepancy is eration to the left-hand one. included in (3).

401 The 100 years anniversary of the Polish Physical Society — the APPA Originators

Another disadvantage of the Einstein subtrac- Statement 1. Axial subtraction of velocity vec- tion (4) is the lack of its antisymmetry and the tors (14) is of the form lack of associativity for the addition coopera- u k v = w| = u − ϕv, (15) tion [6, 11, 12]. As we will see, with the right approach to the Lorentz boosts, the problem of the where the ϕ function results from parallel subtrac- lack of velocity subtraction antisymmetry can be tion (11). eliminated. In a sense, the problem of lack of an- Proof. Using (11) to (15) allows to calculate 2 2 2 2 tisymmetry also does not occur at the differential 1 − ux/c 1 − (uv) /(cv) ϕ = 2 = 2 , (16) level (7) and (8). On the other hand, the prob- 1 − uxv/c 1 − uv/c lem of the lack of velocity addition associativity is which inserted into (15) after simplifications leads more serious and is not covered by this paper. This to axial subtraction (14), Q.E.D. problem has been tackled in [12–14]. It should be emphasized that the lack of associativity explicitly Calculating the differentials of relative axial ve- concerns the velocity composition, not the Lorentz locity is not difficult 2 group. dw|x = γv dux, dw|y = duy, Although the four-vector approach is not considered in the SR for the velocity subtraction, let us dw|z = duz. (17) consider such four-velocities [15, 16]: This form of differentials (which will be confirmed µ µ later) is more general than that of (5) and (6). The µ = (γuc, γuu), ν = (γvc, γvv). (9) Let us try to provisionally write (2) and (3) by sub- perpendicular part of the differential (6) is modified tracting four-velocities here in a similar way as in the paper [20]. µ µ µ − ν (γuc − γvc, γuu − γvv) 2 = 2 , (10) µ ◦ ν/c γuγv (1 − uv/c ) 3. Binary subtraction β of 4D and 3D velocities where µ ◦ ν = µβν is the scalar product of four- vectors. Unfortunately, this formula is not valid for the parallel component (2), although it is cor- Most often, an ordinary operation is a bi- rect for the perpendicular component (3). Never- nary operation. Some velocity operations de- theless, it turns out that (4) can be formally con- pend on additional parameters, for example oper- verted into a four-vector form with the appropriate ation (3) or ternary subtraction, discussed in the tools (see [17, 18]). However, this paper focuses next section. In this article, the terms “binary” on other (as it will turn out to be valid) meth- and “ternary” have the nature of proper names ods of velocity subtraction that are ideologically borrowed from Oziewicz [17, 18] — they are ade- similar to (10). quate in the case of 4D, while in 3D they are less precise. Consider some generalization of the four-vector 2. Axial subtraction of velocities velocity subtraction (10) as a linear combination ωµ := (µ− ν)µ := λ µµ − λ νµ. (18) As indicated above, the orthodox velocity sub- C 2 1 traction method (4) has several natural disadvan- In order to determine the λ1, λ2 form factors, we impose two conditions: orthogonality to the tages. The most artificial is the operation of ux for components perpendicular to the subtrahend ve- subtrahend four-velocity and reduction of the locity, which differs significantly from the operation result to ordinary velocity for zero subtrahend i of for parallel components. Therefore, we can three-velocity (ui ≡ u ): i i i try to standardize the operation of for all com- ω ◦ ν := 0, ω (v = 0) := u = µ /γu. (19) ponents [5, 19]: These conditions with the signature + − −− ux − v i 2 (µi = −µ ) lead to the value λ1 = 1, λ2 = c /(µ ◦ ν) w|x := ux v = 2 , (11) 1 − uxv/c and the final form of the relative binary four- velocity (see [12, 17, 19, 21]): w|y := uy vy = uy 0 = uy, (12) 2 µ µ c µ µ ω = (µ−C ν) = µ − ν . (20) w|z := uz. (13) µ ◦ ν If the x axis goes along v, then the above equa- Statement 2. The norm of the binary subtraction tions define an equivalent vector operation k = ( , , ): of the four-velocities (20) with the accuracy of the sign of the signature is equal to the norm of the u − v + uv (u × v) × v/c2 v2 Einstein subtraction (4): u k v := =: w|. 1 − uv/c2 2 2 ||µ−C ν|| = −|u 0 v| . (21) (14) Proof. The square of the binary four-vector is This subtraction only changes the parallel compo- 6 nent to the velocity subtrahend, so it will be called µ c 2 ω ωµ = 2 − c . (22) axial subtraction and w| relative axial velocity. (µ ◦ ν)

402 The 100 years anniversary of the Polish Physical Society — the APPA Originators

The easiest way to calculate the square of the sub- 4. Ternary subtraction traction of velocity is in a properly directed coordi- of 4D and 3D velocities nate system xyz: (u − v)2 + u2(1 − v2/c2) 02 02 x y Based on (18) and (20), we already know ux + uy = 2 2 = (1 − uxv/c ) that naive subtraction of the four-velocities (10) 2 2 2 2 2 2 2 has little speciﬁc sense. However, we can con- c (1 − uxv/c ) − (c − u )(1 − v /c ) 2 2 , (23) sider such a subtraction projected onto a 3D (1 − uxv/c ) hypersurface orthogonal to a certain reference which with the opposite sign equals (22), Q.E.D. four-velocity σµ: Similarly to the dependence of the four-velocity µ µ µ β β ξ := µ4− σν := λP (σ) µ − ν . (31) on the three-velocity (9) or more exact the (σ) ⊥β four-force on the three-force, we can introduce It is therefore a ternary velocity subtraction dependent on the three four-velocities. The orthog- a three-dimensional equivalent for the binary µ µ four-velocity onal projection operator has the form P⊥β = δβ − µ 2 ωµ = (γ wv/c, γ w) . (24) σ σβ/c . In order to calculate the form factor λ v v for the sake of simplicity, we will limit ourselves The three-dimensional relative binary velocity [22] to the three-dimensional approach, assuming that (jet velocity [19]) can be expressed as follows: σµ = (c2, 0, 0, 0). This leads to a 3D relative ternary u − v + (u × v) × v/c2 velocity w = =: u v. (25) 1 − uv/c2 ⊥ W := u ∧ v := λ γuu − γvv . (32) At the same time, a new three-velocity subtraction The ternarity of this operation is implicitly hidden operation was introduced that is simpler than op- in choice σ. Regardless of the choice of λ, this oper- erations and . 0 k ation is not equal to any of the previously discussed. Statement 3. Binary 3D subtraction of velocity Nevertheless, in the case of axial (parallel vectors), vectors (25) is of the form a condition can be imposed on λ conforming to stan-

u ⊥ v = w = ψu − v, (26) dard subtraction where the function ψ results from the parallel sub- u ∧ v := u v (u k v). (33) traction according to (11). Unfortunately, the condition cannot be ex- tended to each case for a parallel component. Proof. Using (11) with (26) allows to calculate The given definition condition allows, however, 2 2 2 2 1 − v /c 1 − v /c to calculate ψ = 2 = 2 , (27) 1 − uxv/c 1 − uv/c u − v λ = uv = which inserted into (26) after simplifications leads 1 − c2 γuu − γvv to 3D binary subtraction (25), Q.E.D. (u − v)(γ−1 + γ−1) u v = The above statement shows that 3D vector sub- uv γu γv 1 − c2 u − v + γ u − γ v traction differs from the Einstein subtraction in or- v u γ−1 + γ−1 thogonal component u v = 2 2 uv uv 1 − v /c 1 − c2 1 + γuγv 1 + c2 wy = (u ⊥ v)y = uy 2 . (28) 1 − uxv/c γ−1 + γ−1 u v . (34) Seemingly, the difference relative to (3) is not large uv u2v2 1 − c2 + γuγv 1 − c2 and does not seem to solve the non-intuitive prob- We will now make a generalization of this expres- lem (apart from time dilation) of scaling the per- sion for the case of any velocities, assuming that pendicular component. However, the problem of the square of the velocity concerns the square of this scaling disappears if the subtracted velocities the norm (u2 = u2) and the expression of the first have equal components ux = v ≡ vx. degree in a given velocity denotes its component The above property is important in the compu- (u = ux, uv = uxv = uv). The validity of these as- tation of the 3D binary velocity differential (differ- sumptions will be further confirmed in the form of 2 2 −2 ential of jet velocity): the lemma proof. Applying u /c = 1−γu , we get

∂wx 2 dwx := dux = γ dux, (29) −1 −1 v γu + γv ∂ux ux=v λ = = 1 − uv + γ γ 1 − u2v2 ∂w ∂w c2 u v c4 dw := y du + y du = du . y x y y γu + γv ∂ux uy =0 ∂uy ux=v . (35) γ γ 1 − uv + γ2 + γ2 − 1 (30) u v c2 u v The covariant equivalent of this expression is Relative 3D binary velocity differentials coincide c2σ ◦ µ + c2σ ◦ ν with axial velocity differentials as opposed to the λ = . (36) Einstein subtraction differentials. c2µ ◦ ν + (σ ◦ µ)2 + (σ ◦ ν)2 − c4

403 The 100 years anniversary of the Polish Physical Society — the APPA Originators

5 5 5 3 5 Finally, the relative ternary four-velocity becomes + × c − × 0 W = 4 4 4 5 4 = µ µ y 5 5 52 52 ξ(σ) = (µ4− σν) = 4 × 4 (1 − 0) + 42 + 42 − 1

2 µ µ β β 30 σ ◦ (µ + ν) c δβ − σ σβ µ − ν c ≈ 0.5085c, (45) . (37) 59 (σ ◦ µ)2 + (σ ◦ ν)2 + c2µ ◦ ν − c4 which gives it value W ≈ 0.7191c. Thus, neither This velocity, written differently and derived the x components nor the values of the considered differently, was first published by Oziewicz in velocities are equal, Q.E.D. 2004 [17–19]. Ternary subtraction is a generaliza- Statement 4 would suggest that the 3D ternary tion of binary subtraction which is a special case velocity is weakly anchored in the Lorentz transfor- (σ = ν): mation. However, nothing could be more wrong, as shown below (see [23]). µ−C ν = µ4− ν ν. (38) Another special case (σi = 0) is the already intro- Lemma. The Lorentz boost (the Einstein sub- duced three-dimensional ternary relative velocity traction) with a ternary 3D velocity for a veloc- (γ + γ )(γ u − γ v) ity, which is minuend ternary subtraction, restores W = u v = u v u v . ∧ uv 2 2 ternary subtrahend velocity γuγv 1 − 2 + γu + γv − 1 c u W = u (u v) = v. (46) (39) 0 0 ∧ In other words, in the above sense, 3D ternary sub- This velocity, written and derived somewhat dif- traction is a left-inverse operation of the Einstein ferently, first appeared in 2012 in the Dragan [23] subtraction. lectures notes, but has not been published so far either in the form of an article or in the Proof. Proving the lemma thesis is computation- form of a preprint (except for the Polish lectures). ally complicated and it cannot be done just by boost The poster [22] provides a visualized interpreta- in the x direction. A significant difficulty is calcu- tion of the 3D ternary velocity using the so-called lating the Lorentz factor for velocity W : 2 uv 2 laufer/bishop method. Both 4D and 3D ternary 1 W γ γ 2 + γ γ + 1 = 1 − = u v c u v , subtractions are antisymmetric 2 2 2 γW c uv 2 2 γuγv 1 − 2 + γu + γv − 1 µ4− σν = −ν4− σµ, c (47) u ∧ v = −v ∧ u. (40) which can be written more compactly Other than the one-dimensional velocity subtrac- 1 tions considered in this paper do not have this prop- + 1 = (γu + γv)λ. (48) γ erty — including the Einstein subtraction. The dif- W The main calculations can now be made ferences do not end there. 2 1 uW /c u − W + −1 W γW 1+γ Statement 4. Ternary 3D subtraction, with the W u 0 W = uW = exception of parallel velocities, does not generally 1 − c2 coincide with the Einstein subtraction in the sub- Bu + Cv uW . (49) trahend direction, nor do the norms of the results 1 − c2 of these operations Now it is enough to calculate the coefficients B

(u ∧ v)v 6≡ (u 0 v)v, and C: 2 uv γu − 1 − γuγv c2 |u v| 6≡ |u v|. (41) B = −1 + γv + λ = 0, (50) ∧ 0 γu + γv uv Proof. It is enough to indicate one example for γ γ γ + 1 + γ γ 2 uW C = v u v u v c λ = 1 − . (51) which the considered equations do not exist. So let 2 γu γu + γv c us consider perpendicular velocities with equal val- Based on these calculations, the value of the expres- ues: u = v = vx = uy = 0.6c. Einstein’s subtract- sion (49) is v, Q.E.D. ing leads to the components of the velocity vector The ordinarily understood inverse Lorentz trans- 0 0 − 0.6c ux = = −0.6c, (42) formation is the right-hand inverse (according to the 1 − 0 × 0.6 adopted notation convention): √ 2 0 0 1 − 0.6 u 0 v = u → u = u 0 (−v). (52) u0 = 0.6c = 0.48c, (43) y 1 − 0 × 0.6 On the other hand, the ternary operation can be whose value is u0 ≈ 0.7684c. Whereas the compo- understood as a left-inverse operation nents of the 3D ternary velocity are u 0 W = v → W = u ∧ v. (53) 5 5 5 5 3 + × × 0 − × c Like the Einstein subtraction, left-hand application W = 4 4 4 4 5 = x 5 5 52 52 is the left-hand inverse of ternary action (46). It is × (1 − 0) + 2 + 2 − 1 4 4 4 4 worth writing it explicitly in both orders 30 − c ≈ −0.5085c, (44) u (u v) = u (u v) = v. (54) 59 0 ∧ ∧ 0

404 The 100 years anniversary of the Polish Physical Society — the APPA Originators

To sum up, the main reason for the difference be- selected frame of reference, in which the moving tween the 3D ternary velocity subtraction from the time will be measured and the acceleration calcu- Einstein subtraction is the noncommutativity of ve- lated. Time in the inertial frame σ relative to self- locity composition (and the Lorentz group). The time of body with a four-velocity µµ(τ) is described impact of the lack of associative velocity composi- by time dilation dT = (σ◦µ/c2)dτ = dτ/λ(σ, µ, µ). tion is not directly apparent here, and the Lorentz This leads to the definition of the four-acceleration group is associative by definition. from 4D ternary subtraction: µ The differential of the 3D ternary velocity, which µ dξ µ(τ0 + ∆τ)4− σµ(τ0) is crucial for this paper, remains to be calculated. Aµ := (σ) := lim = The ternary velocity is so homogeneous due to its (σ) dT ∆τ→0 ∆τ/λ components that we can immediately calculate the c4 vector differential αµ − (σ ◦ α)σµ/c2 . (59) 2 ∂W γ2 σ ◦ µ v dW := dui = du + 2 v(v du) = ∂ui u=v c The ternary four-acceleration is thus a properly nor- malized projection of the ordinary four-acceleration. 2 µ du⊥ + γv duk. (55) For σ = µ four-acceleration A(σ) coincides with As before, the velocity differential v would only dif- µ four-acceleration α , and for σi = 0 it amounts to fer in sign and names of symbols. Thus, as can be a three-dimensional relativistic acceleration A — seen, the 3D ternary velocity differential is consis- in the same way that the 4D ternary velocity comes tent with the 3D axial velocity differential and the down to the 3D ternary velocity. 3D binary velocity differential. This compatibility It is time to define the 3D relativistic accelera- is not accidental and reflects a class of differentially tion. For definition, we need a velocity differen- equivalent operations to which the Einstein subtrac- tial based on the difference of velocities, that is on tion does not belong. relative velocity. Apart from the ordinary subtraction, we have the Einstein subtraction and three other methods of 3D subtraction: axial, binary, and 5. Relativistic 4D and 3D acceleration ternary. The Einstein subtraction leads to a velocity differential in the system of instantaneous rest We will start with the 4D approach as it will of the body, which only allows the determination 0 pave the way for the main original 3D result. The of the rest acceleration a0 = du /dτ [19]. The rest standard four-acceleration is derivative of the four- acceleration is not an acceleration per se, so they re- velocity respect to self-time [16]: main consistent differentials of axial, 3D binary, and dµµ ternary velocities. This velocity differential will be αµ = = (α0, ~α) = dτ called the relativistic differential (of velocity) and 4 2 4 2 will be denoted as follows: γuau/c, γua + γu(au)u/c . (56) 2 Du := dw| = dw = dW = du⊥ + γu duk. (60) Despite the simple definition, four-acceleration de- pends on the ordinary three-acceleration a in a very Statement 5. The velocity relativistic differenti- complicated way. The same result can be obtained ation operation is a differentiation operation mul- for a binary four-velocity if we identify the sub- tiplied on both sides by the gamma factor and its tracted four-velocities after derivative calculated inverse (on the left): µ −1 dωµ d µ(τ)−C µ(τ0) D = γ × d × γ. (61) µ α = (τ0, τ) = . dτ τ=τ0 dτ τ=τ0 Proof: (57) −1 2 2 Du = γu d(γuu) = du + γuu(udu)/c , (62) This is equivalent to writing the derivative defini- which is equal to (60) by virtue of (55), Q.E.D. tion, in which the ordinary subtraction is replaced The key relativistic 3D acceleration for this paper by a binary 4D subtraction can now be defined: µ µ µ(τ0 + ∆τ)−C µ(τ0) Du u(t + ∆t) u(t) α = lim . (58) A := := lim ∧ , (63) ∆τ→0 ∆τ dt ∆t→0 ∆t Thanks to this notation, it is clear that when cal- where antisymmetric ternary subtraction can be re- culating the acceleration (or differential) from the µ placed by binary or axial subtraction. This defini- relative velocity, the subtracted velocities ν and tion leads to the formula: µµ should be tending to each other. A = a + γ2u(ua)/c2 = a + γ2a , (64) For the ternary four-velocity, there is also the u ⊥ u k third reference four-velocity σµ. If this four-velocity where a = du/dt is the ordinary acceleration. were to take the same value as the previous two, Four-acceleration can now be expressed simply by then the four-acceleration calculations would coin- acceleration A: cide with the binary calculations. Thus, it is worth µ 0 2 2 α = (α , ~α) = (γuAu/c, γuA). (65) treating σµ as the four-velocity of the independent

405 The 100 years anniversary of the Polish Physical Society — the APPA Originators

[4] V. Fock, The Theory of Space, Time and Gravitation, 1st ed. 1959, 2nd rev. ed., Pergamon Press, 1964. [5] M. Fernández-Guasti, Proc. SPIE 8121, 812108 (2011). [6] A.A. Ungar, Results Math. 16, 168 (1989). [7] K. Rębilas, Apeiron 15, 206 (2008). [8] A. Dragan, T. Odrzygóźdź, Am. J. Phys. 81, 631 (2013). [9] J.F. Barrett, “Review of problems of dynamics in the hyperbolic theory of special relativity”, in: Proc. Physical Interpreta- Fig. 1. Diagram showing the relations of the main tions of Relativity Theory Conf., Imperial 3D acceleration vectors. The new relativistic ac- Coll., London 2002, PD Publ., Liverpool celeration A is a parallel projection of ordinary ac- 2002, p. 17. celeration a along the velocity on the direction of force (or spatial part ~α of the four-acceleration αµ). [10] K. Rębilas, Am. J. Phys. 78, 294 (2010). Thanks to this, the construction of acceleration A [11] M. Ahmad, M.S. Alam, Int. J. Reciproc. is simpler than the construction of rest accelera- Symm. Theor. Phys. 1, 9 (2014). tion a0. [12] Z. Oziewicz, Int. J. Geom. Methods Mod. Phys. 4 , 739 (2007). Thanks to this, also the ternary four-acceleration [13] M. Ahmad, Int. J. Reciproc. Symm. Aµ can be expressed as acceleration A. However, (σ) Theor. Phys. 1, 69 (2014). this is a more complex expression that simplifies for [14] J. Kocik, arXiv:1910.06785v1 (2019). σi = 0: [15] H. Minkowski, in: Nachrichten von der Aµ = (0, A) . (66) (σi=0) Gesellschaft der Wissenschaften zu Göttin- A graphical interpretation of (64)–(66) is shown gen, Mathematisch-Physikalische Klasse, in Fig. 1. Berlin 1908, p. 53 (in German). [16] A.K. Wróblewski, J.A. Zakrzewski, Intro- 6. Conclusion duction to Physics, Vol. 1, PWN, Warsaw 1976, (in Polish). It has been shown and proved through the paper [17] Z. Oziewicz, in: Group Theoretical Meth- that within the Lorentz group it is possible to rea- ods in Physics, Conference Series, Vol. 185, sonably subtract velocities in a different way than CRC Press, 2004, p. 439. it was established in SR. First of all, it can be done in an antisymmetric way or in a four-vector way. [18] Z. Oziewicz, arXiv:1104.0682v1 (2011). For example, the antisymmetric operation is the [19] G.M. Koczan, Res. Phys. 24, 104121 left-hand reciprocal of the Lorentz boost. Veloc- (2021). ity subtraction operations made it possible to de- [20] K. Rębilas, Europ. J. Phys. 34, L55 fine different 4D and 3D relative velocities. Due to (2013). the differential equivalence of these subtraction op- [21] V.J. Bolós, Commun. Math. Phys. 273, erations, unambiguous and original 3D relativistic 217 (2007). acceleration was introduced. This relativistic accel- [22] G.M. Koczan, poster for XLVI Extraordi- eration is important for simplifying the dynamics of nary Congress of Polish Physicists on the the SR [19, 22]. Whereas in this article, the rela- 100 Years of the Polish Physical Society, tivistic acceleration has been generalized to the 4D Warsaw, 2020. ternary four-acceleration. [23] A. Dragan, monograph — lecture notes, Unusually special theory of relativity, References Ch. 3. Thomas–Wigner rotation (2012) (in Polish). [1] W. Rindler, Relativity: Special, General and Cosmological, Oxford University Press, 2007. [2] A. Einstein, Ann. Phys. 17, 891 (1905). [3] M. Cannoni, arXiv:1605.00569v2 (2016).

406