ACTA PHYSICA POLONICA A No. 4 Vol. 139 (2021) Special issue, XLVI Extraordinary Congress of Polish Physicists, Warsaw, Poland, October 16–18, 2020 Relativistic Relative Velocities and Relativistic Acceleration 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 differen- tial 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 = wj = u − 'v; (15) tion [6, 11, 12]. As we will see, with the right ap- proach to the Lorentz boosts, the problem of the where the ' 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) ' = 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. dwjx = γv dux; dwjy = duy; Although the four-vector approach is not consid- ered in the SR for the velocity subtraction, let us dwjz = 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 !µ := (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- ! ◦ ν := 0;! (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) wjx := ux v = 2 ; (11) 1 − uxv=c and the final form of the relative binary four- velocity (see [12, 17, 19, 21]): wjy := uy vy = uy 0 = uy; (12) 2 µ µ c µ µ ! = (µ−C ν) = µ − ν : (20) wjz := 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 := =: wj: 1 − uv=c2 2 2 jjµ−C νjj = −|u 0 vj : (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 ! !µ = 2 − c : (22) axial subtraction and wj 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 de- pendent on the three four-velocities.