l 8 (1995 ACTA PHYSICA POLONICA A
GALILEAN TRANSFORMATION EXPRESSED BY THE DUAL FOUR-COMPONENT NUMBERS
V. MAJERNIK In t t t f M th t Sl v A d f S n 1 7 r t l v St f n v 9 Sl v p bl nd p rt nt f h r t l h l Un v r t Sv b d 77 1 Ol Cz h p bl
( ecei e Augus 19 199 ; e ise e sio α ua y 1995
W xpr th p l G l l n tr n f r t n n th l br r n f th d l f r- p n nt n b r ACS n b r 1 h 1 j
. Intr d t n In th pr v p p r [1] h v pr nt d th r l t n b t n d ff r nt t - p n nt n b r t nd th rr p nd n t -d n n l p -t tr r p In rd r t f nd l r r l t n b t n th ph l f r-d - n n l p -t tr r p nd th rr p nd n n b r t n d t xt nd th x t n t - p n nt n b r t t th f r- p n nt n A ll n n th r ntz tr n f r t n f b t f v n th x d r t n n b r tt n r l t n b t n th f r- p n nt n b r ll d t r- n n [21
h r x = ct+ixj j nd l 3 r th t rn n n t r th t n r h th r t d x t h f r- p n nt n b r b h h n n xpr th p l G l l n tr n f r t n n l r f r (1 W ll h th t th n b h v d b n f th d l f r- p n nt n b r t h h r pr nt th xt n n f th d l t - p n nt n b r t v r b f r d n th l t r ll f t r rd n th t - p n nt n b r t [1] It ll n n th t t - p n nt n b r t f r n n l br r n n b r tt n n th f r Z = α d- eb, α b E R, h r rt n " - n r " n t h l br tr t r f r n d nd th t th pr d t f h t - p n nt n b r 1 = α - eb nd = c+ ed,
(919 9 Ma e ik
b l n t th r n ll h r f r = β + εγ , β ε An p rt nt th t l th r t t th t n p bl t - p n nt n b r n b r d d t n f th f ll n thr t p [3] h " n r " n t ll b d n t d b i r r p t v l n h f th thr p f l t d b l
( th pl x n b r = α + e th ε = = —1 ( th b n r n b r [3] ( ll d l th d bl [ ] nd n l- pl x [5] b = α + ε ; th ε = λ = 1 nd
( th d l n b r d = α + e th = µ = h r t r n f r th b r h p t n f th t - p n nt n b r t v n b th n f th xpr n Q = (β + λ/ If Q n t v p t v r z r th n t th pl x b n r r d l n b r r p t v l h l p r t n d n f r th pl x n b r h b l t v l n tr f r nd E l r f r l h v rr p nd n p r t n n th r t - p n nt t r n b r = α + e fr n f th thr n b r t t conjugate v n = α — e nd t b l t v l d f n d I = √ All n b xpr d l n th n tr f r = xp(εϕ ϕ = (α/ h r = r t n(α/b h ld f r th pl x = r t nh(α/b f r th b n r nd = (α f r th d l n b r r p t v l
2. r p n nt n b r t nd G l l n tr n f r t n
h f r-d n n l xt n n f th pl x t - p n nt n b r - t r pr nt th t f lt n n t rn n A lt n n t rn n d f n d [ ]
th th lt pl t n h f r th t rn n n t
h r εi k r pr nt th p n nt f t t ll nt tr t n r lt n n t rn n h v l r f ld f ppl t n n th t l ph ( [ ] h rr p nd n f r- p n nt n b r h h r pr nt th f r-d n n l xt n n f th t - p n nt b n r n b r r th h p rb l t rn n h h v d p ll n p r n fl d h n [ ] An th r f r f t rn n ll d th b t rn n pl n p rt nt r l b th p t nd l nt f r l t n f p l r l t v t l tr d n nd p r tr th r ( [7] xt ll h th t G l l n tr n f r t n n l b r tt n n th fr f th f r-d n n l n b r t h h r pr nt f r-d n n l xt n n f th t - p n nt d l n b r h l nt f th t d l f r- p n nt n b r ( f r h rt d f n d Gα i ea a s o ma io E esse y 921
The product of two DFNs Q and Q is again a DFN so that the set of DFNs forms an algebraic ring. The ratio R of two DFNs
Q = q +q1i+q +q3k and Q = q'0+q'1i = q 2j+q3k can be written in the form where the conjugate DFN is defined as and the norm of Q given as QQ qó DFNs represent the natural number system for the expression of special Galilean transformation where r' and r are the position vectors in the inertial systems moving with velocity v relative to each other and I is time. If we express the space-time in the form of the DFN where i, j, k obey the multiplication rules of the dual units (2) then the special Galilean transformation (3) can be written in the following form where X' = ct' + x'j + z'k and Φl = v1 /c, 0 = v /c and Φ3 = v3/ Taking into account that z' = = kn = O for n > 1, we get
Equation (4) can be written in the familiar form of Galilean transformation (3),
and t' = t, which leaves the expression XX* = (ct) unchanged. This is in agree- ment with the assumption of the Newtonian physics requiring the existence of the absolute time. Equation (4) can be rewritten also in the form where G(0) = xp(- Φ t nd f r th p r t r f G l l n tr n f r t n h v ppl t n f t G l l n tr n f r t n d r b d b th p r t r ),G(Φ 1 nd G(Φ is equivalent to an application of a single operator
h r G( 3 = G(Φ1 G(Φ = G(Φ G(Φ1 = G(Φ1 + Φ2) which implies the addition theorem for the velocities in the Galilean transformation. Equation (4) resembles the formula for the rotation of a complex vector, Z = x+iy, Z' = R(ϕ)Z, where R(ϕ) = exp(iϕ) represents the operator of rotation and is the angle of rotation. As is well known the set of the operators R(0) forms a group. Similarly, 922 Mα e ik
the set of the Galilean operators G( forms a group with the continuous parameter = v/c the "angle" of the rotation of the dual space-time The formulation of Galilean transformation by means of DFNs has not only a certain elegance and aesthetic appeal but it shows that the linkage between the space and time exists also in the Newtonian physics. Moreover, it may have a con- siderable heuristic value for the study of the underlying mathematical formalism of physical laws.
. A p bl f t p n nt n b r n nt r t n
Let us finish with a mathematical curiosity connected with the dual two-com-
ponent ' umbers. As is well known the Euler and Moivre formulas for the complex, binary and dual numbers are
respectively. By means of Eqs. (5a), (5b) we get the identities
However, Eq. (5c) makes possible to express also the variable and the real number 1 in a form containing only the combinations of exponential functions
Therefore, a special function of the following type
where P1 and stand for polynomials, can be expressed as a sum of exponential functions
the general form of which is
The decomposition (5e) makes possible to perform the integration of the functions ( simply as a sum of exponential functions. As an elementary example let us determine the integral
Using Eq. (5d) we obtain Galilean Transformation Expressed by ... 9 3
n nt nt th t (1 + μ (1— μ) = 1 nd n n E (5d t f n ll
W r l z th t h r th nt r t n b n f d l n b r r r l r r pr d r th n th f th r n nt r t n- th d v rth l t h n nt r t n nd r f nt r t n f rt n t p f f n t n n th f r f f pl xp n nt l f n t n W n t th t n th ll t - p n nt n b r t n d r bl l r l f f n t n f th t p F(x) = P1(x ( n x, x 3 ( nh x, h x ( xp (x)),
h r 1 13 3 nd t nd f r p l n l n b r tt n f rt n xp n nt l f n t n v r lt p n nt n b r t h d t l d th t l n l ld x d th p f th p p r ll b bj t f b nt p p r
f r n
[1] M j rn Acta Phys. Pol. A 91 (199
[ ] S n Quaternions, Lorentz Transformation and Conway - Dirac - Eddington Matrices, bl n In t t t f r Adv n d St d bl n 197 [3] I K nt r S l d n v Hypercomplexe Zahlen, bn r pz 197 [ ] W nz Vorlesungen uber Geometrie der Algebren, Spr n r rl n 1973 [5] I M Y l Complex Numbers n Geometry, A d r Y r 19 [ ] M r nt v Ch b t Effects Hydrodynamigues et Modeles Mathematigues, M r M 19 [7] And r n G h Physics Essays 3 (1993