l 8 (1995 ACTA PHYSICA POLONICA A GALILEAN TRANSFORMATION EXPRESSED BY THE DUAL FOUR-COMPONENT NUMBERS V. MAJERNIK Inttt f Mtht Slv Ad f Sn 1 7 rtlv Stfnv 9 Slv pbl nd prtnt f hrtl h l Unvrt Svbd 77 1 Ol Czh pbl (eceie Augus 19 199; eise esio αuay 1995 W xpr th pl Glln trnfrtn n th lbr rn f th dl fr-pnnt nbr ACS nbr 1h 1j . Intrdtn In th prv ppr [1] hv prntd th rltn btn dffrnt t-pnnt nbr t nd th rrpndn t-dnnl p-t tr rp In rdr t fnd lr rltn btn th phl fr-d- nnl p-t tr rp nd th rrpndn nbr t nd t xtnd th xtn t-pnnt nbr t t th fr-pnnt n A ll nn th rntz trnfrtn f bt f v n th x drtn n b rttn rltn btn th fr-pnnt nbr lld tr- nn [21 hr x = ct+ixj j nd l 3 r th trnn nt r th tn r hthr t d xt h fr-pnnt nbr b hh n n xpr th pl Glln trnfrtn n lr fr (1 W ll h tht th n b hvd b n f th dl fr-pnnt nbr t hh rprnt th xtnn f th dl t-pnnt nbr t vr bfr dn th lt rll ft rrdn th t-pnnt nbr t [1] It ll nn tht t-pnnt nbr t frn n lbr rn n b rttn n th fr Z = α d- eb, α b E R, hr rtn "- nr" nt h lbr trtr f rn dnd tht th prdt f h t-pnnt nbr 1 = α - eb nd = c+ ed, (919 9 Maeik bln t th rn ll hrfr = β + εγ , β , ε An prtnt thtl thr tt tht n pbl t-pnnt nbr n b rdd t n f th flln thr tp [3] h "nr" nt ll b dntd b i r rptvl n h f th thr pf ltd bl ( th plx nbr = α + e th ε2 = 2 = —1 ( th bnr nbr [3] (lld l th dbl [] nd nl-plx [5] b = α + ε; th ε2 = λ2 = 1 nd ( th dl nbr d = α + e th 2 = µ 2 = h rtrn fr th brhp t n f th t-pnnt nbr t vn b th n f th xprn Q = (β + λ/ If Q ntv ptv r zr thn t th plx bnr r dl nbr rptvl h l prtn dn fr th plx nbr h blt vl ntr fr nd Elr frl hv rrpndn prtn n thr t-pnnt t r nbr = α + e fr n f th thr nbr t t conjugate vn = α — e nd t blt vl dfnd I = √|ZZ*| All n b xprd l n th ntr fr = xp(εϕ ϕ = (α/ hr = rtn(α/b hld fr th plx = rtnh(α/b fr th bnr nd = (α fr th dl nbr rptvl 2. r - pnnt nbr t nd Glln trnfrtn h fr-dnnl xtnn f th plx t-pnnt nbr - t rprnt th t f ltnn trnn A ltnn trnn dfnd [] th th ltpltn h fr th trnn nt hr εik rprnt th pnnt f ttll nttr tnr ltnn trnn hv lr fld f ppltn n thtl ph ( [] h rrpndn fr-pnnt nbr hh rprnt th fr-dnnl xtnn f th t-pnnt bnr nbr r th hprbl trnn h hv d pll n prn fld hn [] Anthr fr f trnn lld th btrnn pl n prtnt rl b th pt nd lnt frltn f pl rltvt ltrdn nd prtr thr ( [7] xt ll h tht Glln trnfrtn n l b rttn n th fr f th fr-dnnl nbr t hh rprnt fr-dnnl xtnn f th t-pnnt dl nbr h lnt f th t dl fr-pnnt nbr ( fr hrt dfnd Gαiea asomaio Eesse 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 2 = v2 /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) 2 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 tnd fr th prtr f Glln trnfrtn h v ppltn f t Glln trnfrtn drbd b th prtr ),G(Φ1 nd G(Φ is equivalent to an application of a single operator hr G(3 = G(Φ1G(Φ = G(ΦG(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 Mαeik 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 X. 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 pbl f tpnnt nbr n ntrtn 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 x 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 ... 93 n nt nt tht (1 + μ(1— μ) = 1 nd n n E (5d t fnll W rlz tht hr th ntrtn b n f dl nbr rr lrr prdr thn th f thr n ntrtn- thd vrthl t h n ntrtn nd r f ntrtn f rtn tp f fntn n th fr f f pl xpnntl fntn W nt tht n th ll t-pnnt nbr t ndrbl lr l f fntn f th tp F(x) = P1(x2(n x, x3 (nh x, h x4 (xp (x)), hr 1 132 3 nd tnd fr plnl n b rttn f rtn xpnntl fntn vr ltpnnt nbr t h dtld thtl nl ld xd th p f th ppr ll b bjt f bnt ppr frn [1] Mjrn Acta Phys. Pol. A 91 (199 [] Sn Quaternions, Lorentz Transformation and Conway - Dirac - Eddington Matrices, bln Inttt fr Advnd Std bln 197 [3] I Kntr Sldnv Hypercomplexe Zahlen, bnr pz 197 [] W nz Vorlesungen uber Geometrie der Algebren, Sprnr rln 1973 [5] IM Yl Complex Numbers n Geometry, Ad r Yr 19 [] M rntv Chbt Effects Hydrodynamigues et Modeles Mathematigues, Mr M 19 [7] Andrn G h Physics Essays 3 (1993.