Wuming Li and Fan Yang

NEW ZEALAND JOURNAL OF MATHEMATICS Volume 33 (004), 159 164 N DIMENSIONAL MINKOWSKI SPACE AND SPACE TIME ALGEBRA Wuming Li and Fan Yang (Reeived February 003) Abstrat. By using an n Dimensional Minkowski spae, the spae time algebra is introdued. It is used for disussing physial problems of speial relativity. Introdution Clifford algebra was founded by the British mathematiian W.K. Clifford in 1878. Sine 1980 s, it has been applied widely in maths, physis and engineering tehnology fields et [1]. Espeially in geometri alulation (Eulidean geometri alulation, non Eulidean geometri alulation), geometri automati reasoning, differential geometry and theoretial physis field [] [8]. In this paper, by using the hyperboli virtual unit of Clifford algebra to expose n dimensional Minkowski spae and the onept of spae time algebra is introdued. It an be used for disussing physial problems of speial relativity. 1. Definition and Example Definition 1.1. Suppose S is an n dimensional linear spae over R (real field). S is an n dimensional Minkowski spae if there exist a bivariant real valued funtion ρ : S S R and a basi vetor system e 1, e,, e n of S and satisfying and 1. ρ(u, v) =ρ(v, u);. ρ(ku, v) =kρ(u, v); 3. ρ(u + v, w) =ρ(u, w)+ρ(v, w), (1.1) 4. ρ(e i, e j ) = 1, i = j n 1; 1, i = j = n; 0, i j, (1.) for all u, v, w S and k R. The bivariant real valued funtion ρ is said to be spae time inner produt or alled to be Minkowski inner produt (M inner produt) if ρ satisfies (1.1) and (1.). The basi vetor system (1.) is alled to be a Minkowski orthogonal basis (M orthogonal basis) over S. 000 AMS Mathematis Subjet Classifiation: 15A66,83A05. Key words and phrases: Clifford algebra; n dimensional Minkowski spae; spae time algebra; speial relativity, Lorentz transformation.

160 WUMING LI AND FAN YANG In what follows we use the notation ρ(u, v) = u v. (1.3) Example 1.. Creation spae R n 1,1 of Clifford algebra Cl n 1,1 is an n dimensional real linear spae and its basi set {e 1,, e n 1, e n } under the inner produt of Cl n 1,1 satisfy [1,, 3]: 1, i = j n 1; e i e j = 1, i = j = n; 0, i j. (1.4) If the spae time inner produt is that of Cl n 1,1, then R n 1,1 is an n dimensional Minkowski spae. Example 1.3. Suppose i is a imaginary unit of omplex numbers. S = {(x 1,,x n 1, ix n ) x 1,,x n R}, S is an n dimensional Minkowski spae under the spae time inner produt defined as (x 1,,x n 1, ix n ) (y 1,,y n 1, iy n ) = x 1 y 1 + + x n 1 y n 1 + ix n iy n = x 1 y 1 + + x n 1 y n 1 x n y n, (1.5) Example 1.4. Suppose j is hyperboli virtual unit of Clifford algebra (j =1,j = j) [1,, 3]. Let Define a spae time inner produt T = {(x 1,,x n 1,jx n ) x 1,,x n R}. (x 1,,x n 1,jx n ) (y 1,,y n 1,jy n ) = x 1 y 1 + + x n 1 y n 1 + jx n j y n then T is an n dimensional Minkowski spae. = x 1 y 1 + + x n 1 y n 1 x n y n, (1.6) Example 1.5. R n = {(x 1,,x n ) x 1,,x n R} is also an n dimensional Minkowski spae for this inner produt as follows (x 1,,x n 1,x n ) (y 1,,y n 1,y n ) = x 1 y 1 + + x n 1 y n 1 x n y n. (1.7) When n = 4, R 3,1, S, T, R 4 may be applied for disussing physial problems of speial relativity [1,, 3, 4]. In this paper, we denote M n T and disuss some orrelated problems from M n.

N DIMENSIONAL MINKOWSKI SPACE 161. Spae Time Spae In M n = {(x 1,,x n 1,jx n )}, x 1,,x n 1 is said to be spae omponent and x n is time omponent. Definition.1. Suppose S is an n dimensional Minkowski spae. Aording to its spae time inner produt, the spae time inner produt normal number (for short spae time normal number or M normal number) of vetor w(w S) is defined as w M = w w, (.1) n dimensional Minkowski spae S and spae time normal number (.1) be alled together n dimensional spae-time normi spae (n dimensional M normi spae). The spae is denoted by (S, M ). In order to examine some orrelated properties of the spae time normi spae and the appliation in speial relativity. Let x n = t, x n is time omponent in M n. Where denotes light speed, t is time. So we denote or M n = {(x 1,,x n 1, jt)}, (.) M n = {r + jt}, (.3) where r =(x 1,,x n 1 ) is n 1 dimensional real loation vetor. Proposition.. (1) Suppose (M n, ) is a spae time normi spae. For any w = r + jt M n, w M =0iff r = t, r = r r; () kw M = k w M for any k R, w M n. w 1 + w M w 1 M + w M is not neessary for any w 1, w, w 3 M n. Example.3. Choose w 1 = (1, 0, 0,, 0, j), w = (0, 1, 0,, 0, j) M n, then Let w 1 + w M = 14 > 3+ 3= w 1 M + w M. M n + = {w = r + jt M n t r, iff when t =0, the equality sign is orret}. (.4) M + n is said to be a future timelike region of M n, Ṁ + n = M + n \{0} is alled stritly future timelike region. Theorem.4 (Inverted triangle inequality). Choose any w 1, w M n. When w 1, w M n +, we have w 1 + w M w 1 M + w M, (.5) and iff w 1, w is linear dependent, the equality holds.

16 WUMING LI AND FAN YANG 3. Spae Time Algebra Definition 3.1. Let (S, M ) is an n dimensional spae time normi spae. S is an n dimensional spae time algebra (n dimensional spae time normi algebra. If there exists a multipliation operation defined on S and satisfying a (b +) = a b + a, (b + ) a = b a + a; k(a b) =(ka) b = a (kb) for all a, b, S and k R, besides a b M = a M b M. (3.1) Example 3.. Two dimensional Minkowski spae M may be denoted by M = {x + jy}. (3.) Let the multipliation operation over M be (x 1 + jy 1 )(x + jy )=(x 1 x + y 1 y )+ j(x 1 y + x y 1 ), then M is two dimensional real exhange algebra and (M, ) beomes two dimensional spae time algebra. Define a multipliation operation over an n dimensional Minkowski spae as follows :(r 1 + jt 1 ) (r + jt ) = j ( (r 1 r + t 1 t )+j(t 1 r + t r 1 ) ), (3.3) then M n is n dimensional real algebra, but spae time normi spae M n is not spae time algebra under the operation. ( 3 ( Example 3.3. Choose w 1 = ),, 0, 0,, 0,j w = 0, ) 3, 0,, 0,j M n then we get w 1 w M = 1 4 = w 1 M w M. Theorem 3.4. If r 1, r is linear dependent, then for any w 1, w M n. w 1 w M = w 1 M w M (3.4) Theorem 3.5. By using the spae time normal number (.1) and the binary operation (3.3), we introdue a binary operation over M n as what follows :(r 1 + jt 1 ) (r + jt ) = (r 1 + jt 1 ) (r + jt )+ w 1 M r, (3.5) then M n beomes an n dimensional real algebra and w 1 w M = w 1 M w M, where r and r are the omponent whih r is parallel and perpendiular with r 1 respetively. Thus we an say n dimensional spae time normi spae M n is an n dimensional spae time algebra over the binary operation. Theorem 3.6. For all u, w M + n, we have espeially, let u w M + n, (3.6) w = u w, (3.7) when u M = 1. We an dedue a Lorentz transformation of n dimensional Minkowski spae time.

N DIMENSIONAL MINKOWSKI SPACE 163 Proof. Denote u = r u +jt u for all u, w M n +, u = 1. Then u an be denoted by u = j(osh ϕ + jr u sinh ϕ) [1], where ϕ = artanh ( ) r u t u, r u = r r u u. Let v = r t u u, ( ) 1 ( ) then osh ϕ, sinh ϕ an be written as osh ϕ = 1 v 1, sinh ϕ = 1 v. If let w = r + jt = u w, we obtain Let γ = ( 1 v r + jt = j(osh ϕ jr u sinh ϕ) (r + jt)+r. ) 1, by the expansion of it, we have r = γ(r vt)+r,t = γ(t v r/ ), (3.8) whih satisfies (t ) (r ) =(t) r. When n = 4, substituting v =(v x,v y,v z ), r =(r u r)r u /ru, r = r r into (3.8), we obtain linear equation system ( ) x = 1+(γ 1) v x x + ( ) ( v y + (γ 1) v xv z ) v z γvx t, y = ( ) x + ( z = ( (γ 1) vzvx t = γ vxx γ vyy γ vzz Its orresponding matrix form is x y z jt = 1+(γ 1) v x (γ 1) vxvz 1+(γ 1) v y y + ( ) v z γvy t, ) ( v x + (γ 1) zv y ) ( ) v y + 1+(γ 1) v z v z γ z t, The oeffiient matrix [3, 4] 1+(γ 1) v x v A = (γ 1) vxvz satisfies + γt. ) 1+(γ 1) v y 1+(γ 1) v y (γ 1) vxvz 1+(γ 1) v z (γ 1) vxvz 1+(γ 1) v z γ γ x y z jt (3.9). (3.10) AA H = E, (3.11) where A H is the transposed onjugate matrix of A; E is an unit matrix. Espeially, if v x = v, v y = v z = 0 then equation system (3.10) an be hanged into { x = γ(x vt),y = y, z = z, t = γ(t vx (3.1) ),

164 WUMING LI AND FAN YANG its orresponding matrix form is x osh ϕ 0 0 j sinh ϕ y z = 0 1 0 0 0 0 1 0 jt j sinh ϕ 0 0 osh ϕ x y z jt. (3.13) Referenes 1. W.E. Baylis, Clifford (Geometri) Algebra With Appliations to Physis, Mathematis, and Engineering, Birkhäuser, Boston, 1996.. Wuming Li, Clifford algebra and the properties of Minkowski spae, Journal of Jilin University (Natural Siene), 4 (000), 13 16. 3. Wuming Li, Hyperboli Euler formula in the N dimensional Minkowski spae, Advanes in Applied Clifford Algebra, 1(1) (00), 7 11. 4. Xuegang Yu and Wuming Li, The four dimensional hyperboli spherial harmonis, Advanes in Applied Clifford Algebra, 10 () (000), 163 171. 5. Hongbo Li, Clifford algebra and automated geometri theorem proving, Researh of the World tehnology and development, 3 (3) (001), 41 47. 6. Hongbe Li and Minde Cheng, Proving theorems in elementary geometry with Clifford algebrai method, Advanes in mathematis, 6 (4) (1997), 357 371. 7. Hongbo Li, Hyperboli onformal geometry with Clifford algebra, International Journal of Theoretial Physis, 40 (1) (001), 79 91. 8. Hongbo Li and Minteh Cheng, Clifford algebrai redution method for mehanial theorem proving in differential geometry, Journal of Automated Reasoning, 1 (1998), 1 1. Wuming Li Institute of Math Physis Tonghua Teaher s College, Tohghua Jilin 13400 CHINA thwli@mail.jl.n Fan Yang Shool of Siene Northesatern University Shenyang Liaoning 110004 CHINA yangf1918@163.om Current Address: Institute of Math Physis Tonghua Teaher s College Tohghua Jilin 13400 CHINA blank-line break in here!