ENERGY-DEPENDENT MINKOWSKI METRIC IN SPACE-TIME

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1 ENERGY-DEPENDENT MINKOWSKI METRIC IN SPACE-TIME R. Mron, A. Jannusss and G. Zet Abstract The geometrcal propertes of the space-tme endowed wth a metrc dependng on the energy E of the consdered process are obtaned usng the generalzed Lagrange space methods. The tensor of the electromagnetc feld s ntroduced n order to nclude both electromagnetc and gravtatonal felds n the same model. An example of energy-dependent metrc s gven and a comparson wth other results s presented. AMS Subject Classfcaton: 53B40, 53C60 Key words: generalzed Lagrange space, N-connecton, energy, (deformed Mnkowsk metrc. 1 Introducton In a recent paper [1], Cardone and Mgnan startng from the Mnkowsk metrc ds 2 = c 2 dt 2 { ( dx ( dx ( dx 3 2 } = ηj dx dx j ;, j = 0, 1, 2, 3; x 0 = ct, (1.1 have consdered a deformed Mnkowsk metrc of the form: dš 2 = b 2 0(Ec 2 dt 2 { b 2 1(E ( dx b 2 2 (E ( dx b 2 3 (E ( dx 3 2 } = = ˇη j dx dx j. (1.2 Here, E s the energy of the consdered process. The energy E s vewed as a phenomenologcal varable,.e. E s the energy measured by detectors va the electromagnetc nteracton n the Mnkowsk space M, and b (E > 0 ( = 0, 1, 2, 3. Recently, many authors have consdered deformed Lorentz metrcs n space-tme M, ds 2 = a j dx dx j, wth the coeffcents a j dependng on the coordnates ( x, Edtor Gr.Tsagas Proceedngs of The Conference of Appled Dfferental Geometry - General Relatvty and The Workshop on Global Analyss, Dfferental Geometry and Le Algebras, 2001, c 2004 Balkan Socety of Geometers, Geometry Balkan Press

2 110 R. Mron, A. Jannusss, G. Zet the energy E, the velocty and of other characterstcs of the studed process. Such metrcs are, evdently, not Remannan ones, but of Fnsler type or, more generally, they are generalzed Lagrange metrcs [2]. The generalzed Lagrange metrcs have been ntroduced by R. Mron [2], and appled by G.S. Asanov, S. Ikeda, R. Tavakol, J. Roxburgh, R. Mron, T. Kawaguch, V. Balan, P. Stavrnos, G. Zet and many others to the Relatvstc Optcs and also to the General Relatvty [3], [7], [10 19]. In ths paper we study the geometrcal propertes of the space-tme M endowed wth the metrc (1.2 dependng of the energy E of the consdered process. We adopt the defnton: dx dx j E = η j dt dt. (1.3 In addton, we ntroduce the tensor ˇF j = g k F kj of the electromagnetc feld n order to nclude n the same model both the gravtatonal and electromagnetc propertes based on the deformed metrc ( Generalzed Lagrange Space GL n If M s a dfferentable manfold and (T M, π, M s the tangent bundle, then the coordnates of x M wll be denoted by ( x,, j = 1, 2,..., n = dm(m, and those of u T M, π(u = x, by ( x, y. In the dfferentable manfold T M we have the followng local transformatons of coordnates: ( x x = x (x, det x j 0, (2.1 ỹ = x x j yj. The { natural } bass of the tangent space T u (T M at a pont u = (x, y T M s x, y j,, j = 1, 2,..., n. Takng nto account (2.1, we have: x = xs x x s + ỹs x ỹ s, (2.2 y = ỹs y ỹ s. { } Therefore, the vector felds y, = 1, 2,..., n, generate locally a dstrbuton V. As t results from (2.2, the dstrbuton V s defned everywhere on the tangent manfold T M and s ntegrable, too. V s named the vertcal dstrbuton on T M. Let N be a dstrbuton on T M supplementary to V,.e. T u (T M = N u V u, u T M. (2.3

3 Energy-dependent Mnkowsk metrc n space-tme 111 Then, N s named a horzontal dstrbuton, or a nonlnear connecton on T M. An addapted bass to the dstrbutons { } N and V (or { addapted } to the drect sum decomposton (2.3 s of the form δ δx n N and y n V, where: δ δx = x N j (x, y y j. (2.4 Here, N j (x, y are the coeffcents of the nonlnear connecton N. δ Because the vector felds are contaned n the dstrbuton N, t follows then δx [ ] δ that N s an ntegrable dstrbuton f and only f the brackets δx, δ δx j,, j = 1, 2,..., n, determne vector felds ncluded n N. But, we can wrte [3]: where [ δ δx, ] δ = R s j δxj R = δn j δx k y s, (2.5 δn k δx j (2.6 s a feld of d-tensors (or dstngushed tensors. Ths means that the components R transform lke the components of a tensor of (1, 2-type on the base space M under the coordnates transformaton (2.1 on T M: R = x x s x r x j x p x k Rs rp. The tensor of torson of the nonlnear connecton s: t = N j y k N k y j. Remark. Ths wll be consdered the general rule of defnton for dstngushed tensors (d-tensors on the manfold T M. From (2.5 t follows: Proposton 2.1. The horzontal dstrbuton N (the nonlnear connecton s ntegrable f and only f the d-tensor R on T M s vanshng. In that follows, we wll consder the dual bass { { } dx, δy j} of the addapted bass δ δx, y j. We obtan: δy = dy + N jdx j. (2.7 { } { } δ It s mportant to remark that δx s a d-feld of covarant vectors and y has the same property. Analogous, { dx } s a d-feld of contravarant vectors and { δy } has the same property.

4 112 R. Mron, A. Jannusss, G. Zet A generalzed Lagrange space s a par GL n = (M, g j (x, y, where g j (x, y s a d- feld of tensors on the manfold T M (or, sometmes, on the manfold T M = T M \{0}, covarant, symmetrc, non-degenerate and of constant sgnature. The d-feld g j (x, y s non-degenerate f rank g j (x, y = n. In the next secton we wll show that the space endowed wth the metrc dš 2 n (1.2 s a generalzed Lagrange space. Clearly, f the d-tensor g j (x, y do not depend on the varables y, then GL n = (M, g j (x, y s a pseudo-remannan space (or a Remannan one. When g j (x, y depends only of the varables y (n prefered charts, ths space s a generalzed Lagrange space, locally Mnkowsk. A functon L : (x, y T M L(x, y R, (2.8 dfferentable on T M and contnuous on the null secton of π s named a regular Lagrangan f the Hessan of L wth respect to the varables y s non-sngular. A generalzed Lagrange space GL n = (M, g j (x, y s named reducble to a Lagrange space f there s a regular Lagrangan L such that: g j = L y y j. (2.9 on T M. A necessary condton that GL n be reducble to a Lagrange space s that the d-tensor g j y k s totally symmetrc. If the prevous condton s satsfed and g j are 0-homogeneous n the varables y, then the functon L = g j (x, yy y j s a soluton of the system of equatons (2.9 (wth partal dervatves. In ths case the par (M, L s a Fnsler space (M, F, wth F 2 = L. We say that GL n s reducble to a Fnsler space. A lnear N-connecton on T M (or on T M s defned by a par of geometrcal objects CΓ(N = ( L, C on T M wth the property that L (x, y transform wth respect to (2.1 lke the coeffcents of a lnear connecton on the base manfold M, and C (x, y transform under (2.1 lke a d-tensor of (1, 2-type. We can defne then two types of covarant dervatves: a covarant h-dervatve denoted by and a covarant v-dervatve denoted by. For example, n the case of the feld of d-tensors g j (x, y, we have respectvely: g j k = δg j δx k g sjl s k g s L s, g = g j y k g sjc s k g s C s. We remark that g j k and g j k are d-tensors of type (0, 3. (2.10

5 Energy-dependent Mnkowsk metrc n space-tme 113 Proposton 2.2. The followng Rcc denttes hold: s s g j k h g j h k = g sj R kh g s R j kh T s khg j s R s khg js, g j kh g s s s jh k = g sj P kh g s P j kh C k h g j s P s khg js, g h g s s jhk = g sj S kh g s S j kh S s khg js. (2.11 Here, R j kh, S j kh and P j kh are the d-tensors of curvature: R j kh = δl δx h δl jh δx k + L r L rh L r jh L rk + C jrr r kh, P j kh = L y h C jh k + C jrp r kh, (2.12 S j kh = C y h C jh y k + Cr C rh C r jhc rk, and T, S, R, C, P are the d-tensors of torson of the connecton CΓ(N: T = L L kj, S = C C kj, P = N j y k L kj, (2.13 and R s gven n (2.6. The quanttes C are the v-coeffcents of the connecton CΓ(N. Theorem For every generalzed Lagrange space GL n = (M, g j (x, y ( there s only one lnear N-connecton CΓ(N = L, C whch satsfes the followng axoms 1 0. N s a nonlnear connecton a pror gven; 2 0. g j k = 0, ( CΓ(N s a h-metrc; 3 0. g = 0, ( CΓ(N s a v-metrc; 4 0. T h j = 0, ( CΓ(N s h-symmetrc; 5 0. S h j = 0, ( CΓ(N s v-symmetrc The coeffcents of connecton CΓ(N n 1 0 are gven by the generalzed Chrstoffel symbols: L = 1 ( δgsj 2 gs δx k + δg ks δx j δg δx s C = 1 2 gs ( gsj y k + g ks y j g y s,. (2.14 The tensors of deflecton assocated to the connecton CΓ(N are defned by: { D j = y j = Nj + ys L sj, d j = y = δ j + y s C (2.15 j sj,

6 114 R. Mron, A. Jannusss, G. Zet where y s the Louvlle vector feld on T M,.e. y y. We wll use the connecton CΓ(N, whch s named the metrcal canoncal N- connecton of the generalzed Lagrange space GL n, to study the deformed Mnkowsk metrc ( Energy-Dependent Metrc of Mnkowsk Space Le us consder the deformed Mnkowsk metrc (1.2 (1.3: dš 2 = ˇη j (E dx dx j, (, j,... = 0, 1, 2, 3; x 0 = t, (3.1 where the d-tensor ˇη j (E s gven by the matrx: ˇη j (E = b 2 0 (E c b 2 1 (E b 2 2 (E b 2 3 (E. (3.2 The functons b (E are postve and the varable E s the energy of the consdered process: dx dx j E = η j dt dt = ( dx = c { (dx 1 2 ( dx 2 2 ( dx 3 2 } + +. dt dt dt dt (3.3 It s supposed that the metrc (3.1 s locally defned n the regon of the space-tme where the process occurs. ( dx The dervatve defnes a contravarant vector tangent to the manfold M dt n the pont x. We wll denote ths vector by y = ( y. Then, (x, y s a pont of the tangent bundle T M. Now, let us consder the generalzed Lagrange space GL n = (M, g j (x, y where { g j (x, y = ˇη j ( E(x, y, E(x, y = η j y y j. (3.4 Then, we can prove the followng theorem: Theorem 3.1. The par GL n = (M, g j (x, y, wth g j gven by (3.4 s a generalzed Lagrange space whch s not reducbe to a Remann space, or to a Fnsler space, or to a Lagrange space.

7 Energy-dependent Mnkowsk metrc n space-tme 115 To prove the theorem let us observe that g j n (3.4 depend essentally of y and therefore GL n s not a Remann space. In order to show that GL n s not reducble to a Lagrange space, we must prove that the d-tensor feld g j y k.e. the equaton g j y k s not totally symmetrc, = g k y, (3.5 does not holds. We assume, by absurdum, that the equalty (3.5 s verfed. Then, for = j k we have g j = 0, k. (3.6 yk Now, usng (3.4, the equaton (3.6 becomes: ˇη = 0,, k; k. (3.7 yk Ths result mples the property = 0 whch s mpossble. Therefore, the theorem yk s proved. The followng two partcular cases are nterestng: 1 0. The space GL n has spatal symmetry: b 1 (E = b 2 (E = b 3 (E The space s local conform wth Mnkowsk spaces: b 0 (E = b 1 (E = b 2 (E = b 3 (E. Ths s a partcular case of the Mron-Tavakol metrc [4] whch satsfy the Ehlers- Pran-Schld axomatcs of Specal Relatvty. Another mportant result s: Theorem 3.2. The generalzed Lagrange space GL n endowed wth the fundamental metrc tensor (3.4 s a local Mnkowsk space. Indeed [5], the fundamental tensor g j (x, y do not depend on the varables x (n the exstent chart because = 0, q.e.d. x Therefore, we have g j x k = 0, = 0. (3.8 xk Let us consder then the Mnkowsk space M endowed wth the fundamental tensor η j and denote by F j (x ts electromagnetc tensor. We wll put ˇF j(x, y = ˇη s F sj (x, x M, (3.9 where F j (x = A j x A x j, (3.10

8 116 R. Mron, A. Jannusss, G. Zet wth A (x the potentals of the electromagnetc feld. Therefore ˇF j(x, y from (3.9 s a d-tensor of type (1, 1, called the electromagnetc tensor of the generalzed Lagrange space GL n. Consequently, n GL n, we can a pror gve the nonlnear connecton N wth local coeffcents: { } N j = y k ˇF j(x, y, (3.11 { } where are the Chrstoffel symbols of the Remannan metrc tensor η j. That { } means = 0. Therefore, we have to consder the non-lnear connecton N n the generalzed Lagrange space GL n = (M, ˇη j havng the local coeffcents: N j(x, y = ˇF j(x, y. (3.12 { } δ The addapted bass of the dstrbuton N, δx, = 0, 1, 2, 3, has the form: δ δx = x + ˇF j(x, y y j. (3.13 The bass ( ( dx, δy δ, whch s dual to the bass δx, y, has the local covector felds δy gven by: δy = dy ˇF j(x, y dx j. (3.14 The ntegrablty tensor R of the non-lnear connecton N, defned by the equaton (2.6, s determned by the followng proposton: Proposton 3.1. The tensor R of the non-lnear connecton (3.12 has the expresson: R = δ ˇF k δx j δ ˇF j δx k. ( The Canoncal Metrc N-Connecton of the Space GL n The generalzed Lagrange space GL n = (M, g j studed n ths paper has the fundamental tensor g j (x, y = g j (y: g j (y = ˇη j ( E(y, E(y = ηj y y j. (4.1 We consder then the contravarant components: g j (y = ˇη j( E(y, (4.2

9 Energy-dependent Mnkowsk metrc n space-tme 117 where the matrx ˇη j s of the form: b 2 0 (E/c ˇη j ( E(y = The dervaton operators results: 0 b 2 1 (E b 2 2 (E b 2 3 (E δ δx and δg j δx k = ˇη j x k + ˇF r ˇη j k y r g j y k. (4.3 y appled to the components gj (y gve the = ˇη j = ˇF r k ˇη j y r, (4.4 y k. (4.5 Then, the coeffcents of the canoncal metrc connecton CΓ(N, defned by (2.14, can be wrtten under the form: L = 1 ( ˇηsj ˇηs 2 y r ˇF k r + ˇη sk ˇF j r ˇη ˇF s r, (4.6 C = 1 ( ˇηsj ˇηs 2 y r δr k + ˇη sk δr j ˇη δr s, As we can see, the vanshng of the electromagnetc feld tensor,.e. F j = 0, mples L = 0. In addton, we used the expresson y s = 2η sj y j, (4.7 n order to obtan the last equaltes n (4.4 and (4.5. Of course, we have g j k = 0, g j k = 0, T = 0, S = 0. Now, we can calculate the deflecton tensors D j and d j defned by the equatons (2.15: D j = y j = δy δx j + ys L sj = ˇF j + ys L sj, d j = y = δ j j + y s Csj. (4.8 The covarant components of these tensors are: D j = g r D r ( j = ˇη r ˇF r j + y s L r sj = = F j (x + 1 ( ˇηs ys 2 y p ˇF p j + ˇη j d j = g r d r j = ˇη j + 1 ys 2 y p ˇF p s ˇη sj ˇF p, ( ˇηs δp j + ˇη j δp s ˇη sj δp. (4.9

10 118 R. Mron, A. Jannusss, G. Zet Let ˇF j be the covarant components of the electromagnetc tensor wth respect to the energy-dependent metrc ˇη r : where Therefore, we have or, equvalently (ˇη r η rs = ˇF j = ˇη r F r j = ˇη r η rs F sj, ( b b b b 3 = ( b 2 δ s. (4.11 ˇF 0j = b 0 2 F 0j, ˇF1j = b 1 2 F 1j, ˇF2j = b 2 2 F 2j, ˇF3j = b 3 2 F 3j, ˇF j = b 2 F j, = 0, 1, 2, 3. (4.12 As a consequence, the tensor ˇF j s not antsymmetrc, that s ˇF j ˇF j. (4.13 We defne now the h-electromagnetc nternal tensor on the space GL n by the relaton: F j = 1 2 (D j D j = F j (x + 1 ( ˇηs ys 2 y p ˇF p j ˇη js ˇF p. (4.14 Analogous, the v-electromagnetc nternal tensor on the space GL n has the components f j = 1 2 (d j d j = 1 ( ˇηs ys 2 y p δp j ˇη js δp. (4.15 If we use (4.12, then the components of the tensor F (horzontal and f (vertcal defned n the equatons (4.14 and (4.15 become: F j = 1 ( 2 b + b ( 2 ˇηk j F j + 2 F rj ˇη F r y r y k, (4.16 ( ˇηk f j = η rj ˇη η r y r y k. Havng these components, we can wrte the correspondng Maxwell equatons. Frst of all, let us observe that applyng the Rcc denttes to the Louvlle vector feld y and takng nto account the expressons of the deflecton tensors D j = y j, d j = y j, we obtan [7]: D j k D k j = y m R m d mr m, D j k d k j = y m P m D mc m d mp m, d j d k k = y m S m. j

11 Energy-dependent Mnkowsk metrc n space-tme 119 If we multply these equatons by g h and take the sum over (contracton, then we have: D j k D k j = y m R m d m R m, D d k j = y m P m D m C m d m P m, (4.17 d d kj = y m S m. Consderng the cyclc permutatons of the frst equaton (4.17 and addng the results, we deduce: 2 ( F j k + F + F k j = y m (R m + R m + R mkj (d m R m + d jm R m k + d km R m j. (4.18 Analogous, from the second and thrd equatons (4.17, we obtan 2 ( F j k + F + F k j 2 ( fj k + f kj + f j k = = y m (P m P m + c.p. ( D m C m D jm C m k + c.p. ( d m P m d jm P m k + c.p., (4.19 and, respectvely: 2 ( f j k + f + f k j = y m (S m + S m + S mkj. (4.20 The equatons (4.18, (4.19 and (4.20 are the generalzed Maxwell equatons satsfed by the nternal electromagnetc felds F j and f j. We can wrte these equatons n a more usual form f we consder the Banch denttes satsfed by the canoncal metrc connecton and take nto account the vanshng of the tensors T and S. The relevant Banch denttes are: R j kh + c.p. ( R m Cmh + c.p. = 0, S j kh + c.p. = 0, C j s k C k s j + Pjs mc km P ks mc jm = ( (4.21 P j ks P k js. But, the propretes g j k = 0, g j k = 0 and Rcc denttes gve the followng relatons: R h + R h = 0, P h + P h = 0, S h + S h = 0. Then, we can transform the frst equaton n (4.21 as follows: R h + R khj + R h = R m C mh + R m khc mj + R m hjc mk. Contractng ths equaton by y we obtan: y m (R mh + R mkhj + R mh = y (R m C mh + R m khc mj + R m hjc mk.

12 120 R. Mron, A. Jannusss, G. Zet Fnally, usng ths result, the frst generalzed Maxwell equaton (4.18 becomes: 2 ( F j k + F + F k j = y m (R s jc smk + R s C sm + R s kc smj, (4.22 where R = η s F. The thrd generalzed Maxwell equaton (4.20 can be wrtten xs under the form [usng (4.21]: f j k + f + f k j = 0. (4.23 Fnally, we transform the second generalzed Maxwell equaton (4.19 as follows. Frst, we wrte the last dentty n (4.21 n the form: C js k C ks j + P m ksc jm = P s P kjs. (4.24 Then, contractng (4.24 wth y we have: y ( C js k C ks j + y ( P m js C km P m ksc jm = y m (P ms P mkjs, (4.25 or, equvalently (changng wth m and s wth : y m( C jm k C km j + y m ( P s jc kms P s kc jms = = y m (P m P mkj. (4.26 Therefore, we obtan: y m (P m P m + c.p. = y m ( C jm k C mj k + c.p. + + y m ( P s jc kms P s kc jms + c.p.. (4.27 Now, ntroducng (4.27 n (4.19 we have: 2 ( F j k + F + F k j 2 ( fj k + f + f k j = = y m ( C jm k C mj k + c.p. + y m ( P s jc kms P s jc kms + c.p. ( D m C m D jm C m k + c.p. ( d m P m d jm P m k + c.p.. (4.28 On the other hand, usng the expressons: P = N j y k L kj = L kj = L j k, (4.29 C mj = C jm, C j k = Ckj, we obtan: D m C m D jmck m + c.p. = 0, d m P m d jmpk m + c.p. = 0, C jm k C mj k + c.p. = 0, Pj s C kms Pj s C kms + c.p. = 0, (4.30

13 Energy-dependent Mnkowsk metrc n space-tme 121 Introducng (4.30 n (4.28 we can wrte the second generalzed Maxwell equaton n the form: F j k + F + F k j = f j k + f + f k j. (4.31 Dfferent authors [8, 9] provded a geometrc descrpton of the nteractons between partcles or felds, n the sense that each nteracton produces ts own metrc. In partcular, usng the geometrcal propretes of the generalzed Lagrange spaces endowed wth energy-dependent metrcs, we can study the electromagnetc nteracton and, possble, other types of nteractons. We can also obtan solutons for the generalzed Maxwell equatons and establsh a geometrcal sgnfcance of the energy. 5 An example of energy-dependent metrc We consder, as a smple example, the metrc ds 2 = a(edt 2 [(dx (dx (dx 3 2 ], (5.1 where a(e s an arbtrary functon of the energy E, and the unts c = 1 are understood. Then, the non-vanshng coeffcents C of the canoncal metrc connecton CΓ(N defned n (4.6 are: C00 0 = a a y0, C01 0 = a a y1, C02 0 = a a y2, C03 0 = a a y3, (5.2 C00 1 = a y 1, C00 2 = a y 2, C00 3 = a y 3, wth a = da. We suppose also that there s no electromagnetc feld n the model we de consdered. The ndependent Ensten s equatons correspondng to the metrc (5.1 have the form: a = 0, (5.3 2aa a 2 = 0. (5.4 The equaton (5.3 has the soluton a = const. whch reproduces the Mnkows-k metrc, wthout energy-dependence. The equaton (5.4 has the soluton a(e = 1 ( α 0 + E 2, (5.5 4 E 0 where α 0 and E 0 are two constants of ntegraton. It s mportant to remark that for α 0 = 0 ths soluton concdes wth those gven n Ref. 5 for the strong nteractons: a(e = 1 ( 2 E. On the other hand, f we 4 choose α 0 = 1, then we obtan a(e = 1 4 E 0 ( 1 + E E 0 2, (5.6 whch s the soluton gven n Ref. 5 for the gravtatonal nteracton.

14 122 R. Mron, A. Jannusss, G. Zet References [1] Cardone F. and Mgnan R., Broken Lorentz nvarance and metrc descrpton of nteractons n a deformed Mnkowsk space, Found. Phys. (n press. [2] Mron R., Introducton to theory of Fnsler spaces, Proc. of the Nat. Semnar on Fnsler soaces, Unv. Brasov ( [3] Mron R. and Anastase M., The geometry of Lagrange spaces. Theory and applcatons, Kluwer Acad. Publ., FTPH No.59,1994. [4] Mron R., Tavakol R., Balan V. and Roxbourgh I., Geometry of space-tme and generalzed Lagrange gauge theory, Publ.Math. Debreceen, Vol. 42/3-4 ( [5] Cardone F., Francavgla M. and Mgnan R., Fve-dmensonal relatvty wth energy as extra dmenson, Gen.Rel. and Gravtaton. Vol. 31, No. 7 ( [6] Mron R., Lagrange geometry, Mathl. and Comput. Modellng Vol.20, No. 4/5 ( [7] Mron R. and Zet G., Post-Newtonan approxmaton n relatvstc optcs, Tensor N.S. Vol. 53 ( [8] Cardone F. and Mgnan R., Energy-dependent metrc for gravtaton from clockrate experments, Int. J. Modern Physcs A, Vol. 14, No.24 ( [9] Enders A. and Nmtz G. J.Phys.I (France Vol.2 ( [10] Jannusss A., J. Phys. A26 ( [11] Mron R. and Zet G., Relatvstc optcs of nondspersve meda, Found. Physcs Vol. 25, No. 9 ( [12] Zet G., Lagrange geometrcal models n physcs, Mathl. and Comput. Modellng Vol.20, No. 4/5 ( [13] Zet G. and Manta V., Post-Newtonan estmaton n relatvstc optcs, Int. J. Theor.Phys. Vol. 32 (1993 p [14] Tavakol R.K. and Van der Berg N., Vablty crtera for the theores of gravty and Fnsler spaces, G.R.G. 18 (1986, p [15] Bel R.G., Electrodynamcs from metrc, Int. J. Theor. Physcs 26 ( [16] Ikeda S., Found. of Phys. 10 ( [17] Mron R. and Kawaguch T., Relatvstc geometrcal optcs, Int. J. Theor. Physcs 30, No.11 ( [18] Mron R., Rosca R., Anastase M. and Buchner K., Aspects of Lagrangan Relatvty, Found. Physcs Lett. Vol. 5, No. 2 (1992 p [19] Mron R. and Tavakol R., Geometry of space-tme and generalzed Lagrange spaces, Publ. Math. Debreceen, No. 4 (1994. Authors addresses:

15 Energy-dependent Mnkowsk metrc n space-tme 123 Radu Mron Al.I.Cuza Unversty, Iaş, 6600, ROMANIA E-mal: rmron@uac.ro A. Jannusss Department of Physcs Unversty of Patras 26500, Patras, GREECE G. Zet Al.I.Cuza Unversty, Iaş, 6600, ROMANIA