Complex Spacetime Frame: Four-Vector Identities and Tensors

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1 Advances in Pure Mathematics, 014, 4, Published Online November 014 in SciRes. Complex Spacetime Frame: Four-Vector Identities and Tensors Joseph Aeyo Omolo Department of Physics and Materials Science, Maseno University, Maseno, Kenya Received 9 July 014; revised 9 August 014; accepted 15 September 014 Copyright 014 by author and Scientific Research Publishing Inc. This wor is licensed under the Creative Commons Attribution International License (CC BY. Abstract This paper provides derivation of some basic identities for complex four-component vectors defined in a complex four-dimensional spacetime frame specified by an imaginary temporal axis. The resulting four-vector identities tae exactly the same forms of the standard vector identities established in the familiar three-dimensional space, thereby confirming the consistency of the definition of the complex four-vectors and their mathematical operations in the general complex spacetime frame. Contravariant and covariant forms have been defined, providing appropriate definitions of complex tensors, which point to the possibility of reformulating differential geometry within a spacetime frame. Keywords Complex Spacetime Frame, Four-Vector Identities, Contravariant and Covariant Forms, Complex Tensors 1. Introduction In a recent derivation [1], the present author identified the unit wave vector to be the temporal unit vector within four-dimensional spacetime frame. The temporal direction is specified as an imaginary axis with unit vector i, where i = 1 is the imaginary number. General spacetime frame is then defined as a complex four-dimensional coordinate system spanned by the temporal unit vector i and the three mutually perpendicular spatial unit vectors x, y, z specifying the x, y, z axes, respectively. We tae the temporal unit vector to have general orientation relative to the spatial unit vectors according to x 0, y 0, z 0 (1 The usual assumption, implicit in conventional four-vector mathematics that the temporal axis is perpendicu- How to cite this paper: Omolo, J.A. (014 Complex Spacetime Frame: Four-Vector Identities and Tensors. Advances in Pure Mathematics, 4,

2 lar to all the three mutually perpendicular spatial axes, with x = 0, y = 0, z = 0, may occur only as a special case to be specified. The basic elements of the complex spacetime frame are complex four-component vectors, which we generally call four-vectors. We define a general four-vector V in the form V= V + V = V + Vx + Vy + Vz (a with the imaginary temporal component x y z V defined by V = icφ, φ = φ r, t (b where φ is a scalar quantity specifying the nature of the temporal component of the four-vector. As usual, t denotes time. In general, the temporal component of each four-vector occurs along the imaginary temporal axis, i, multiplied by a factor c, where c is the speed of light. We follow standard convention denoting fourvectors by uppercase letters, e.g., V, while the usual three-component spatial vectors are denoted with arrows over letter symbols, e.g., r. According to the general definition in Equations (a-(b, the spacetime displacement four-vector X taes the form X = x + r = x + xx + yy + zz, x = ict (3a ( while the corresponding event interval dx between two neighboring spacetime points, taes the form dx = dx + dr = dx + dxx + dyy + d zz, dx = icdt (3b The spacetime derivative four-vector, is defined as where x i = + = + (4a x c = ict and is the usual three-component spatial gradient vector defined by = x + y + z (4b x y z Interpreting in Equation (4a in terms of the general four-vector V in Equations (a-(b gives i 1 V = V =, φ =, V = (4c c c We observe that the concept of imaginary temporal axis developed here, represents a rediscovery of the idea of imaginary time first introduced independently by Poincare [], Lorentz [3] and Einstein [4] in their original theories of electrodynamics or special relativity in a four-dimensional spacetime frame. These authors did not identify the temporal unit vector and therefore could not completely specify the complex spacetime frame and develop the full mathematical operations using complex four-vectors in the manner presented in this paper.. Mathematical Operations with Four-Vectors The general four-component vector form in Equation (a with all unit vectors specified allows us to carry out four-vector mathematical operations in the complex spacetime frame in exactly the same manner as the standard mathematical operations with the familiar three-component vectors in three-dimensional space. In developing the mathematical operations in general form, we shall tae the temporal unit vector to be of general orientation relative to the spatial unit vectors x, y, z, satisfying the conditions in Equation (1. We denote four-vectors with arrows, while the three-component spatial vectors are written in boldface as stated above. We use two general four-vectors V and U defined by V = V + V, V = icφ, U = U + U, U = icϕ (5 to develop the mathematical operations with four-vectors. The basic mathematical operations are essentially addition, subtraction, dot product, cross product, divergence and curl..1. Addition and Subtraction Four-vector addition and subtraction is straightforward, taing the form 568

3 .. The Dot Product ( ( The dot product of the four-vectors U and V is obtained as W = U ± V = U ± V + U ± V (6 ( ( U V= U+ U V+ V (7a which we expand term by term, maintaining the order of components in the products and then substitute U = icϕ, V = icφ from Equation (5, to obtain the dot product in the final form.3. The Cross Product The cross product of the four-vectors U and V is obtained as ( U V = U V c ϕφ ic ϕv + U φ (7b ( ( U V= U+ U V+ V (8a which we expand term by term, using = 0, U = U and then substitute U from Equation (5 to obtain the cross product in the final form.4. Divergence of a Four-Vector ( Setting U equal to the spacetime derivative four-vector according to with = icϕ, V = icφ U V = U V ic ϕv U φ (8b i 1 U = = + = ic icϕ + = + U (9a c c 1 ϕ =, U = (9b c in the general four-vector dot product obtained in Equation (7b, we obtain the divergence of a general fourvector V in the final form.5. Curl of a Four-Vector φ 1 V = + V + i ( cφ (9c c Setting U equal to the spacetime derivative four-vector according to Equations (9a, (9b in the general four-vector cross product obtained in Equation (8b, we obtain the curl of a general four-vector V in the final form 1 V = V i + ( cφ (10 c 3. Four-Vector Identities We now derive some basic four-vector identities in complex four-dimensional spacetime frame, which generalize standard vector identities in three-dimensional Euclidean space [5] [6] Curl of Gradient Four-Vector A gradient four-vector Φ generated through application of the spacetime derivative four-vector on a scalar function Φ is obtained as 569

4 with i 1 Φ Φ= ic + Φ= + Φ c c (11a Setting the general four-vector V equal to the gradient four-vector according to 1 Φ V = Φ= ic icφ + Φ= + V (11b c 1 Φ φ =, V = Φ (11c c in the general curl of a four-vector obtained in Equation (10, we obtain the curl of a gradient four-vector Φ in the form 1 Φ 1 Φ Φ = Φ i (11d c c which on using the standard three-dimensional space vector analysis results gives the final result 1 Φ 1 Φ Φ= 0, = (11e c c Φ = 0 (1 This shows that the curl of a gradient four-vector vanishes. This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given in the first part of Equation (11e. 3.. Divergence of Curl of a Four-Vector Taing the divergence of the curl of the general four-vector V in Equation (10, we obtain which on expanding term by term becomes i 1 V i ( cφ = + V + (13a c c i V ( c = V + φ i ( cφ + + V + (13b c c c c Applying standard three-dimensional space vector analysis results we express Equation (13b in the form 1 V = 0, + ( cφ = 0 (13c c 1 1 V i ( cφ = + (13d c c Application of standard three-dimensional space vector identity gives which on using ( Q R R ( Q Q ( R = (13e = + + (13f c c c ( cφ ( c ( c ( φ φ 570

5 taes the final form ( cφ = 0, = 0 (13g 1 ( c φ 1 + = (13h c c Substituting Equation (13h into Equation (13d gives the final result V = 0 (14 This shows that the divergence of curl of a four-vector vanishes. This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given in the first part of Equation (13c General Vanishing Four-Vector Dot Product: U ( U V The important identity on the vanishing of the divergence of curl of a four-vector in Equation (14 can be generalized by taing the dot product of the four-vector U and the cross product of the four-vectors U and V which on using the general result in Equation (8b is obtained as ( ( { ( } U U V = U+ U U V + UV U V (15a which we expand term by term and use standard three-dimensional space vector identities to obtain { } ( ( U V U U V = 0, V U = 0 (15b { } ( ( ( U U V = U U V + U U V U V (15c Applying a three-dimensional space vector identity and using gives { ( U } V {( U V } U V U = V U U (15d UV U = V U U = 0 (15e { ( U } V ( U which we substitute into Equation (15c to obtain the final result U V U = U V (15f U ( U V = 0 (15g This result generalizes the divergence of curl of a four-vector obtained in Equation (14. It is a generalization of the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given in the first part of Equation (15b Divergence and Curl of fv For a scalar function f( rt,, we use the definitions of V and from Equations (a and (4a to obtain i ( { ( i }, ( fv f V { ( V fv f V V } = + + = + + (16a c c Expanding these term by term gives i f ( i f i i fv f V = f + f + V + f + c c V V (16b c c 571

6 i f ( i f i f i f fv f V = f + f + V + f + c c V V (16c c c which we reorganize to obtain the four-vector identities (, ( fv = f V + f V fv = f V + f V (16d These four-vector identities generalize the corresponding vector identities in standard three-dimensional Euclidean space [5] [6] Divergence of Four-Vector Cross Product: ( U V We use the general form of the curl of a four-vector from equation (10 to obtain 1 U ( V ( i cϕ V = U V V + + ( cφ U (17a c 1 V ( U ( i cφ U = V U U + + ( cϕ V (17b c after applying standard three-dimensional space vector identities 1 1 ( c 0, U φ ( cϕ + = + = 0 (17c c c 1 ( c 1 U + φ = + ( cφ U (17d c c 1 U ( c 1 U V + ϕ = + ( cϕ V (17e c c Subtracting Equation (17b from Equation (17a and applying standard three-dimensional vector identities, together with appropriate rules of differentiation of vector products, we obtain the final result ( ( ( 1 V U U V = U V + i ( U V + ( cϕv ( cφu (17f c We now use the four-vector cross product from Equation (8b to obtain ( i ( U V = ic + U V ( ϕv Uφ (18a c which we expand as appropriate and apply standard three-dimensional space vector identities to obtain the final result ( ϕv Uφ 0, ( ϕv Uφ ( ϕv Uφ = = (18b ( ( 1 U V = U V + i ( U V + ( cϕv ( cφu (18c c Substituting Equation (17f into Equation (18c gives the four-vector identity ( U V = V ( U U ( V (18d This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given earlier in Equation (13e The Curl of a Four-Vector: ( V Let us start by taing the four-vector curl of the curl of the general complex four-vector in Equation (10 to obtain 57

7 which we on expansion taes the form i 1 ( V i = + V + + ( cλ c c ( V i ( c ( i 1 = + Λ + V + + ( cλ (19b c c c c Next, we tae the four-vector gradient of the divergence of the general complex four-vector in Equation (9c to obtain which we expand in the form (19a i 1 ( Λ V i = + + V + ( cλ (19c c c i ( Λ 1 1 Λ 1 V ( c i = + V Λ + + V + ( cλ (19d c c c c We apply standard three-dimensional Euclidean space vector identities giving ( c ( c ( c Λ = Λ Λ ( cλ c c c c c = ( cλ + ( cλ. c c c c (19e ( c ( Λ = + ( cλ (19f c c c ( = ( + V V V (19g which we substitute into Equation (19d as appropriate to obtain the final form i ( Λ Λ 1 1 ( V = + V + + V + V ( cλ c c c ( c i i ( Λ + + ( cλ c c c c. Subtracting Equation (19b from Equation (19h and using standard three-dimensional Euclidean space vector identities giving we obtain (19h ( c ( c ( + Λ = + Λ + ( cλ (0a c c c ( V ( V ( ( ( ( ( cλ = cλ cλ (0b V ( ( ( i Λ 1 V 1 = + V + V + cλ + iv i + cλ. c c c c which on reorganizing taes the form ( V (, V ( V (0c 1 1 ( c c c i i ic + Λ = + Λ Λ + = Λ + = iv (0d c c 573

8 ( c 1 Λ 1 1 ( ( ( V ( V V = i cλ + V + i (0e c c c We easily obtain ( c 1 Λ ( 1 V 1 1 i c ( ic Λ + V = Λ + V = V (0f c c c c which we substitute into Equation (0c to obtain Setting 1 1 V V = + i V (0g c c ( ( ( in the general four-vector dot product in Equation (7b gives 1 U = V = ϕ =Λ=, U = V = (1a c ( 1 1 = = + i c c (1b which we substitute into Equation (4e and reorganize to obtain the four-vector identity ( V = ( V V ( This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6] given earlier in Equation (19g Gradient of Dot Product of Four-Vectors: ( U V For general complex four-vectors U, V, we have Expanding these term by term gives ( ( 1 U V = icϕ + U V + i + ( cλ (3a c ( ( 1 U V U = icλ + V U + i + ( cϕ (3b c 1 ( U V ϕ i cϕ = U + ( icλ U + V (3c c 1 ( V U = V +Λ i cλ ( icϕ V + U ( ( U V + V U (3d c ( ic c ϕ Λ U = ϕ V +Λ U + + c c { ( ( ( } 1 V 1 U ϕ ϕ U V c c { U ( ( V ( ( ϕ } U ( V V ( V. + c cλ +Λ c + i + + i cλ + c + + (3e 574

9 ( + ( U V V U ( U ( ( cλ V ( cϕ ( U ( V { } {( ( } Λ ϕ = ic U Λ+ V ϕ ic ϕ +Λ cϕ( ( cλ + cλ( ( cϕ + + ic +Λ c c { ϕ ( V ( U} U U + i + + ϕ +Λ + ( U V + ( V U. c c We apply three-dimensional Euclidean space vector identities to obtain ϕ Λ U c c ϕ Λ U ϕ Λ U + = + c + c c c c c c { ( ( ( } ( ( ( { } ( ( ( (3f (4a c ϕ cλ +Λ V cϕ = c ϕ cλ +ΛV cϕ c ϕ cλ +ΛV cϕ (4b ( ( i V U i V U i V U U + V = U + V U + V (4c c c c c c c ϕ { ϕ } {( ( ( ( ϕ } { ϕ V U} { ϕ ( V ( U } { ϕ( V ( U} { ( ( ( ( } ( ( i U cλ + V c = i U cλ + V c i U cλ + V c (4d ic +Λ = ic +Λ + ic +Λ (4e Substituting Equations (4a-(4e into Equation (3e, adding the result to Equation (3f and reorganizing gives i i U V + V U + U V + V U = c Λ c c ( ( ( ( ( U V ( ϕ ϕ Λ U 1 ϕ 1 Λ ( V ( U c c c c { ϕ ( V ( U } {( V ϕ ( U } + c + + c + ic +Λ ic + Λ. where we have applied a standard three-dimensional Euclidean space vector identity to introduce Rewriting ( = ( + ( + ( + ( (4f U V U V V U U V V U (4g 1 ϕ 1 i ( ( { ( ϕ } ϕ Λ U c c Λ ic + + V + U = V + U Λ (5a c c c c c { ϕ ( V ( U } {( V ϕ ( U } { ( ϕv U } ic +Λ ic + Λ = ic + Λ (5b we express Equation (4f in the form ( i ( ( ( ( U V + V U + U V + V U = c ϕ ic + Λ ( ϕ + Λ c U V V U (5c which on substituting the definition of the spacetime derivative four-vector from Equation (6a and the fourvector dot product obtained in Equation (11c, gives the desired four-vector identity in the final form ( U V = U ( V + V ( U + ( U V + ( V U (6 This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Eucli- 575

10 dean space [5] [6] given earlier in Equation (4g Triple Cross Product of Four-Vectors: W ( U V We introduce a third four-vector defined by and use the four-vector cross product to obtain W = icχ +W (7 ( ( χ ( ( ϕ W U V = ic + W U V ic V U Λ (8a which we expand as W U V = icχ U V c χ ϕv UΛ icw ϕv UΛ + W U V (8b ( ( ( { } { ( } ( We apply standard three-dimensional Euclidean space vector identities to write icχ ( = icχ{ ( ( } { ( } ( { ( ( } { ( ϕ } { ( ( ϕ ( ϕ ( } U V U V V U (8c c χ ϕ Λ = c χ ϕ Λ ϕ Λ V U V U V U (8d icw V UΛ = ic W V UΛ V UΛ W (8e W ( U V = U( W V V ( W U (8f which we substitute into Equation (8b and collect lie terms to obtain ( χ χ ( χϕ ( χ ϕ icϕ ( W V ic χv + icλ ( W U ic χu. ( ( W U V = U W V c Λ ic V + WΛ V W U c ic U + W We rewrite the last two terms in Equation (8g by subtracting and adding appropriate terms according to ( ( (8g W V ic χv = W V c χλ ic χv ic WΛ + c χλ+ ic W Λ (9a W U ic χu = W U c χϕ ic χu ic Wϕ + c χϕ + ic W ϕ (9b which we substitute bac and collect lie terms to express Equation (8g in the form ( W U V ( icϕ U ( W V c χ ic ( χv W ( ic V ( W U c χϕ ic ( χu Wϕ = + Λ + Λ Λ + +. Finally, we apply the usual definitions of the four-vectors U, V and introduce the four-vector dot products (, ( W V c ic W U c ic (9c = W V χλ χv + WΛ = W U χϕ χu + W ϕ (9d in Equation (9c to obtain the desired four-vector identity in the form W ( U V = U( W V V ( W U (30a This four-vector identity generalizes the corresponding vector identity in standard three-dimensional Euclidean space given earlier in Equation (8f. We easily apply the four-vector identity obtained in Equation (30a to establish the cyclic property of the triple four-vector cross product in the form W ( U V + U ( V W + V ( W U = 0 (30b which generalizes the corresponding vector identity in standard three-dimensional Euclidean space [5] [6]. The four-vector identities derived in Equations (1, (14, (15g, (16d, (18d, (, (6, (30a and (30b con- 576

11 firm the consistency of the definitions of the complex four-component vectors and corresponding mathematical operations within the complex four-dimensional spacetime frame. This means that complex four-dimensional spacetime frame characterized by complex four-component vectors is a consistent mathematical extension of the standard three-dimensional space characterized by the usual three-component vectors. The other four-vector identities can be derived following similar procedure. 4. Contravariant and Covariant Four-Vectors To complete the mathematical formalism within complex four-dimensional spacetime frame, we introduce contravariant and covariant forms, which are useful in carrying out general mathematical operations with four-vectors. A contravariant four-vector is specified by positive spatial components, while a covariant four-vector is specified by negative spatial components. We represent the four-vector V in a contravariant form by V and in a covariant form by V, where = 0, 1,, 3 with 0 labeling the temporal component, while 1,, 3 label the spatial ( xyz,, components, respectively. We define V and V below. Denoting the four unit vectors by, x, y, z as presented in this paper, we define the contravariant coordinates x of the general four-dimensional complex spacetime frame in the form [7] [8] x, 0,1,,3, x ict, x x, x y, x z = = = = = (31a The corresponding covariant coordinates x are defined in the form x x x ict x x x x x y x x z 0 1 3, = 0,1,,3, 0 = =, 1 = =, = =, 3 = = (31b We then express the contravariant spacetime displacement four-vector X and the corresponding covariant form X as = = (31c X x xx x y xz, X x 0 xx 1 xy xz 3 which we introduce the position vector r = xx + yy + zz according to Equations (31a-(31b to express in the final forms X = ( ict, X ( ict = + 0 The spacetime event interval taes the contravariant and covariant forms ( dx = dx0 dx = dx 0 + d = ( icdt d, dx dx 0 d ( icdt = = + d with r r (31d r r r r (31e A general four-vector V as defined earlier is expressed in contravariant and covariant forms according to which we express in the final forms V = icφ, V = V, V = V, V = V (3a x y z V = V, V = V, V = V, V = V (3b = + + +, = (3c V V V x V y V z V V0 V1x Vy V3z ( φ, ( φ V = ic V V = ic + V (3d The contravariant and covariant four-vectors are related through complex conjugation in the form V = V, V = V (3e We use this contravariant-covariant four-vector conjugation relation to obtain (, ( V V = V V V = V (3f which provides the definition of the invariant length V of the general four-vector V ( ( V V V or V according to = = (3g 577

12 We express this in the general form V V V V V = = (3h Using V, V from Equation (3d, noting the relation in Equation (3e, we apply Equation (3g or (3h to obtain the invariant length in the explicit form ( cφ V ( c φ V ( φ V η( φ V V = 1+ c = c which is modified by a factor η arising from the general orientation of the temporal unit vector relative to the spatial unit vectors according to the definition ( cφ V ( c φ V η = 1 +, V = V V The invariant length ds of the spacetime event interval is then modified according to ( s η ( ct ( ( v (3i (3j 4 dr d = ( d d r, η = 1 +, v = (3 dt v c 1 c where v is the velocity, with speed v = v defined as usual. This result has important implications for current theoretical and experimental research in physics [9] [10]. Tensors in the Complex Spacetime Frame We now develop the procedure for defining tensors [7] [8] within the general four-dimensional complex spacetime frame. To put the presentation in familiar form, we adopt the standard contravariant and covariant fourvector notation to express V and V from Equation (0d in the form with complex conjugates taing the form ( φ, ; ( φ, V = ic V V = ic V (33a ( φ, ; ( φ, V = ic V = V V = ic V = V (33b where the usual four-vector mathematics is applied, but now taing account of the general orientation of the temporal unit vector relative to the spatial unit vectors x, y, z to obtain the general results presented above. Using the complex conjugation relation from Equation (33b gives V V V V ν ; V = V = V V (34a ν ν ν from which a definition of complex contravariant and covariant ran- tensors T ν and T ν follows according to In addition, T ν V V V V ν ; T V = = = V = V V (34b ν ν ν ν V V ν V V ; V = V ν = V V ν (34c ν provides a definition of complex ran- mixed tensors T ν and T ν in the form T V V ν V V ; T ν V = = = V ν = V V ν (34d ν ν 578

13 The definition of more general tensors of higher ran follows easily. Some mathematical properties of the ran- tensors defined above can be obtained by interchanging the indices, ν or taing complex conjugation or carrying out both operations simultaneously. The complete definition of contravariant and covariant complex four-vectors, which can be used to define tensors of general rans in contravariant, covariant or mixed forms, provides the necessary foundation for more general vector and tensor analysis, leading to reformulation of differential geometry a complex four-dimensional spacetime frame. This is indeed the origin of a new framewor for studying physics, mathematics and related disciplines in the 1st-century and beyond. Some important implications for physics are presented in [1]. 5. Conclusion All the basic four-vector identities which we have derived in this wor tae exactly the same form as the standard vector identities established in the familiar three-dimensional space. This confirms the consistency of the definition of complex four-component vectors and corresponding mathematical operations within a complex four-dimensional spacetime frame with an imaginary temporal axis. The contravariant and covariant forms introduced here lead to consistent definitions of complex tensors, which are the basic quantities for reformulation of differential geometry within complex spacetime frame. This new mathematical framewor has important implications for various models of relativistic mechanics, quantum field theory and general relativity as a theory of gravitation and cosmology. Acnowledgements I than Maseno University, Kenya, for supporting this wor by providing facilities and woring environment. References [1] Aeyo Omolo, J. (014 On a Derivation of the Temporal Unit Vector of Space-Time Frame. Journal of Applied Mathematics and Physics, in Press. [] Poincare, H. (1906 Sur la dynamique de l e lectron (On the Dynamics of the Electron. Rendiconti del Circolo Matematico di Palermo, 1, 19. [3] Lorentz, H.A. (1904 Electromagnetic Phenomena in a System Moving with Any Velocity Smaller than That of Light. Proceedings of the Royal Netherlands Academy of Arts and Sciences, 6, 809 [4] Einstein, A. (003 The Meaning of Relativity. Routledge Classics, London and New Yor. [5] Charbonneau, P. (013 Solar and Stellar Dynamics. Saas-Fee Advanced Course, 39, 15. [6] Arfen, G.B. and Weber, H.J. (1995 Mathematical Methods for Physicists. Academic Press Inc., San Diego. [7] Landau, L.D. and Lifshitz, E.M. (1975 The Classical Theory of Fields. Pergamon Press Ltd., Oxford. [8] Dirac, P.A.M. (1975 General Theory of Relativity. John Wiley and Sons Inc., New Yor. [9] Bogoslovsy, G.Y. (007 Some Physical Displays of the Space Anisotropy Relevant to the Feasibility of Its Being Detected at a Laboratory. arxiv: [gr-qc] [10] Gibbons, G.W., Gomis, J. and Pope, C.N. (007 General Very Special Relativity in Finsler Geometry. Physical Review D, 76, Article ID:

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