Ratio Mathematica Volume 42, 2022 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean, or Lorentz, or generally Isotropic Metric Spyridon Vossos* Elias Vossos† Christos G. Massouros‡ Abstract This paper is the first in a series of documents showing that Newtonian physics and Einsteinian relativity theory can be unified, by using a Generalized Real Boost (GRB), which expresses both the Galilean Transformation (GT) and the Lorentz Boost. Here, it is proved that the Closed Linear Transformations (CLTs) in Spacetime (ST) correlating frames having parallel spatial axes, are expressed via a 4x4 matrix ΛI, which contains complex Cartesian Coordinates (CCs) of the velocity of one Observer / Frame (O/F) wrt another. In the case of generalized Special Relativity (SR), the inertial Os/Fs are related via isotropic ST endowed with constant real metric, which yields the constant characteristic parameter ωI that is contained in the CLT and GRB of the specific SR. If ωI is imaginary number, the ST can only be described by using complex CCs and there exists real Universal Speed (cI). The specific value ωI=±i gives the Lorentzian-Einsteinian versions of CLT and GRB in ST endowed with metric: -gI00η and cI=c, where i; c; gI00; η are the imaginary unit; speed of light in vacuum; time-coefficient of metric; Lorentz metric, respectively. If ωI is real number, the corresponding ST can be described by using real CCs, but does not exist cI. The specific value ωI=0 gives GT with infinite cI. GT is also the reduction of the CLT and GRB, if one O/F has small velocity wrt another. The results may be applied to any ST * Core department, National and Kapodistrian University of Athens, Euripus Campus, GR 34400, Psahna, Euboia, Greece; svossos@uoa.gr. † Core department, National and Kapodistrian University of Athens, Euripus Campus, GR 34400, Psahna, Euboia, Greece; evossos@uoa.gr. ‡ Core department, National and Kapodistrian University of Athens, Euripus Campus, GR 34400, Psahna, Euboia, Greece; ChrMas@uoa.gr. 305 S. Vossos, E. Vossos, Ch. G. Massouros endowed with isotropic metric, whose elements (four-vectors) have spatial part (vector) that is element of the ordinary Euclidean space. Keywords: 5th Euclidean postulate; complex space; electromagnetic tensor; Euclidean metric; Euclidean space; Galilean Transformation, general relativity; isometry; linear transformation; Lorentz boost, Lorentz metric, Lorentz transformation, Minkowski spacetime, Newtonian physics, spacetime; special relativity; universal speed. 2010 AMS subject classification: 15A04; 83A05.§ Abbreviations-Annotations CCs: Cartesian Coordinates CILToCST: Closed Isometric Linear Transformation of Complex Spacetime CLT: Closed Linear Transformation cI: Universal Speed E3: three-dimensional Euclidean Space E4: Euclidean Spacetime ERT: Einsteinian Relativity Theory ESR: Einsteinian Special Relativity GR: General Relativity GRB: Generalized Real Boost GSR: Generalized Special Relativity GT: Galilean Transformation IO: Inertial Observer LT: Linear Transformation LB: Lorentz Boost M4: Minkowski Spacetime NPs: Newtonian Physics O/F: Observer / Frame QMs: Quantum Mechanics RT: Relativity Theory SR: Special Relativity ST: Spacetime (four-dimensional Space) TPs: Theory of Physics U: Invariant Speed wrt: with respect to § Received on June 7th, 2022. Accepted on June 29th, 2022. Published on June 30th, 2022. doi: 10.23755/rm.v41i0.810. ISSN: 1592-7415. eISSN: 2282-8214. © Spyridon Vossos et al. This paper is published under the CC-BY licence agreement. 306 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric 1 Introduction Linear transformations (LTs) are very important in Relativity Theory (RT) and Quantum Mechanics (QMs) [1]. Moreover, there exist many different approaches of RT, which emerge the corresponding QMs. For instance, Galilean Transformation (GT) endowed with the corresponding metric of Spacetime (ST) produces Newtonian Physics (NPs), which gives the classic QMs (Schrödinger Equation). Thus, many low-velocity phenomena, like the atomic spectra (without fine structure) were explained. On the other hand, Lorentz Transformation (endowed with the Lorentz metric of ST) produces Einsteinian Special Relativity (ESR), which gives relativistic QMs (Klein- Gordon Equation). Thus, many high-velocity phenomena and the fine structure of atomic spectra were explained [2]. In this paper, we prove that there exist two types of Closed Isometric Linear Transformation of Complex Spacetime (CILToCST) with common solution the GT. These can apply not only to Special Relativity (SR), but also to General Relativity (GR), because they are reached without adopting one specific metric of spacetime. In addition, any complex Cartesian Coordinates (CCs) of the theory may be turned to the corresponding real CCs, in order to be perceived by human senses [3] (pp. 5-6). SR relates the frames of Inertial Observers (IOs), via LTs of linear spacetime. ESR uses real spacetime (Minkowski spacetime) (M4) endowed with Lorentz Metric (η) and the frames of two IOs with parallel spatial axes are always related via Lorentz Boost (LB). But is known that LB is not Closed Linear Transformation (CLT). In contrast, Lorentz Transformation (combination of spatial Euclidean Rotation with LB) is CLT (e.g. see [4], p. 41, eq. 1.104). Thus, if three Observers / Frames (Os/Fs): Oxyz, O΄x΄y΄z΄ and O΄΄x΄΄y΄΄z΄΄ are related, where the axes of O΄x΄y΄z΄ are parallel not only to the corresponding axes of Oxyz, but also to the corresponding axes of Ο΄΄x΄΄y΄΄z΄΄, then the axes of Oxyz and Ο΄΄x΄΄y΄΄z΄΄ are not parallel (Figure 1). Thus, the transitive attribute in parallelism (which is equivalent to the 5th Euclidean postulate) is cancelled, when more than two Os/Fs are related. This consideration leads to successful results, such as Thomas Precession, which explains the fine structure of atomic spectra. But this happens only if we take successive observers O, O΄ and O΄΄ with Thomas’ order [5]. The reversed order of this sequence yields a result with 200% relative error. 307 S. Vossos, E. Vossos, Ch. G. Massouros Figure 1: Correlation of three successive observers (frames), by using Lorentz Boost. The frame O΄x΄y΄z΄ has parallel axes to the corresponding of frame Oxyz, moving with velocity (β1 c, 0, 0) wrt Oxyz. The frame O΄΄x΄΄y΄΄z΄΄ has parallel axes to the corresponding of frame O΄x΄y΄z΄, moving with velocity (0, β2 c, 0) wrt O΄x΄y΄z΄. The correlation of the observers, by using Lorentz Boost, cancels the absolute character of parallelism. Thus, the axes of frame O΄΄x΄΄y΄΄z΄΄ are not parallel to the corresponding of frame Oxyz (Thomas Rotation). In this paper, we prove that there exists CLT, which relates Os/Fs with parallel spatial axes (in case of IOs, or observers that have the same acceleration). Thus, the transitive attribute in parallelism is valid in complex three-dimensional Euclidean Space (E3) and the axes rotation that happens in real space, when more than two observers are related, is the equivalent phenomenon of the corresponding Generalized Real Boost (GRB) [3] (pp. 5- 6). The CLT is divided into two cases: one, where time depends on the position where the event happens, which can have real Invariant Speed (U) and another, where time is independent from the position and has U=∞. Moreover, the demand that the CLT is isometric, gives the CILToCST. If the metric of ST is independent from the position of the event in ST, we have the case of SR and the CILToCST may be applied globally, relating IOs. Thus, infinite number of SR-theories is produced (each one of which with the corresponding metric of ST), keeping the ESR-formalism. In the case that the metric of ST depends on the position of the event in ST, we have the case of GR and the CILToCST may be applied locally, relating Os/Fs with the same acceleration. Thus, infinite number of GR-theories is produced (each one of which with the corresponding metric of ST of IOs), all of them keeping Einsteinian GR- formalism. Of course, zero acceleration leads to the corresponding SR. Finally, we present the improper isometric LT in ST endowed with Euclidean, or Lorentz, or generally any isotropic metric. 308 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric 2 The Matrix of Closed Linear Transformation of Complex Spacetime Initially, we determine the matrix Λ of active interpretation of the CLT of complex ST endowed with any metric. Figure 2: Two frames Oxyz and O΄x΄y΄z΄ initially coincide. The second is moving with velocity (βc, 0, 0) wrt to Oxyz. 2.1 Motion in the x-Direction We consider one unmoved O/F Oxyz, measuring real spacetime and another O/F O΄x΄y΄z΄ with parallel spatial axes, moving to the right, along x-axis with velocity cc  ==  wrt O/F Oxyz (Figure 2), where c=299,792,458 m s−1 is the speed of light in vacuum and the frames initially coincide. Supposing the next linear transformation cdt΄ = bcdt + adx + kdy + νdz (1) dx΄ = gcdt + fdx + δdy + θdz (2) dy΄ = g1cdt + f1dx + hdy + λdz (3) dz΄ = g2cdt + f2dx + ξdy + μdz, (4) we determine the coefficients with the following condition: the space has isotropy. Rotating the coordinates system about the x-axis, by one negative right angle (Figure 1), we correspond the new axes to the initial axes: t→t, t΄→t΄, x→x, x΄→x΄, y→-z, y΄→-z΄, z→y and z΄→y΄. Thus, from (1), we have cdt΄ = bcdt + adx - kdz + νdy. (5) (1) compared to (5), gives k=ν=0. Besides, from (2) we have dx΄= gcdt + fdx - δdz + θdy. (6) (2) compared to (6), gives δ=θ=0. Besides, from (3) we obtain -dz΄= g1cdt + f1dx - hdz + λdy. (7) (4) compared to (7), gives g2=-g1, f2 =-f1, ξ=-λ and μ=h. Besides, from (4), we have dy΄= g2cdt + f2dx - ξdz + μdy. (8) (3) compared to (8), gives g2=g1, f2=f1, ξ=-λ and μ=h. So, k=ν=δ=θ=g1=g2=f1=f2=0, ξ=-λ, μ=h and the transformation becomes 309 S. Vossos, E. Vossos, Ch. G. Massouros cdt΄ = bcdt +adx (9) dx΄ = gcdt + fdx (10) dy΄ = hdy + λdz (11) dz΄ = -λdy + hdz. (12) Using matrices we have the active interpretation of the LT [4] (p. 6):                          − =                 z y x t h h fg ab z y x t d d d cd 00 00 00 00 d d d cd   , (13) or equivalently, dΧ΄ = Λ1(x) dΧ , (14) where the base and the coordinates are     3210 eeeee  = ;             =               = z y x t x x x x X d d d cd d d d d d 3 2 1 0 (15) respectively. Besides, the velocities are related in the following way: c c c x x x ab fg    + + = ; c c y x zy ab h    + + = ; c c z x zy ab h    + +− = . (16) 2.2 General Linear Transformation (Motion in a random direction) We then consider one unmoved O/F Oxyz and another O/F O΄x΄y΄z΄ with parallel spatial axes, moving with velocity (υx, υy, υz) wrt Oxyz, where they initially coincide (Figure 3). We rotate Oxyz, in order to parallelize the unitary vector x̂ to the velocity vector   of the moving O΄x΄y΄z΄. This is sequentially achieved as following (Figure 4). We firstly rotate the coordinate system Oxyz about z-axis, through an angle θ: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjizyx → .We then rotate the coordinate system )ˆ,ˆ,ˆ( kji about ĵ , by an angle ω: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjikji → . Thus, we have the transformation                      −− −=           z y x z y x    cossinsincossin 0cossin sinsincoscoscos R R R , (17) where 22 sin yx y    + = ; 22 cos yx x    + = ;     z=sin ;     22 cos yx + = . (18) As a result, the 3x3 matrix of (17) becomes 310 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric                         + + − + − ++ −=                   22 2222 2222 0 yx yx zy yx zx yx x yx y zyx R (19) and we define       = R R 0 01~ . (20) The unit means that time is not affected by the spatial rotation. Moreover, the transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is analyzed to the following sequence of successive transformations: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjizyx → ; )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjikji → ; )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxkji → . The above simple transformations have active interpretations: XRX ~ R = ; ( ) R1R XX x= ; R T~ XRX = , (21) respectively, where T~ R is the transpose matrix of R ~ . Thus, the transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is actively interpreted: ( ) ( ) XXRRX x T dd ~~ d 1 == . (22) So, we calculate                               +−−−+− +−+−−− −−+−+− = hhfhfhf g hfhhfhf g hfhfhhf g aaa b zxzyyzx z xzyyzyx y yzxzyxx x zyx 2 2 22 22 2 2 222 2 )( )()()( )()()( )()()(                                                . (23) 311 S. Vossos, E. Vossos, Ch. G. Massouros Figure 3: Two frames Oxyz and O΄x΄y΄z΄, which initially coincide. The second is moving with random velocity (υx, υy, υz) wrt to Oxyz. Figure 4: Rotation of the initial frame Oxyz, in order to achieve parallelization of vector x̂ to the velocity vector   of the moving observer O΄x΄y΄z΄ [ )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjiOkjiOzyxO →→ ]. 2.3 Solution of the proper Closed Linear Transformation of Complex Spacetime (Correlation of two perpendicular moving Observers / Frames) We consider one unmoved O/F Oxyz, another O/F O΄x΄y΄z΄ with parallel spatial axes, moving to the right, along x-axis with velocity (βc, 0, 0) wrt Oxyz and also a third O/F Ο΄΄x΄΄y΄΄z΄΄ with parallel spatial axes, moving upward, along y-axis with velocity (0, βc, 0) wrt Οxyz (Figure 5). All of them initially coincide and also β > 0, because   = . (24) 312 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric Figure 5: Two frames O΄x΄y΄z΄ and O΄΄x΄΄y΄΄z΄΄ moving with corresponding velocities (βc, 0, 0) and (0, βc, 0) wrt Oxyz. The transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is analyzed to the following sequence: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → ; )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → . The above simple transformations have active interpretations, respectively: XX x = −1 )(1 ; XX y )(2 = . Thus, the transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is actively interpreted: XXX xy == −1 )(1)(2 . According to equation (23), it is             − = h h fg ab x   00 00 00 00 )(1 (25) and             − = h fg h ab y 00 00 00 00 )(2   . (26) Thus, we have 313 S. Vossos, E. Vossos, Ch. G. Massouros                       ++ + − + −− − − − − == − 2222 2222 )(2 1 )(1)(2 00 00 00 00      h h h hh h agbf b agbf g agbf a agbf f Π yxy . (27) or equivalently,                       ++−− − + − +− − − + − + − −− − + − +− − − = 22 2 22 2222 2222 2 2222             h h h h agbf b agbf g h f h fh agbf ag agbf gf h h hagbf bh agbf gh h a h ah agbf ab agbf bf Π . (28) Now, we calculate the velocity factor 4    of observer Ο΄΄x΄΄y΄΄z΄΄ wrt Ο΄x΄y΄z΄. Equation (16) can be applied, because O/F Ο΄x΄y΄z΄ is moving in the x- direction and observer Ο΄΄ can be considered as the observed body. So, it is x4 = b g , b h y   = 4 , b z   −= 4 (29) and we obtain 22 2 22222 4 )(     ++= ++ = h g bb hg    . (30) Replacing the above to (23), yields 4)( 4 Λ=  . The condition that the transformation is closed, gives Π = Λ4. (31) Comparing the matrices, element by element, we shall calculate the parameters α, f and g. The transformation must be reduced to GT, if one IO has small velocity wrt another IO. So, it must be b, g, f, h ≠ 0. We have two cases: (i) λ=0 and (ii) λ≠0. 2.3.1 The case of proper Closed Linear Transformations of Complex Spacetime with λ=0 (time independent from the position, i.e. a=0). When λ=0, we compare matrices Π and Λ4 element by element and we also take into account (29). Thus, we have: h4=1 (from element Π33) and λ4=0 (from element Π13). We then obtain f4=1 (from element Π12). So, 314 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric h4=f4=1 ; λ4=0. (32) From elements Π10 and Π20, we get agbf gh h g gg − −= + 2 2 2 4     ; agbf gf h g hg − = + 2 2 2 4   (33) respectively. Thus, f h g 2   −= . (34) From elements Π01 and Π02, we have agbf ab h g ga − −= + 2 2 2 4     ; h a h g ha = + 2 2 2 4   (35) respectively. So, agbf bh h g − −=   . (36) Replacing (34) to the above, implies α=0, (37) for the CLT, or g=0 for the non-closed LT (because (34) gives h=0 and matrix (25) cannot be identical). Thus, element Π11 gives f=h (38) and (34) becomes hg   −= . (39) Finally, (23) yields the general matrix of CLT:       − =             − − − = 3 T )( I O 00 00 00 000 hh b hh hh hh b z y x      ;           =           = 3 2 1        z y x ;           = 0 0 0 . (40) and the typical matrix CLT (along x-axis):             − = h h hh b x 000 000 00 000 )(  , (41) where b=b(β) and h=h(β). 315 S. Vossos, E. Vossos, Ch. G. Massouros Next, we calculate the corresponding CILToCST. The representation of the non-degenerate inner product in basis   e  =  3210 eeee  = ]ˆ,ˆ,ˆ,[ ^ zyxct is the real matrix of metric             = 33 22 11 00 000 000 000 000 g g g g g . (42) In this paper, we consider g00<0 [signature of spacetime: (-+++), or (----)]. The fundamental equation of isometry - Killing’s equation in a linear space - (see e.g. [4], p.10, eq.1.15) is g΄= ΛΤ g Λ. (43) The element by element comparison of the above matrices gives 00 2 00332211 ,0 gbgggggg iiii ====== . (44) The isometry of spacetime [see e.g. [4], (p. 240)] is dS΄2= dS2, (45) or equivalently, j iji j iji xgxtgxgxtg dddcdddc 22 00 22 00 +=+ , (46) which combined with (44) and (40) gives b=1 for the CLT, (47) or b = -1, ±i for the non-closed LT (because matrix (25) cannot be identical). So, since b=1, CLT keeps time invariant. The Einstein’s summation convention [4] (p. 3) was used in (46) and will be used in the equations that follow. Besides, (44ii) becomes 0000 gg = . (48) Thus, for any O/F the metric of the ST in accordance with the complex LT is             =  0000 0000 0000 000 00 g g . (49) We observe that detgΓ=0. So, this spacetime is degenerate [6] (p. 174). In order to calculate function h, we consider the unmoved O/F Oxyz, another O/F O΄x΄y΄z΄ moving to the right, along the x-axis with velocity ( )0,0c, wrt Oxyz and a third O/F O΄΄x΄΄y΄΄z΄΄ moving to the left, along the x-axis with velocity ( )0,0c,− wrt Oxyz. Thus, X΄=Λ(x)(β)X and X΄΄=Λ(x)(-β)X give XX xx = − − 1 ))(())((  . (50) Also, the typical transformation of velocities (16) becomes 316 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric ( ) xx h  +−= c , yy h = , zz h = . (51) Thus, the calculation of the velocity factor of observer O΄΄ wrt O/F O΄x΄y΄z΄ gives  h x 2 3 −= , 0 3 = y  , 0 3 = z  . (52) As the transformation is closed, we have ))(( 1 ))(())(( 3 xxx = − − , (53) or equivalently,             =                                3 3 33 000 000 002 0001 1 000 0 1 00 00 1 0001 000 000 00 0001 h h hhh h h h h h hh    , (54) from which it derives that 1 )(3 3 == hh for any value of β. As h depends only on the norm of velocity factor β, the only solution is h=1. Hence, there derives the GT, which is expressed by the general matrix ( )       − =             − − − =  3 T I O1 100 010 001 0001      z y x ;           =           = 3 2 1        z y x , (55) and typical matrix along x-axis             − =  1000 0100 001 0001 )(  x , (56) which produces NPs with invariant time and infinite universal speed. As unmoved O/F Oxyz measures real velocity, the transformation matrix (ΛΓ) contains only real numbers. So, the spacetime is limited to the real domain 4 R . Moreover, this ST (Galilean spacetime) endowed with the Galilean metric (49), is degenerate. 2.3.2 The case of proper Closed Linear Transformations of Complex Spacetime with λ≠0 (time dependent on the position). When λ≠0, we compare matrices Π and Λ4 element by element and we also take into account (29). We start with (Λ4)21+(Λ4)12=Π21+Π12 and we obtain 317 S. Vossos, E. Vossos, Ch. G. Massouros ( ) 22 2 22 2 2 44 2      + − − −=           ++ − hagbf ag h g hfgh   . (57) We then use (Λ4)32+(Λ4)23=Π32+Π23, which gives ( ) 22 22 2 2 44 2    + − = ++ − h hf h g hfh  . (58) The combination of (57) with the above equation implies ( ) ( ) 22 2 22    + − − −= + − hagbf ag h hfg  . (59) Also, (Λ4)31+(Λ4)13=Π31+Π13 gives ( ) 22 22 2 2 44 2     + − − =           ++ −− h h agbf b h g hfg   . (60) The combination of the above equation with (58) also gives ( ) ( ) 22 2 22  + − − = + −− h h agbf bh h hfg  . (61) We then add (59) and (61) and get f=h ; f4=h4. (62) Moreover, from (Λ4)11=Π11 and (Λ4)00=Π00, we have 44 b agbf bf agbf bh h = − = − = , (63) which combined with (62) gives f = h = b. (64) We then use (Λ4)22=Π2, and we obtain 224 + = h fh h . (65) The combination of the above equation with (63) gives 22 + = − h f agbf b . (66) Furthermore, (Λ4)01=Π01 and (Λ4)02=Π02 give respectively: 318 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric agbf ab h g ga − −= ++ 22 2 2 4      ; (67) 22 22 2 2 4    + = ++ h a h g a  . (68) The substitution of (68) to (67) gives ( ) agbf b h g − −= + 22   . (69) Moreover, the combination of the above equation with (66) gives g = h  − . (70) Finally, (66) combined with (64) and (70) yields     h 2 = . (71) The replacement of (64), (70), (71) and b  = , (72) makes the general matrix (23) equivalent to               −− −− −− =                             −− −− −− = 1 1 1 1 222 2 2 2 2 2 2 xyz xzy yzx zyx xy z xz y yz x zyx b bb bb bb bbb b Λ                                 . (73) We also define           =           = 3 2 1        z y x ;           =           = 3 2 1        z y x ; ( )           − − − =           − − − = 0 0 0 0 0 0 12 13 23        xy xz yz . (74) It is noted that the antisymmetric matrix A(β) is related to the cross product (external product) [7] (p. 1048), because Α(β) δ =      =− . (75) 319 S. Vossos, E. Vossos, Ch. G. Massouros Thus, the four-vectors of two observers are related, by using the general Matrix:         +− =               −− −− −− = )(3 2 222 ),( AI 1 1 1 1 1 Λ         T xyz xzy yzx zyx bb . (76) Besides, the typical Matrix along x-axis is               − − = 100 100 001 001 2 ),)((      bx . (77) So, the proper Closed Linear Transformation of Complex Spacetime (22) is                            −− −− −− =                 z y x t b z y x t xyz xzy yzx zyx d d d cd 1 1 1 1 d d d cd 222     . (78) The pure mathematical approach is simply obtained by replacing ct→x0. Thus,                              −− −− −− =                   3 2 1 0 123 132 231 322212 3 2 1 0 d d d d 1 1 1 1 d d d d x x x x b x x x x     . (79) Below, we calculate the corresponding CILToCST. For simplicity reasons, when we write i (the imaginary unit), we mean ±i: ii → ; ii →− . (80) The combination of the fundamental equation of isometry (43) (the Killing’s equation in a linear space) with the above, gives ;)1()1(;; 2 2 22 00 222 00200332211    iiii ii g bgbg g ggggg  +=+===== (81) iiii gbg )1( 2 22   += . (82) So, for any O/F, the metric of spacetime in accordance with the CILToCST is isotropic:             = ii ii ii g g g g g 000 000 000 000 00 (83) 320 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric and also 00 2 g g ii= . (84) Thus, ω2 is a real number. So, ω is a real or an imaginary number, which only depends on the metric of spacetime. Besides, the metrics of the spacetime of two observers (frames) Oxyz and O΄x΄y΄z΄, are related using the formulas gbg )1( 2 22   += ; (85) iiii gbg )1( 2 22   += . (86) So, 2 2 2 1 1   +  = ii ii g g b . (87) Using the well-known Lorentz γ-factor function )(2T )( 1 1 1 1 1 1      = − = − = − = , (88) equation (87) may be written as 2 )(i 2    ii ii g g b  = . (89) Besides, the isometry of spacetime (45) combined with (89) and (78) gives iiii gg = . (90) Thus, (89) gives 2 )( 2    i b = and we obtain 0 )(i = b . (91) Moreover, (81iii) gives 0000 gg = . (92) This means that the metric of ST must be affected in the same way for any O/F, in the case of CILToCST. Equivalently, the observers that are related must be IOs or must have the same acceleration. Thus, the metric of spacetime in accordance with the CILToCST is             − − − − −=               =             = 2 2 2 00 2 00 000 000 000 0001 1000 0100 0010 000 1 000 000 000 000     gg g g g g g ii ii ii ii . (93) In case of SR (the frames are moved with constant velocity / the observes are IOs), equation (84) becomes 321 S. Vossos, E. Vossos, Ch. G. Massouros 00I I2 I g g ii= . (94) So, the time and space metric’s coefficients are independent from the position and they are combined to produce ωΙ, which is the characteristic parameter of SR. The continuity of the metric of spacetime at the point λ=ω=0 and the matrices (49) and (93) gives 0limlim 0 I 0I == →→ iiii gg  . (95) We also observe that if the metric’s coefficients have the same sign [signature of spacetime: (----)], then the characteristic parameter ωΙ is a real number, in contrast with the case that the coefficients of metric have different signs [signature of spacetime: (-+++)], where the characteristic parameter ωΙ is an imaginary number. The representation of the non-degenerate inner product in basis     ]ˆ,ˆ,ˆ,[ ^ 3210 zyxcteeeee ==   for IOs is the matrix             − − − − −=                 =               = 2 I 2 I 2 I 00I 2 I I I I I 00I I 000 000 000 0001 1000 0100 0010 000 1 000 000 000 000     gg g g g g g ii ii ii ii . (96) Generally, equation (78) gives the active transformation of O/F Oxyz to O/F O΄x΄y΄z΄ (if they are accelerated with the same acceleration):                            −− −− −− =                 z y x t z y x t xyz xzy yzx zyx d d d cd 1 1 1 1 d d d cd 222 )(i       . (97) The replacement ct→x0 gives the pure mathematical approach. Thus,                              −− −− −− =                   3 2 1 0 123 132 231 322212 )(i 3 2 1 0 d d d d 1 1 1 1 d d d d x x x x x x x x       . (98) Using vectors, the above transformation becomes ( )xtt    dcdcd 2 )(i +=   ; ( ) xtxx    dcddd )(i −−=   . (99) Moreover, the general and typical matrices of CILToCST are, respectively: 322 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric         +− =               −− −− −− = )(3 2 )(i 222 )(i),( AI 1 1 1 1 1 Λ           T xyz xzy yzx zyx ; (100)           = z y x     ; ( )           − − − = 0 0 0 xy xz yz     ;               − − = 100 100 001 001 2 )(i),)((      x . (101) The above matrices Λ have the following properties: 4),( I=  ;           = 0 0 0 ; (102) ),( 1 ),(  − − = ; (103) det ),(  = 1. (104) In case of SR, the matrices form a new group (which corresponds to Lorentz group) with elements d = ( ),( I   , B) ;             =               = z y x b b b t b b b b B 0 3 2 1 0 c , (105) and operation d1*d2 = ( ),( 2I  ),( 1I  , ),( 2I  B1+ B2), (106) where:  1 b is the μ-coordinate which is measured in O΄x΄y΄z΄, when all the coordinates  x , for ν=0, 1, 2, 3, in Oxyz are equal to zero and  2 b is the μ-coordinate which is measured in O΄΄x΄΄y΄΄z΄΄, when all the coordinates  x , for ν=0, 1, 2, 3, in O΄xyz are equal to zero. The above operation expresses the successive transformations: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → ; )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → . (107) These have active interpretations: 1),( 1I BXX +=  ; 2),( 2I BXX + =  . (108) respectively. Thus, the transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is actively interpreted: 21),(),(),( 2I1I2I BBXX ++=  . (109) 323 S. Vossos, E. Vossos, Ch. G. Massouros As ω2 is a real number, we observe that we always have real time. Besides, the norm of the position four-vector for Os/Fs with the same acceleration / the same metric of ST, is the corresponding invariant quantity ( )2222 00 222 2 222 00 2 ddcddc 1 ddcddd xtgxtgxgtgXgXS iiii T    −−−=      +=+== . (110) In the case of SR, the above equation becomes ( )22 I 22 00 222 2 I I 2 I 22 00I 2 ddcddc 1 ddcddd xtgxtgxgtgXgXS iiii T    −−−=         +=+== . (111) If ω is a real number [the coefficients of metric of time and space have the same sign: signature of spacetime: (----)], then s g g ii == 00  (112) with Rs . Thus, the transformation matrix (Λ) contains only real numbers and the ST is limited to the real domain R4. Finally, the four-vectors of two Os/Fs have the same metric             − − − − −=               =             = 2 2 2 00 2 00 000 000 000 0001 1000 0100 0010 000 1 000 000 000 000 s s s g s g g g g g g ii ii ii ii (113) and their CCs are related via the matrix: ( )         +− =               −− −− −− = )(3 2 )(i 222 )(i, AI 1 1 1 1 1           s s ss ss ss sss T s xyz xzy yzx zyx s . (114) The typical matrix along x-axis is               − − = 100 100 001 001 2 )(i),)((       s s s sx . (115) If ω is an imaginary number [the coefficients of metric of time and space have different sign: signature of spacetime: (-+++)], then ω = 00 i g g ii − = ξ i ; 00 g g ii − = (116) with +  R . Thus, the transformation matrix (Λ) contains complex numbers and the spacetime is represented by the complex domain 3 CR . Finally, the four-vectors of two Os/Fs have the same metric 324 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric            − −=               − =             = 2 2 2 00 200 000 000 000 0001 1000 0100 0010 000 1 000 000 000 000     gg g g g g g ii ii ii ii (117) and their CCs are related via the matrix: ( )         +− − =               −− −− −− −−− = )(3 2 )( 222 )(i, AiI 1 1ii i1i ii1 1           T xyz xzy yzx zyx . (118) Besides, the typical Matrix along x-axis is               − − − = 1i00 i100 001 001 2 )()i,)((      x . (119) The substitution of (64), (70), (71) and (72) to (16) gives the velocities typical transformation of CILToCST: c c c 2 x x x    + − = ; c c 2 x zy y    + + = ; c c 2 x yz z    + − = . (120) For the purpose of finding a possible Invariant Speed (U) for Os/Fs with the same ω (or equivalently the same acceleration), we assume that a particle is moving to the right with velocity 1 eU  = ; U > 0. So, we have c c c 2 U U x    + − = ; 0= y  ; 0= z  . (121) According to the Euclidean metric in the ordinary space E3, the norm of U is 2 22242 222 2 2 2 c c2c cc2 00c c c UU UU U U U     ++ +− =++      + − = , (122) which may be written as ( ) ( ) 0cc2c 3222444 =++−  UUU . (123) So, we obtain 2 2 2 c  −=U , (124) or equivalently, 2 2 2 c U −= . (125) 325 S. Vossos, E. Vossos, Ch. G. Massouros Since norm U > 0, ω is an imaginary number ( + = Ri, ) independent from the velocity (i.e. depends on the acceleration or equivalently the gravitation). Thus, we have c 1  =U . (126) So, it is       =       − == − = + = + = U u U u       2)(2 2 22 )(i 1 1 1 1 1 1 1 1 (127) and (110) can also be written as                 +−−=+−== 2 2 22 00 2222 d c dcddddd x U tgxtUgXgXS ii T  . (128) In the case of spacetime endowed with constant metric (or equivalently IOs), equation (126) becomes c 1 I I  =c . (129) and we obtain the Universal Speed (cI) of the specific SR. Besides, we have          =         − == − = + = + = I II 2 I )(2 2 I 2 2 I 2 I )(i 1 1 1 1 1 1 1 1 c u c u       (130) and                   +−−=+−== 2 2 I 22 00 222 II 2 d c dcddddd x c tgxtcgXgXS ii T  . (131) Now, let us find the corresponding Euclidean CILToCST. We initially define               =               =               = 3 2 1 0 3 2 1 0 d d d d 1 d d d cd 1 d d d d d x x x x z y x t x x x X X    ; 00 d 1 d xX   = , (132) where x0 ; 0 X are the zeroth-coordinates, by using the bases   e  =  3210 eeee  ;  e  =   3210 eeeE  (133) of ST endowed with metric (93) and Euclidean metric, respectively. Thus, 0000 gee =  ; 1 00 = EE  , (134) where dot “∙” is Euclidean inner product [4] (p. 7). So, we understand that 326 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric 0 00 0 i e g E  − = . (135) Then, the CILToCST (97) can be written as                              −− −− −− =                   z y x t z y x t xyz xzy yzx zyx d d d cd 1 1 1 1 1 d d d cd 1 )(i         , (136) or equivalently, dΧ΄ω= )( ~ R dΧω, (137) where               −− −− −− = 1 1 1 1 ~ )(i)( xyz xzy yzx zyx R       ;           =           = 3 2 1        z y x , (138) or equivalently, } A 0 {I AI 1~ )( 4)(i )(3 )(i)(         − +=         +− =           TT R . (139) The above matrix R ~ is a rotation matrix with the following properties: 4)( I ~ = R ;           = 0 0 0 , (140) )()( 1 )( ~~~  − − == RRR T , (141) det )( ~ R = 1. (142) The corresponding typical matrix along the x-axis in Euclidean spacetime (E4) is             − − = 100 100 001 001 ~ )(i))((      xR . (143) Now, using vectors the CILToCST of equation (136) becomes )d cd ( cd )(i x tt   +=       ;       −      −= x t xx   d cd dd )(i     , (144) or equivalently, 327 S. Vossos, E. Vossos, Ch. G. Massouros )d(dd 0 )(i 0 xXX   +=   ; ( ) xXxx   dddd 0 )(i −−=   . (145) Thus, we have the Euclidean metric of the position four-vector Xω in E 4, and the corresponding invariant quantity is ( ) 2220222 2E 2 d 1 ddddc 1 ddd S g xXxtXgXS ii T =+=+==    , (146) according to (110ii). Also, we observe that β-factor can be written as    iii i B X x x x === 00 d d d d ; 0 d d X x B i i = , (147) by using (132ii). The quantity Bi is called Β-factor and it can substitute the β- factor, in E4. Then, equations (136-139) are rewritten:                              −− −− −− =                   3 2 1 0 123 132 231 321 )(i 3 2 1 0 d d d d 1 1 1 1 d d d d x x x X BBB BBB BBB BBB x x x X B   , (148) dΧ΄ω= )( ~ B R dΧω, (149)               −− −− −− = 1 1 1 1 ~ 123 132 231 321 )(i)( BBB BBB BBB BBB R BB  ;           = 3 2 1 B B B B , (150) } A 0 { AI 1~ )( 4)(i )(3 )(i)(         − +=         +− =   B B I B B R T B B T BB . (151) The above matrix R ~ is a rotation matrix having the following properties: 4)( I ~ =  R ; (152) )()( 1 )( ~~~ B T BB RRR − − == ; (153) det )( ~ B R = 1. (154) Besides, the corresponding typical matrix along the x-axis in E4 is             − − = 100 100 001 001 ~ )(i))(( B B B B R BBx  . (155) Note that the above transformation can be limited in the real spacetime (R4), because the corresponding Lorentz γ-factor is positive for any real Β-factor. 328 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric We observe that the above results could be obtained from the initial equations of E4: (136-146), when ω→1 and ct→Xω. We also observe that R ~ reminds us of the contravariant electromagnetic tensor [3] (p. 14), [4] (p. 414):               −− −− −− = 0cc c0c cc0 0 12 m 3 1 m 3 m 2 2 m 3 m 1 321 ),( m m BBE BBE BBE EEE F BE , (156) where E and Bm are the intensity of electric field and induction of magnetic field, respectively [4] (p. 396). Actually, they are correlated via the formula 4)(i),()( I ~ m BBEB FR += ; j B j BE )(i = ; c )(i m j Bj B B  = . (157) Thus, it is jj BE m c= ; ( ) j j j j BBEE mm 2 c= (158) where (158ii) is the same as the electromagnetic waves in vacuum, while (158i) means that the vectors of the induction of magnetic field and intensity of electric field are parallel. This reveals a hidden correlation between the spacetime and electromagnetism (Maxwell equations). Moreover, for any constant value of ωI (or more precisely for any constant metric, i.e. constant values of gI00 and gIii), we have a specific CILToCST which correlates IOs and the corresponding SR-theory. Furthermore, the limit s → sI → 0 in the equations (113-115) and their combination with (95) gives GT of complex spacetime with infinite universal speed. In the same way, the limit ξ → ξI → 0 in the equations (117-119) and their combination with (95), gives again GT. Thus, the result when λ=0 (GT) is embedded to the case when λ≠0, if we take the corresponding limit to zero (λ→0, or equivalently, ω→0). Besides, if one O/F has small velocity wrt another, the CILToCST (even been complex) is reduced to GT. The replacement ξ → ξI=1 to the equations (118) and (119), produces the Lorentzian-Einsteinian version of CILToCST (ΛΒ) [7] (pp.1047-1048), which is expressed via the general matrix               −− −− −− −−− =  1ii i1i ii1 1 )( xyz xzy yzx zyx      (159) and the typical matrix along the x-axis 329 S. Vossos, E. Vossos, Ch. G. Massouros             − − − =  1i00 i100 001 001 ))((      x . (160) From (96), we take the corresponding metric of complex spacetime  iiii ggg II 1000 0100 0010 0001 =            − =  , (161) which for gIii=1 becomes the Lorentz metric. Thus, we have the SR-theory with universal speed being the speed of light in vacuum (cI=c) [7,8]. This theory gives results that are exactly the same as ΕSR, when only two Os/Fs are related. But the results are different, when more than two Os/Fs are related. Besides, it calculates the fine structure peeks of atomic hydrogen’s spectrum [8] (p. 4) more accurately than ESR. The explicit form of forward Lorentzian- Einsteinian CILToCST is ct΄ = γ(ct – βxx – βyy – βzz) (162) x΄ = γ(– βx ct + x + iβzy - iβyz) (163) y΄ = γ(– βyct - iβz x + y +iβxz) (164) z΄ = γ(– βzct + iβyx - iβxy + z) (165) The explicit form of reverse Lorentzian-Einsteinian CILToCST is ct = γ(ct΄ + βxx΄ + βyy΄ + βzz΄) (166) x = γ(βx ct΄ + x΄ - iβzy΄ + iβyz΄) (167) y = γ(βyct΄ + iβz x΄ + y΄ - iβxz΄) (168) z = γ(βzct΄ - iβyx΄ + iβxy΄ + z΄) (169) When the metric of ST depends on the position of the event in spacetime (GR), the transformation is applied locally, not globally (correlating Os/Fs with the same acceleration / gravitation). A metric is in accordance with the CILToCST, if only the limit of vanishing acceleration leads to the corresponding SR. Thus, the usage of (84) and (94) leads to iiii a gg I 0 lim = →  ; (170) 00I2 I I 2 0 00 0 limlim g gg g iiii aa === →→   . (171) 3 Proper Time – Special and General Relativity Let P be a particle moving with velocity P   wrt observer O ( P  wrt observer O΄) in spacetime. The generalized definition of proper time (τ) is 330 Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean or Lorentz or generally Isotropic Metric 2 00 2 2 c d d g S = . (172) Using (84) and (110), we have       +=      += 222 22 2 222 22 00 2 ddc 1 c ddc 1 c d xtxt g g ii      , (173) or equivalently,         +=+= 2 2 2 22 2 2 22 c 1dd c dd P txt     . (174) Thus, the relation between the time and the proper time is 2 )(i2 2 d d P t   = . (175) For ω=s with Rs , there does not exists real Invariant Speed (U) and the γ-factor is always positive. So, )(i d d Ps t   = . (176) When ω=ξi with R , there exists a real U. If the speed of particle is less than the invariant speed ( P 