Euclidean Closed Linear Transformations of Complex Spacetime and Generally of Complex Spaces of Dimension Four Endowed with the Same Or Different Metric

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This content was downloaded from IP address 131.169.4.70 on 11/01/2018 at 22:43 5th International Conference on Mathematical Modeling in Physical Sciences (IC-MSquare 2016) IOP Publishing Journal of Physics: Conference Series 738 (2016) 012048 doi:10.1088/1742-6596/738/1/012048

Euclidean Closed Linear Transformations of Complex Spacetime and generally of Complex Spaces of dimension four endowed with the Same or Different Metric

Spyridon Vossos1, 3 and Elias Vossos2 1Chemist, 2Physicist, School of Science, National and Kapodistrian University of Athens, Panepistimiopolis, GR15784 Zografos, Athens, Hellas

E-mail: [email protected]

Keywords: 5th Euclidean postulate, complex space, Einstein equations, Euclidean complex relativistic mechanics, Euclidean metric, general relativity, human senses, isometry, linear spacetime transformation, Lorentz matrix, Lorentz metric, Lorentz transformation, Minkowski space, Schwarzschild metric, spacetime, special relativity, real space, universal speed, velocity, Vossos matrix, Vossos transformation.

PACS: 02.10.Ud, 02.40.Dr, 03.30.+p, 03.50.De, 03.65.-w, 04.20.-q

Abstract . Relativity Theory and the corresponding Relativistic Quantum Mechanics are the fundamental theories of physics. Special Relativity (SR) relates the frames of Relativistic Inertial observers (RIOs), through Linear Spacetime Transformation (LSTT) of linear spacetime. Classic Special Relativity uses real spacetime endowed with Lorentz metric and the frames of two RIOs with parallel spatial axes are always related through Lorentz Boost (LB). This cancels the transitive attribute in parallelism, when three RIOs are related, because LB is not closed transformation, causing Thomas Rotation. In this presentation, we consider closed LSTT of Complex Spacetime, so there is no necessity for spatial axes rotation and all the frames are chosen having parallel spatial axes. The solution is expressed by a 4x4 matrix (Λ) containing components of the complex velocity of one Observer wrt another and two functions depended by the metric of Spacetime. Demanding isometric transformation, it emerges a class of metrics that are in accordance with the closed LSTT and the transformation matrix contains one parameter ω depended by the metric of Spacetime. In case that we relate RIOs with steady metric, it emerges one steady number (ωI) depended by the metric of Spacetime of the specific SR. If ωI is an imaginary number, the elements of the Λ are complex numbers, so the corresponding spacetime is necessarily complex and there exists real Universal Speed (UI). The specific value ωI=±i gives Vossos transformation (VT) endowed with Lorentz metric (for gii=1) of complex spacetime and invariant spacetime interval (or equivalently invariant speed of light in vacuum), which produce the theory of Euclidean Complex Relativistic Mechanics (ECRMs). If ωI is a real number (ωI≠0) the elements of the Λ are real numbers, so the corresponding spacetime is real, but there exist imaginary UI. The specific value ωI=0 gives Galileo Transformation (GT) with the invariant time, in which any other closed LSTT is reduced, if one RIO has small velocity wrt another RIO. Thus, we have infinite number of closed LSTTs, each one with the corresponding SR theory. In case that we relate accelerated

3Spyridon Vossos, 39 Epidavrou Str., GR-15233 Halandri, Athens, Hellas, e-mail: [email protected].

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observers with variable metric of spacetime, we have the case of General Relativity (GR). For being that clear, we produce a generalized Schwarzschild metric, which is in accordance with any SR based on this closed complex LSTT and Einstein equations. The application of this kind of transformations to the SR and GR is obvious. But, the results may be applied to any linear space of dimension four endowed with steady or variable metric, whose elements (four- vectors) have spatial part (vector) with Euclidean metric.

1. Introduction Relativity Theory (RT) is the foundation stone of modern Physics which combined with Quantum Mechanics (QMs), leads to the Relativistic Quantum Mechanics (RQMs). Moreover, there exist many different approaches of RT, each one with the corresponding RQMs. For instance, Galileo Transformation (GT) endowed with the corresponding metric, produces Newtonian Physics (NP), which is associated with the classic QMs producing Schrödinger Equation. Thus, many low velocities phenomena, like the atomic spectra without fine structure, were explained. On the other hand, Lorentz Transformation (LT) endowed with the corresponding metric, produces Classic Special Relativity (CSR), which is associated with the Classic RQMs, producing Klein-Gordon Equation. Thus, many high velocities phenomena and the fine structure of atomic spectra were explained [1,2]. In this presentation, we prove that there exist two types of complex Linear Spacetime Transformation (LSTTs) with common solution the GT, which can be applied not only to the Special Relativity (SR), but also to General Relativity (GR), because the production of the corresponding matrices has become without adapting one specific metric. In addition, any complex Cartesian Coordinates (CCs) of the theory may be turned to the corresponding real CCs, in order to can be realized through human senses. Moreover, we produce a generalized Schwarzschild metric, which is in accordance with any SR based on this closed complex LSTT and Einstein equations. This new modeling of study gives us the capability studying at the same time Einstein RT, NP, or any other Theory of Physics that is in accordance with closed Linear Spacetime Transformations. This is achieved because the coefficients of spacetime metric are contained in the transformation matrix. Besides NP is obtained, not only by the low velocity limit, but also by the zero limit of the space coefficient of spacetime metric.

2. The Matrix (Λ) and the Metric (g) of the Closed Linear Transformation In a 3D complex ‘space’ endowed with Euclidean metric, there exists a frame Oxyz having real CCs. Another real independent variable (‘time’) and the aforementioned coordinates produce a real four- vector. There exist two types of closed linear ‘spacetime’ transformation to this real four-vector: one with the ‘time’ being depended by the ‘spatial’ position where the ‘event’ happens and another with the ‘time’ being independent by the position. The first type has real or imaginary Invariant ‘Speed’ (U), in contrast to the second type which has only infinite U. Moreover, demanding the transformation having isometry [3], the first type transformation matrix is totally calculated and contains except for the ‘velocity’ of the frame O΄x΄y΄z΄, a parameter ω, with 2 ω = gii/g00 (1) where g00 and gii are the metric’s coefficients of time and space respectively. The second type is turn to GT. Taking the limit ω→0 to the first type, may emerges GT. So, in case of isometry, the second type is embedded to the first type transformation. Besides the first type is divided to two cases: one case that the ‘time’ and the ‘space’ have coefficients of metric of ‘spacetime’ with different sign, where the transformation leads to complex 3D ‘space’ with real U and the other case that ‘time’ and ‘space’ have coefficients of metric with the same sign, where the transformation leads to real 3D ‘space’ with imaginary U. Time remains real, in any case. Below, we present the typical matrix (Λτυπ), the general matrix (Λ(β)), the covariant matrix of spacetime metric (g), the universal speed (U) and the domain of spacetime (C4) that corresponds to the transformation of a contravariant four-vector: dX΄=ΛdX (2)

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4 (Λτυπ, Λ(β), g, U, C ) | r | = βωλ b = ? λ≠0 | λ=0 ______|______| |  1 2βω 00   b 000     hh  − β 001  − β 00  τυπ =Λ b , Λτυπ = ,  100 ωβ   h 000       00 −ωβ 1   000 h  1 ω 2β T   b O  ()β =Λ b  , ∈ ℜ IU },{ . Λ β )( =   , U =+∞.  +− ωβ AI β )(  − β hh I | | | isometry | isometry | |  1 2 βω 00   0001     001  − β 001  − β  τυπ =Λ γ ωβ )(i , Γ βτυπ )( =Λ , ω,( β )  100 ωβ   0100       00 −ωβ 1   1000   1 ω 2β T  O1 r r   ()β =Λ γ βω )(i   , Γ β )( =Λ   ,  +− ωβ AI β )(  − β I   1  000 g00 000  ω 2      0000 0010 .   , U =+∞, R4. = gg ii   gΓ =  0000   0100      0000  000 1   | | ω=? ω=ζi | ω=ζ∈ℜ ______|______| |  1 − 2 βζ 00   1 2βζ 00      r − β 001  − β 001  ζτυπ β ),(i =Λ γ , τυπ =Λ γ ζβ )(i , βζ )(  i100 ζβ  ζ β ),(  100 ζβ       − ζβ 1i00   00 −ζβ 1   1 −ζ 2β T   1 ζ 2 β T  =Λ γ rr , r =Λ γ r , () βζβζ )(,i   () βζβζ )(i,    +− ζβ AiI β )(   +− ζβ AI β )(   1   1  − 000 000  ζ 2  ζ 2    1 3   4 = gg  0010  ,U = c , X ∈RC . = gg 0010 , ∈ IU , X ∈R . ii ζ ii    0100   0100       000 1  000 1

3 5th International Conference on Mathematical Modeling in Physical Sciences (IC-MSquare 2016) IOP Publishing Journal of Physics: Conference Series 738 (2016) 012048 doi:10.1088/1742-6596/738/1/012048 where

β x  δ x   0 z − ββ y      β = β , δ = δ  , A = − β 0 β (3)  y   y  (β)  z x        β z  δ z   y − ββ x 0  We observe that r r r r Α(β) δ = [ ] [ ×=×− βδδβ ] (4) Using vectors, the transformation becomes r r r r r r r ′ r 2 ′ r cdt = γ βω )(i (cd + βω ⋅ xt )d , d x = γ βω )(i ( cdd βωβ ×−− d xtx ) (5) The norm of the position four-vector for observers with the same ω is the corresponding invariant quantity: r  g  r   1  r  2 = T = 22 + ddcddd 2 = gxgtgXgXS  00  22 + )1(c d 2 = gxt 22 + d)1(dc xt 2 (6) 00 ii ii    ii  2    gii    ω   Researching for a possible Invariant Speed (U) for observers with the same ω, it emerges c 2 ω 2 −= (7) U 2 In case of SR, the matrices form a group of elements d = ( Λ r , b) (8) I βω ),( with operation r r r d1*d2 = ( Λ Λ , Λ b1+ b2) (9) βω 2I ),( βω 1I ),( βω 2I ),( where bµ is µ-coordinate measured by Ο΄, if O measures xν = 0 and bµ is µ- coordinate measured by 1 2 Ο΄΄, if O΄ measures x′ν = 0 . M

The Universal Speed (UI) for RIOs, is 2 2 c U I −= 2 (10) ωI

3. Time – Proper Time r r Let have a particle P, moving with velocity υP wrt observer O (υ P′ wrt observer O΄) in a spacetime. The generalized definition of Proper Time (PT) is 2 2 d S′ dτ = 2 (11) g00 c Using (6) and (1), we have 2 2 2 g  1 r  ω  1 r  ω r  ω r 2  dτ 2 = ii ′ + ddc xt ′222 = ′ + ′222 = dddc ′2 + ′2 = txtxt ′2 1dd + υ′  (12)  22   22  2  2 P  g00 c ω  c  ω  c  c  Thus, the relation between the time and the proper time is dt′ = γ ′ r (13) dτ βω P )(i For GT with ω→0, it is γ ′ r d=1. So dτ= t΄=dt. i βω P )( In case of RIOs, (13) becomes dt′ = γ ′ r (14) dτ I βω P )(i

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r r So, for Vossos Transformation (VT) with ωI=±i, it emerges ′ = γγ ′ . Thus, we have the same I βω P β P )()(i result as CSR [4]. In any case using PT, we can define four-velocity, four-momentum etc building the whole structure of SR and GR.

4. Generalized Schwarzschild metric in accordance with closed complex Linear Spacetime Transformation For being clear the meaning of this theory, we apply it to the unique solution of Einstein equations in vacuum with spherical symmetry that is Schwarzschild metric, according to Birkoff’s theorem [5]. Thus, it emerges a generalized Schwarzschild metric that is in accordance with any acceptable SR having real, imaginary or infinite universal speed. We define the relativistic potential Φ around a center of gravity as U 2  r  Φ = I 1ln − SI  (15) 2  r  where r is the distance between the center of the gravity and the spatial position of the event and rSI is the generalized Schwarzschild radius G2 M rSI = 2 (16) U I The definition of Schwarzschild radius is 2G M r = (17) S c2 Thus (16) combined with (17) and (10) gives 2 −= ωISI rr S (18) and (15) becomes − c2  ω 2r  c2 r G M Φ = 1ln + I S  S ... +−=+−= ... (19) 2   2ωI  r  2 r r

We observe that if ωI=±i (VT), it emerges the original Schwarzschild potential. Moreover, if ωI → 0 (GT), we compute   2ω rSI − c2  1  ω 2r  − c2  1  c2 r G M lim Φ = lim 1ln + I S  = lim  r  S =−= − (20)  2   2 ωI →0 2 ωI →0 ω r 4 ωI →0ω ω r  2 r r  I   I 1+ I S  r  In figure 1, we show the parametric plot of the relativistic potential Φ wrt r/rs, for different values of ωI. We observe that for ωI→±∞, the relativistic potential becomes zero–function.

Now, we examine the case that g I00 ≤ 0 and gii I ≥ 0 , it is grr I ≥ 0 , too. Besides ω=ζi, with ζ∈ℜ . The generalized Schwarzschild metric is A − (− 0 I0 )eg 000  cdt     0 eg B 00 d r 2 = T = [ rtXgXS dddcdddd φθ ]⋅  rr I  ⋅   (21)  0 0 r 2 0  dθ   22     0 r sin00 θ  dφ  where 2 A = 2 Φ (22) U I This combined with Einstein equations in vacuum gives

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2  rSI  22 1 222222 d gS I00  −=  dc1 t − θ ++ rrr dsindd φθ (23)  r   rSI  g I00 1−   r 

Figure 1. Parametric plot of the relativistic potential Φ wrt r/rs, for different values of ωI. The isotropic form of the generalized Schwarzschild metric is 2 2 2 ~ ~ ~ 222222 = 00 ii (ddcd dθ +++ rrrgtgS dsin φθ ) (24) with d ~ d ~ ~ dθ ++= ~rrrx dsin φφ 2222222 (25) ~ = ~rx θ cossin φ , ~ = ~ry θ sinsin φ , ~ = ~rz cosθ (26) 1

  − g I00 ζ r 2r  2 r  ~r = SI  1 1−+ ζ S  −1 (27) 4  2  I r    ζ I rS    2 −g 2  −g00 I  I00 ζ r  ζ r   4r~  r I S 1+=  SI     (28) 4   4~r    ζ r       SI  2  − g I00    ζ r   1−  SI    4~r   = gg     (29) I0000 − g   ζ r  I00  1+  I S     ~     4r  

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4 2 − g 4 2  − g00 I  I00 ζ r   ζ r    4r~  g~ = I S 1+  SI    (30) ii ~2  ~  16r   4r    ζ r       SI  3 − g  I00  ~ g 00I −− 1 g −− 1 2 −− g   ζ r    r  I00 I00  SI    4 ζ I 1+  ~       4r    rS  ζ =   (31) − g  ζ r  I00 g  SI  I00 1−−  ~   4r  Besides  p g I00 ≤− 01,0  2 gii I =  ζ I , g 0 I0 = −1 (32)  +∞ , g p −1  I00 The universal speed is 1 U = c (33) ζ so, the relative change of U wrt U +∞)( is

r −UU ()( +∞) ζ I U r =∆ −= 1 (34) U +∞)( ζ The transformation inside the infinitesimal area around the surface of a sphere with center the same as the center of the gravity and radius r is cdt′  1 − 2 2 2βζβζβζ  cdt  x y z  d ~x′ − β 1 −ζβζβ d ~x    = γ r  x z y  ⋅   (35) ~ ( βζ )   ~ d y′ y −− ζββ z 1 ζβ x d y        ~′ ~  d z  − z y −ζβζββ x 1  d z 

In case of Lorentz metric ( g I00 −= 1and gii I =1), it is ζI=±1 and we have the SR which corresponds to VT with the metric of spacetime 2  r  1− S 4  ~  r ~  4r  ~  S  g00 −= , gii = 1+ ~  (36)  rS   4r  1+   4r~  so, it emerges the original isotropic form of Schwarzschild metric [6] with 3  r  1+ S  4~r ζ =   (37) r 1− S 4~r and  2rr  r   ~r = S   + − S  −111  (38)  r  r   4  S   

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5. Experimental Appling the above results of the original isotropic form of Schwarzschild metric: (i) to the Earth with mass M=5.9742×1024 kg, on the sea level with r=6378140 m and taking into account that G=6.67428(67)×10−11m3kg−1s−2 and c=299792458 ms−1 (exact) [7], it emerges -3 -9 ~ rS=8.873578523×10 m≈0.88cm. Thus rs/r=1.391248628×10 , r =6378139.995 m, ζ=±1.000000001, −1 −9 U=299792457.6 m s and ∆Ur=-1.36×10 and (ii) to the Sun, with mass is M=1.9891×1030 kg, on the surface with radius is r=6.9599×108 m, -6 ~ 8 rS=2954.443279 m, we have rs/r=4.244950759×10 m, r =6.959885228×10 m, ζ=±1.000004245, −1 -6 U=299791185.4 m s and ∆Ur=-4.244912×10 .

References [1] Kleppner D 1999 A short history of atomic physics in the twentieth century Rev. Mod. Phys. 71 (2) S78 http://www.cstam.org.cn/Upfiles/200732678933.pdf [2] Thomas L H 1926 The Motion of the Spinning Electron Nature (London) 117 514 [3] Tsamparlis M 2010 Special relativity: An introduction with 200 problems and solutions (Berlin Heidelberg: Springer-Verlag) p 10, eq 1.15, p 240 [ISBN: 978-3-642-03836-5, e-ISBN: 978-3-642-03837-2] [4] Vossos S and Vossos E 2015 Euclidean Complex Relativistic Mechanics: A New Special Relativity Theory J. Phys.: Conf. Ser. 633 012027 http://dx.doi.org/10.1088/1742-6596/633/1/012027 [5] Rindler W 2006 Relativity: Special, General and Cosmological (New York: Oxford University Press) p 228-231 [ISBN: 978-0-19-856732-5] [6] Rindler W 2006 Relativity: Special, General and Cosmological (New York: Oxford University Press) p 237, eq 11.26 [ISBN: 978-0-19-856732-5] [7] Lide D R ed 2009 CRC Handbook of Chemistry and Physics 89th Edition (Internet Version) (FL: CRC Press/Taylor and Francis, Boca Raton) pp 1-1,14-2