METHOD OF DETERMINING THE TRANSPORT PROPERTIES OF LOCAL FRAMES OF TIMELIKE WORLDLINES IN FOUR DIMENSIONAL SPACE WITH MINKOWSKI METRIC η USING MATRIX PARTITIONING FIELD OF THE INVENTION [0001] The invention relates to the theories of special and general relativity. In this application we use for the Minkowski metric the following convention: Greek indices run from 0 to 3, latin indices run from 1 to 3. We use units c = G = 1 with c being the speed of light and with G being the gravitation constant. Symbols L,R,D and A denote contravariant tensors of rank 2 with components and . Consequently products and are mixed tensors of rank 2 with components Finally products are covariant tensors of rank 2 with components For an antisymmetric contravariant tensor A of rank two the symbol denotes the contravariant Hodge dual tensor with components , consequently constitutes the mixed Hodge dual tensor with components and constitutes the covariant Hodge dual tensor with components The Hodge Dual tensor is defined in various textbooks as outlined in annex 3. [0002] Throughout the application documents three-dimensional quantities are al- ways denoted by bold symbols like r,u, a,R,L in contrast to the associated four- dimensional quantities r, u, a, R, L. An exception is the unfortunate resemblance between proper time τ and 3D-torsion , it should however always be possible to distinguish the two symbols and also the context should always prevent any con- fusion. A bold 1 denotes the unity matrix in three dimensional Euclidian space and a bold 0 denotes either a three dimensional vector with three zero compo- nents or a 3 × 3-matrix with nine zero components depending on the context. A point ∙ between four-vectors denotes always the Minkowski pseudo scalar product , a point ∙ between three-dimensional vectors denotes always a normal Euclidian scalar product and a point ∙ between a matrix and another matrix or a vector denotes always matrix multiplication, however the latter point is most often omitted. In an analogous manner the square of a three dimensional vector a is defined as a ² = a ∙ a = a ^T a and the square of a four-vector a is defined as a ² = a ∙ a = a ^T ηa = a _ν a ^ν = −(a ⁰ ) ² + a ² . The bases of the Minkowski space used in the following will all be Minkowski-orthogonal and right handed. BACKGROUND OF THE INVENTION [0003] An arbitrary timelike worldline is given by and parametrised by proper time τ . The four-velocity u(τ) and the four-acceleration are given by (u ⁰ > 0 applies, since u must be future-directed, see [4, p.35 and p.16]) [0004] Note that equ.(4) applies only, if one uses either, as we do (see above), the units c = G = 1 with c being the speed of light and with G being the gravitation _{constant, or if one defines} as done by textbook [4, equ. 2.12 on p.35 and remark 2.9 on p.36], otherwise the condition u ² = −c ² would apply with c being the speed of light. [0005] A local frame along the worldline is defined (compare textbook [4, p.77]) by a 4-tupel of proper time dependent four-vectors conveniently written as the column vectors of a 4× 4-matrix representing a mixed tensor with the four four-vectors being defined at any point on the worldline and having the following properties: [0006] 1. The four four-vectors form a right handed Minkowski-orthonormal basis, such that the matrix is a right handed Minkow- ski-orthonormal matrix, i.e. (in the following the explicit dependence on τ is often omitted to increase the readability of the formulas) and (11) This applies only, if one uses either, as we do (see above), the units c = G = 1 with c being the speed of light and with G being the gravitation constant, or if one _defines d as done by textbook [4, equ. 2.12 on p.35 and remark 2.9 on p.36]. _{Otherwise a factor} has to be supplemented. [0008] 3. All four four-vectors ) and thus also are differentiable. [0009] The above definition of a local frame is thus equivalent to the requirement that the matrix representing is a differentiable restricted Lorentz transforma- _{tion matrix for all proper times In the trivial case} , where is a straight line in Minkowski space, the above definition of a local frame reduces to the definition of a conventional restricted Lorentz transformation independent from proper time τ . [0010] A crucial property of a local frame is its transport property defined in inertial cartesian coordinates by (see [19, equ.(11) on p.4]) (12) [0011] Derivation of equation (9) yields that matrices representing the contravari- ant tensor D or the covariant tensor are skew symmetric for all local frames (see [19, annex 1 "Symmetry of transport property . Thus mixed tensor can always be partitioned in the form (see [19, equ.(12) on p.5]) (13) with D ₁ and D ₂ being three-dimensional vectors and with matrix [a] _× associated with a three-dimensional vector a being defined by (14) [0012] Transport property of a local frame as defined in equ.(7) can thus be written as follows: (15) (16) (17) [0013] In textbooks the last formula is normally provided without the components of the two columns vectors being transposed, but then this equation looses its char- acter of a simple matrix equation and becomes a "vectrix equation", which is why we prefer the formulation of equ.(17). [0014] For skew symmetric (= antisymmetric) 4 × 4-matrices this type of parti- tioning has been used to calculate certain matrix properties like the determinant, the inverse matrix or the exponential matrix, see for example [6, fact 4.12.1.xliii)-xlv) on p.385, fact 6.9.19 on p.528, fact 8.9.18vi) on p.665 and fact 15.12.18 on p.1208]. This sort of partitioning has also been used in four-dimensional Minkowski space for antisymmetric tensors of second order for example to determine the Hodge dual ten- sor, see for example annex 3 or [10, equ.(4.83), (4.83’) on p.110 and p.116, 1st para.], [4, equ.17.19, 17.22 on p.551 and Remark 3.11 on p.85] or [7, equ.11.137, 11.138 and 11.140 on p.556]. The matrix partitioning of equ.(13) has already been used to calculate the matrix exponential ([17, equ.(4) on p.99] and [18, equ.(1) on p.2]). [0015] From equ.(15) one can derive (18) for all local frames. A corresponding equation can also be found in [1, equ.(55) on p.730 and equ.(69) on p.732] and [2, equ.(24) on p.1583 and equ.(29) on p.1584]. [0016] Dη is the transport property in coordinates of the inertial cartesian basis vectors. However often the transport property is given in the basis of the moving column vectors of the local frame. In order to construct the corresponding mixed tensor matrix Aη one has to multiply Dη from the left with Lη and from the right with (Lη) ⁻¹ (see for example textbook [11, equ.(4.8.5) and (4.8.6) on p.253]): (19) (20) (21) [0017] In the prior art Aη is generally given in orthogonally decomposed form (see [1, equ.(67) on p.732], [2, equ.(28) on p.1584], [4, equ.(3.52) on p.86, see also equ.(3.37) on p.83, equ.(14.80) and equ.(14.83) on p.493 and the sign change in the second line above equ.(3.50) on p.86] and [9, equ.(6.19) and (6.20) on p.174 and equ.(3.50d), (3.50e) and (3.50f) on p.87]): (22) (23) (24) (25) with being the Hodge dual of an antisymmetric contravariant tensor T (i.e. (see [4, p.490-492], [9, equ.(3.51), (3.52) on p.88] and annex 3 below) and with being the components of the Levi-Civita tensor ([4, equ.(14.51) on p.486 and equ.(14.63) on p.489] and [9, equ.(3.50d), (3.50e), (3.50f) on p.87]) given by (26) [0018] An alternative equation to calculate the four-rotation ω is given in [1, equ.(68), (69) on p.732] and [2, equ.(26) on p.1583 and equ.(29) on p.1584]: (27) [0019] Note that the sign of the transport property Dη or Aη is chosen differently by different authors (compare equ.(16) and (21) with [4, equ.(3.29) on p.81], [1, equ.(50) on p.729], [2, equ.(22) on p.1583], [5, equ.(18.117) and (18.118) on p.420] and [9, equ.(6.19) on p.174]) and that while the transport property is given in [4, equ.(3.51) on p.86] in mixed tensor form (Aη) as in equ.(22) and (23) above, the transport property is given in [1, equ.(67) on p.732], [2, equ.(28) on p.1584] and [4, equ.(3.50) on p.86, equ.(3.37) on p.83 and equ.(14.80) on p.493] in covariant form (ηAη) and is given in [9, equ.(6.20) on p.174, note the minus sign in equ.(6.19)] in contravariant form (A). [0020] That the first summand in equ.(22) and the first summand in equ.(23) are identical is immediately apparent. That the second summands are also identical can be deduced from [4, equ.(3.52) on p.86, equ.(3.37) on p.83, equ.(14.80) on p.493, equ.(17.20) on p.551]. Another proof of the identity of the second summands is also given below (see paragraph entitled "Identity of equations equ.(22), (23) and (41)"). [0021] Some authors define a cross product of four-vectors in the three dimensional subspace Minkowski-orthogonal to four-velocity u in order to formulate the transport property Aη (see for example [4, lower half of p.86] or [5, equ.(18.123), (18.125) on p.421]). The paragraph entitled "Generalised cross product in three dimensional subspace of four dimensional Minkowski space" below shows as to how this cross product of four-vectors is related to equ.(22) and (23). [0022] The most discussed local frames in the prior art are the Frenet-Serret frame and the Fermi-Walker frame. For the Frenet-Serret frame (sometimes also called Serret-Frenet frame) there are a more narrow and a broader definition. [0023] The more narrow definition is applicable only for those timelike worldlines r _{, for which the first, second, third and fourth derivatives} are linearly independent and thus form a basis of four dimensional Minkowski space, such that the Gram-Schmidt procedure ([11, chapter 5.5 on p.307], [5, section "18.17 Covari- ant Serret-Frenet Theory" on p.417-419], [4, section "2.7.3 Curvature and Torsions" on p.59-62]) can be used to construct a Minkowski-orthonomal basis of the Minkowski space from said four derivatives. This procedure ensures that the transport property of the Frenet-Serret frame ( ) in inertial cartesian coordinates assumes the form (28) with all curvatures curvatures ρ ₁ , ρ ₂ , ρ ₃ being non-vanishing ( see for example [1, equ.(57) on p.731] or [19, equ.(26)] or [5, equ.(18.115) on p.419] or [13, equ.(3.1) (mixed tensor), equ.(3.2) (covariant tensor) on p.39, 40], for see for example [19, equ.(27)] or [4, equ.(2.63) on p.62, equ.(3.29) on p.81 and Example 3.1. on p.88] or [8, equ.(11.122)-(11.126) on p.390] or [3, equ.(55a),(55b),(55c),(55d) on p.10] or [15, last equ. on p.1550109-5; there is a sign error in the first column of said equ., see equ.(1) on p.1550109-4 and the first two lines of the paragraph "proof" on p.1550109- 6]). We recall that according to equ.(18) holds. Within the framework of this more narrow definition the Frenet-Serret frame of a given timelike worldline is always uniquely determined. [0024] We adapt here the broader definition, which is applicable to arbitrary time- like worldlines and according to which every local frame with a transport property as defined in equ.(28) is considered to be a Frenet-Serret frame even if one or more of the curvatures ρ ₁ , ρ ₂ , ρ ₃ are vanishing, i.e. even if the first, second, third and fourth _{derivatives u} of the timelike worldline r are linearly dependent. Within the framework of this broader definition a Frenet-Serret frame of a given timelike worldline is uniquely determined only if the second curvature ρ ₂ 6= 0 is not vanish- ing, which implies that the first curvature ρ ₁ 6= 0 is likewise not vanishing. In [19] an analytical expression is given for the Frenet-Serret frame, which covers also the cases that one or more of the curvatures ρ ₁ , ρ ₂ , ρ ₃ are vanishing. Said analytical ex- pression contains parameters which ensure that said analytical expression provides in those cases, in which the Frenet-Serret frame is not uniquely determined, all possible Frenet-Serret frames. [0025] For a Frenet-Serret frame the four-rotation , which according to equ.(22) or (23) above fully determines the transport property in the moving coordinate system, is given by (see for example [4, equ.(3.58) on p.88 and equ.(2.63) on p.62] or [5, equ.(18.124) on p.421 and equ.(18.115) on p.419] or [3, equ.(145) on p.140 and equ.(55a)-(55d) on p.10] or [13, equ.(3.11) on p.41]): (29) [0026] A Fermi-Walker frame of a given timelike worldline is any local frame, for which the transport property in inertial cartesian coordinates assumes the form (see for example [3, equ.(77) on p.14 and the paragraph containing equ.(83) on p.15], [12, equ.(13) on p.4 and paragraph containing equ.(22),(23),(24) on p.7] or [15, first equ. on p.1550109-6]): (30) We recall that according to equ.(18) holds. [0027] For a Fermi-Walker frame the four-rotation , which according to equ.(22) or (23) above fully determines the transport property in the moving coordinate system, is given by (see for example [4, equ.(3.52) on p.86 and p.88, first para.] or [9, equ.(6.13) on p.171 and equ.(6.20) on p.174] or [5, equ.(18.117) on p.420] or [3, equ.(72) on p.13] or [12, equ.(6) on p.3] or [13, equ.(4.1),(4.2) and (3.12) on p.41]): (31) i.e., the Fermi-Walker frame is by definition non rotating. [0028] Construction of a Fermi-Walker frame for a given timelike worldline is gen- erally non-trivial ([12, chapter "Construction of Fermi-Walker frames" on p.7ff], [5, chapter "18.19 Example of Fermi-Walker Transport" on p.421,422] or [15]). [0029] The Fermi Walker frame is sometimes called Bishop frame or "parallel frame" (see for example [15, title and para. "Introduction"] due to the author Richard L. Bishop of article [16], which promoted the analogous frame in three dimensional Euclidian space and called it "relatively parallel adapted frame". The term "parallel frame" is unfortunate, since it might lead to confusion with "parallel transport", which differs from Fermi-Walker transport and does not even involve a local frame within the meaning of equ.(9)-(11) above (see for example [3, §4. PARALLEL TRANSPORT AND FERMI-WALKER TRANSPORT" on p.12]): in case of the Fermi-Walker trans- port the derivatives of the three spacelike basis vectors are at each proper time τ parallel to the four-velocity N ₀ = u, while in parallel transport each basis vector remains parallel to itself at different proper times. [0030] A further example of a local frame given in [19, equ.(53)] is defined as (32) [0031] The transport property in inertial cartesian coordinates is given by (33) with ( ³⁴⁾ being the conventional 3D-curvature κ and 3D-torsion (see for example [19, annex 4]) of the three dimensional curve r constituting the three dimensional spatial part ( ) of four dimensional worldline [0032] The local frame defined in equ.(32) is well defined only if Reference [19, equ.(43),(53),(64),(70)] discloses an analytical expression for frame , which encompasses also the case and which contains a parameter which ensures that said analytical expression provides in those cases, in which frame is not uniquely determined, all possible frames with transport property Dη. The four-rotation which according to equ.(22) or (23) above fully determines the transport property Aη in the moving coordinate system, is not disclosed in [19]. SUMMARY OF THE INVENTION [0033] The problem to be solved by the present invention is to provide alternative methods for calculating the transport property Dη in inertial cartesian coordinates, the transport property Aη in moving coordinates and the four-rotation ω. The prob- lem is solved by the methods defined in the claims. BRIEF DESCRIPTION OF THE DRAWINGS [0034] Not Applicable DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS [0035] In order to solve this problem the first step is to perform the following decomposition of Dη as defined in equ.(13): (35) It is easy to see that Equ.(35) can be written in the form (36) with (37) (38) (39) [0036] Starting from this decomposed form of Dη and using equ.(19) it is shown in annex 1 that Aη assumes the following form: (40) (41) with (42) Thus one can determine Aη in the form of equ.(41) by replacing in Dη in the form of equ.(36) the three four-vectors n,D ₁ , D ₂ by their images u, a, ω under Lη. Equ.(42) corresponds to equ.(27), i.e. to [1, equ.(68), (69) on p.732] and [2, equ.(26) on p.1583 and equ.(29) on p.1584]. It follows directly from formula (42) that , since the last three column vectors of Lη are by definition Minkowski-orthogonal to the first column vector formed by the four-velocity u according to equ.(9a) and (11). [0037] Formula (41) provides an alternative formula to equ.(22) and (23) for de- termining the transport propertyAη and thus constitutes a first solution of the afore- mentioned problem. [0038] Generalised cross product in three dimensional subspace of four dimensional Minkwoski space From equ.(35) and (14) it is apparent that the second summand of equ.(35) acts in the three dimensional subspace Minkowski-orthogonal to the cartesian four-vector ( ) ( ) like a cross product and that four-vectors n and belong to the nullspace of the second summand. The second summand of equ.(41) also acts like a cross product in the three dimensional subspace Minkowski-orthogonal to the four-velocity u = Lηn. That four-vectors u and ω belong to the nullspace of the second summand of equ.(41) can easily be shown (see annex 2). [0039] Since thus the second summand of Aη in any of equ.(23), (22), (40) or (41) acts like a cross product in the three dimensional subspace Minkowski-orthogonal to the first column four-vector u of Lη and spanned by the last three column four-vectors of Lη, some authors ([4, equ.(3.52) on p.86], [5, equ.(18.125) on p.421]) formulate the transport property not as in equ.(19) and (22) (or equ.(23), (40) or (41)) for the ( ) entire matrix Lη, but for each column four-vector N _μ of matrix Lη = N ₀ N ₁ N ₂ N ₃ separately in the form with ω× _u denoting said second summand in equ.(41) (or equ.(22), (23), (40)), i.e. (43) Thus the role played matrix [a] _× of equ.(14) for the cross product in three dimensional Euclidian space is played by matrix ω× _u for the generalised cross product in the three dimensional subspace Minkowski-orthogonal to u of four dimensional Minkowski space. [0040] Identity of equations equ.(22), (23) and (41) The derivation of equ.(41) in annex 1 and the derivations of equ.(22) and (23) in the cited prior art prove in combination that equ.(22), (23) and (41) are identical. An additional direct proof that equ.(22) is identical to equ.(41) is given in annex 3. An additional direct proof that equ.(23) is identical to equ.(41) is given in annex 4. [0041] Advantages of equ.(41) An advantage of Aη in the form of equ.(41) is that the structure of transport property Aη can easier be grasped when compared to equ.(22) and (23). [0042] A further advantage of our derivation is as follows: if one multiplies equ.(35) or (36) from the left with η, i.e. if one transformed the two equations from mixed tensor form to covariant tensor form, then one gets a trivial case of an orthogonal decomposition of an antisymmetric bilinearform (see [4, chapter "3.5.2 Orthogonal Decomposition of Antisymmetric Bilinear Forms" on p.83ff, chapter "14.5.4 Orthogo- nal Decomposition of 2-Forms" on p.493ff and chapter "17.2.3 Electric and Magnetic fields" on p.547ff]). It is further apparent that step (35) decomposes Dη into the symmetric and the antisymmetric (skew symmetric) part. In order to calculate the transport properties (Dη or Aη), which represent an endomorphism and are thus in mixed tensor form [4, equ.(3.29)-(3.31) on p.81], the prior art switches first from the mixed tensor form to the covariant form (bilinear form) ([4, (equ.(3.32) on p.82]) or to the contravariant form ([9, equ.(6.20)]) in order to exploit the orthogonal de- composition and the antisymmetry of the covariant or contravariant tensor, and then switches back to mixed tensor form ([4, equ.(3.51), (3.52) on p.86]) in order to formu- late the final result of the transport property. The above decomposition of equ.(35) obviates the need to switch between mixed tensor and covariant tensor form. The above decomposition of equ.(35) also obviates the need to derive the general form of an orthogonal decomposition ([4, p.84-86]), since in the special form of equ.(35) the orthogonal decomposition is trivial. [0043] The above decomposition has the further advantage that - as is apparent from the second antisymmetric summand - one has to deal effectively only with a three dimensional antisymmetric matrix and not with a four dimensional antisymmetric matrix, which simplifies the calculation (see annex 1). [0044] Prior equ.(24) determining ω form Aη can also be simplified by using matrix partitioning (see equ.(79) in annex 3). Equ.(14.81) on p.493 of [4] can in view of [4, paragraph above equ.(3.50) on p.86] be written in the form (44) Annex 5 proves the identity of equ.(44) and (24) by explicitly calculating each com- ponent of ω both by equ.(44) and by equ.(24). Equ.(44) thus solves the problem of the invention by providing an alternative to equ.(24) and (27) of the prior art. [0045] It is - for example from equ.(36) and equ.(41) - apparent that Dη can also be brought in the forms of equ.(22), (23) and that also formulas (24), (25), (27) and (44) apply correspondingly as explicitly shown in the following: (45) (46) (47) (48) (49) (50) In the last equation, which is analog to equ.(27), we used [0046] Partitioning of the matrix Lη(τ) for a given local frame can also be used to simplify the calculation of the vectors D ₁ ,D ₂ ,A ₁ ,A ₂ defining the two forms of the transport property Dη and Aη. As proven in annex 12 a matrix represents a local frame if and only if the matrix can be partitioned in the following way: with (51) (52) Note that all the information of Lη is contained in L, i.e. Lη can be reconstructed from L alone using the preceding equations (for example u can be determined from L by using the equation above). [0047] Using this partitioned form of Lη the vectors D ₁ ,D ₂ ,A ₁ ,A ₂ can be deter- mined from three dimensional quantities. In the following the second summand of equ.(41) is denoted with Bη, i.e. (53) which defines the vectors B ₁ ,B ₂ . [0048] The vectors D ₁ ,D ₂ can be determined from three dimensional quantities by the following formulas as shown in annex 6: (54) (55) (56) (57) (58) (59) (60) [0049] The following relationship applies to all local frames of all timelike world- lines: [0050] The vectors A ₁ ,A ₂ can be determined from three dimensional quantities by the following formulas as shown in annex 7: (61) (62) (63) (64) (65) [0051] The four-rotation ω can be determined in many ways from three dimensional quantities, for example by equ.(44). Another simple formula proven in annex 7 is: (66) [0052] The vector B ₂ can be determined from three dimensional quantities by the following formulas as shown in annex 6 and annex 7: (67) (68) [0053] The reason as to why use of - the matrix [a] _× defined in equ.(14) for skew symmetric (antisymmetric) three dimensional matrices, - the partitioned form of A defined in equ.(78) for skew symmetric (antisymmetric) four dimensional matrices and - the partitioned form of the matrices Dη and Aη defined in equ.(13) and (79) produces particularly simple expressions not only above, but also for many other quantities derived from these matrices (for skew symmetric three dimensional ma- trices see for example [6, fact 4.12.1 on p.384-387; fact 8.9.18 on p.664; fact 15.12.6 on p.1206] and for skew symmetric four dimensional matrices see for example [6, fact 4.12.1 xliii), xliv), xlv) on p.385; fact 8.9.18 vi) on p.665, fact 15.12.17 and fact 15.12.18 on p.1208]) appears to lie in the theory of Lie groups and Lie algebras: [0054] The three dimensional skew symmetric (antisymmetric) matrices [y] _× with form the Lie algebra so(3) ([14, paragraph 5.5. on p.38]) of the Lie group SO(3) formed by the three dimensional rotation matrices with determinant=+1 ([14, paragraph 2.3. on p.10,11]). The components y ₁ ,y ₂ ,y ₃ of y are the components of [y] _× in the canonical basis of the Lie algebra so(3) (see for example [14, p.88, last paragraph]). [0055] The four dimensional skew symmetric (antisymmetric) matrices with form the lie algebra so(4) ([14, paragraph 5.5. on p.38]) of the Lie group SO(4) formed by the four dimensional rotation matrices with determinant=+1 ([14, paragraph 2.3. on p.10,11]). The six components of the two vectors A are the components of in the canonical basis of the Lie algebra so(4). [0056] The four dimensional matrices with form the Lie algebra so(3,1) ([4, paragraph containing equ.(7.6) on p.222]) of the Lie group SO ₀ (3, 1) formed by the four dimensional restricted Lorentz transformation matrices ([4, p.221,222,230]). The six components of the two vectors are the components of in the canonical basis of the Lie algebra so(3,1) with the canonical basis vectors being called generators ([4, p.225]) or infinitesimal generators ([7, chapter 11.7 on p.543-548]). [0057] As an exemplary use of the above formulas we determined the transport properties of the frame defined in equ.(32). [0058] The vector has already been determined in [19] using four dimensional quantities and is given in equ.(33) above. In annex 8 it is shown that the same result is obtained using any of formulas (54),(55),(56) or (57). [0059] The vector has also already been determined in [19] using four dimen- sional quantities and is likewise given in equ.(33) above. In annex 9 it is shown that the same result is obtained using any of formulas (58) or (59). [0060] The vector is determined in annex 9 using equ.(67) with the following result: [0061] The vector is determined in annex 10 using equ.(61) or (63) with the following result: [0062] The vector is determined in annex 10 using equ.(65) with the following result: [0063] The four-rotation is determined in annex 11 using any of equ.(42), (44) or (66) with the following result: [0064] All described calculations are preferably performed using software installed on a computer.

Annex 1 [0065] In the following the two summands of equ.(40) are evaluated separately. [0066] First summand of equ.(40): (69) (70) [0067] Second summand of equ.(40) In order to evaluate the second summand of equ.(40) we need the following theorem proven in annex 12: a matrix represents a local frame if and only if the matrix can be partitioned in the following way: (71) with (72) (73) (74) [0068] Now we can calculate the second summand of equ.(40): [0069] For the further transformation we need the following relationship: (75) (76) (77) [0070] For the further transformation we need additionally the following relation- ship: [0071] Now we can resume the transformation: with [0072] It is easy to see that the second condition of equ.(25) applies:

Annex 2 [0073] This annex shows that four-vectors u and ω belong to the nullspace of the s ^{econd summand of equ.(41):}

Annex 3 [0074] That the first summands of equ.(22) and equ.(41) are identical can be seen from equ.(69) and (70) in annex 1. [0075] In the following we prove that the second summands of equ.(22) and equ.(41) are identical. [0076] We recall here the statement of para.[0001] above that for an antisymmetric contravariant tensor A of rank two the symbol denotes the contravariant Hodge dual tensor with components denotes the mixed Hodge dual tensor with components denotes the covariant Hodge dual tensor with components An antisymmetric contravariant tensor A of rank two can always be partitioned in the following way: (78) (79) (80) Thus (81) (82) [0077] The Hodge dual tensor of an antisymmetric contravariant tensor A of rank two is usually defined by one of the two equivalent equations (see for example textbooks [4, equ.(17.20) on p.551 and equ.(14.75c) on p.490], [9, equ.(3.51) on p.88], [10, equ.(9.49),(9.50),(9.51) on p.307,308 and equ.(4.111) on p.115] or [7, equ.(11.140) on p.556]), but the same textbooks disclose - albeit al- most exclusively for the electric field strength tensor - that an equivalent definition i ^{s to replace A} _{1 by A2 and A2 by −A1. Note that all cited textbooks use} denoting the electric field and denoting the magnetic field (see [4, last para. of p.551], [9, box 4.3 on p.108], [10, first para. of p.116] and [7, last para. of p.556]). Thus the Hodge dual tensor of an antisymmetric tensor A of rank two can equivalently be defined as follows: (83) (84) (85) [0078] With this definition of the Hodge dual tensor of a two rank antisymmetric tensor it is easy to show that the second summand of equ.(22) is identical to the second summand of equ.(41). First it is apparent that is an antisymmetric tensor, which can be expressed as follows: Annex 4 [0079] That the first summands of equ.(23) and equ.(41) are identical can be seen from equ.(69) and (70) in annex 1. [0080] That the second summands of equ.(23) and equ.(41) are identical can be seen as follows:

Annex 5 [0081] Equ.(24) of the description reads and is with b = −ω ([4, paragraph above equ.(3.50) on p.86]) identical to equ.(14.83) on p.493 of [4], which reads : [0082] For the further transformation we need the relationship: [0083] Now we can resume the transformation: [0084] In the following each of the four components b ⁰ , b ¹ , b ² , b ³ are determined separately: (86) equ. (87) equ.(81),(82) second component of equ.(90) (88) (89) (90) [0085] From equ.(86), (87), (88), (89) and (90) follows which proves again the identity of equ.(44) and (24). Annex 6 [0086] This annex proves equ.(54),(55),(56),(57),(58),(59),(60) and equ.(68) in the description: (91) [0087] The two expressions in equ.(91) for D ₁ , which correspond to equ.(56) and (57) in the description, must be equal, this can also be proven explicitly: [0088] The expression in equ.(91) for [D ₂ ] _× , which corresponds to equ.(58), must be skew symmetric, this can also be proven explicitly by showing that the symmetric part is zero: [0089] The expressions in the following three boxes correspond to equ.(54), (55) and (56) in the description: (92) [0090] From annex 1 one can derive: The preceding expression corresponds to equ.(68) in the description. [0091] The next two expressions correspond to equ.(59) and (60) in the description: Annex 7 [0092] This annex proves equ.(61),(62),(63),(64),(65),(66) and (67) in the descrip- tion. Following the definition of Aη in equ.(19) we get: (93) [0093] The two expressions in equ.(93) for A ₁ , which correspond to equ.(61) and (63) in the description, are identical: [0094] The expression in equ.(93) for [A ₂ ] _× , which corresponds to equ.(64) in the description, is skew symmetric, as can be seen as follows: [0095] In the following equ.(62) in the description is proven: [0096] In the following equ.(65) in the description is proven: (94) [0097] In the following equ.(67) in the description is proven: That this expression for [B ₂ ] _× is skew symmetric is shown in an analogous manner to the corresponding proof for [A ₂ ] _× above. [0098] The formula of equ.(66) for the four-rotation ω can be determined from Bη in the following way: This equation is structurally similar to [19, equ.(32)].

Annex 8 [0099] In this annex is determined using formulas (54),(55),(56) or (57). [0100] Determining using formula (54): [0101] Determining using formula (55): (55)

[0102] Determining using formula (56):

[0103] Determining using formula (57):

Annex 9 [0104] In this annex is determined using formulas (58) or (59) and is determined using equ.(67): [0105] Before we evaluate the formulas for we recall some properties of discussed in [19]: (95) [0106] For the 3D Frenet-Serret frame the following applies ([19, annex 4]): with 3D-curvature and 3D-torsion being defined in equ.(34) above. [0107] From this follows: (96) [0108] In the following we evaluate the two summands of equ.(58): (97) [0109] Finally we get from equ.(58): [0110] Before we evaluate the formula (59) we have to determine by equ.(67). First we determine only the second summand of equ.(67):

[0111] For the further transformation we need the identity: [0112] Now we can continue the transformation: [0113] Now we can evaluate equ.(67):

(98) [0114] Now we can determine via formula (59): Annex 10 [0115] In this annex is determined using formulas (61),(63) and is deter- mined using formula (65). [0116] Determining using formula (61): [0117] Determining using formulas (63):

[0118] Determining using formulas (65): (99) Annex 11 [0119] In this annex the four-rotation is determined using any of equ.(42),(44) or (66). [0120] Determining using equ.(42): [0121] Determining using equ.(44): [0122] Determining using equ.(44):

Annex 12 [0123] A local frame Lη can always be partitioned in the following way: [0124] From conditions (9a),(9b),(9c) and (10) then the following relations can be derived (most of the following relations have already been disclosed in [19, annex 7, paragraph 0093]): (100) (101) (102) (103) [0125] According to textbook [6, equ.(3.9.11) on p.303 and fact 3.17.2 on p.334] :

[0126] The following paragraph lists all cited documents:

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