A Method of Successive Approximations in the Framework of the Geometrized Lagrange Formalism

October, 008 PROGRESS IN PHYSICS Volume 4 A Method of Successive Approximations in the Framework of the Geometrized Lagrange Formalism Grigory I. Garas ko Department of Physics, Scientific Research Institute of Hyper Complex Systems in Geometry and Physics Zavodskoy proezd, Frjazino, Moscow Region 490, Russia E-mail: gri9z@mail.ru It is shown that in the weak field approximation the new geometrical approach can lead to the linear field equations for the several independent fields. For the stronger fields and in the second order approximation the field equations become non-linear, and the fields become dependent. This breaks the superposition principle for every separate field and produces the interaction between different fields.the unification of the gravitational and electromagnetic field theories is performed in frames of the geometrical approach in the pseudo-riemannian space and in the curved Berwald-Moor space. Introduction In paper [] the new (geometrical) approach was suggested for the field theory. It is applicable for any Finsler space [] for which in any point of the main space x ; x ; : : : ; x n the indicatrix volume V ind (x ; x ; : : : ; x n ) can be defined, ev provided the tangent space is Euclidean. Then the action I for the fields present in the metric function of the Finsler space is defined within the accuracy of a constant factor as a volume of a certain n-dimensional region V : I const Z V (n) dx dx : : : dx n (V ind (x ; x ; : : : ; x n )) ev : () Thus, the field Lagrangian is defined in the following way L const (V ind (x ; x ; : : : ; x n )) ev : () In papers [, 4] the spaces conformally connected with the Minkowski space and with the Berwald-Moor space were regarded. These spaces have a single scalar field for which the field equation was written and the particular solutions were found for the spherical symmetry and for the rhombododecaeon symmetry of the space. The present paper is a continuation of those papers dealing with the study and development of the geometric field theory. Pseudo-Riemannian space with the signature ( ) Let us consider the pseudo-riemannian space with the signature ( ) and select the Minkowski metric tensor g in the metric tensor g (x); of this space explicitly g (x) g h (x) : () Let us suppose that the field h (x) is weak, that is jh (x)j : (4) According to [], the Lagrangian for a pseudo-riemannian space with the signature ( ) is equal to L q det(g )] within the ac- Let us calculate the value of [ curacy of jh (x)j : det(g ) : (5) det(g ) ' L L ; (6) L g h h 00 h h h ; (7) L h 00 (h h h ) h h h h h h h h h h 0 h 0 h 0 : The last formula can be rewritten in a more convenient way then L h 00 h 0 h 0 h h 00 h 0 h 0 h h 00 h 0 h 0 h h h h h h h h h h h h h ; L ' L L 4 L (8) (9) : (0) To obtain the field equations in the first order approximation, one should use the Lagrangian L ; and to do the same in the second order approximation the Lagrangian L L 4 L : Grigory I. Garas ko. A Method of Successive Approximations in the Framework of the Geometrized Lagrange

Volume 4 PROGRESS IN PHYSICS October, 008 Scalar field For the single scalar field '(x) the simplest representation of tensor h (x) has the form : That is why L ' L q h (x) h (') (x) @x j : () det(g ) p L ' L @x 0 @x @x 8 L ; () @x : () In the first order approximation, we can use the Lagrangian L to obtain the following field equation @x 0 @x 0 @x @x @x @x @x @x 0 ; (4) which presents the wave equation. The stationary field that depends only on the radius will satisfy the equation r p (x ) (x ) (x ) ; (5) d r the integration of which gives 0 ; (6) C r ) '(r) C 0 C r : (7) In the second order approximation one should use the Lagrangian L 4 L to obtain the field equation in the second order approximation g @ L @x j 0 ; (8) this equation is already non-linear. The strict field equation for the tensor h (x) () is 0 g @ B @ p @x j C L A 0 ; (9) then the stationary field depending only on the radius must satisfy the equation d 0 B @ r s C A 0 : (0) Integrating it, we get C p r 4 C Z ) '(r) C 0 r ) C p r 4 C : () The field with the upper sign and the field with the lower sign differ qualitatively: the upper sign in Eq. () gives a finite field with no singularity in the whole space, the lower sign in Eq. () gives a field defined every but for the spherical region in which there is no field, while 0 r p jc j ; () r > p jc j ; r! p jc j )! C : () At the same time in the infinity (r! ) both solutions '(r) behave as the solution of the wave equation Eq. (7). If we know the Lagrangian, we can write the energymomentum tensor Tj i for the these solutions and calculate the energy of the system derived by the light speed c : Z () P 0 const T0 0 dv : (4) To obtain the stationary spherically symmetric solutions, we get T0 0 r p r 4 C ; (5) that is why for both upper and lower signs P 0!. The metric tensor of Eq. (,) is the simplest way to insert the gravity field into the Minkowski space the initial flat space containing no fields. Adding several such terms as in Eq. () to the metric tensor, we can describe more and more complicated fields by tensor h h (grav) : 4 Covariant vector field To construct the twice covariant symmetric tensor h (x) with the help of a covariant field A i (x) not using the connection objects, pay attention to the fact that the alternated partial derivative of a tensor is a tensor too F @A j @A i @x j ; (6) but a skew-symmetric one. Let us construct the symmetric tensor on the base of tensor F. To do this, first, form a scalar L A g g km F ik F jm g g km @Ak @A m @x j @A k @A j @x m ; (7) Grigory I. Garas ko. A Method of Successive Approximations in the Framework of the Geometrized Lagrange

October, 008 PROGRESS IN PHYSICS Volume 4 which gives the following expressions for two symmetric tensors h () g km @A k @A m @A k @A j @A k @A i @x j @x m @x j @x m ; (8) h () g km @A i @x k @A j @x m @A i @A m @x k @x j @A j @A m @x k : (9) Notice, that not only F and L A but also the tensors h (), h() are gradient invariant, that is they don t change with transformations A i! A i @f(x) ; (0) f(x) is an arbitrary scalar function. Let h h (A k) (x) h () [ (x)] h () ; () (x) is some scalar function. Then in the first order approximation we get L g g km Ak @A j @x m L A ; () and the first order approximation for the field A i (x) gives Maxwell equations g @ @x j A k For Lorentz gauge the equations Eqs. () take the form @ g @A j @x k 0 : () g @A j 0 ; (4) A k 0 : (5) It is possible that Eq. () is not the most general form for tensor h which in the first order approximation gives the field equations coinciding with Maxwell equations. To obtain Maxwell equations not for the free field but for the field with sources j i (x); one should add to h (Ak) Eq. () the following tensor 6 h (jk) c (A j A j j i ) : (6) This means that the metric tensor Eq. () with tensor h h (Max) h (Ak) h (jk) (7) describes the weak electromagnetic field with source j k (x): We must bear in mind that we use the geometrical approach to the field theory, and we have to consider j k (x) to be given and not obtained from the field equations. So, the metric tensor Eq. () with tensor h h (Ak) h (grav) ; (8) ; are the fundamental constants in frames of the unique pseudo-riemannian geometry describes simultaneously the free electromagnetic field and the free gravitational field. To include the sources, j k (x); of the electromagnetic field, the metric tensor must either include not only j k (x) but the partial derivatives of this field too or the field j k (x) must be expressed by the other fields as shown below. If the gravity field is inserted in the simplest way as shown in the previous section, then the sources of the electromagnetic field can be expressed by the scalar field as follows j i (x) q : (9) In this case the first order approximation for Lorentz gauge gives A k 4 c j k ; (40) ' 0 : (4) Since the density of the current has the form of Eq. (9), the Eq. (4) gives the continuity equation 5 Several weak fields g @j i @x j 0 : (4) The transition from the weak fields to the strong fields may lead to the transition from the linear equations for the independent fields to the non-linear field equations for the mutually dependent interacting fields '(x) and (x) including gravity field in the Minkowski space. Let h " ' @x j " @ @ @x j ; (4) " ', " are independent sign coefficients. Then the strict Lagrangian can be written as follows q L '; L L ; (44) and L g " ' @x j " @ @ @x j ; (45) Grigory I. Garas ko. A Method of Successive Approximations in the Framework of the Geometrized Lagrange

Volume 4 PROGRESS IN PHYSICS October, 008 L " ' " " @ @x 0 @x @ @x 0 @x @ @x 0 @x @ @x @x @ @x @x @ @x @x @ @x @x 0 @ @x @x 0 @ @x @x 0 @ @x @x @ @x @x # @ @x @x 9 > >; : (46) This formula, Eq. (46), can be obtained from Eq. (9) most easily, if one uses the following simplifying formula h i j h ii h i j h jj @ @x j @x j @ @x j : @ @ @x j In the first order approximation for the Lagrangian, the expression L should be used. Then the field equations give the system of two independent wave equations @x 0 @x 0 @x @x @x @x @ @ @ @ @x 0 @x 0 @x @x @x @x @x @x 0 @x @x 0 9 > >; Here the fields '(x) and (x) are independent and the superposition principle is fulfilled. Using the strict Lagrangian for the two scalar fields Eq. (44) we get a system of two non-linear differential equations of the second order g @ 4 ' ;j g rs ;r ;s ;j g rs ' ;r ;s p 5 0 ; L L g @ 4 ;j g rs ' ;r ' ;s ' ;j g rs ' ;r ;s p 5 0 ; L L comma means the partial derivative. Here the fields '(x), (x) depend on each other, and the superposition principle is not fulfilled. The transition from the last but one equations to the last equations may be regarded as the transition from the weak fields to the strong fields. : 6 Non-degenerate polynumbers Consider a certain system of the non-degenerate polynumbers P n [5], that is n-dimensional associative commutative non-degenerated hyper complex numbers. The corresponding coordinate space x ; x ; : : : ; x n is a Finsler metric flat space with the length element equal to ds nq gi i :::i n dx i dx i : : : dx in ; (47) g i i :::i n is the metric tensor which does not depend on the point of the space. The Finsler spaces of this kind can be found in literature (e.g. [6, 7, 8, 9]) but the fact that all the non-degenerated polynumber spaces belong to this type of Finsler spaces was established beginning from the papers [0, ] and the subsequent papers of the same authors, especially in [5]. The components of the generalized momentum in geometry corresponding to Eq. (47) can be found by the formulas p i g :::j n dx j : : : dx jn gii :::i n dx i dx i : : : dx i n n n : (48) The tangent equation of the indicatrix in the space of the non-degenerated polynumbers P n can be always written [5] as follows g i i :::i n p i p i : : : p in n 0 ; (49) is a constant. There always can be found such a basis (and even several such bases) and such a > 0 that g i i :::i n g ii :::i n : (50) Let us pass to a new Finsler geometry on the base of the space of non-degenerated polynumbers P n. This new geometry is not flat, but its difference from the initial geometry is infinitely small, and the length element in this new geometry is ds n r hgi i:::i n "h i i:::i n (x) i dx i dx i : : : dx in ; (5) " is an infinitely small value. If in the initial space the volume element was defined by the formula dv dx i dx i : : : dx i n ; (5) in the new space within the accuracy of " in the first power we have dv h ' h " C 0 g i i :::i n h ii :::i n (x) i dx i dx i : : : dx i n : That is according to [], the Lagrangian of the weak field in the space with the length element Eq. (5) in the first order 4 Grigory I. Garas ko. A Method of Successive Approximations in the Framework of the Geometrized Lagrange

October, 008 PROGRESS IN PHYSICS Volume 4 approximation is L g ii:::in h i i :::i n (x) : (5) This expression generalizes formula Eq. (7). 7 Hyper complex space H 4 In the physical ( orthonormal [5]) basis in which every point of the space is characterized by the four real coordinates x 0 ; x, x ; x the fourth power of the length element ds H 4 is defined by the formula (ds H 4 )4 g kl dx 0 dx dx dx (dx 0 dx dx dx )(dx 0 dx dx dx ) (dx 0 dx dx dx )(dx 0 dx dx dx ) (dx 0 ) 4 (dx ) 4 (dx ) 4 (dx ) 4 8dx 0 dx dx dx (dx 0 ) (dx ) (dx 0 ) (dx ) (dx 0 ) (dx ) (dx ) (dx ) (dx ) (dx ) (dx ) (dx ) : (54) Let us compare the fourth power of the length element ds H 4 in the space of polynumbers H 4 with the fourth power of the length element ds Min in the Minkowski space (ds Min ) 4 (dx 0 ) 4 (dx ) 4 (dx ) 4 (dx ) 4 (dx 0 ) (dx ) (dx 0 ) (dx ) (dx 0 ) (dx ) (dx ) (dx ) (dx ) (dx ) (dx ) (dx ) : This means (ds H 4 )4 (ds Min ) 4 8dx 0 dx dx dx 4(dx ) (dx ) 4(dx ) (dx ) 4(dx ) (dx ) ; and in the covariant notation we have g (ds H 4 )4 gkl gkl 0 Gkl dx i dx j dx k dx l ; gkl 0 ; if i; j; k; l are all different 0 ; else Gkl 8 >< >: ; if i; j; k; l 0 and i j k l; or i k j l; or i l j k 0 ; else (55) (56) (57) (58) (59) The tangent equation of the indicatrix in the H 4 space can be written in the physical basis as in [5] : (p 0 p p p )(p 0 p p p ) (p 0 p p p )(p 0 p p p ) 0 ; (60) p i are the generalized momenta p i @ ds H4 @(dx i ) : (6) Comparing formula Eq. (60) with formula Eq. (6), we have g kl p i p j p k p l 0 : (6) Here g kl g g kl g 0 kl G kl ; (6) and g kl gkl g 0 kl g 0 kl G kl Gkl 9 > >; : (64) To get the Lagrangian for the weak field in the first order approximation, we have to get tensor h kl in Eq. (5). In the simplified version it could be splitted into two additive parts: gravitational part and electromagnetic part. The gravitational part can be constructed analogously to Sections and 5 with regard to the possibility to use the two-index number tensors, since now tensors g kl ; h kl have four indices. The construction of the electromagnetic part should be regarded in more detail. Since we would like to preserve the gradient invariance of the Lagrangian and to get Maxwell equations for the free field in the H 4 space, let us write the electromagnetic part of tensor h kl in the following way h A k kl (x) h() kl [ (x)] h() kl ; (65) the tensors h () kl ; h() kl are the tensors present in the round brackets in the r.h.s. of formulas Eqs. (8,9). Then L A g kl h A k kl g g km @Ak @A m @x j @A k @A j @x m : (66) To obtain Maxwell equations not for the free field but for the field with a source j i (x); one should add to the tensor h (Ak) kl Eq. (65) the following tensor h (j k) kl 8 A i j j gkl A i gjk j l j i gjk A l ; symmetrized in all indices, that is tensor h kl h Max kl h (Ak) kl h (jk) (kl) ; Grigory I. Garas ko. A Method of Successive Approximations in the Framework of the Geometrized Lagrange 5

Volume 4 PROGRESS IN PHYSICS October, 008 describes the weak electromagnetic field with the sources j i (x), X @ (b) j i q (a) ; (67) b and (b) are the scalar components of the gravitational field. To obtain the unified theory for the gravitational and electromagnetic fields one should take the linear combination of tensor h (Max) kl corresponding to the electromagnetic field in the first order approximation, and tensor h (grav) kl corresponding to the gravitational field in the first order approximation h kl h (Max) kl h (grav) kl ; (68) ; are constants. Tensor h (grav) kl may be, for example, constructed in the following way h grav kl MX b NX a " (a) (a) (a) @x j (b) @ (b) @x (i @ (b) @x j gkl) ; (a) @x k (a) @x l (69) " (a) ; (b) are the sign coefficients, and ' (a) ; (b) are the scalar fields. The whole number of scalar fields is equal to (N M). 8 Conclusion In this paper it was shown that the geometrical approach [] to the field theory in which there usually appear the non-linear and non-splitting field equations could give a system of independent linear equations for the weak fields in the first order approximation. When the fields become stronger the superposition principle (linearity) breaks, the equations become non-linear and the fields start to interact with each other. We may think that these changes of the equations that take place when we pass from the weak fields to the strong fields are due to the two mechanisms: first is the qualitative change of the field equations for the free fields in the first order approximation; second is the appearance of the additional field sources, that is the generation of the field by the other fields. In frames of the geometrical approach to the field theory [] the unification of the electromagnetic and gravitational fields is performed both for the four-dimensional pseudo-riemannian space with metric tensor g (x) and for the fourdimensional curved Berwald-Moor space with metric tensor g kl (x): References. Garas ko G.I. Field theory and Finsler spaces. Hyper Complex Numbers in Geometry and Physics, 006, v. (), 6 0.. Rashevsky P. K. Geometrical theory of equations with partial derivatives. OGIZ, Moscow-Leningrad, 947.. Garas ko G.I. Particular stationary solution of the field equation for the space conformally connected with Minkowski space. Hyper Complex Numbers in Geometry and Physics, 007, v. 4 (), 6 7. 4. Garas ko G.I. The space conformally connected with Berwald- Moor space. Hyper Complex Numbers in Geometry and Physics, 007, v. 4 (), 8 5. 5. Pavlov D.G., Garas ko G.I. Geometry of the non-degenerated polynumbers. Hyper Complex Numbers in Geometry and Physics, 007, v. 4 (), 5. 6. Matsumoto M., Numata S. On Finsler spaces with -form metric. Tensor, 978, v., 6 69. 7. Matsumoto M., Numata S. On Finsler spaces with -form metric II. Berwald Moor s metric L (y y : : : y n ) n. Tensor, 978, v., 75 77. 8. Matsumoto M., Numata S. On Finsler spaces with a cubic metric, Tensor, 979, v., 5 6. 9. Shimada H. On Finsler spaces with the metric L of m-th root metric. Tensor, 979, v., 65 7. 0. Pavlov D.G. Generalized axioms of the scalar product. Hyper Complex Numbers in Geometry and Physics, 004, v. (), 6 0.. Garas ko G.I. Generalized analytical functions of the polynumber variable. Hyper Complex Numbers in Geometry and Physics, 004, v. (), 75 88. Acknowledgements This work was supported in part by the Russian Foundation for Basic Research under grant RF BR 07 0 968 RA a and by the Non-Commercial Foundation for Finsler Geometry Research. Submitted on August 0, 008 / Accepted on August, 008 6 Grigory I. Garas ko. A Method of Successive Approximations in the Framework of the Geometrized Lagrange