arXiv subjects

Bogdan G. Dimitrov

Publications and source records attributed to Bogdan G. Dimitrov.

18 recordsLinked to original sources

arXiv2025

Topological, Differential Geometry Methods and Modified Variational Approach for Calculation of the Propagation Time of a Signal, Emitted by a GPS-Satellite and Depending on the Full Set of 6 Kepler Parameters Parameters

Previously a mathematical approach has been developed for calculation of the propagation time of a signal, emitted by a moving along an elliptical orbit satellite, with account also for the General Relativity Theory (GRT) effects. The formalism was restricted to one dynamical parameter (the true anomaly or the eccentric anomaly angle). In this paper the aim is to extend the formalism to the case, when also the other five Kepler parameters will be changing.The following problem can be formulated: if two satellites move on two space-distributed orbits and they exchange signals, how can the propagation time be calculated? In this paper approaches from differential geometry and topology were implemented.The action functional for the propagation time is represented in the form of a quadratic functional in the differentials of the Kepler elements. The known mapping from celestial mechanics is used, when by means of a transformation the 6 Kepler parameters are mapped into the cartesian coordinates X, Y, Z. This is in fact a submersion of a manifold of 6 parameters into a manifold of 3 parameters. If a variational approach is applied with respect to a differential form in terms of the differentials of the Kepler parameters, the second variation will be different from zero and the Stokes theorem can be applied, provided that the second partial derivatives of the Cartesian coordinates with respect to the Kepler parameters are assumed to be different from zero. From topology viewpoint this requirement is equivalent to the existence of the s.c. Morse functions (non-degenerate at the critical points). In the given case it has been shown that Morse function cannot exist with respect to each one of the Kepler parameters- Morse function cannot be defined with respect to the omega angle.

physics.gen-ph↗

arXiv2022

Elliptic Integrals for Calculation of the Propagation Time of a Signal, Emitted and Received by Satellites on One Orbit

The propagation time of a signal, emitted by a moving along an elliptical orbit satellite from the GPS (or GLONASS) satellite confi gurations is a very important ingredient of the theory, based on the formalism of the null cone and accounting for the effects of the General Relativity Theory. For the case of satellites, orbiting along a plane elliptic orbit, it has been proved that the propagation time for the signal between the satellites is given by a combination of elliptic integrals of the first, second and third kind. For the more general case of satellites on a space-distributed elliptic orbit, the propagation time is expressed by higher (fourth) order elliptic integrals, which according to the standard theory can be expressed recurrently by means of lower-order elliptic integrals. In the concrete case, the elliptic integrals of the second and the fourth order are expressed by means of a combination of irrational functions and the zero-order elliptic integral in the Legendre form. It has been proved that for the investigated case, second-order elliptic integrals can be expressed by elementary functions.

physics.gen-ph↗

arXiv2022

New methods for analytical calculation of elliptic integrals, applied in various physical problems

A short review will be made of elliptic integrals, widely applied in GPS (Global Positioning System) communications (accounting for General Relativity Theory-effects), cosmology, Black hole physics and celestial mechanics. Then a novel analytical method for calculation of zero-order elliptic integrals in the Legendre form will be presented, based on the combination of several methods from the theory of elliptic functions: 1. the recurrent system of equations for higher-order elliptic integrals in two different representations. 2. uniformization of four-dimensional algebraic equations by means of the Weierstrass elliptic function 3.a variable transformation, inversely (quadratically) proportional to a new variable. The developed method is a step forward towards constructing analytical methods, which can improve the precision of the calculation of elliptic integrals, necessary both for theoretical and experimental problems.

gr-qc↗

arXiv2019

New Mathematical Models of GPS Intersatellite Communications in the Gravitational Field of the Near-Earth Space

Several space missions such as GRACE, GRAIL, ACES and others rely on intersatellite communications (ISC) between two satellites at a large distance one from another. The main goal of the theory is to formulate all the navigation observables within the General Relativity Theory (GRT). The same approach should be applied also to the intersatellite GPS-communications (in perspective also between the GPS, GLONASS and Galileo satellite constellations). In this paper a theoretical approach has been developed for ISC between two satellites moving on (one-plane) elliptical orbits based on the introduction of two gravity null cones with origins at the emitting-signal and receiving-signal satellites. The two null cones account for the variable distance between the satellites during their uncorrelated motion. This intersection of the two null cones gives the space-time interval in GRT. Applying some theorems from higher algebra, it was proved that this space-time distance can become zero, consequently it can be also negative and positive. But in order to represent the geodesic distance travelled by the signal, the space-time interval has to be "compatible" with the Euclidean distance. So this "compatibility condition", conditionally called "condition for ISC", is the most important consequence of the theory. The other important consequence is that the geodesic distance turns out to be the space-time interval, but with account also of the "condition for ISC". The geodesic distance is proved to be greater than the Euclidean distance - a result, entirely based on the "two null cones approach" and moreover, without any use of the Shapiro delay formulae. Application of the same higher algebra theorems shows that the geodesic distance cannot have any zeroes, in accord with being greater than the Euclidean distance.

gr-qc↗

arXiv2019

Two null gravitational cones in the theory of GPS-intersatellite communications between two moving satellites. I. Physical and mathematical theory of the space-time interval and the geodesic distance on intersecting null cones

Several space missions such as GRACE, GRAIL, ACES and others rely on intersatellite communications (ISC) between two satellites at a large distance one from another. The main goal of the theory is to formulate all the navigation observables within the General Relativity Theory. The same approach should be applied also to the intersatellite GPS-communications (in perspective also between the GPS, GLONASS and Galileo satellite constellations). In this paper a theoretical approach has been developed for ISC between two satellites moving on (one-plane) elliptical orbits, based on the introduction of two gravity null cones with origins at the emitting-signal and receiving-signal satellites. The two null cones account for the variable distance between the satellites during their uncorrelated motion.The intersection of the two null cones defines a distance, which can be found from a differential equation in full derivatives. This distance is the space-time interval in GRT. Applying some theorems from higher algebra, it was proved that this space-time distance can become zero, consequently it can be also negative and positive. But in order to represent the geodesic distance travelled by the signal, the space-time interval has to be "compatible" with the Euclidean distance. So this "compatibility condition", conditionally called "condition for ISC", is the most important consequence of the theory. The other important consequence is that the geodesic distance turns out to be the space-time interval, but with account also of the "condition for ISC". This interpretation enables the strict mathematical proof that the geodesic distance is greater than the Euclidean distance - a result, entirely based on the "two null cones approach" and moreover, without any use of the Shapiro delay formulae. Consistency between several other newly derived numerical parameters is noted.

gr-qc↗

arXiv2009

Elliptic Curves and Algebraic Geometry Approach in Gravity Theory I.The General Approach

Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and also on the choice of the contravariant components, it was shown that a wide variety of third, fourth, fifth, sixth, seventh - degree algebraic equations exists in gravity theory. This fact, together with the derivation of the algebraic equations for a generally defined contravariant tensor components in this paper, are important in view of finding new solutions of the Einstein's equations, if they are treated as algebraic ones. Some important properties of the introduced in hep-th/0107231 more general connection have been also proved - it possesses affine transformation properties and it is an equiaffine one. Basic and important knowledge about the affine geometry approach and about gravitational theories with covariant and contravariant connections and metrics is also given with the purpose of demonstrating when and how these theories can be related to the proposed algebraic approach and to the existing theory of gravity and relativistic hydrodynamics.

math-ph↗

arXiv2009

Elliptic Curves and Algebraic Geometry Approach in Gravity Theory II. Parametrization of a Multivariable Cubic Algebraic Equation

In a previous paper, the general approach for treatment of algebraic equations of different order in gravity theory was exposed, based on the important distinction between covariant and contravariant metric tensor components. In the present second part of the paper it has been shown that a multivariable cubic algebraic equation can also be parametrized by means of complicated, irrational and non-elliptic functions, depending on the elliptic Weierstrass function and its derivative. As a model example, the proposed before cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been investigated. This is quite different from the standard algebraic geometry approach, where only the parametrization of two-dimensional cubic algebraic equations has been considered. Also, the possible applications in modern cosmological theories has been commented.

math-ph↗

arXiv2009

Elliptic Curves and Algebraic Geometry Approach in Gravity Theory III. Uniformization Functions for a Multivariable Cubic Algebraic Equation

The third part of the present paper continues the investigation of the solution of the multivariable cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian. The main result in this paper constitutes the fact that the earlier found parametrization functions of the cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian can be considered also as uniformization functions. These functions are obtained as solutions of first - order nonlinear differential equations, as a result of which they depend only on the complex (uniformization) variable z. Further, it has been demonstrated that this uniformization can be extended to two complex variables, which is particularly important for investigating various physical metrics, for example the ADS metric of constant negative curvature (Lobachevsky spaces).

math-ph↗

arXiv2009

Algebraic Geometry Approach in Gravity Theory and New Relations between the Parameters in Type I Low-Energy String Theory Action in Theories with Extra Dimensions

On the base of the distinction between covariant and contravariant metric tensor components, a new (multivariable) cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been derived and parametrized with complicated non - elliptic functions, depending on the (elliptic) Weierstrass function and its derivative. This is different from standard algebraic geometry, where only two-dimensional cubic equations are parametrized with elliptic functions and not multivariable ones. Physical applications of the approach have been considered in reference to theories with extra dimensions. The s.c. "length function" l(x) has been introduced and found as a solution of quasilinear differential equations in partial derivatives for two different cases of "compactification + rescaling" and "rescaling + compactification". New physically important relations (inequalities) between the parameters in the action are established, which cannot be derived in the case $l=1$ of the standard gravitational theory, but should be fulfilled also for that case.

hep-th↗

arXiv2009

Some Algebraic Geometry Aspects of Gravitational Theories with Covariant and Contravariant Connections and Metrics (GTCCCM) and Possible Applications to Theories with Extra Dimensions

On the base of the distinction between covariant and contravariant metric tensor components, an approach from algebraic geometry will be proposed, aimed at finding new solutions of the Einstein's equations both in GTCCCM and in standard gravity theory, if these equations are treated as algebraic equations. As a partial case, some physical applications of the approach have been considered in reference to theories with extra dimensions. The s.c. "length function" l(x) has been introduced and has been found as a solution of quasilinear differential equations in partial derivatives for two different cases, corresponding to "compactification + rescaling" and "rescaling + compactification" of the type I low-energy string theory action. New (although complicated) relations between the parameters in the action have been found, valid also for the standard approach in theories with extra dimensions.

hep-th↗

arXiv2008

Block-Structure Method for the Solution of the Matrix System of Equations g{ij}g{jk}=delta{i}{k} in the N-dimensional Case

In this paper a new block-structure method is presented for the solution of the well-known from gravity theory matrix system of equations g{ij}g{jk}=delta{i}{k} (with respect to the unknown covariant components g{ij} and by known contravariant ones g{jk}) by transforming this matrix system into a linear algebraic system of equations in the general N-dimensional case. Although powerful computer methods exist for the solution of this problem for a given (fixed) dimension of the matrices g{ij} and especially for numerical elements of g{ij}, the structure of the obtained linear algebraic system in the general N-dimensional case and for arbitrary elements of g{ij} (functions) has not been known. The proposed new analytical block-structure method for the case of symmetrical matrices g{ij} and g{jk} (the standard case in gravity theory) is based on the construction of a block-structure matrix, whose "elements" are again matrices. The method allows to obtain the structure of this linear system in the general N-dimensional case, after multiplication (to the left) with the transponed matrix. Some arguments are given why the proposed method may be applied, after some refinement and generalization for the case of non-symmetrical matrices g{ij} and g{jk}, for finding the graviton modes in the Kaluza-Klein expansion in theories with extra dimensions.

hep-th↗

arXiv2008

Algebraic Geometry Approach in Theories with Extra Dimensions I. Application of Lobachevsky Geometry

This present paper has the purpose to find certain physical appications of Lobachevsky geometry and of the algebraic geometry approach in theories with extra dimensions. It has been shown how the periodic properties of the uniformization functions-solutions of cubic algebraic equations in gravity theory enable the orbifold periodic identification of the points pr{c} and -pr{c} under compactification. It has been speculated that corrections to the extradimensional volume in theories with extra dimensions should be taken into account due to the non-euclidean nature of the Lobachevsky space. It has been demonstrated that in the Higgs mass generation model with two branes (a "hidden" and a "visible" one), to any mass on the visible brane there could correspond a number of physical masses. Algebraic equations for 4D Schwarzschild Black Holes in higher dimensional brane worlds have been obtained.

hep-th↗

arXiv2008

Algebraic Geometry Approach in Theories with Extra Dimensions II. Tensor Length Scale, Compactification and Rescaling

In this second part of the paper, dedicated to theories with extra dimensions, a new physical notion about the "tensor length scale" is introduced, based on the gravitational theories with covariant and contravariant metric tensor components. Then the notion of "compactification" in low energy type I string theory is supplemented by the operation of "rescaling" of the contravariant metric components. For both the cases of "rescaling + compactification" and "compactification + rescaling", quasilinear differential equations in partial derivatives have been obtained and the corresponding solutions have been found for the scale (length) function and for the case of a flat 4D Minkowski space, embedded into a 5D space with an exponential warp factor. A differential equation has been obtained and investigated also from the equality of the "rescaled" scalar curvature with the usual one.

hep-th↗

arXiv2008

Elliptic Curves, Algebraic Geometry Approach in Gravity Theory and Uniformization of Multivariable Cubic Algebraic Equations

Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and the s.c. "gravitational theories with covariant and contravariant connection and metrics", it is shown that a wide variety of third, fourth, fifth, seventh, tenth- degree algebraic equations exists in gravity theory. This is important in view of finding new solutions of the Einstein's equations, if they are treated as algebraic ones. Since the obtained cubic algebraic equations are multivariable, the standard algebraic geometry approach for parametrization of two-dimensional cubic equations with the elliptic Weierstrass function cannot be applied. Nevertheless, for a previously considered cubic equation for reparametrization invariance of the gravitational Lagrangian and on the base of a newly introduced notion of "embedded sequence of cubic algebraic equations", it is demonstrated that in the multivariable case such a parametrization is also possible, but with complicated irrational and non-elliptic functions. After finding the solutions of a system of first - order nonlinear differential equations, these parametrization functions can be considered also as uniformization ones (depending only on the complex uniformization variable z) for the initial multivariable cubic equation.

hep-th↗

arXiv2005

Elliptic Curves, Algebraic Geometry Approach in Gravity Theory and Some Applications in Theories with Extra Dimensions I

Motivated by the necessity to find exact solutions with the elliptic Weierstrass function of the Einstein's equations (see gr-qc/0105022),the present paper develops further the proposed approach in hep-th/0107231, concerning the s.c. cubic algebraic equation for effective parametrization. Obtaining an ''embedded'' sequence of cubic equations, it is shown that it is possible to parametrize also a multi-variable cubic curve, which is not the standardly known case from algebraic geometry. Algebraic solutions for the contravariant metric tensor components are derived and the parametrization is extended in respect to the covariant components as well. It has been speculated that corrections to the extradimensional volume in theories with extra dimensions should be taken into account, due to the non-euclidean nature of the Lobachevsky space. It was shown that the mechanism of exponential "damping" of the physical mass in the higher-dimensional brane theory may be more complicated due to the variety of contravariant metric components for a spacetime with a given constant curvature. The invariance of the low-energy type I string theory effective action is considered in respect not only to the known procedure of compactification to a four-dimensional spacetime, but also in respect to rescaling the contravariant metric components. As a result, instead of the simple algebraic relations between the parameters in the string action, quasilinear differential equations in partial derivatives are obtained, which have been solved for the most simple case. In the Appendix, a new block structure method is presented for solving the well known system of operator equations in gravity theory in the N-dimensional case.

hep-th↗

arXiv2004

Integral Geometry on the Lobachevsky Plane and the Conformal Wess-Zumino-Witten Model of Strings on an ADS3 Background

The main purpose of the report is to provide some argumentation that three seemingly distinct approaches of 1. Giveon, Kutasov and Seiberg (hep-th/9806194); 2. Hemming, Keski-Vakkuri (hep-th/0110252); Maldacena, Ooguri (hep-th/0001053) and 3. I. Bars (hep-th/9503205) can be investigated by applying the mathematical methods of integral geometry on the Lobachevsky plane, developed previously by Gel'fand, Graev and Vilenkin. All these methods can be used for finding the transformations, leaving the Kac-Moody and Virasoro algebras invariant. The near-distance limit of the Conformal Field Theory of the SL(2, R) WZW model of strings on an ADS3 background can also be interpreted in terms of the Lobachevsky Geometry : the non - euclidean distance is conserved and the Lobachevsky formulae for the angle of parallelism is recovered. Some preliminary technique from integral geometry for inverting the modified integral representation for the Kac- Moody algebra has been demonstrated.

hep-th↗

arXiv2002

Some Algebro-Geometric Aspects of The SL(2, R) Wess-Zumino-Witten Model of Strings on an ADS$_{3}$ Background

The SL(2, R) WZW model of strings on an ADS3 background is investigated in the spirit of J.Maldacena's and H.Ooguri's approach (hep-th/0001053) and (hep-th/0005183). Choosing a standard, but most general three-variable parametrization of the SL(2, R) group element g, the system of equations for the Operator Product Expansion (OPE) relations is analysed. In the investigated SL(2, R) case, this system is consistent if each three points on the complex plane lie on a certain hypersurface in CP3. A system of three nonlinear first-order differential equations has been obtained for the parametrization functions. It was demonstrated also how the mathematical apparatus of generalized functions and integral geometry can be implemented in order to modify the integral operators, entering the Kac-Moody and Virasoro algebras.

hep-th↗

arXiv2002

Cubic Algebraic Equations in Gravity Theory, Parametrization with the Weierstrass Function and Non-Arithmetic Theory of Algebraic Equations

A cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle.It was suggested that such an equation may find application in gravity theory, brane, string and Rundall-Sundrum theories. The obtained algebraic equation was brought by means of a linear-fractional transformation to a parametrizable form, expressed through the elliptic Weierstrass function, which was proved to satisfy the standard parametrizable form, but with $g_{2}$ and $g_{3}$ functions of a complex variable instead of the definite complex numbers (known from the usual arithmetic theory of elliptic functions and curves). The generally divergent (two) infinite sums of the inverse first and second powers of the poles in the complex plane were shown to be convergent in the investigated particular case, and the case of the infinite point of the linear-fractional transformation was investigated. Some relations were found, which ensure the parametrization of the cubic equation in its general form with the Weierstrass function.

hep-th↗