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Abstract
Path and path deviation equations for charged, spinning and spinning charged objects in different versions of Kaluza-Klein (KK) theory using a modified Bazanski Lagrangian have been derived. The significance of motion in five dimensions, especially for a charged spinning object, has been examined. We have also extended the modified Bazanski approach to derive the path and path deviation equations of a test particle in a version of non-symmetric KK theory.
... From this perspective, we ought to study the problem of spinning objects in depth, as it is very close to the reality, rather than examining its simplicity by means of determining the equation of motion of test particles, i.e. the geodesic equation. The spinning object has been studied by many authors long time ago, Mathisson [1] started the idea; Papapetrou amended its content [2] and then it was developed to include charged objects by Dixon [3], which led many of their followers to obtain the corresponding equations of motion of moving objects in different types of geometries [4][5][6][7][8][9]. Not only these path equations but also their deviation equations play a fundamental role in regulating the stability of objects [10]. ...
... Equations of geodesic and geodesic deviation equations in Riemannian geometry are required to examine many problems of motion for different test particles in gravitational fields. This led many authors to derive them by various methods, one of the most applicable ones is the Bazanski approach [27] in which from one single Lagrangian one can obtain simultaneously equation of geodesic and geodesic deviations which has been applied in different theories of gravity [4][5][6][7][8][9],and [28][29][30]. Thus, by analogy this technique in case of Poly-vectors to become [31], ...
... In our approach, we have obtained the relevant equations of spinning and spinning of charged poly-vectors, as well as their deviation equations in C-space i.e. (4.84), (5.97) and (5.98). This type of work is regarded an extension to a previous work, obtaining equations of spinning and spinning objects in Riemannian and non-Riemannian geometries, using the Bazanski approach [4][5][6][7][8][9]. Throughout, this study, it has been found the necessity to regard extended objects, the most reliable ones to express the actual nature of objects, rather than relying to a point-like system. ...
... In order to give a systematic treatment of the extra forces in the presence of a scalar field we will use the Bazanski approach for obtaining the geodesic equation [64]. According to this approach, the equation of motion in any dimensions can be obtained by applying the action principle to the Lagrangian [65] ...
... where u A = (dx µ /dS, dξ/dS) is the five-velocity andΓ A BC are the Christoffel symbols formed with the 5D metric [58,61,62,65]. ...
We investigate the possibility that the observed behavior of test particles outside galaxies, which is usually explained by assuming the presence of dark matter, is the result of the dynamical evolution of particles in higher dimensional spacetimes. Hence, dark matter may be a direct consequence of the presence of an extra force, generated by the presence of extra dimensions, which modifies the dynamic law of motion, but does not change the intrinsic properties of the particles, like, for example, the mass (inertia). We discuss in some detail several possible particular forms for the extra force, and the acceleration law of the particles is derived. Therefore, the constancy of the galactic rotation curves may be considered as an empirical evidence for the existence of the extra dimensions.
... Yet, one may find out that the Weyssenhoff tensor [6] is most eligible candidate to express a spin fluid element i.e. which indicates equation of spin density tensor. Accordingly, we may obtain equations analogously by means of its corresponding Bazanski Lagrangian stemmed from its original formalism [3] and its modification in GR [12] to become ...
Equations of motion of spinning density for extended objects, and corresponding deviation equations are derived. The problem of motion for a variable mass to a spinning extended object is obtained. Spinning fluids may be considered as a special case to express the motion of spinning density for extended objects. Meanwhile, spinning density tensor can be expressed in terms of tetrad formalism of General Relativity to be regarded as a gauge theory of gravity. Equations of spinning and spinning deviation density tensors have been derived using a specific type of Bazanski Lagrangian is performed.
... The importance of the Bazanski approach appears when it is extended to non Riemannan geometries, that contain simultaneously non vanishing curvature and torsion [9], [5]. This approach has also been generalized in Riemannian geometry to express the Papapetrou equation for a rotating object and the Dixon equation for a charged rotating object together with their corresponding deviation equations [6]. ...
The Bazanski approach for deriving paths is applied to Finsler geometry. The approach is generalized and applied to a new developed geometry called "Absolute parallelism with a Finslerian Flavor" (FAP). A sets of path equations is derived for the FAP. This is the horizontal (h) set. A striking feature appears in this set, that is: the coefficient of torsion term, in the set, jumps by a step of one-half from one equation to the other. This is tempting to believe that the h-set admits some quantum features. Comparisons with the corresponding sets in other geometries are given. Conditions to reduce the set of path equations obtained, to well known path equations in some geometries are summarized in a schematic diagram.
... non-Riemannian geometries admitting non-vanishing curvature and torsion tensors simultaneously [20][21][22]. This approach helps to implement the concept of geometrization to include not only physics but also biological epidemic curves [23] as well as economic complex systems in terms of information geometry [24]. Also, this Lagrangian has been modified to describe the path equation of charged object to take the following form [25]; ...
The problem of motion of different test particles, charged and spinning objects with a constant spin tensor in different versions of the bimetric theory of gravity is considered by deriving their corresponding path and path deviation equations using a modified Bazanski Lagrangian. Such a Lagrangian, as in the framework of Riemannian geometry, has a capability to obtain path and path deviations of any object simultaneously. This method enables us to derive the path and path deviation equations of different objects orbiting in very strong gravitational fields.
... And taking the variation with respect to velocity vector U σ to get the corresponding components of path deviation equation [14] : ...
Recently, the behavior of different epidemic models and their relation both to different types of geometries and to some biological models has been revisited. Path equations representing the behavior of epidemic models and their corresponding deviation vectors are examined. A comparison between paths and their deviation vectors in Riemannian and Finslerian Geometries is presented.
... Does the particle leave the brane or does it stay moving around it? The motion of test particle in higher-dimensional spaces has been investigated under different approaches [9,10,11,12,13,14,15,16,17,18,19]. In Ref. [13] necessary general geometric conditions for the geodesic motion to be stable around a hypersurface were established. ...
We consider the motion of test particles in thick branes of ‘Randall–Sundrum II’-type. It is known that gravity alone cannot explain the confinement of test particles in this kind of brane. In this Letter we show that a stable confinement in a domain wall is possible by admitting a direct interaction between test particles and a scalar field. This interaction is implemented by a modification of the Lagrangian of the particle which is inspired by a Yukawa-type interaction between fermions and scalar fields.
Adopting Bazanki's action, two new classes of path equations are derived in Einstein non-symmetric geometry. The first class is the path equations of a test particle moving in a gravitational field, while the second class represents path equations of charged particles. The quantum features of this geometry appear in both classes. The path equations of charged particles give rise to Lorentz force. Moreover, these path equations may represent an interpretation of some interactions between torsion and electromagnetic potential even if the electromagnetic force vanishes. It is to be noted that the above two classes of paths are formulated in terms of Einstein non-symmetric connection. An explicit formula of such a connection, satisfying the Einstein metricity condition, is obtained by localizing the global formula given recently by Ivanov-Zlatanovic.
Spinning equations of bi-metric type theories of gravity, the counterpart of the Papapetrou equations of motion are derived as well as their corresponding spinning deviation equations, by means of introducing different types of bi-metric theories. The influence of different curvatures based on different connections is illustrated. A specific Lagrangian function for each type theory is proposed, in order to derive the set of spinning motion and their corresponding spinning deviation equations.
Equations of non-geodesic and non-geodesic deviations for different particles are obtained, using a specific type of classes of the Bazanski Lagrangian. Such type of paths has been found to describe the problem of variable mass in the presence of Riemannian geometry. This may give rise to detect the effect of dark matter which reveals the mystery of motion of celestial objects that are not responding neither to Newtonian nor Einsteinian gravity. An important link between non-geodesic equations and the dipolar particle or fluids has been introduced to apply the concept of geometization of physics. This concept has been already extended to represent the hydrodynamic equations in a geometric way. Such an approach, demands to seek for an appropriate theory of gravity able to describe different regions, eligible for detecting dark matter. Using different versions of bi-metric theory of gravity, to examine their associate non-geodesic paths. Due to implementing the geometrization concept, the stability problem of non-geodesic equations are essential to be studied for detecting the behavior of those objects in the presence of dark matter.
The problem of spinning and spin deviation equations for particles as defined by their microscopic effect has led many authors to revisit non-Riemannian geometry for being described torsion and its relation with the spin of elementary particles. We obtain a new method to detect the existence of torsion by deriving the equations of spin deviations in different classes of non-Riemannian geometries, using a modified Bazanski method. We find that translational gauge potentials and rotational gauge potentials regulate the spin deviation equation in the presence of Poincare gauge field theory of gravity.
We study the classical geodesic motions of nonzero rest mass test particles and photons in five-dimensional warped product spaces. We show that it is possible to obtain a general picture of these motions using the natural decoupling that occurs in such spaces between the motions in the fifth dimension and the motion in the hypersurfaces. This splitting allows the use of phase space analysis in order to investigate the possible confinement of particles and photons to hypersurfaces in five-dimensional warped product spaces. Using such an analysis, we find a novel form of quasiconfinement which is oscillatory and neutrally stable. We also find that this class of warped product spaces locally satisfy the Z2 symmetry by default. The importance of such a confinement is that it is purely due to the classical gravitational effects, without requiring the presence of brane-type confinement mechanisms.
The geodesic deviation equation, describing the relative accelerations of
nearby particles, and the Raychaudhury equation, giving the evolution of the
kinematical quantities associated with deformations (expansion, shear and
rotation) are considered in the framework of modified theories of gravity with
an arbitrary curvature-matter coupling, by taking into account the effects of
the extra force. As a physical application of the geodesic deviation equation
the modifications of the tidal forces due to the supplementary curvature-matter
coupling are obtained in the weak field approximation. The tidal motion of test
particles is directly influenced not only by the gradient of the extra force,
which is basically determined by the gradient of the Ricci scalar, but also by
an explicit coupling between the velocity and the Riemann curvature tensor. As
a specific example, the expression of the Roche limit (the orbital distance at
which a satellite will begin to be tidally torn apart by the body it is
orbiting) is also obtained for this class of models.
Path and path deviation equations for neutral, charged, spinning and spinning
charged test particles, using a modified Bazanski Lagrangian, are derived. We
extend this approach to strings and branes. We show how the Bazanski Lagrangian
for charged point particles and charged branes arises `a la Kaluza-Klein from
the Bazanski Lagrangian in 5-dimensions.
Recently, the behavior of different epidemic models and their relation both to different types of geometries and to some biological models has been revisited . Path equations representing the behavior of epidemic models and their corresponding deviation vectors are examined. A comparison between paths and their deviation vectors in Riemannian and Finslerian Geometries is presented. Comment: A revised version, 12 LaTeX pages
We study the classical geodesic motions of nonzero rest mass test particles and photons in (3+1+n)- dimensional warped product spaces. An important feature of these spaces is that they allow a natural decoupling between the motions in the (3+1)-dimensional spacetime and those in the extra n dimensions. Using this decoupling and employing phase space analysis we investigate the conditions for confinement of particles and photons to the (3+1)- spacetime submanifold. In addition to providing information regarding the motion of photons, we also show that these motions are not constrained by the value of the extrinsic curvature. We obtain the general conditions for the confinement of geodesics in the case of pseudo-Riemannian manifolds as well as establishing the conditions for the stability of such confinement. These results also generalise a recent result of the authors concerning the embeddings of hypersurfaces with codimension one.
Recently, path equations have been obtained for charged, spinning objects in brane world models, using a modified Bazanski Lagrangian. In this study, path deviation equations of extended objects are derived. The significance of moving extended objects in brane world models is examined. Motion in non- symmetric brane world models is also considered.
We investigate the dynamics of particles moving in a spacetime augmented by one extra dimension in the context of the induced matter theory of gravity. We examine the appearance of a fifth force as an effect caused by the extra dimension and discuss two different approaches to the fifth force formalism. We then give two simple examples of application of both approaches by considering the case of a Ricci-flat warped-product manifold and a generalized Randall–Sundrum space.
In this paper we calculate the emission of gravity waves by the binary pulsar in the framework of five-dimensional spacetime. We consider spacetimes with one compact extra dimension. We show that the presence of additional degrees of freedom, especially the 'gravi-scalar', leads to a modification of Einstein's quadrupole formula. We compute the induced change for the binary pulsar PSR 1913 + 16 in the simple example of a 5D Minkowski background. In the example of a cylindrical braneworld it amounts to about 20% which is by far excluded by present experimental data.
We examine the question of whether violation of 4D physics is an inevitable consequence of existence of an extra non-compactified dimension. Recent investigations in membrane and Kaluza–Klein theory indicate that when the metric of the spacetime is allowed to depend on the extra coordinate, i.e., the cilindricity condition is dropped, the equation describing the trajectory of a particle in one lower dimension has an extra force with some abnormal properties. Among them, a force term parallel to the four-velocity of the particle and, what is perhaps more surprising, uμfμ≠uμfμ. These properties violate basic concepts in 4D physics. In this Letter we argue that these abnormal properties are not consequence of the extra dimension, but result from the formalism used. We propose a new definition for the force, from the extra dimension, which is free of any contradictions and consistent with usual 4D physics. We show, using warp metrics, that this new definition is also more consistent with our physical intuition. The effects of this force could be detected observing objects moving with high speed, near black holes and/or in cosmological situations.
This is an introductory review of gravity on branes with an emphasis on codimension 1 models. However, for a new result it is also pointed out that the cosmological evolution of the 3-brane in the model of Dvali, Gabadadze and Porrati may follow the standard Friedmann equation. Contents: 1. Introduction 2. Conventions 3. The Lanczos-Israel matching conditions 4. The action principle with codimension 1 hypersurfaces: Need for the Gibbons-Hawking term 5. The Newtonian limit on thin branes 6. A remark on black holes in the model of Dvali, Gabadadze and Porrati 7. The cosmology of codimension 1 brane worlds Comment: 25 pages
Paths in an appropriate geometry are usually used as trajectories of test particles in geometric theories of gravity. It is shown that non-symmetric geometries possess some interesting quantum features. Without carrying out any quantization schemes, paths in such geometries are naturally quantized. Two different non-symmetric geometries are examined for these features. It is proved that, whatever the non-symmetric geometry is, we always get the same quantum features. It is shown that these features appear only in the pure torsion term (the anti-symmetric part of the affine connection) of the path equations. The vanishing of the torsion leads to the disappearance of these features, regardless of the symmetric part of the connection. It is suggested that, in order to be consistent with the results of experiments and observations, torsion term in path equations should be parametrized using an appropriate parameter.
We study the problem of test-particle motion in the Nonsymmetric
Gravitational Theory (NGT) assuming the four-velocity of the particle is
parallel-transported along the trajectory. The predicted motion is studied on a
static, spherically symmetric background field, with particular attention paid
to radial and circular motions. Interestingly, it is found that the proper time
taken to travel between any two non-zero radial positions is finite. It is also
found that circular orbits can be supported at lower radii than in General
Relativity for certain forms of motion.
We present three interactions which could be used as alternate methods for
coupling a test-particle to the antisymmetric components of the NGT field. One
of these takes the form of a Yukawa force in the weak-field limit of a static,
spherically symmetric field, which could lead to interesting phenomenology.
The Bazanski approach, for deriving the geodesic equations in Riemannian geometry, is generalized in the absolute parallelism geometry. As a consequence of this generalization three path equations are obtained. A striking feature in the derived equations is the appearance of a torsion term with a numerical coefficients that jumps by a step of one half from equation to another. This is tempting to speculate that the paths in absolute parallelism geometry might admit a quantum feature. Comment: 4 pages Latex file Journal Reference: Astrophysics and space science 228, 273, (1995)
We study the dynamics of test particles and pointlike gyroscopes in 5D manifolds like those used in the Randall-Sundrum brane world and non-compact Kaluza-Klein models. Our analysis is based on a covariant foliation of the manifold using 3+1 dimensional spacetime slices orthogonal to the extra dimension, and is hence similar to the ADM 3+1 split in ordinary general relativity. We derive gauge invariant equations of motion for freely-falling test particles in the 5D and 4D affine parameterizations and contrast these results with previous work concerning the so-called ``fifth force''. Motivated by the conjectured localization of matter fields on a 3-brane, we derive the form of the classical non-gravitational force required to confine particles to a 4D hypersurface and show that the resulting trajectories are geometrically identical to the spacetime geodesics of Einstein's theory. We then discuss the issue of determining the 5D dynamics of a torque-free spinning body in the point-dipole approximation, and then perform a covariant (3+1)+1 decomposition of the relevant formulae (i.e. the 5D Fermi-Walker transport equation) for the cases of freely-falling and hypersurface-confined point gyroscopes. In both cases, the 4D spin tensor is seen to be subject to an anomalous torque. We solve the spin equations for a gyroscope confined to a single spacetime section in a simple 5D cosmological model and observe a cosmological variation of the magnitude and orientation of the 4D spin. Comment: 13 pages, 2 figures, in press in Phys. Rev. D
We study in detail the equations of the geodesic deviation in multidimensional theories of Kaluza-Klein type. We show that their 4-dimensional space-time projections are identical with the equations obtained by direct variation of the usual geodesic equation in the presence of the Lorentz force, provided that the fifth component of the deviation vector satisfies an extra constraint derived here. Comment: 5 pages, Revtex, 1 figure. To appear in Phys. Rev. D (Brief Report)
A formalism of integrating the equations of geodesics and of geodesic deviation is examined based upon the Hamilton–Jacobi equation for geodesics. The latter equation has been extended to the case of geodesic deviation and theorems analogous to Jacobi’s theorem on the complete integral has been proved. As a result, a straightforward algorithm of integrating the geodesic deviation equations on Riemannian (or pseudo‐Riemannian) manifolds is obtained.
Definitions are proposed for the total momentum vector p^alpha and spin tensor Salphabeta of an extended body in arbitrary gravitational and electromagnetic fields. These are based on the requirement that a symmetry of the external fields should imply conservation of a corresponding component of momentum and spin. The particular case of a test body in a de Sitter universe is considered in detail, and used to support the definition p_betaSalphabeta = 0 for the centre of mass. The total rest energy M is defined as the length of the momentum vector. Using equations of motion to be derived in subsequent papers on the basis of these definitions, the time dependence of M is studied, and shown to be expressible as the sum of two contributions, the change in a potential energy function Phi and a term representing energy inductively absorbed, as in Bondi's illustration of Tweedledum and Tweedledee. For a body satisfying certain conditions described as 'dynamical rigidity', there exists, for motion in arbitrary external fields, a mass constant m such that M = m + 1/2S^kappaOmega_kappa + Phi, where Omega_kappa is the angular velocity of the body and S^kappa its spin vector.
A method for the derivation of the equations of motion of test particles in a given gravitational field is developed. The equations of motion of spinning test particles are derived. The transformation properties are discussed and the equations of motion are written in a covariant form.
A classical Lagrangian is proposed for a relativistic particle with spin. It is supersymmetric under transformations between position and spin variables. The theory can be quantized and becomes identical with conventional Dirac theory. This correspondence continues to be valid when the particle interacts with external electromagnetic or gravitational fields as long as its coupling to these fields conserves the supersymmetry.
This book is an introduction to particle physics and cosmology. It discusses the interaction of these two fundamental branches of physics, and considers recent advances beyond the standard models. Eight chapters comprise an introduction to the gauge theories of the strong and the electroweak interactions, the so-called grand unified theories, and general relativity. The other chapters address recent concepts such as composite fermions and bosons, supersymmetry, quantum gravity, supergravity, and string theory, and relate them to modern cosmology and experimental astronomy.
This paper achieves a unification of two theories of gravitation, Einstein-Cartan theory and the Moffat theory. Our unification, here called ''Einstein-Cartan-Moffat theory,'' has all the advantages of both previous theories. The possible connection to supergravity is discussed.
A new theory of gravity is proposed in which the geometry of space-time is determined by a nonsymmetric field structure. The theory satisfies the following requirements: (1) general covariance, (2) (weak) principle of equivalence, (3) the field equations are derivable from a Lagrangian action principle, (4) the theory agrees with all the classical (weak gravitational field) tests of Einstein's general relativity. The field equations for the nonsymmetric Hermitian gμν lead to a rigorous static spherically symmetric solution for the gravitational field in empty space that excludes the essential singularity at r=0 from physical space-time. It is expected that the predictions of the theory will differ significantly from Einstein's theory of gravitation for compact sources or supermassive stars. Matter undergoing gravitational collapse is prevented from forming a black hole (in physical space-time) of the kind predicted in Einstein's theory, when a new gravitational parameter l that appears as a constant of integration in the solution satisfies l>2m.
Brane worlds and large extra dimensions have attracted a lot of
attention as possible new paradigms for spacetime. I review the theory
of gravity on 3-branes with a focus on the cododimension-one models.
However, for a new result it is also pointed out that the cosmological
evolution of the 3-brane in the model of Dvali, Gabadadze and Porrati
may follow the standard Friedmann equation.
We solve exactly the equations of 5D general relativity for the case of a spinning test object moving in a circular orbit in a Schwarzschild-like spacetime augmented by one extra dimension. All of the classical tests give their standard results, but the spin axis precesses at a different rate in 4D and 5D. This suggests that the dimensionality of the world could be tested using a gyroscope in Earth orbit.
Heisenberg's uncertainty relation is commonly regarded as defining a level of unpredictability that is fundamentally incompatible with the deterministic laws embodied in classical field theories such as Einstein's general relativity. We here show that this is not necessarily the case. Using 5D as an example of dimensionally-extended relativity, we employ a novel metric to derive the standard quantum rule for the action and a form of Heisenberg's relation that applies to real and virtual particles. The philosophical implications of these technical results are somewhat profound.
We review the implications of modern higher-dimensional theories of gravity for astrophysics and cosmology. In particular,
we discuss the latest developments of STM theory in connection with dark matter, particle dynamics and the cosmological constant,
as well as related aspects of quantum theory. There are also more immediate tests of extra dimensions, notably involving perturbations
of the cosmic 3K microwave background and the precession of a supercooled gyroscope in Earth orbit. We also outline some general
features of embeddings, and include pictures of the big bang as viewed from a higher dimension.
By using the relativistic top theory, we derive a relativistic top deviation equation. This equation turns out to be a generalization of the geodesic deviation equation for a pair of nearby point particles. In fact, we show that when the spin angular momentum tensor associated to the top vanishes, such a relativistic top deviation equation reduces to the geodesic deviation equation for spinless point particles. Just as the geodesic deviation equation for spinless particles can be used to investigate the detection of gravitational waves, our generalized formula for a relativistic top can be used to study the gravitational wave background. Our formulation may be of special interest to detect the inflationary gravitational waves via the polarization of the cosmic background radiation.
We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identified and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time.
We extend the classical general relativistic theory of measurement to include the possibility of existence of higher dimensions. The intrusion of these dimensions in the spacetime interval implies that the inertial mass of a particle in general varies along its worldline if the observations are analyzed assuming the existence of only the four spacetime dimensions. The variations of mass and spin are explored in a simple 5D Kaluza-Klein model. Comment: LaTeX, 12 pages, no figures, 2 references added, to appear in Phys. Lett. A
Summer School on Astrophysics and Cosmology
- G Gabadaze