Standard model of fermions on the domain wall in five-dimensional space-time

The mechanism of generation of the Standard Model for fermions on the domain wall in five-dimensional space-time is presented. As a result of self-interaction of five-dimensional fermions and gravity induced by matter fields, in the strong coupling regime, in the model there arises a spontaneous translational symmetry breaking, which leads to localization of light particles on a 3 + 1-dimensional domain wall (“3-brane”) that is embedded into a five-dimensional anti-de Sitter space-time (AdS₅). Appropriate low-energy, effective action, classical kink-like vacuum configurations for the gravity and scalar fields are investigated. Mass spectra for light composite particles and their coupling constants interaction in ultra-low-energy, which localize on the brane, are explored. We establish estimates of characteristic scales and constants interactions of the model and also a relation between the bulk five-dimensional gravitational constant, curvature of AdS₅ space-time, and brane Newton’s constants. The induced cosmological constant on the brane exactly vanishes in all orders of the theory perturbation. We find out that scalar interaction is strongly suppressed at ultra-low-energy, and the brane fluctuations (branons) are suitable “sterile” canditates for explanation of the phenomenon of Dark Matter. Bibliography: 21 titles. Dedicated to the 100th birthday of M. P. Bronstein __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 5–29.     

Tensor-tensor model of gravity

The main problem with the standard gauge theory of the Poincaré group realized as a subgroup of GL(5, R) is that fields, whose physical sense is unclear, appear in connection with the non-Lorentz symmetries. Here, the Poincaré fields are treated as new Yang-Mills-type tensor fields and gravity is treated as a Higgs-Goldstone field. In this case, the effective metric tensor for matter is a hybrid of two tensor fields. In the linear approximation, the massive translation gauge field gives the Yukawa-type correction to the Newtonian potential. Also, corrections to the standard Einstein post-Newtonian formulas for light deflection and radar echo delay are obtained. A spherically symmetric solution to the equations of translation gauge fields is also found. The translation gauge field leads to the existence of a singular surface, which is impenetrable to matter and can prevent gravitational collapse of a large body, inside the Schwarzschild sphere. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 3, pp. 448–460, December, 1997. This work was supported in part by the Georgian Government and the International Science Foundation (ISF Grant No. MXL 200).     

Linearized gravity in the Randall-Sundrum model with stabilized distance between branes

We consider linearized gravity in the Randall-Sundrum model in which the distance between branes is stabilized by introducing the scalar Goldberger-Wise field. We construct the second variation Lagrangian for fluctuations of gravitational and scalar fields over the background solution and investigate its gauge invariance. We obtain, separate, and solve the corresponding equations of motion. For physical degrees of freedom, we obtain the effective four-dimensional Lagrangian describing the massless graviton, massive gravitons, and the set of massive scalar fields. We also find masses and coupling constants of these fields to the matter on the negative-tension brane. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 339–353, December, 2006.     

Induced gravity and universe creation on the domain wall in five-dimensional space-time

We present a model with a strong fermion self-interaction and an induced gravitational interaction in five-dimensional space-time. In the strong-coupling regime, this model develops a spontaneous breaking of translational invariance, which results in localizing light particles on the (3+1)-dimensional domain wall embedded in the anti-de Sitter five-dimensional space-time. We obtain the corresponding low-energy effective action and investigate kink-type vacuum solutions in a quasiflat Riemannian metric. We discuss the physics of light particles in 3+1 dimensions and establish the fundamental relations between the induced gravitational constant and both the curvature in the five-dimensional anti-de Sitter space-time and the Newton gravitational constant of our universe. Dedicated to Yu. V. Novozhilov on his 80th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 4–22, July, 2006.     

Cauchy constraints and particle content of fourth-order gravity in n dimensions

The full class of purely metrical gravitational theories in n⩾3 dimensions which follows from a Lagrangian composed of linear and quadratic curvature terms is analyzed. The type of the field equations in a suitable gauge is discussed. The principal symbol and the particle content of the linearized field equations are investigated. The space+time decomposition and the ADM formalism are used to derive the constraints and evolution equations for the variational derivative tensor.     

The n-Wave Procedure and Dimensional Regularization for the Scalar Field in a Homogeneous Isotropic Space

We obtain expressions for the vacuum expectations of the energy–momentum tensor of the scalar field with an arbitrary coupling to the curvature in an N-dimensional homogeneous isotropic space for the vacuum determined by diagonalization of the Hamiltonian. We generalize the n-wave procedure to N-dimensional homogeneous isotropic space–time. Using the dimensional regularization, we investigate the geometric structure of the terms subtracted from the vacuum energy–momentum tensor in accordance with the n-wave procedure. We show that the geometric structures of the first three subtractions in the n-wave procedure and in the effective action method coincide. We show that all the subtractions in the n-wave procedure in a four- and five-dimensional homogeneous isotropic space correspond to a renormalization of the coupling constants of the bare gravitational Lagrangian.     

Clifford tetrads,null zig zags,and quantum gravity

Quantum gravity has been so elusive because we have tried to approach it by two paths which can never meet: standard quantum field theory and general relativity. These contradict each other, not only in superdense regimes, but also in the vacuum, where the divergent zero-point energy would roll up space to a point. The solution is to build in a regular, but topologically nontrivial distribution of vacuum spinor fields right from the start. This opens up a straight road to quantum gravity, which we map out here. The gateway is covariance under the complexified Clifford algebra of our space-time manifold and its spinor representations, which Sachs dubbed the Einstein group, E. The 16 generators of E transformations obey both the Lie algebra of Spin^c -4, and the Clifford (SUSY) algebra of We derive Einstein’s field equations from the simplest E-invariant Lagrangian density, contains effective electroweak and gravitostrong field actions, as well as Dirac actions for the matter spinors. On microscales the massive Dirac propagator resolves into a sum over null zig-zags. On macroscales, we see the energy-momentum current, *T, and the resulting Einstein curvature, G. For massive particles, *T flows in the “cosmic time” direction—centri-fugally in an expanding universe. Neighboring centrifugal currents of *T present opposite radiotemporal vorticities G_or to the boundaries of each others’ worldtubes, so they advect, i.e. attract, as we show here by integrating by parts in the spinfluid regime. This boundary integral not only explains why stress-energy is the source for gravitational curvature, but also gives a value for the gravitational constant, κ (T), that couples them. κ turns out to depend on the dilation factor T = y⁰, which enters kinematically as “imaginary time”: the logradius of our expanding Friedmann 3-brane. On the microscopic scale, quantum gravity appears as the statistical mechanics of the null zig-zag rays of spinor fields in imaginary time T. Our unified field/particle action also contains new couplings of gravitomagnetic fields to strong fields and weak potentials. These predict new physical phenomena: Axial jets of nuclear decay products emitted with left helicity along the axis of a massive, spinning body. This paper gives the derivations of the results I reported at the PIMS conference entitled “Brane World and Supersymmetry” in July, 2002 at Vancouver, B.C. It also contains new results on spin-gravity coupling, on how a topologically-nontrivial distribution of vacuum spinors removes singularities and divergences, and how the amplitude of the vacuum spinors determines the gravitational constant and the rate of cosmic expansion     

Dynamical symmetry breaking in hyperbolic 4D spacetime and in extra dimensions

We study the dynamical symmetry breaking in quark matter within two different models. First, we consider the effect of gravitational catalysis of chiral and color symmetries breaking in strong gravitational field of ultrastatic hyperbolic spacetime ℝ ⊗ H ³ in the framework of an extended Nambu-Jona-Lasinio model. Second, we discuss the dynamical fermion mass generation in the flat 4-dimensional brane situated in the 5D spacetime with one extra dimension compactified on a circle. In the model, bulk fermions interact with fermions on the brane in the presence of a constant abelian gauge field A ₅ in the bulk. The influence of the A ₅-gauge field on the symmetry breaking is considered both when this field is a background parameter and a dynamical variable.     

Special relativity and theory of gravity via maximum symmetry and localization

Like Euclid,Riemann and Lobachevski geometries are on an almost equal footing,based on the principle of relativity of maximum symmetry proposed by Professor Lu Qikeng and the postulate on invariant universal constants c and R,the de Sitter/anti-de Sitter(dS/AdS)special relativity on dS/AdS-space with radius R can be set up on an almost equal footing with Einstein's special relativity on the Minkowski-space in the case of R→∞. Thus the dS-space is coin-like:a law of inertia in Beltrami atlas with Beltrami time simultaneity for the principle of relativity on one side,and the proper-time simultaneity and a Robertson-Walker-like dS-space with entropy and an accelerated expanding S~3 fitting the cosmological principle on another side. If our universe is asymptotic to the Robertson-Walker-like dS-space of R(?)(3/Λ)~(1/2),it should be slightly closed in O(A)with entropy bound S(?)3πc~3k_B/ΛGh.Contrarily,via its asymptotic behavior, it can fix on Beltrami inertial frames without‘an argument in a circle’and acts as the origin of inertia. There is a triality of conformal extensions of three kinds of special relativity and their null physics on the projective boundary of a 5-d AdS-space,a null cone modulo projective equivalence[N](?)_p(AdS~5). Thus there should be a dS-space on the boundary of S~5×AdS~5 as a vacuum of supergravity. In the light of Einstein's‘Galilean regions’,gravity should be based on the localized principle of relativity of full maximum symmetry with a gauge-like dynamics.Thus,this may lead to the theory of gravity of corresponding local symmetry.A simple model of dS-gravity characterized by a dimensionless constant g(?)(AGh/3c~3)~(1/2)～10~(-61)shows the features on umbilical manifolds of local dS-invariance. Some gravitational effects out of general relativity may play a role as dark matter. The dark universe and its asymptotic behavior may already indicate that the dS special relativity and dS-gravity be the foundation of large scale physics.     

Relativistic theory of gravity and a torsion field

The fundamentals of gravity theory are stated in a Minkowski space with an effective nonzero-torsion Riemann-Cartan space-time, which is more general than the Riemannian space. The theory presented thus includes a torsion field of the Einstein-Cartan type in the general concept of the relativistic theory of gravity. Expressions for the metric and canonical energy-momentum tensors of the gravitational field and nongravitational matter in the Minkowski space are found. Noncoordinate gauge transformations are introduced under which the variation of the density of the gravitational Lagrangian is a divergence expression. Translated from Teoreticheskaya i Matematischeskaya Fizika, Vol. 118, No. 1, pp. 126–132, January, 1999.     

Theory of Dark Energy and Dark Matter

In (T. Ma and S. Wang. Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34(2): 335-366, 2014; arXiv:1206.5078v2), a new set of gravitational field equations are derived based only on 1) the Einstein principle of general relativity, and 2) the principle of interaction dynamics, due to the the presence of dark energy and dark matter. With the field equations, we show that gravity can display both attractive and repulsive behavior, and the dark matter and dark energy are just a property of gravity caused by the nonlinear interactions of the gravitational potential $g_{μv}$and its dual field. The main objectives of this paper are two-fold. The first is to study the PID-induced cosmological model, and to show explicitly, as addressed in (T. Ma and S. Wang, Astrophysical dynamics and cosmology, Journal of Mathematical Study, 47(4): 305-378, 2014), that 1) dark matter is due to the curvature of space, and 2) dark energy corresponds to the negative pressure generated by the dual gravitational potential in the field equations, and maintains the stability of geometry and large scale structure of the Universe. Second, for the gravitational field outside of a ball of centrally symmetric matter field, there exist precisely two physical parameters dictating the two-dimensional stable manifold of asymptotically flat space-time geometry, such that, as the distance to the center of the ball of the matter field increases, gravity behaves as Newtonian gravity, then additional attraction due to the curvature of space (dark matter effect), and repulsive (dark energy effect). This also clearly demonstrates that both dark matter and dark energy are just a property of gravity.     

Universally coupled massive gravity

We derive Einstein’s equations from a linear theory in flat space-time using free-field gauge invariance and universal coupling. The gravitational potential can be either covariant or contravariant and of almost any density weight. We adapt these results to yield universally coupled massive variants of Einstein’s equations, yielding two one-parameter families of distinct theories with spin 2 and spin 0. The Freund-Maheshwari-Schonberg theory is therefore not the unique universally coupled massive generalization of Einstein’s theory, although it is privileged in some respects. The theories we derive are a subset of those found by Ogievetsky and Polubarinov by other means. The question of positive energy, which continues to be discussed, might be addressed numerically in spherical symmetry. We briefly comment on the issue of causality with two observable metrics and the need for gauge freedom and address some criticisms by Padmanabhan of field derivations of Einstein-like equations along the way. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 311–336, May, 2007.     

New mechanism of energy release in astrophysical objects

Conclusions The above qualitative analysis of the evolution of astrophysical objects shows that in the field theory of gravitation with minimal coupling objects in the regionM/a<1/3 of values of the mean gravitational potential are stable to small perturbations of their radius with unchanged rest mass.However, the mean gravitational potential of these objects increases when they capture matter surrounding them. When the mean potential reaches the valueM/a=1/3, the object passes abruptly from an infinitely stable state to an infinitely unstable state (with respect to small perturbations of its radius). Therefore, even small perturbations in the radius of the object once the critical value of the mean gravitational potential has been reached necessarily lead to expansion of the matter, which may be accompanied by the ejection of mass of this object and the release of energy.Therefore, instead of gravitational collapse, the outcome of the instability of astrophysical objects in general relativity, in the present theory there is a new mechanism of energy release.Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 48, No. 3, pp. 275–283, September, 1981.     

Weyl-Eddington-Einstein affine gravity in the context of modern cosmology

We propose new models of the “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposed method for obtaining the geometry using the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein theory with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) meson, and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the Lagrangian determines further details of the theory, for example, the nature of the fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, dark energy must also arise. The basic parameters of the theory (cosmological constant, mass, possible dimensionless constants) are theoretically indeterminate, but in the framework of modern “multiverse” ideas, this is more a virtue than a defect. We consider further extensions of the affine models and in more detail discuss approximate effective (“physical”) Lagrangians that can be applied to the cosmology of the early Universe.     

Progress in Clifford Space Gravity

Clifford-space Gravity is revisited and new results are found. The Clifford space (C-space) generalized gravitational field equations are obtained from a variational principle and which is based on an extension of the Einstein-Hilbert-Cartan action. One of the main results of this work is that the C-space connection requires torsion in order to have consistency between the Clifford algebraic structure and the zero nonmetricity condition ${\nabla{\kappa g}^{MN} = 0}$ . A discussion on the field equations, the cosmological constant, dark energy and multi-metric theories of gravity follows. We continue by pointing out the relations of Clifford space gravity to Lanczos-Lovelock-Cartan (LLC) higher curvature gravity with torsion. We finalize by describing the Clifford bundle approach to C-space gravity and by pointing out that C-space gravity involves higher-spins beyond spin 2 and argue why one could view the LLC higher curvature actions, and other extended gravitational theories based on f(R), f(R _μν ), . . . actions, for polynomial-valued functions, as effective actions after integrating the C-space gravitational action with respect to all the poly-coordinates, except the vectorial ones x ^μ .     

The Two-Parameter Higher-Order Differential Calculus and Curvature on a Quantum Plane

We construct an associative differential algebra on a two-parameter quantum plane associated with a nilpotent endomorphism d in the two cases d ² = 0 and d ³ = 0 (d ² ≠ 0). The correspondent curvature is derived and the related non-commutative gauge field theory is introduced.     

Multiplicity results for the two-vortex Chern-Simons Higgs model on the two-sphere

We consider a Ginzburg-Landau type functional on S ² with a 6 ^th order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly detect a rich and varied asymptotic behavior depending on the position of the vortices. We exploit analogies with the Nirenberg problem for the prescribed Gauss curvature equation on S ². Received: December 3, 1997     

Self–gravitating fluid flows with Gowdy symmetry near cosmological singularities

We consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus T³ and, in this class, we solve the singular initial value problem for the Einstein–Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the notion of sub-critical, critical, and super-critical regimes. Solutions to the Einstein–Euler systems when the fluid is governed by a linear equation of state are constructed in the first two regimes, while additional difficulties arise in the latter one. All previous studies on inhomogeneous spacetimes concerned vacuum cosmological spacetimes only.     

Optimal pinching constants of odd dimensional homogeneous spaces

In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B ¹³=SU(5)/(Sp(2)×_ℤ2S¹) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W ⁷ _1,1=SU(3)/S¹ _1,1. We prove that the optimal pinching constants are precisely in both cases. So far B ¹³ and W ⁷ _1,1 were only known to admit Riemannian metrics with pinching constants .?We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W ⁷ _k,l =SU(3)/S¹ _k,l . Our computations cover the cone of invariant T²-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.?In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W ⁷ _k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B ¹³ is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ³/ℤ₂ and inherits an Sp(2)-invariant Riemannian metric, and the other is W ⁷ _1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W ⁷ _1,1 and the Berger space B ¹³. Oblatum 9-XI-1998 & 3-VI-1999 / Published online: 20 August 1999     

Gravitation and Electromagnetism as Geometrical Objects of a Riemann-Cartan Spacetime Structure

In this paper we first show that any coupled system consisting of a gravitational plus a free electromagnetic field can be described geometrically in the sense that both Maxwell equations and Einstein equation having as source term the energy-momentum of the electromagnetic field can be derived from a geometrical Lagrangian proportional to the scalar curvature R of a particular kind of Riemann-Cartan spacetime structure. In our model the gravitational and electromagnetic fields are identified as geometrical objects of the structure.We show moreover that the contorsion tensor of the particular Riemann-Cartan spacetime structure of our theory encodes the same information as the one contained in Chern-Simons term ${{\bf A} \wedge {\it d}{\bf A}}$ that is proportional to the spin density of the electromagnetic field. Next we show that by adding to the geometrical Lagrangian a term describing the interaction of a electromagnetic current with a general electromagnetic field plus the gravitational field, together with a term describing the matter carrier of the current we get Maxwell equations with source term and Einstein equation having as source term the sum of the energy-momentum tensors of the electromagnetic and matter terms. Finally modeling by dust charged matter the carrier of the electromagnetic current we get the Lorentz force equation. Moreover, we prove that our theory is gauge invariant. We also briefly discuss our reasons for the present enterprise.