Canonical formulation of the embedded theory of gravity equivalent to Einstein’s general relativity

We study the approach in which independent variables describing gravity are functions of the space-time embedding into a flat space of higher dimension. We formulate a canonical formalism for such a theory in a form that requires imposing additional constraints, which are a part of Einstein’s equations. As a result, we obtain a theory with an eight-parameter gauge symmetry. This theory becomes equivalent to Einstein’s general relativity either after partial gauge fixing or after rewriting the metric in the form that is invariant under the additional gauge transformations. We write the action for such a theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 271–288, November, 2007.     

Perturbation Method for Particle-like Solutions of the Einstein–Dirac Equations

The aim of this work is to prove by a perturbation method the existence of solutions of the coupled Einstein–Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. We relate the solutions of our equations to those of the nonlinear Choquard equation and we show that the nondegenerate solution of Choquard’s equation generates solutions of the Einstein–Dirac equations.     

Two-time Green’s functions in superconductivity theory

Based on the method of the equations of motion for two-time Green’s functions, we derive superconductivity equations for different types of interactions related to the scattering of electrons on phonons and spin fluctuations or caused by strong Coulomb correlations in the Hubbard model. We derive an exact Dyson equation for the matrix Green’s function with the self-energy operator in the form of the multiparticle Green’s function. Calculating the self-energy operator in the approximation of noncrossing diagrams leads to a closed system of equations corresponding to the Migdal-Eliashberg strong-coupling theory. We propose a theory of high-temperature superconductivity due to kinematic interaction in the Hubbard model. We show that two pairing channels occur in systems with a strong Coulomb correlation: one due to the antiferromagnetic exchange in interband hopping and the other due to the coupling to spin and charge fluctuations in hopping within one Hubbard band. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 129–146, January, 2008.     

Realizability of greedy algorithms

We discuss new models of an “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 proposal to specify the space-time geometry by the use of 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, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein gravity with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) vector field (vecton), 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 geometric Lagrangian determines further details of the theory, for example, the nature of the vector and scalar fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, which are discussed here, dark energy must also arise. We mainly focus on intricate relations between geometry and dynamics while only very briefly considering approximate cosmological models inspired by the geometric approach.     

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.     

Einstein’s equations and Clifford algebra

Einstein’s equations of the general theory of relativity are rewritten within a Clifford algebra. This algebra is otherwise isomorphic to a direct product of two quaternion algebras. A multivector calculus is developed within this Clifford algebra which differs from the corresponding complexified algebra used in the standard spacetime algebra approach.     

Future Stability of the Einstein-Maxwell-Scalar Field System

Ringstr?m managed (in Invent Math 173(1):123–208, 2008) to prove future stability of solutions to Einstein’s field equations when matter consists of a scalar field with a potential creating an accelerated expansion. This was done for a quite wide class of spatially homogeneous space–times. The methods he used should be applicable also when other kinds of matter fields are added to the stress-energy tensor. This article addresses the question whether we can obtain stability results similar to those Ringstr?m obtained if we add an electromagnetic field to the matter content. Before this question can be addressed, more general properties concerning Einstein’s field equation coupled to a scalar field and an electromagnetic field have to be settled. The most important of these questions are the existence of a maximal globally hyperbolic development and the Cauchy stability of solutions to the initial value problem.     

Equations for two-temperature Green’s functions for a Heisenberg ferromagnet in the mode-coupling approximation

We use the mode-coupling approximation to derive a closed system of equations for the Green’s functions for the transverse and longitudinal spin components taking the sum rules related to the finiteness of the spin into account exactly. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 197–207, January, 2008.     

Convergent Null Data Expansions at Space-Like Infinity of Stationary Vacuum Solutions

We present a characterization of the asymptotics of all asymptotically flat, stationary solutions with non-vanishing ADM mass to Einstein’s vacuum field equations. This characterization is given in terms of two sequences of symmetric trace free tensors (we call them the ‘null data’), which determine a formal expansion of the solution, and which are in a one to one correspondence to Hansen’s multipoles. We obtain necessary and sufficient growth estimates on the null data to define an absolutely convergent series in a neighborhood of spatial infinity. This provides a complete characterization of all asymptotically flat, stationary vacuum solutions to the field equations with non-vanishing ADM mass. Submitted: December 9, 2008.; Accepted: January 21, 2009.     

The two-time Green’s function and the diagram technique

Based on constructing the equations of motion for the two-time Green’s functions, we discuss calculating the dynamical spin susceptibility and correlation functions in the Heisenberg model. Using a Mori-type projection, we derive an exact Dyson equation with the self-energy operator in the form of a multiparticle Green’s function. Calculating the self-energy operator in the mode-coupling approximation in the ferromagnetic phase, we reproduce the results of the temperature diagram technique, including the correct formula for low-temperature magnetization. We also consider calculating the spin fluctuation spectrum in the paramagnetic phase in the framework of the method of equations of motion for the relaxation function.     

New dynamical variables in Einstein’s theory of gravity

We describe an alternative formalism for Einstein’s theory of gravity. The role of dynamical variables is played by a collection of ten vector fields f _μ ^A , A = 1,..., 10. The metric is a composite variable, g _μν = f _μ ^A f _ν ^A . The proposed scheme may lead to further progress in a theory of gravity where Einstein’s theory is to play the role of an effective theory, with Newton’s constant appearing by introducing an anomalous Green’s function.     

Future stability of the Einstein-non-linear scalar field system

We consider the question of future global non-linear stability in the case of Einstein’s equations coupled to a non-linear scalar field. The class of potentials V to which our results apply is defined by the conditions V(0)>0, V’(0)=0 and V”(0)>0. Thus Einstein’s equations with a positive cosmological constant represents a special case, obtained by demanding that the scalar field be zero. In that context, there are stability results due to Helmut Friedrich, the methods of which are, however, not so easy to adapt to the presence of matter. The goal of the present paper is to develop methods that are more easily adaptable. Due to the extreme nature of the causal structure in models of this type, it is possible to prove a stability result which only makes local assumptions concerning the initial data and yields global conclusions in time. To be more specific, we make assumptions in a set of the form for some r ₀>0 on the initial hypersurface, and obtain the conclusion that all causal geodesics in the maximal globally hyperbolic development that start in are future complete. Furthermore, we derive expansions for the unknowns in a set that contains the future of . The advantage of such a result is that it can be applied regardless of the global topology of the initial hypersurface. As an application, we prove future global non-linear stability of a large class of spatially locally homogeneous spacetimes with compact spatial topology.     

On the well-posedness of the vacuum Einstein��s equations

The Cauchy problem of the vacuum Einstein’s equations aims to find a semi-metric g _αβ of a spacetime with vanishing Ricci curvature R _α,β and prescribed initial data. Under the harmonic gauge condition, the equations R _α,β  = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric h _ab and a second fundamental form K _ab . A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h _ab , K _ab ) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of the present article is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.     

Spacetime and the Geometry behind it

This is the Leonardo da Vinci Lecture given in Milan in March 2006. It is a survey on the concept of space-time over the last 3000 years: it starts with Euclidean geometry, discusses the contributions of Gauss and Riemannian geometry, presents the dynamic concept of space-time in Einstein’s general relativity, describes the importance of symmetries, and ends with Calabi-Yau manifolds and their importance in today’s string theories in the attempt for a unified theory of physics. Leonardo da Vinci Lecture held on March 28, 2006     

Freud’s Identity of Differential Geometry, the Einstein-Hilbert Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR

We reveal in a rigorous mathematical way using the theory of differential forms, here viewed as sections of a Clifford bundle over a Lorentzian manifold, the true meaning of Freud’s identity of differential geometry discovered in 1939 (as a generalization of results already obtained by Einstein in 1916) and rediscovered in disguised forms by several people. We show moreover that contrary to some claims in the literature there is not a single (mathematical) inconsistency between Freud’s identity (which is a decomposition of the Einstein indexed 3-forms in two gauge dependent objects) and the field equations of General Relativity. However, as we show there is an obvious inconsistency in the way that Freud’s identity is usually applied in the formulation of energy-momentum “conservation laws” in GR. In order for this paper to be useful for a large class of readers (even those ones making a first contact with the theory of differential forms) all calculations are done with all details (disclosing some of the “tricks of the trade” of the subject).      

On the restricted evolution problem of the Einstein relativistic mechanics

This work is focused on the interior Cauchy problem for the Einstein’s field equations. Precisely, in the relativistic study of the evolution of a continuum reversible system, the Cauchy problem is broken into two separate problems: the initial data problem and the restricted problem of evolution.     

On a class of wave solutions to einstein’s gravity equations

A class of wave solutions to Einstein’s gravity equations is found and a covariant conservation law is derived for it. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 484–492, March, 1997.     

A Geometric Theory of Growth Mechanics

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. The time dependence of the metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and “shape”. We show that the time dependency of the material metric will affect the energy balance and the entropy production inequality; both the energy balance and the entropy production inequality have to be modified. We then obtain the governing equations covariantly by postulating invariance of energy balance under time-dependent spatial diffeomorphisms. We use the principle of maximum entropy production in deriving an evolution equation for the material metric. In the case of isotropic growth, we find those growth distributions that do not result in residual stresses. We then look at Lagrangian field theory of growing elastic solids. We will use the Lagrange–d’Alembert principle with Rayleigh’s dissipation functions to derive the governing equations. We make an explicit connection between our geometric theory and the conventional multiplicative decomposition of the deformation gradient, F=F _e F _g, into growth and elastic parts. We linearize the nonlinear theory and derive a linearized theory of growth mechanics. Finally, we obtain the stress-free growth distributions in the linearized theory.     

Guarding galleries where no point sees a small area

In this paper, we explain the existence of certain modular equations discovered by S. Ramanujan via function field theory. We will prove some of these modular equations and indicate how new equations analogous to those found in Ramanujan’s notebooks can be constructed.     

Maxwell–Dirac Equations in Four-Dimensional Minkowski Space

Abstract

I derive the global existence and asymptotic behavior of small amplitude solutions to the system of massive coupled classical Maxwell–Dirac equations in the four-dimensional Minkowski space. Because the physically defined energy of the system is not positive definite, I transform it into an equivalent system of Maxwell–Klein–Gordon equations, which I study with a method based on gauge invariant energy estimates and geometric properties of the equations.