Yang-Mills gauge theory and gravitation

We point out that Yang's and Einstein's gravitational equations can be obtained from a geometric approach of Yang-Mills gauge theory in a sourceless case, under a decomposition of the Poincaré algebra. Otherwise, Einstein's equations cannot be derived from a Yang-Mills gauge equation when sources are inserted in the rotational sector of that algebra. A gauge Lagrangian structure is also discussed.     

Geometry of the Einstein and Yang-Mills equations

It is shown that the Einstein and Yang-Mills equations arise from the conditions for the space-time to be a submanifold of a pseudo-Euclidean space with dimension greater than 5. Some possible applications to cosmology, spin-2 fields, and geometrodynamics are discussed.     

Mechanism for induction of Einstein gravitation

A quantum theory of Weyl gravitation, conformally bound to material, is considered. Direct solution of the renormalization group equation is used to obtain the effective action to the accuracy of terms linear in curvature. It is shown that the theory admits a phase transition of the first sort, caused by curvature, as a result of which Einstein gravitation is induced.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 77–81, March, 1986.The author is indebted to G. A. Vilkovyskii, I. V. Tyutin, and E. S. Fradkin for their numerous discussions.     

Finite Group Discretization of Yang-Mills and Einstein Actions

Discrete versions of the Yang-Mills and Einstein actions are proposed for any finite group. These actions are invariant respectively under local gauge transformations and under the analogues of Lorentz and general coordinate transformations. The case Z_n×Z_n×···×Z_n is treated in some detail, recovering the Wilson action for Yang-Mills theories and a new discretized action for gravity.     

A connection between the Einstein and Yang-Mills equations

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unifield equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie-algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie algebra as that of the volume preserving 3-dimensional diffeomorphisms.) When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einsteinvacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of anSO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations.Andrew Mellon Postdoctoral fellow and Fulbright ScholarSupported in part by NSF grant no. PHY 80023     

Self-dual Yang-Mills fields in an Einstein universe

We find exact solutions of the self-consistent Einstein-Yang-Mills system of equations. These solutions are self-dual Yang-Mills fields inI ¹×S³ space-time.     

Einstein gravitation from scalar nonsingular sources alone

In the light of Einstein's equations a system only containing two scalar fields is considered: One is of long range and attractive, the other is of short range and repulsive. The sources of these fields are taken to be nonsingular and spherically symmetric. All components of the energy-momentum tensor are continuous. A static solution of the equations is obtained in the weak-field approximation. The source of the gravitational field shows a finite concentration on the center of symmetry and dilutes monotonically to zero outwards. A Schwarzschild-type gravitation is found at infinity.     

Yang-Mills vacuum and the gluon mass

The vacuum sector of the pureSU(3) Yang-Mills theory in temporal gauge has been investigated. A variational calculation based on a BCS trial wave functions is performed. It is shown that this is equivalent to solving the Dyson-Schwinger equation in the Hartree approximation. By extending the Dyson-Schwinger approach, we shown using simple examples that it is possible to obtain an effective mass for the transverse gluons which is generated dynamically by the exchange of longitudinal gluons in a non perturbative vacuum.     

Smooth static solutions of the Einstein/Yang-Mills equations

We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions withSU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.Research supported in part by the NSF, Contract No. DMS 89-05205Research supported in part by the ONR, Contract No. DOD-C-N-00014-88-K-0082Research supported in part by the DOE, Grant No. DE-FG02-88ER25065Research supported in part by the U.K. Science and Engineering Research Council     

A relation between the Einstein and the Yang-Mills field equations

We show that any solution of the vacuum Einstein equations is a double self dual solution of the O(4) Yang-Mills equations.     

Increasing vacuum solutions in induced gravitation

It is shown that a solution of the form R(ν)=0 (R is the space-time curvature) exists for a vacuum solution (the field fluctuations are assumed zero, and only the ground state with the minimum effective potential energy remains) in the initial stage with consideration of the effect of spontaneous symmetry breaking for a scalar field with the “wrong sign” of the mass term and the conformal factor. For a spherically symmetric metric in vacuum, a solution increasing as a square of the distance and proportional to the square of the Higgs boson mass exists. A. A. Fridman Theoretical Physics Laboratory. Translated from Izvestiya Vysshikh Uchebhykh Zavedenii, Fizika, No. 4, pp. 34–38, April, 2000.     

The O(3,1) Yang-Mills equations and the Einstein equations

The vacuum Einstein equations (with cosmological constant) written in a slightly unconventional manner, can be decomposed into three parts: the first two parts are the ordinary self dual Yang-Mills equations and the anti-self dual Yang-Mills equations for anO(3,1) gauge group, on an unspecified background space-time, the third part are equations that solder or relate these two Y-M fields and connections to the curvature and connection of that unknown space-time. It is the purpose of this note to take this point of view seriously and concentrate on the first two parts in their own right. We apply to them generalizations of solution construction techniques which have arisen from the study of self dual Yang-Mills equations on Minkowski space. At the end we discuss how to solder or bootstrap these results to the determination of the space-time itself.     

Exact solutions of vacuum n-dimensional Einstein equations

A class of exact solutions of the vacuum n-dimensional Einstein equations for the case in which the components of the metric tensor depend on two variables is obtained by the method of separation of variables. Particular cases of this class of solutions are considered: a plane-symmetric metric; an axisymmetric metric; metrics of the Casner type.M. V. Lomonosov State University, Moscow. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 120–124, October, 1995.     

Exact vacuum solutions of the Einstein equations

The vacuum Einstein equations for the Kerr-Schild metric are investigated. It is shown that they admit representation in the form of the double four-dimensional curl of the perturbation of the Euclidean metric, whereupon it is possible to note certain general directions in which to seek exact solutions. For spaces with a normal isotropic geodesic congruence the GR equations are rewritten with the application of a dyadic splitting of the metric; cases of two-dimensional subspaces of constant curvature are discussed. The investigation is illustrated by the exact nonstationary algebraic type N and anti-Schwarzschild solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 23–26, September, 1982.     

Application of Lie Algebroid Structures to Unification of Einstein and Yang-Mills Field Equations

Yang-Mills field equations describe new forces in the context of Lie groups and principle bundles. It is of interest to know if the new forces and gravitation can be described in the context of algebroids. This work was intended as an attempt to answer last question. The basic idea is to construct Einstein field equation in an algebroid bundle associated to space-time manifold. This equation contains Einstein and Yang-Mills field equations simultaneously. Also this equation yields a new equation that can have interesting experimental results.     

The path of a sound signal in Einstein and Einstein-Cartan theory of gravitation

The path equation of a sound signal in general relativity is studied, It is shown that the sound path deflects towards stronger gravitational fields and is expected to defect less in a medium with polarization than in one without it, using the Einstein-Cartan theory of gravitation.