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.     

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.     

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.     

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     

Existence of infinitely-many smooth,static, global solutions of the Einstein/Yang-Mills equations

We prove the existence of infinitely-many globally defined singularity-free solutions, to the EYM equations withSU(2) gauge group. The solutions are indexed by a coupling constant, have distinct winding numbers, and their corresponding Einstein metrics decay at infinity to the flat Minkowski metric. Each solution has a finite (ADM) mass; these masses are derived from the solutions, and arenot arbitrary constants.Both authors supported in part by the NSF, Contract No. DMS 89-05205     

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.     

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.     

Equivalence of vacuum Yang-Mills gravitation and vacuum Einstein gravitation

In the Yang-Mills formulation of gravitational dynamics based uponSL(2,C) spin transformations acting on Dirac spinors, the vacuum field equations are R +C R = 0 and and . HereR is the Ricci curvature andC is the Weyl conformal curvature; is a coupling constant. We show the equivalence between solutions of these equations and the vacuum Einstein equationsR = 0. The proof uses the Newman-Penrose formalism.Supported by a NATO fellowship.Supported by a SRC fellowship.     

A remark on the connection between affine Lie algebras and soliton equations

The Lax equations of Drinfeld-Sokolov are derived in the framework of the Fock representation of Clifford algebras. The derivation is based on the bilinear identities for -functions.     

Linear Einstein equations

Formulation of Cauchy and boundary problems for the equations of a weak gravitational field is considered. The effect of coordinate conditions on their validity is considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 3–6, January, 1982     

A new classical solution of the Yang-Mills field equations

The MIT bag theory is used to calculate the quarl anti-quark annihilation amplitude for the ??γ transition and produces m_?²/∫_? = 0.13 compared to the experimental result of 0.11.     

Averaging out the Einstein equations

A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.     

Null solutions of the Yang-Mills equations

We investigate the correspondence between null solutions of the Yang-Mills equations and shearfree geodesic null congruences. We give an example of a non-Abelian null solution with twisting rays.Alexander von Humboldt Fellow. Address after 31 May 1986: Institute of Theoretical Physics, University of Warsaw, Hoa 69, 00-681 Warsaw, Poland.     

Yang-Mills equations on the two-dimensional sphere

We construct a 1:1 correspondence between the equivalence classes of Yang-Mills fields overS ² and the conjugacy classes of closed geodesics of the structure group. Furthermore, we give an explicit isolation theorem for any Yang-Mills field overS ².     

Pseudoparticle solutions of the Yang-Mills equations

We find regular solutions of the four dimensional euclidean Yang-Mills equations. The solutions minimize locally the action integrals which is finite in this case. The topological nature of the solutions is discussed.