The Einstein 3-form G_a and its equivalent 1-form L_a in Riemann-Cartan space

The definition of the Einstein 3-form G_a is motivated by means of the contracted 2nd Bianchi identity. This definition involves at first the complete curvature 2-form. The 1-form L_a is defined via G_a = L^b \wedge #(o_b \wedge o_a). Here # denotes the Hodge-star, o_a the coframe, and \wedge the exterior product. The L_a is equivalent to the Einstein 3-form and represents a certain contraction of the curvature 2-form. A variational formula of Salgado on quadratic invariants of the L_a 1-form is discussed, generalized, and put into proper perspective.Comment: LaTeX, 13 Pages. To appear in Gen. Rel. Gra

Linear Einstein equations and Kerr-Schild maps

We prove that given a solution of the Einstein equations $g_{ab}$ for the matter field $T_{ab}$, an autoparallel null vector field $l^{a}$ and a solution $(l_{a}l_{c}, \mathcal{T}_{ac})$ of the linearized Einstein equation on the given background, the Kerr-Schild metric $g_{ac}+\lambda l_{a}l_{c}$ ($\lambda$ arbitrary constant) is an exact solution of the Einstein equation for the energy-momentum tensor $T_{ac}+\lambda \mathcal{T}_{ac}+\lambda ^{2}l_{(a}\mathcal{T}_{c)b}l^{b}$. The mixed form of the Einstein equation for Kerr-Schild metrics with autoparallel null congruence is also linear. Some more technical conditions hold when the null congruence is not autoparallel. These results generalize previous theorems for vacuum due to Xanthopoulos and for flat seed space-time due to G\"{u}rses and G\"{u}rsey.Comment: 9 pages, accepted by Class. Quant. Gra

A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold

A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30(9), (1991)], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation.Comment: 20 page

Post-Newtonian extension of the Newton-Cartan theory

The theory obtained as a singular limit of General Relativity, if the reciprocal velocity of light is assumed to tend to zero, is known to be not exactly the Newton-Cartan theory, but a slight extension of this theory. It involves not only a Coriolis force field, which is natural in this theory (although not original Newtonian), but also a scalar field which governs the relation between Newtons time and relativistic proper time. Both fields are or can be reduced to harmonic functions, and must therefore be constants, if suitable global conditions are imposed. We assume this reduction of Newton-Cartan to Newton`s original theory as starting point and ask for a consistent post-Newtonian extension and for possible differences to usual post-Minkowskian approximation methods, as developed, for example, by Chandrasekhar. It is shown, that both post-Newtonian frameworks are formally equivalent, as far as the field equations and the equations of motion for a hydrodynamical fluid are concerned.Comment: 13 pages, LaTex, to appear in Class. Quantum Gra

Mathisson-Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation

We present a simple method to derive the semiclassical equations of motion for a spinning particle in a gravitational field. We investigate the cases of classical, rotating particles (pole-dipole particles), as well as particles with intrinsic spin. We show that, starting with a simple Lagrangian, one can derive equations for the spin evolution and momentum propagation in the framework of metric theories of gravity and in theories based on a Riemann-Cartan geometry (Poincare gauge theory), without explicitly referring to matter current densities (spin and energy-momentum). Our results agree with those derived from the multipole expansion of the current densities by the conventional Papapetrou method and from the WKB analysis for elementary particles.Comment: 28 page

3-dimensional Cauchy-Riemann structures and 2nd order ordinary differential equations

The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs

On certain relationships between cosmological observables in the Einstein-Cartan gravity

We show that in the Einstein-Cartan gravity it is possible to obtain a relation between Hubble's expansion and the global rotation (vorticity) of the Universe. Gravitational coupling can be reduced to dimensionless quantity of order unity, fixing the scalar mass density and the resulting negative cosmological constant at spacelike infinity. Current estimates of the expansion and rotation (see also astro-ph/9703082) of the Universe favour the massive spinning particles as candidate particles for cold and hot dark matter. Nodland and Ralston vorticity (Phys. Rev. Lett. 78 (1997) 3043) overestimates the value favoured by the Einstein-Cartan gravity for three orders of magnitude.Comment: 7 pages, LaTeX styl

Observables, gauge invariance, and the role of the observers in the limit from general relativity to special relativity

Some conceptual issues concerning general invariant theories, with special emphasis on general relativity, are analyzed. The common assertion that observables must be required to be gauge invariant is examined in the light of the role played by a system of observers. Some features of the reduction of the gauge group are discussed, including the fact that in the process of a partial gauge fixing the reduction at the level of the gauge group and the reduction at the level of the variational principle do not commute. Distinctions between the mathematical and the physical concept of gauge symmetry are discussed and illustrated with examples. The limit from general relativity to special relativity is considered as an example of a gauge group reduction that is allowed in some specific physical circumstances. Whether and when the Poincar\'e group must be considered as a residual gauge group will come out as a result of our analysis, that applies, in particular, to asymptotically flat spaces.Comment: 17 page

Plane torsion waves in quadratic gravitational theories

The definition of the Riemann-Cartan space of the plane wave type is given. The condition under which the torsion plane waves exist is found. It is expressed in the form of the restriction imposed on the coupling constants of the 10-parametric quadratic gravitational Lagrangian. In the mathematical appendix the formula for commutator of the variation operator and Hodge operator is proved. This formula is applied for the variational procedure when the gravitational field equations are obtained in terms of the exterior differential forms.Comment: 3 May 1998. - 11

Volterra Distortions, Spinning Strings, and Cosmic Defects

Cosmic strings, as topological spacetime defects, show striking resemblance to defects in solid continua: distortions, which can be classified into disclinations and dislocations, are line-like defects characterized by a delta function-valued curvature and torsion distribution giving rise to rotational and translational holonomy. We exploit this analogy and investigate how distortions can be adapted in a systematic manner from solid state systems to Einstein-Cartan gravity. As distortions are efficiently described within the framework of a SO(3) {\rlap{\supset}\times}} T(3) gauge theory of solid continua with line defects, we are led in a straightforward way to a Poincar\'e gauge approach to gravity which is a natural framework for introducing the notion of distorted spacetimes. Constructing all ten possible distorted spacetimes, we recover, inter alia, the well-known exterior spacetime of a spin-polarized cosmic string as a special case of such a geometry. In a second step, we search for matter distributions which, in Einstein-Cartan gravity, act as sources of distorted spacetimes. The resulting solutions, appropriately matched to the distorted vacua, are cylindrically symmetric and are interpreted as spin-polarized cosmic strings and cosmic dislocations.Comment: 24 pages, LaTeX, 9 eps figures; remarks on energy conditions added, discussion extended, version to be published in Class. Quantum Gra