Biframe bundle geometry and an extension of RMW theory: Application to a charged perfect fluid

An extension of the original Rainich-Misner-Wheeler (RMW) theorem to include Einstein-Maxwell spacetimes with geometrical sources has recently been accomplished by generalizing the geometrical arena from the linear frame bundleLM to the bundle of biframesL ² M. The assumptions of a Riemannian connection one-form onLM and a general connection one-form onL ² M necessarily implies the existence of a difference formK. We provide new algebraic and differential conditions on an arbitrary triple (M, g, K), in addition to those already imposed by the generalization of the RMW theorem, which guarantee the form of the coupled Einstein-Maxwell field equations associated with a charged perfect fluid spacetime. All physical quantities associated with these field equations, namely the Maxwell field strength, the mass-energy density, the pressure, the electric and magnetic charge to mass ratios, and the unit four velocity of the fluid, can be recovered from the geometry as they are constructible entirely from the metricg, the difference formK, and their derivatives.     

New geometrical approach to Rainich-Misner-Wheeler theory

We introduce a new principal fiber bundle, the bundle of biframes, associated with the geometry of bivectors on spacetime. It is shown that the biframe bundle is a natural geometric arena for modeling the already unified theory of Rainich, Misner, and Wheeler (RMW). The structure equations for the bitorsion inherent in the biframe bundle lead to a generalization of Rainich's algebraic conditions for electromagnetic-type stress tensors which includes sources in a natural way. Besides the usual complexion vector of the RMW theory, an additional new complexion-type vector is found. The generalized algebraic conditions reduce to the usual RMW conditions in the special case of no sources.     

Affine connection structure of the charged symplectic 2-form

It is shown that the charged symplectic form in Hamiltonian dynamics of classical charged particles in electromagnetic fields defines a generalized affine connection on an affine frame bundle associated with spacetime. Conversely, a generalized affine connection can be used to construct a symplectic 2-form if the associated linear connection is torsion-free and the antisymmetric part of theR ^4* translational connection is locally derivable from a potential. Hamiltonian dynamics for classical charged particles in combined gravitational and electromagnetic fields can therefore be reformulated as aP(4)=O(1, 3)R ^4* geometric theory with phase space the affine cotangent bundleAT ^* M of spacetime. The sourcefree Maxwell equations are reformulated as a pair of geometrical conditions on the ^4* curvature that are exactly analogous to the source-free Einstein equations.     

On a covariant version of Caianiello’s model

Caianiello’s derivation of Quantum Geometry through an isometric embedding of the spacetime (M, g̃) in the pseudo-Riemannian structure (T*M, g* _AB ) is reconsidered. In the new derivation, using a non-linear connection and the bundle formalism, we obtain a Lorentzian-type structure in the 4-dimensional manifold M that is covariant under arbitrary local coordinate transformations in M. We obtain that if models with maximal acceleration are non-trivial, gravity should be supplied with other interactions in a unification framework.     

The construction from initial data of spacetimes with nontrivial spatial and bundle topology

We show how to use the 3 + 1 construction program to build globally heperbolic spacetimes with topologically nontrivial Cauchy surfaces. Spacetimes in which the classical fields are sections of a nontrivial bundle are handled as well. In evolving the initial data in these spacetimes, one must work with an atlas of overlapping patches. Data must be transfered from patch to patch during the evolution, so the transition functions on patch intersections must be evolved as well. We describe how to do this. Often the evolution of the Cauchy data is considerably simplified by choosing the coordinate-shift field $\text{M}$ and the gaugeshift field A_↓ to be patch dependent. We give examples of this phenomenon and show how to incorporate the patch dependence of $\text{M}$ and A_↓ into a consistent evolution program for the spacetime.     

Neutral perfect fluids of Majumdar-type in general relativity

We consider the extension of the Majumdar-type class of static solutions for the Einstein-Maxwell equations proposed by Ida to include charged perfect fluid sources. We impose the equation of state ρ+3p = 0 and discuss spherically symmetric solutions for the linear potential equation satisfied by the metric. In this particular case the fluid charge density vanishes and we locate the arising neutral perfect fluid in the intermediate region defined by two thin shells with respective charges Q and −Q. With its innermost flat and external (Schwarzschild) asymptotically flat spacetime regions, the resultant condenser-like geometries resemble solutions discussed by Cohen and Cohen in a different context. We explore this relationship and point out an exotic gravitational property of our neutral perfect fluid. We mention possible continuations of this study to embrace non-spherically symmetric situations and higher dimensional spacetimes.     

M factor invariance through the geometric etendue for Gaussian Schell model sources

Taking advantage of the relation of the M² factor for Gaussian Schell model sources in terms of the global coherence parameter, derived by Santarsiero et al., we have shown in this paper the invariance of the M² factor through its connection with geometrical Etendue of the pencil, along each independent coordinate.     

Construction of Sources for Majumdar-Papapetrou Spacetimes

We study Majumdar-Papapetrou solutions for the 3 + 1 Einstein-Maxwell equations, with charged dust acting as the external source for the fields. The spherically symmetric solution of Gürses is considered in detail. We introduce new parameters that simplify the construction of class C ¹, singularity-free geometries. The arising sources are bounded or unbounded, and the redshift of light signals allows an observer at spatial infinity to distinguish these cases. We find out an interesting affinity between the conformastatic metric and some homothetic and matter collineations. The associated geometric symmetries provide us with distinctive solutions that can be used to construct non-singular sources for Majumdar-Papapetrou spacetimes.     

Second-order tangent structures

Second-order differential processes have special significance for physics. Two reasonable generalizations of the procedure for constructing a tangent bundle over a smoothn-manifoldM yield different second-order structures, each projecting onto the standard first-order structureTM. One approach, based on the work of Ehresmann generalizes the notion of a tangent vector as a derivation. The other, based on the work of Yano and Ishihara generalizes the notion of a tangent vector as the velocity of a curve. The former leads toJ ² M, the 2-jet vector bundle consisting of second-order derivations, the latter leads toT ⁽²⁾ M, the bundle of curves agreeing up to acceleration. Both project naturally ontoTM because the 1-jet bundle of first-order derivations and the bundle of curves agreeing up to velocity are isomorphs ofTM. Both generalizations admit extension to higher orders but the second-order case illustrates their differences and is important in applications. It is always true thatJ ² M is a vector bundle; butT ⁽²⁾ M is a vector bundle if and only ifM has a linear connection and thenT ⁽²⁾ MTMTM with fiber ²ⁿ, whereasJ ² M always has fiber . We compare these constructions and give some results aboutT ⁽²⁾ M and the principal bundleL ⁽²⁾ M to which it is associated. In a space-time there is a distinguished linear connection induced by the Lorentz metric, so both second-order tangent structures are available and the reduction ofJ ² M toT ⁽²⁾ M is a considerable simplification in the casen=4. We show also that both second-order bundles have applications to the study of space-time boundaries.     

The GHP conformai Killing equations and their integrability conditions,with application to twisting type N vacuum spacetimes

The conformai Killing equations in resolved form and their first and second integrability conditions are obtained in the compact spin coefficient formalism for arbitrary spacetimes. To facilitate calculations an operatorL is introduced which agrees with the Lie derivative only when operating on quantities with GHP weights (0,0). The resulting equations are used to find the conditions for the existence of a two dimensional non-Abelian group of homothetic motions in a twisting typeN vacuum spacetime. The equivalence of two such sets of metrics is established, metrics that were recently the subject of independent investigations by Herlt on the one hand and by Ludwig and Yu on the other.     

Gravitation theory in the spacetimeR×S 3

A geometric formulation of the gravitation theory in the spacetime R × S ³ is given. A linear connection is introduced on the tangent bundle T(R × S ³ ) and then the connection coefficients and the Riemann curvature tensor are calculated. It is shown that their expressions differ from those of Carmeli and Malin [Found. Phys.17, 407 (1987)] by supplementary terms due to the noncommutativity of derivatives used on the spacetime R × S ³ . The Einstein field equations are written as usually and a comparison with other results is given. Finally, some observations about a possible gauge theory of gravitation in the spacetime R × S ³ are made.     

Homothetic transformations with fixed points in spacetime

A study is made of homothetic motions with fixed points in spacetime. Some general properties of such spacetimes are established, and a characterization of generalized plane wave spacetimes is proved. A general discussion of homothetic motions in Einstein's theory is given.This is in the sense that no open subset ofM is flat.     

A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

A program is outlined which addresses the problem of thereduction of Einstein's equations, namely, that of writing Einstein's vacuum equations in (3+1)-dimensions as anunconstrained dynamical system where the variables are thetrue degrees of freedom of the gravitational field. Our analysis is applicable for globally hyperbolic Ricci-flat spacetimes that admit constant mean curvature compact orientable spacelike Cauchy hypersurfaces M with degM=0 andM not diffeomorphic toF ₆, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincaré conjecture and the use of local cross-sections rather than a global cross-section, (3+1)-reduction can be completed much as in the (2+1)-dimensional case. In both cases, one gets as the reduced phase space the cotangent bundleT ^* T _M of theTeichmüller space T _M of conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, g _V ), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT ^* T _M . For these reduced systems, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact orientable spacelike Cauchy hypersurfaces in a neighborhood of a given initial one. In the (2+1)-dimensional case, the Hamiltonian is the area functional of these hypersurfaces, and in the (3+1)-dimensional case, the Hamiltonian is the volume functional of these hypersurfaces.     

Quantum Structures in Einstein General Relativity

We introduce the notion of a quantum structure on an Einstein general relativistic classical spacetime M. It consists of a line bundle over M equipped with a connection fulfilling certain conditions. We give a necessary and sufficient condition for the existence of quantum structures and classify them. The existence and classification results are analogous to those of geometric quantisation (Kostant and Souriau), but they involve the topology of spacetime, rather than the topology of the configuration space. We provide physically relevant examples, such as the Dirac monopole, the Aharonov–Bohm effect and the Kerr–Newman spacetime. Our formulation is carried out by analogy with the geometric approach to quantum mechanics on a spacetime with absolute time, given by Jadczyk and Modugno.     

Lorentzian characteristic classes

Universal classifying spaces and characteristic classes for O(p,q)-bundles are constructed and applied in the case (p,q) =(1,3) to the Lorentz bundle of a relativistic spacetime. The classes are used to investigate the existence of Uspinor structure on space and time orientable spacetimes, and spinor structures on nonorientable spacetimes.     

General relativity is a gauge type theory

It is shown that the Einstein-Maxwell theory of interacting electromagnetism and gravitation, can be derived from a first-order Lagrangian, depending on the electromagnetic field and on the curvature of a symmetric affine connection on the space-time M. The variation is taken with respect to the electromagnetic potential (a connection on a U(1) principal fiber bundle on M) and the gravitational potential (a connection on the GL(4, R) principal fiber bundle of frames on M). The metric tensor g does not appear in the Lagrangian, but it arises as a momentum canonically conjugated to . The Lagrangians of this type are calculated also for the Proca field, for a charged scalar field interacting with electromagnetism and gravitation, and for a few other interesting physical theories.     

Cosmological Horizons and Reconstruction of Quantum Field Theories

As a starting point, we state some relevant geometrical properties enjoyed by the cosmological horizon of a certain class of Friedmann-Robertson-Walker backgrounds. Those properties are generalised to a larger class of expanding spacetimes M admitting a geodesically complete cosmological horizon common to all co-moving observers. This structure is later exploited in order to recast, in a cosmological background, some recent results for a linear scalar quantum field theory in spacetimes asymptotically flat at null infinity. Under suitable hypotheses on M, encompassing both the cosmological de Sitter background and a large class of other FRW spacetimes, the algebra of observables for a Klein-Gordon field is mapped into a subalgebra of the algebra of observables constructed on the cosmological horizon. There is exactly one pure quasifree state λ on which fulfills a suitable energy-positivity condition with respect to a generator related with the cosmological time displacements. Furthermore λ induces a preferred physically meaningful quantum state λ _M for the quantum theory in the bulk. If M admits a timelike Killing generator preserving , then the associated self-adjoint generator in the GNS representation of λ _M has positive spectrum (i.e., energy). Moreover λ _M turns out to be invariant under every symmetry of the bulk metric which preserves the cosmological horizon. In the case of an expanding de Sitter spacetime, λ _M coincides with the Euclidean (Bunch-Davies) vacuum state, hence being Hadamard in this case. Remarks on the validity of the Hadamard property for λ _M in more general spacetimes are presented. Dedicated to Professor Klaus Fredenhagen on the occasion of his 60th birthday.     

Non-holonomic and semi-holonomic frames in terms of Stiefel and Grassmann tangent bundles

We prove that the bundles of non-holonomic and semi-holonomic second-order frames of a real or complex manifold M can be obtained as extensions of the bundle F²(M) of second-order jets of (holomorphic) diffeomorphisms of into M, where or . If and is the bundle of -linear frames of M we will associate to the tangent bundle two new bundles and with fibers of type the Stiefel manifold and the Grassmann manifold , respectively, where . The natural projection of onto defines a -principal bundle. We have found that the subset of given by the horizontal n-planes is an open sub-bundle isomorphic to the bundle of semi-holonomic frames of second-order of M. Analogously, the subset of given by the horizontal n-bases is an open sub-bundle which is isomorphic to the bundle of non-holonomic frames of second-order of M. Moreover the restriction of the former projection still defines a -principal bundle. Since a linear connection is a horizontal distribution of n-planes invariant under the action of it therefore determines a -reduction of the bundle , in a bijective way. This is a new proof of a theorem of Libermann.