We argue that all Einstein-Maxwell or Einstein-Proca solutions to general relativity may be used to construct a large class of solutions (involving torsion and non-metricity) to theories of non-Riemannian gravitation that have been recently discussed in the literature.
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IntroductionNon-Riemannian geometries feature in a number of theoretical descriptions of the interactions between fields and gravitation. Since the early pioneering work by Weyl, Cartan, Schroedinger and others such geometries have often provided a succinct and elegant guide towards the search for unification of the forces of nature [1]. In recent times interactions with supergravity have been encoded into torsion fields induced by spinors and dilatonic interactions from low energy effective string theories have been encoded into connections that are not metric-compatible [2], [3], [4], [5]. However theories in which the non-Riemannian geometrical fields are dynamical in the absence of matter are more elusive to interpret. It has been suggested that they may play an important role in certain astrophysical contexts [6]. Part of the difficulty in interpreting such fields is that there is little experimental guidance available for the construction of a viable theory that can compete effectively with general relativity in domains that are currently accessible to observation. In such circumstances one must be guided by the classical solutions admitted by theoretical models that admit dynamical non-Riemannian structures [7], [8], [9], [10], [11]. A number of recent papers have pursued this approach and have found static spherically symmetric solutions to particular models [12], [13]. In [6] it was pointed out that in a particularly simple model all EinsteinMaxwell solutions to general relativity could be used to generate dynamic non-Riemannian geometries and a tentative interpretation was offered for the matter couplings in such a model. Particular solutions have also been found to more complex models, provided the coupling constants in the action are correlated [12], [13], [14], [15]. It is the purpose of this note to point out that, if such correlations are maintained then solutions may be generated from all Einstein-Maxwell solutions of general relativity. Furthermore the correlations may be discarded and solutions generated from all Einstein-Proca solutions of general relativity with or without the inclusion of a cosmological term in the action.
The authors exhibit a family of Hilbert subspaces describing solutions to the Wheeler-DeWitt equation. Each subspace describes a distinct quantum theory in which the parameters of the potential are required to satisfy a particular 'quantization' condition. Tentative evidence suggests that within each Hilbert subspace there exist states that may be identified with nondispersive wave packets peaking in the vicinity of submanifolds in superspace corresponding to classical cosmological eras. These submanifolds admit parameterizations corresponding to metric solutions of Einstein's equations that admit a transition between a Euclidean and a Lorentzian signature.
As an extension of the intertwining operator idea, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced. By realization of the method in two dimensions, two infinite families of superintegrable and isospectral stationary potentials are generated. The method makes it possible to perform Darboux transformations in such a way that, in addition to the isospectral property, they acquire the superintegrability preserving property. Symmetry generators are second and fourth order in derivatives and all potentials are isospectral with one of the Smorodinsky-Winternitz potentials. Explicit expressions of the potentials, their dynamical symmetry generators and the algebra they obey as well as their degenerate spectra and corresponding normalizable states are presented.
It has been shown that, for all dimensions and signatures, the most general first-order linear symmetry operators for the Dirac equation including interaction with Maxwell field in curved background are given in terms of Killing-Yano (KY) forms. As a general gauge invariant condition it is found that among all KY-forms of the underlying (pseudo) Riemannian manifold, only those which Clifford commute with the Maxwell field take part in the symmetry operator. It is also proved that associated with each KY-form taking part in the symmetry operator, one can define a quadratic function of velocities which is a geodesic invariant as well as a constant of motion for the classical trajectory. Some geometrical and physical implications of the existence of KY-forms are also elucidated.
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