Einstein-like Equation Research Articles

A geometrization on dual 1-jet spaces of the time-dependent Hamiltonian of electrodynamics

In this paper, we develop the distinguished Riemannian differential geometry (in the sense of d-connections, d-torsions, d-curvatures, and the geometrical Maxwell-like and Einstein-like equations) for the time-dependent Hamiltonian of momenta which governs the electrodynamics phenomena.

Spinor fields in -gravity

We present a tetrad–affine approach to gravity coupled to spinor fields of spin-. After deriving the field equations, we derive the conservation law of the spin density, showing that the latter ensures the vanishing of the antisymmetric part of the Einstein-like equations, just as it happens in theories with torsion and metricity. We then focus on Bianchi type-I cosmological models proposing a general procedure to solve the corresponding field equations and providing analytical solutions in the case of gravitational Lagrangian functions of the kind . At late time such solutions are seen to isotropize and, depending on the value of the exponent n, they can undergo an accelerated expansion of the spatial scale factors.

Maxwell-modified metric affine gravity

We present a gauge formulation of the special affine algebra extended to include an antisymmetric tensorial generator belonging to the tensor representation of the special linear group. We then obtain a Maxwell modified metric affine gravity action with a cosmological constant term. We find the field equations of the theory and show that the theory reduces to an Einstein-like equation for metric affine gravity with the source added to the gravity equations with cosmological constant mu contains linear contributions from the new gauge fields. The reduction of the Maxwell metric affine gravity to Riemann–Cartan one is discussed and the shear curvature tensor corresponding to the symmetric part of the special linear connection is identified with the dark energy. Furthermore, the new gauge fields are interpreted as geometrical inflaton vector fields which drive accelerated expansion.

On a possible quantum manifestation of the perihelion advance in a hydrogen-like atom

In a static and spherically symmetric case, by following an analog procedure for obtaining a Reissner–Nordstrom solution, thanks to the use of both the Einstein-type equation for an electromagnetic field as well as the Maxwell equation in curved space–time, we managed to find the Schwarzschild-like solution outside a charged sphere Q in interaction with a test charged particle q at an atomic scale where the gravitational force is negligibly small compared to the electrostatic force. Then, in the context of an extension of the equivalence principle to the electromagnetic interaction, we have studied the hydrogen atom. At the atomic scale, where quantum effects can no longer be ignored and the concept of trajectory is abandoned, the corrective terms derived from the geodesic equation for the electron are used within a perturbation theory. At the first order, these radial corrections of the Hamiltonian can partially remove the degeneracy of Bohr’s levels through a Lamb-like shift of the energy levels (2s) and (2p), estimated to E(2s) – E(2p) = 2.4595087102 × 10−4 eV. To match the experimental value of the Lamb shift, 4.372 × 10−6 eV, we have postulated the existence of an electromagnetic background that would describe statistically the vacuum energy by the introduction of a “cosmological”-like constant in the Einstein-like equation for electromagnetism. We have pointed out a divergence when n → +∞ in energy corrections due to a “cosmological” constant that technically could be overcome by the use of a cutoff as well as the renormalization of the electron’s mass.

M2- and M5-branes in E11 current algebra formulation of M-theory

Equations of motion for M2- and M5-branes are written down in the [Formula: see text] current algebra formulation of M-theory. These branes correspond to currents of the second and the fifth rank antisymmetric tensors in the [Formula: see text] representation, whereas the electric and magnetic fields (coupled to M2- and M5-branes) correspond to currents of the third and the sixth rank antisymmetric tensors, respectively. We show that these equations of motion have solutions in terms of the coordinates on M2- and M5-branes. We also discuss the geometric equations, and show that there are static solutions when M2- or M5-brane exists alone and also when M5-brane wraps around M2-brane. This situation is realized because our Einstein-like equation contains an extra term which can be interpreted as gravitational energy contributing to the curvature, thus avoiding the usual intersection rule.

Holographic fluctuations and the principle of minimal complexity

We discuss, from a quantum information perspective, recent proposals of Maldacena, Ryu, Takayanagi, van Raamsdonk, Swingle, and Susskind that spacetime is an emergent property of the quantum entanglement of an associated boundary quantum system. We review the idea that the informational principle of minimal complexity determines a dual holographic bulk spacetime from a minimal quantum circuit U preparing a given boundary state from a trivial reference state. We describe how this idea may be extended to determine the relationship between the fluctuations of the bulk holographic geometry and the fluctuations of the boundary low-energy subspace. In this way we obtain, for every quantum system, an Einstein-like equation of motion for what might be interpreted as a bulk gravity theory dual to the boundary system.

Dynamical foundations of the brane-induced gravity

We present a comprehensive formalism to derive precise expressions for the induced gravity of the braneworld, assuming the dynamics of the Dirac–Nambu–Goto type. The quantum fluctuations of the brane at short distances give rise to divergences, which should be cut off at the scale of the inverse thickness of the brane. It turns out that the induced-metric formula is converted into an Einstein-like equation via quantum effects. We determine the coefficients of the induced cosmological and gravity terms, as well as those of the terms including the extrinsic curvature and the normal connection gauge field. The latter is the characteristic of the brane-induced gravity theory, distinguished from ordinary non-brane-induced gravity.

Rigorous formulation of duality in gravitational theories

In this paper we evince a rigorous formulation of duality in gravitational theories where an Einstein-like equation is valid, by providing the conditions under which and may be considered as the torsion and curvature 2-forms associated with a connection D′, part of a Riemann–Cartan structure , in the cases and , once and are the torsion and curvature 2-forms associated with a connection D part of a Riemann–Cartan structure . A new form for the Einstein equation involving the dual of the Riemann tensor of D is also provided, and the result is compared with others appearing in the literature.

Understanding Correlation Effects for Ion Conduction in Polymer Electrolytes

Polymer electrolytes typically exhibit diminished ionic conductivity due to the presence of correlation effects between the cations and anions. Microscopically, transient ionic aggregates, e.g., ion-pairs, ion-triplets, or higher order ionic clusters, engender ionic correlations. Employing all-atom simulation of a model polymer electrolyte comprising of poly(ethylene oxide) and lithium iodide, the ionic correlations are explored through construction of elementary functions between pairs of the ionic species that qualitatively explains the spatio-temporal nature of these correlations. Furthermore, commencing from the exact Einstein-like equation describing the collective diffusivity of the ions in terms of the average diffusivity of the ions (i.e., the self-terms) and the correlations from distinct pairs of ions, several phenomenological parameters are introduced to keep track of the simplification procedure that finally boils down to the recently proposed phenomenological model by Stolwijk and Obeidi (SO) [Stolwijk, N. A.; Obeidi, S. Phys. Rev. Lett. 2004, 93, 125901]. The approximation parameters, which can be retrieved from simulations, point to the necessity of additional information in order to fully describe the correlation effects apart from the mere fraction of ion-pairs that apparently accounts for the correlations originating from only the nearest neighbor structural correlations. These parameters are close to, but are not exactly unity, as assumed in the SO model. Finally, as an application of the extended SO model, one is able to estimate the dynamics of the free and non-free ions as well as their fractions from the knowledge of the single particle diffusivities and the collective diffusivity of the ions.

Universally coupled massive gravity

We derive Einstein’s equations from a linear theory in flat space-time using free-field gauge invariance and universal coupling. The gravitational potential can be either covariant or contravariant and of almost any density weight. We adapt these results to yield universally coupled massive variants of Einstein’s equations, yielding two one-parameter families of distinct theories with spin 2 and spin 0. The Freund-Maheshwari-Schonberg theory is therefore not the unique universally coupled massive generalization of Einstein’s theory, although it is privileged in some respects. The theories we derive are a subset of those found by Ogievetsky and Polubarinov by other means. The question of positive energy, which continues to be discussed, might be addressed numerically in spherical symmetry. We briefly comment on the issue of causality with two observable metrics and the need for gauge freedom and address some criticisms by Padmanabhan of field derivations of Einstein-like equations along the way.

Matter with dilaton charge in Weyl–Cartan spacetime and evolution of the universe

The perfect dilaton–spin fluid (as a model of the dilaton matter, the particles of which are endowed with intrinsic spin and dilaton charge) is considered as the source of the gravitational field in a Weyl–Cartan spacetime. The variational formalism of the gravitational field in a Weyl–Cartan spacetime is developed in the exterior form language. A homogeneous and isotropic universe filled with the dilaton matter as the dark matter is considered and one of the field equations is represented as the Einstein-like equation which leads to the modified Friedmann–Lemaître equation. From this equation the absence of the initial singularity in the cosmological solution follows. Also the existence of two points of inflection of the scale factor function is established, the first of which corresponds to the early stage of the universe and the second to the modern era when the expansion with deceleration is replaced by the expansion with acceleration. Possible equations of state for the self-interacting cold dark matter are found on the basis of the modern observational data. An inflation-like solution is obtained.

THE EINSTEIN EQUATION IN A THEORY OF STRINGS

We show, using purely classical considerations and logical extrapolation of results belonging to point particle theories, that the metric background field in which a string propagates must satisfy an Einstein or an Einstein-like equation. Additionally, there emerge restrictions on the worldsheet curvature, which seems to act as a source for space–time gravity, even in the absence of other matter fields.

The T(3)-Gauge Model, the Einstein-Like Gauge Equation, and Volterra Dislocations with Modified Asymptotics

Three-dimensional Lagrangian field theory is investigated as the T(3)-gauge model of static defects in continuous solids. The gauge Lagrangian is proposed in the Hilbert–Einstein form, and the non-conventional incompatibility law is given by an Einstein-like gauge equation with the elastic stress tensor as “matter” source. The stress function method is used, and two solutions to the linearized gauge equation (i.e., two modified stress functions) are obtained which are short-ranged in comparison with the Airy and Prandtl stress functions of classical Volterra dislocations. The modified stress functions obtained enable two ways of influencing the asymptotical behaviour of the edge and screw dislocations. Namely, the classically known rule 1/ρ is changed either at spatial extension (in favour of more fast decay of the modified stress components), or within the defect's cores (linear stresses of background Volterra defects become smoothed out due to formation of a core region).

Einstein-Like Equations with Coupled-to-Matter Λ Term

We consider Einstein-like gravitational equations with Λ term proportional to the trace of the energy tensor. The possibility of Lagrangian formulation is shortly discussed. The theory is consistent with the present-day observations, both at the local and cosmological level. Horizon-free cosmological solutions are found and discussed. The initial condition S(0) = 0 (where S(t) is the expansion factor and t the cosmic time) requires hyperbolic space.

Einstein-like field equations with conserved source and decreasing ? term; Their cosmological consequences

The consequences of a cosmological ∧ term varying asS−2 in a spatially isotropic universe with scale factorS and conserved matter tensor are investigated. One finds a perpetually expanding universe with positive ∧ and gravitational ‘constant’G that increases with time. The ‘hard’ equation of state 3P>U (U mass-energy density,P scalar pressure) applied to the early universe leads to the expansion lawS∝t (t cosmic time) which solves the horizon problem with no need of inflation. Also the flatness problem is resolved without inflation. The model does not affect the well known predictions on the cosmic light elements abundance which come from standard big bang cosmology.

Kaluza-Klein theory and Dirac equation in higher dimensional Riemann-Cartan space

The higher dimensional Kaluza-Klein theory in Riemann-Cartan space is discussed. To clarify its implications, we investigate the simplest five-dimensional case of the theory in detail. The Einstein-like, Maxwell, and Dirac equations in four-dimensional space-time are obtained by reducing the corresponding five-dimensional field equations. The effect of spin-spin interaction induced by torsion is revealed by analyzing the Dirac equation in this case.

Qualitative analysis of ten-dimensional lovelock cosmological models

The Einstein-like field equations obtained from the variation of a Lagrangian containing linear, quadratic (Gauss-Bonnet) and quartic terms for a ten-dimensional cosmological model cannot be solved analytically. However, we can reduce them to a system of dynamical equations for the “Hubble parameters”. The study of the mathematical properties of the fixed points of this system gives a qualitative picture of the behaviour of a class of possible solutions. The inclusion of a quartic term generates an extremely rich structure for the corresponding dynamical system. Some solutions are shown to exhibit the interesting property of dimensional reduction, which has been proposed as a possible explanation of the three-dimensional character of our universe.

Microscopic anisotropy in a Robertson-Walker cosmology

It is shown that in spinor-tetrad theory, with both left- and right-acting connection groups, there exists an exact Robertson-Walker cosmological solution with a matter field which possesses a polarized (as opposed to random) spin. The solution arises because the torsion vector, which appears anisotropically in the microscopic Dirac-like equation of the theory, enters the macroscopic Einstein-like gravitational equations in a purely scalar, isotropic form. The energy density of the cosmology is shown to be one quarter of that for the Einstein-de Sitter dust-filled universe. A pressure-density relation for the solution is also derived, although a conclusive physical interpretation for the solution remains unclear.

Confinement of tensor gluons — a classical approach

We look at the confinement of tensor gluons (f μν (c) field) in a strong gravity background and find that the strong gravity provides a trap for the confinement of colour waves of selected frequencies. We assume that the tensorf μν (c) field (mediating quanta: tensor 2+ f-meson) satisfies Einstein-like equations with a cosmological constant. The colour field satisfy equations resembling Maxwell form of the linear theory of gravitation and see the effect off μν (c) field as playing the role of a medium having space dependent dielectric permeabilities. The solution of colour field equations resemble harmonic oscillator type wave functions with equispaced energy levels (no continuum) leading to confinement.

A steeply-rising potential in tensor gauge theory

Massive tensor gauge mesons produce a steeply rising “potential” of the type M 2 r 2 in a non-perturbative classical solution of an Einstein-like equation. Such a potential is shown to give rising Regge trajectories.