Generalized Einstein Equation Research Articles

Violation of the equivalence principle in curvature-based extended gravity at finite temperature

We review the possible violation of the Equivalence Principle at finite temperature [Formula: see text] in the framework of curvature-based Extended Theories of Gravity. Specifically, we first show how it is possible to derive Equivalence Principle violation from Quantum Field Theory at [Formula: see text]. Subsequently, we exhibit how this result can be precisely recovered by following an alternative path that envisages the employment of generalized Einstein equations with a temperature-dependent energy-momentum tensor. Finally, we adopt the latter formalism in the context of some Extended Gravity models to quantify the amount of Equivalence Principle violation. Specifically, Brans–Dicke Theory, Standard Model Extension and Conformal Gravity are considered in detail.

Towards a geometrical interpretation of rainbow geometries

In the literature, there are several papers establishing a correspondence between a deformed kinematics and a nontrivial (momentum dependent) metric. In this work, we study in detail the relationship between the trajectories given by a deformed Hamiltonian and the geodesic motion obtained from a geometry in the cotangent bundle, finding that both trajectories coincide when the Hamiltonian is identified with the squared distance in momentum space. Moreover, following the natural structure of the cotangent bundle geometry, one can obtain generalized Einstein equations. Since the metric is not invariant under momentum diffeomorphisms (changes of momentum coordinates) we note that, in order to have a conserved Einstein tensor (in the same sense of general relativity), a privileged momentum basis appears, a completely new result, that cannot be found in absence of space-time curvature which settles a long standing ambiguity of this geometric approach. After that, we consider in an expanding Universe the geodesic motion and the Raychaudhuri’s equations, and we show how to construct vacuum solutions to the Einstein equations. Finally, we make a comment about the possible phenomenological implications of our framework.

Application and Popularization of Energy Formula

In physics, there are two main energy formulas. One is kinetic energy formula and the another is Einstein equation. But kinetic energy formula can only calculate low speed motion. Einstein equation can only calculate light speed motion. The two formulas are not unified. We hope to get a unified formula. But it didn’t work. According to the principle of Lorentz contraction, we generalize the contraction of length to the contraction of mass, and obtain a unified energy formula. This is the generalized Einstein equation and the new Einstein kinetic energy formula.

D=4 supergravity from the Maxwell-Weyl superalgebra

We present the construction of the first-order D=4, {mathcal {N}}=1 supergravity action by gauging the Maxwell-Weyl superalgebra. The four-form lagrangian is constructed by using the curvatures of the algebra and the local scale invariance of the action is achieved through the introduction of a compensating scalar field. Finally, we find the generalized Einstein equation with a coordinate dependent cosmological term.

Einstein's Viscosity Equation for Nanolubricated Friction.

The generalized Einstein equation for the viscosity of a dispersion/suspension, μ = (1 + αfϕ)μ0, where μ0 is the liquid viscosity, ϕ is the solid volume fraction, and αf is a coefficient, is applied to the viscosity of a nanofluid lubricant. The coefficient of lubricated friction in the hydrodynamic regime is proportional to the viscosity of the lubricant. Therefore, an equation for the coefficient of friction with nanofluid lubrication can be formulated. We present such an equation and show its validity for common types of bearings (journal, rolling, and ball bearings). The equation, which may be viewed as one of nanofriction laws, is compared with experimental results for WS2 nanoparticle-enhanced oil lubrication, showing agreement within 7% accuracy.

تبدیلات دوگانگی آبلی استاندارد در گرانش f(T)

According to the perturbation order, the equations of motion of low-energy string effective action are the generalized Einstein equations. Thus, by making use of the conformal transformation of the metric tensor, it is possible to map the low-energy string effective action into f(T) gravity, relating the dilaton field to the torsion scalar. Considering a homogeneous and isotropic universe and writing the canonical Lagrangian for f(T) gravity, we show that the invariance under the duality transformation holds for the cosmic scale factor a(t) at the level of the Lagrangian. Finally, by use of the dualized Lagrangian and also the invariance of torsion scalar under the scale factor duality a(t)→1/a(t), the specific form of the f(T) function is obtained.

Maximal symmetry and mass generation of Dirac fermions and gravitational gauge field theory in six-dimensional spacetime**Supported by National Science Foundation of China (NSFC) (11690022, 11475237, 11121064) and Strategic Priority Research Program of the Chinese Academy of Sciences (XDB23030100) as well as the CAS Center for Excellence in Particle Physics (CCEPP)

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincaré symmetry P(1,5)=SO(1,5)⋉P1,5 as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.

Nonholonomic jet deformations, exact solutions for modified Ricci soliton and Einstein equations

Let [Formula: see text] be a pseudo-Riemannian metric of arbitrary signature on a manifold [Formula: see text] with conventional [Formula: see text]-dimensional splitting, [Formula: see text] determined by a nonholonomic (nonintegrable) distribution [Formula: see text] defining a generalized (nonlinear) connection and associated nonholonomic frame structures. We work with an adapted linear metric compatible connection [Formula: see text] and its nonzero torsion [Formula: see text], both completely determined by [Formula: see text]. Our first goal is to prove that there are certain generalized frame and/or jet transforms and prolongations with [Formula: see text] into explicit classes of solutions of some generalized Einstein equations [Formula: see text], [Formula: see text], encoding various types of (nonholonomic) Ricci soliton configurations and/or jet variables and symmetries. The second goal is to solve additional constraint equations for zero torsion, [Formula: see text], on generalized solutions constructed in explicit forms with jet variables and extract Levi-Civita configurations. This allows us to find generic off-diagonal exact solutions depending on all space time coordinates on [Formula: see text] via generating and integration functions and various classes of constant jet parameters and associated symmetries. Our third goal is to study how such generalized metrics and connections can be related by the so-called “half-conformal” and/or jet deformations of certain subclasses of solutions with one, or two, Killing symmetries. Finally, we present some examples of exact solutions constructed as nonholonomic jet prolongations of the Kerr metrics, with possible Ricci soliton deformations, and characterized by nonholonomic jet structures and generalized connections.

Geometry of area without length

To define a free string by the Nambu-Goto action, all we need is the notion of area, and mathematically the area can be defined directly in the absence of a metric. Motivated by the possibility that string theory admits backgrounds where the notion of length is not well defined but a definition of area is given, we study space-time geometries based on the generalization of metric to area metric. In analogy with Riemannian geometry, we define the analogues of connections, curvatures and Einstein tensor. We propose a formulation generalizing Einstein's theory that will be useful if at a certain stage or a certain scale the metric is ill-defined and the space-time is better characterized by the notion of area. Static spherical solutions are found for the generalized Einstein equation in vacuum, including the Schwarzschild solution as a special case.

Equivalent off-diagonal cosmological models and ekpyrotic scenarios in $$f(R)$$ f ( R ) -modified, massive, and einstein gravity

We re-investigate how generic off-diagonal cosmological solutions depending, in general, on all spacetime coordinates can be constructed in massive and f-modified gravity using the anholonomic frame deformation method. There are constructed new classes of locally anisotropic and (in) homogeneous cosmological metrics with open and closed spatial geometries. By resorting such solutions, we show that they describe the late time acceleration due to effective cosmological terms induced by nonlinear off-diagonal interactions, possible modifications of the gravitational action and graviton mass. The cosmological metrics and related St\" uckelberg fields are constructed in explicit form up to nonholonomic frame transforms of the Friedmann-Lama\^{\i}tre-Robertson-Walker (FLRW) coordinates. The solutions include matter, graviton mass and other effective sources modelling nonlinear gravitational and matter fields interactions with polarization of physical constants and deformations of metrics, which may explain dark energy and dark matter effects. However, we argue that it is not obligatory always to modify gravity if we consider effective generalized Einstein equations with nontrivial vacuum and/or non-minimal coupling with matter. Indeed, we state certain conditions when such configurations mimic interesting solutions in general relativity and modifications, for instance, when we can extract the general Painlev\' e-Gullstrand and FLRW metrics. In a more general context, we elaborate on a reconstruction procedure for off-diagonal cosmological solutions which describe cyclic and ekpyrotic universes. Finally, there are discussed open issues and further perspectives.

Geometry and general relativity in the groupoid model with a finite structure group

In a series of papers (M. Heller et al. J. Math. Phys. 38, 5840 (1997). doi:10.1063/1.532186 ; M. Heller and W. Sasin. Int. J. Theor. Phys. 38, 1619 (1999). doi:10.1023/A:1026617913754 ; M. Heller et al. Int. J. Theor. Phys. 44, 619 (2005). doi:10.1007/s10773-005-3992-7 ) we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra, [Formula: see text], defined on a transformation groupoid Γ determined by the action of the Lorentz group on the frame bundle (E, πM, M) over space–time M. In the present work, we construct a simplified version of the gravitational sector of this model in which the Lorentz group is replaced by a finite group, G, and the frame bundle is trivial E = M × G. The model is fully computable. We define the Einstein–Hilbert action, with the help of which we derive the generalized vacuum Einstein equations. When the equations are projected to space–time (giving the “general relativistic limit”), the extra terms that appear due to our generalization can be interpreted as “matter terms”, as in Kaluza–Klein-type models. To illustrate this effect we further simplify the metric matrix to a block diagonal form, compute for it the generalized Einstein equations and find two of their “Friedman-like” solutions for the special case when G = [Formula: see text]. One of them gives the flat Minkowski space–time (which, however, is not static), another, a hyperbolic, linearly expanding universe.

Generalized derivations and general relativity

We construct differential geometry (connection, curvature, etc.) based on generalized derivations of an algebra [Formula: see text]. Such a derivation, introduced by Brešar in 1991, is given by a linear mapping [Formula: see text] such that there exists a usual derivation, d, of [Formula: see text] satisfying the generalized Leibniz rule u(ab) = u(a)b + ad(b) for all [Formula: see text]. The generalized geometry “is tested” in the case of the algebra of smooth functions on a manifold. We then apply this machinery to study generalized general relativity. We define the Einstein–Hilbert action and deduce from it Einstein’s field equations. We show that for a special class of metrics containing, besides the usual metric components, only one nonzero term, the action reduces to the O’Hanlon action that is the Brans–Dicke action with potential and with the parameter ω equal to zero. We also show that the generalized Einstein equations (with zero energy–stress tensor) are equivalent to those of the Kaluza–Klein theory satisfying a “modified cylinder condition” and having a noncompact extra dimension. This opens a possibility to consider Kaluza–Klein models with a noncompact extra dimension that remains invisible for a macroscopic observer. In our approach, this extra dimension is not an additional physical space–time dimension but appears because of the generalization of the derivation concept.

Dark energy and matter perturbations in scalar-tensor theories of gravity

We solve analytically and numerically the generalized Einstein equations in scalar-tensor cosmologies to obtain the evolution of dark energy and matter linear perturbations. We compare our results with the corresponding results for minimally coupled quintessence perturbations. We find that Scalar-Tensor dark energy density perturbations are amplified by a factor of about 104 compared to minimally coupled quintessence perturbations on scales less than about 1000h−1Mpc (sub-Hubble scales). On these scales dark energy perturbations constitute a fraction of about 10% compared to matter density perturbations. Scalar-Tensor dark energy density perturbations are anti-correlated with matter linear perturbations on sub-Hubble scales. This anti-correlation of matter with negative pressure perturbations induces a mild amplification of matter perturbations by about 10% on sub-Hubble scales. The evolution of scalar field perturbations on sub-Hubble scales is scale independent and therefore corresponds to a vanishing effective speed of sound (csΦ = 0). We briefly discuss the observational implications of our results which may include predictions for galaxy and cluster halo profiles which are modified compared to ΛCDM. The observed properties of these profiles are known to be in some tension with the predictions of ΛCDM.

Evolution of dark energy perturbations in scalar-tensor cosmologies

We solve analytically and numerically the generalized Einstein equations in scalar-tensor cosmologies to obtain the evolution of dark energy and matter linear perturbations. We compare our results with the corresponding results for minimally coupled quintessence perturbations. Our results for natural (O(1)) values of parameters in the Lagrangian which lead to a background expansion similar to LCDM are summarized as follows: 1. Scalar-Tensor dark energy density perturbations are amplified by a factor of about 10^6 compared to minimally coupled quintessence perturbations on scales less than about 100 h^{-1} Mpc. On these scales dark energy perturbations constitute a fraction of about 10% compared to matter density perturbations. 2. Scalar-Tensor dark energy density perturbations are anti-correlated with matter linear perturbations on sub-Hubble scales. Thus clusters of galaxies are predicted to overlap with voids of dark energy. 3. This anti-correlation of matter with negative pressure perturbations induces a mild amplification of matter perturbations by about 10% on sub-Hubble scales. 4. The evolution of scalar field perturbations on sub-Hubble scales, is scale independent and therefore it corresponds to a vanishing effective speed of sound (c_{s\Phi}=0). It also involves large oscillations at early times induced by the amplified effective mass of the field. This amplification is due to the non-minimal coupling of the field to the Ricci curvature scalar and (thus) to matter. No such oscillations are present in minimally coupled quintessence perturbations which are suppressed on sub-Hubble scales (c_{s\Phi}=1). Implications of our results include modified predictions for cluster halo profiles observed properties. These profiles are known to be in some tension with the predictions of LCDM.

A new formulation of general relativity

In Sections 1–5 of this paper an axiomatic formulation of general theory of rativity (GR) is given and studied. Here use is made of the concept of pre-radar charts. These charts have “infinitesimally” the same properties as the true radar charts used in space–time theory. Their existence in GR has far-reaching consequences which are discussed throughout the paper. For the sake of simplicity and convenience I consider only such material systems the state of which is defined by a velocity field, a mass density and a temperature field. But the main results hold also for more complex systems. It follows from the axiomatics that the pre-radar charts define an atlas for the space–time manifold and that, in addition, they generate the metric, the velocity field and the displacement of the matter. Therefore, they are called generating functions. They act like “potentials”. In Section 6 it is shown that the existence of pre-radar charts allows to simplify the original axiomatics drasticly. But the two versions of GR are equivalent. In Sections 7 and 8 the so-called inverse problem is treated. This means the question whether it is possible to define preradar charts, i.e., generating functions in arbitrary space–times. This problem is subtle. A local general constructive solution of it is presented. Sufficient conditions for the existence of global solutions are given. The aim of the last sections is formulating GR as a scalar field theory. The basic structural elements of it are a generating function, a generalized density and a generalized temperature. One of the axioms of this theory is a generalized Einstein equation that determines the generating function directly. It is shown that basic concepts like orientation, time orientation and isometry are expressible in terms of generating functions. At the end of the paper six problems are formulated which are still unsolved and can act as a stimulant for further research.

Noncommutative closed Friedman universe

Heller et al. (J Math Phys 46:122501, 2005; Int J Theor Phys 46:2494, 2007) proposed a model unifying general relativity and quantum mechanics based on a noncommutative algebra \({\mathcal{A}}\) defined on a groupoid having the frame bundle over space–time as its base space. The generalized Einstein equation is assumed in the form of the eigenvalue equation of the Einstein operator on a module of derivations of the algebra \({\mathcal{A}}\) . No matter sources are assumed. The closed Friedman world model, when computed in this formalism, exhibits two interesting properties. First, generalized eigenvalues of the Einstein operator reproduce components of the perfect fluid energy-momentum tensor for the usual Friedman model together with the corresponding equation of state. One could say that, in this case, matter is produced out of pure (noncommutative) geometry. Second, owing to probabilistic properties of the model, in the noncommutative regime (on the Planck level) singularities are irrelevant. They emerge in the process of transition to the usual space–time geometry. These results are briefly discussed.

Geometry and dynamics of higher-spin frame fields

We give a systematic account of unconstrained free bosonic higher-spin fields on D-dimensional Minkowski and (Anti-)de Sitter spaces in the frame formalism. The generalized spin connections are determined by solving a chain of torsion-like constraints. Via a generalization of the vielbein postulate these allow to determine higher-spin Christoffel symbols, whose relation to the de Wit-Freedman connections is discussed. We prove that the generalized Einstein equations, despite being of higher-derivative order, give rise to the AdS Fronsdal equations in the compensator formulation. To this end we derive Damour-Deser identities for arbitrary spin on AdS. Finally we discuss the possibility of a geometrical and local action principle, which is manifestly invariant under unconstrained higher-spin symmetries.

Noncommutative unification of general relativity and quantum mechanics

We present a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry is developed in terms of a noncommutative algebra A which is defined on a transformation groupoid Γ given by the action of a noncompact group G on the total space E of a principal fiber bundle over space-time M. The case is important since to obtain physical effects predicted by the model we should assume that G is a Lorentz group or some of its representations. We show that the generalized Einstein equation of the model has the form of the eigenvalue equation for the generalized Ricci operator, and all relevant operators in the quantum sector of the model are random operators; we study their dynamics. We also show that the model correctly reproduces general relativity and the usual quantum mechanics. It is interesting that the latter is recovered by performing the measurement of any observable. In the act of such a measurement the model “collapses” to the usual quantum mechanics.

Onsager's coefficients and diffusion laws—a Monte Carlo study

For a numerical description of multicomponent diffusion processes, the coefficients of the Onsager diffusion matrix are needed. We compare a number of models relating these parameters to experimentally accessible quantities, such as tracer diffusion coefficients. Since these models present different levels of approximation, we investigate the differences in Onsager parameters when they are determined from tracer diffusion. Moreover, we study an ideal solution alloy by Monte Carlo methods to determine the Onsager coefficients directly and using the model assumptions. Measuring atomic fluxes and site fraction gradients, our simulation method is more closely related to a real physical experiment than the usual simulation method of generalized Einstein equations. Onsager's variation principle for the calculation of the kinetic coefficients is extended and it is shown that a reasonable description, even of a simple alloy system, requires consideration of a non-diagonal dissipation matrix in the derivation of the diffusion equations.

COSMIC STRING SPACETIME IN DILATON GRAVITY AND FLAT ROTATION CURVES

In dilaton gravity theories, we consider a string-like topological defect formed during U (1) gauge symmetry-breaking phase transition in the early Universe, and far from the cosmic string we have vacuum solutions of the generalized Einstein equation. We discuss how they can be related to the flatness of galactic rotation curves.