pseudo-Riemannian Space-time Research Articles

Falsifiable analog model predictions of W mass in CDF II and ATLAS

The CDF II measurement of [Formula: see text] boson mass is at significant ([Formula: see text]) tension with the Standard Model, and moderate ([Formula: see text]) tension with recent results from ATLAS. Any attempt to interpret this as a signature of new physics requires high-precision, robust, falsifiable predictions. The Classical Analogue to the Standard Model In pseudo-Riemannian space–time (CASMIR) is an analogue model predicting [Formula: see text] and [Formula: see text] boson masses of [Formula: see text] and [Formula: see text], respectively. During baryon collisions satisfying [Formula: see text], CASMIR predicts a color-mediated enhancement of [Formula: see text] and [Formula: see text] boson masses, becoming [Formula: see text] and [Formula: see text], respectively. The unenhanced masses are consistent with ATLAS data collected at [Formula: see text], and the enhanced masses are consistent with CDF II data collected at [Formula: see text]. According to CASMIR, operation of the Large Hadron Collider (LHC) at center-of-mass energies small compared with [Formula: see text] but large enough for [Formula: see text] boson formation should permit ATLAS to replicate the result from CDF II.

Coordinate-Independent Definition of Relative Velocity in Pseudo-Riemannian Space-Time: Implications for Special Cases

Coordinate-Independent Definition of Relative Velocity in Pseudo-Riemannian Space-Time: Implications for Special Cases

Relative velocity in pseudo-Riemannian spacetime

We give a coordinate independent definition of relative velocity of test particle in pseudo-Riemannian spacetime as measured along the observer’s line-of-sight in general and several instructive cases. In doing this, the test particle is considered as a luminous object, otherwise, if it is not, we assume that a light source is attached to it, which has neither mass nor volume. Then we utilize the general solution of independent definition of relative velocity of a luminous source in generic pseudo-Riemannian spacetime. As a corollary, we discuss the implications for the Minkowski metric, the test particle and observer at rest in an arbitrary stationary metric, the uniform gravitational field, the rotating reference frame, the Schwarzschild metric, the Kerr-type metrics, and the spatially homogeneous and isotropic Robertson-Walker (RW) spacetime of standard cosmological model. In the last case, it leads to cosmological consequence that the resulting, so-called, kinetic recession velocity of an astronomical object is always subluminal even for large redshifts of order one or more, so that it does not violate the fundamental physical principle of causality. We also calculate the measure of carrying away of a galaxy at redshift z by the expansion of space, which proves, in particular, that cosmological expansion of a flat 3D–space is fundamentally different from a kinematics of galaxies moving in a non-expanding flat 3D-space. So, it is impossible to mimic the true cosmological redshift by a Doppler effect caused by motion of galaxies in a non-expanding 3D-space, flat or curved. We also give a reappraisal of the `standard ́ kinematic interpretation of redshifts in RW spacetime as accumulated Doppler-shifts.

Quantity in Quantum Mechanics and the Quantity of Quantum Information

The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a particular case of “unitary” qubits. The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation.

Unique definition of relative speed along the line of sight of a luminous object in a Riemannian space-time

Using a way of separating the spectral shifts into infinitesimally displaced `relative´ spectral bins and sum over them, we overcome the ambiguity of the parallel transport of four-velocity, in order to give an unique definition of the so-called kinetic relative velocity of luminous source as measured along the observer’s line-of-sight in a generic pseudo-Riemannian space-time. The ubiquitous relationship between the spectral shift and the kinetic relative velocity is utterly distinct from a familiar global Doppler shift rule (Synge, 1960). Such a performance of having found a kinetic relative velocity of luminous source, without subjecting it to a parallel transport, manifests its virtue in particular case when adjacent observers are being in free fall and populated along the null geodesic, so that it is reduced to a global Doppler velocity as studied by Synge. We discuss the implications for the instructive case of spatially homogeneous and isotropic Robertson-Walker space-time, which leads to cosmological consequences that the resulting kinetic recession velocity of a galaxy is always subluminal even for large redshifts of order one or more, and thus, it does not violate the fundamental physical principle of causality.

Relativistic potential of Weyl: a gateway to quantum physics in the presence of gravitation?

The well-noted correspondence between gravitation and electrodynamics emphasizes the importance of the Lanczos tensor—the potential of the Weyl tensor—which is an inherent structural element of any metric theory of gravity formulated in a 4-dimensional pseudo Riemannian spacetime. However, this ingenious discovery has gone largely unnoticed. We elucidate this important quantity by deriving its expressions in some particularly chosen spacetimes and try to find out what it represents actually. We find out that the Lanczos potential tensor does not represent the potential of the gravitational field, as is ascertained by various evidences. Rather, it is enriched with various signatures of quantum character, which provides a novel insight into the heart of a geometric embodiment of gravity.

A measure for quantum paths, gravity and spacetime microstructure

The number of classical paths of a given length, connecting any two events in a (pseudo) Riemannian spacetime is, of course, infinite. It is, however, possible to define a useful, finite, measure [Formula: see text] for the effective number of quantum paths [of length [Formula: see text] connecting two events [Formula: see text]] in an arbitrary spacetime. When [Formula: see text], this reduces to [Formula: see text] giving the measure for closed quantum loops of length [Formula: see text] containing an event [Formula: see text]. Both [Formula: see text] and [Formula: see text] are well-defined and depend only on the geometry of the spacetime. Various other physical quantities like, for e.g. the effective Lagrangian, can be expressed in terms of [Formula: see text]. The corresponding measure for the total path length contributed by the closed loops, in a spacetime region [Formula: see text], is given by the integral of [Formula: see text] over [Formula: see text]. Remarkably enough [Formula: see text], the Ricci scalar; i.e. the measure for the total length contributed by infinitesimal closed loops in a region of spacetime gives us the Einstein–Hilbert action. Its variation, when we vary the metric, can provide a new route towards induced/emergent gravity descriptions. In the presence of a background electromagnetic field, the corresponding expressions for [Formula: see text] and [Formula: see text] can be related to the holonomies of the field. The measure [Formula: see text] can also be used to evaluate a wide class of path integrals for which the action and the measure are arbitrary functions of the path length. As an example, I compute a modified path integral which incorporates the zero-point-length in the spacetime. I also describe several other properties of [Formula: see text] and outline a few simple applications.

Compact objects in conformal nonlinear electrodynamics

In this paper we consider a special case of vacuum nonlinear electrodynamics with a stress–energy tensor conformal to the Maxwell theory. Distinctive features of this model are the absence of a dimensional parameter for the nonlinearity description and a very simple form of the dominant energy condition, which can easily be verified in an arbitrary pseudo-Riemannian space-time with the consequent constraints on the model parameters. In this paper we analyze some properties of astrophysical compact objects coupled to conformal vacuum nonlinear electrodynamics.

Embeddings for general relativity

We present a systematic approach to embed n-dimensional vacuum general relativity in an (n+1)-dimensional pseudo-Riemannian spacetime whose source is either a (non)zero cosmological constant or a scalar field minimally coupled to Einstein gravity. Our approach allows us to generalize a number of results discussed in the literature. We construct all the possible (physically distinct) embeddings in Einstein spaces, including the Ricci-flat ones widely discussed in the literature. We examine in detail their generalization, which—in the framework under consideration—are higher-dimensional spacetimes sourced by a scalar field with flat (constant 0) potential. We use the Kretschmann curvature scalar to show that many embedding spaces have a physical singularity at some finite value of the extra coordinate. We develop several classes of embeddings that are free of singularities, have distinct non-vanishing self-interacting potentials and are continuously connected (in various limits) to Einstein embeddings. We point out that the induced metric possesses scaling symmetry and, as a consequence, the effective physical parameters (e.g., mass, angular momentum, cosmological constant) can be interpreted as functions of the extra coordinate.

Exact solutions for isometric embeddings of pseudo-Riemannian manifolds

Embeddings into higher dimensions are of direct importance in the study of higher dimensional theories of our Universe, in high energy physics and in classical general relativity. Theorems have been established that guarantee the existence of local and global codimension-1 embeddings between pseudo-Riemannian manifolds, particularly for Einstein embedding spaces. A technique has been provided to determine solutions to such embeddings. However, general solutions have not yet been found and most known explicit solutions are for embedded spaces with relatively simple Ricci curvature. Motivated by this, we have considered isometric embeddings of 4-dimensional pseudo-Riemannian spacetimes into 5-dimensional Einstein manifolds. We have applied the technique to treat specific 4-dimensional cases of interest in astrophysics and cosmology (including the global monopole exterior and Vaidya-de Sitter-class solutions), and provided novel physical insights into, for example, Einstein-Gauss-Bonnet gravity. Since difficulties arise in solving the 5-dimensional equations for given 4-dimensional spaces, we have also investigated embedded spaces, which admit bulks with a particular metric form. These analyses help to provide insight to the general embedding problem.

Quantum-theoretical origin of the Lorentz invariance of the elementary-particle theory

The reason why the elementary-particle theory must be Lorentz invariant is investigated under the principle of quantum priority, which implies that the existence of any space–time manifold such as the Minkowski space should not be assumed logically prior to the formulation of quantum-theoretical framework. The consideration is based on the canonical operator formalism of quantum Einstein gravity, which is formulated without introducing any classical background space–time nor any particular metric signature. The Lorentz invariance of the elementary-particle theory is derived as a consequence of the spontaneous breakdown of general linear invariance on the basis of quantum Einstein gravity. The pseudo-Riemannian space–time can appear not as a consequence of the presence of gravitation, but as that of the spontaneous breakdown of translational invariance.

The tetralogy of Birkhoff theorems

We classify the existent Birkhoff-type theorems into four classes: First, in field theory, the theorem states the absence of helicity 0- and spin 0-parts of the gravitational field. Second, in relativistic astrophysics, it is the statement that the gravitational far-field of a spherically symmetric star carries, apart from its mass, no information about the star; therefore, a radially oscillating star has a static gravitational far-field. Third, in mathematical physics, Birkhoff's theorem reads: up to singular exceptions of measure zero, the spherically symmetric solutions of Einstein's vacuum field equation with Lambda = 0 can be expressed by the Schwarzschild metric; for Lambda unequal 0, it is the Schwarzschild-de Sitter metric instead. Fourth, in differential geometry, any statement of the type: every member of a family of pseudo-Riemannian space-times has more isometries than expected from the original metric ansatz, carries the name Birkhoff-type theorem. Within the fourth of these classes we present some new results with further values of dimension and signature of the related spaces; including them are some counterexamples: families of space-times where no Birkhoff-type theorem is valid. These counterexamples further confirm the conjecture, that the Birkhoff-type theorems have their origin in the property, that the two eigenvalues of the Ricci tensor of two-dimensional pseudo-Riemannian spaces always coincide, a property not having an analogy in higher dimensions. Hence, Birkhoff-type theorems exist only for those physical situations which are reducible to two dimensions.

Majorana–Oppenheimer Approach to Maxwell Electrodynamics. Part III. Electromagnetic Spherical Waves in Spaces of Constant Curvature

Complex formalism of Riemann–Silberstein–Majorana–Oppenheimer in Maxwell electrodynamics extended to the case of arbitrary pseudo-Riemannian space-time applied to solve the Maxwell equations are solved exactly on the background of simplest static cosmological models, spaces of constant curvature of Riemann and Lobachevsky parameterized by spherical coordinates. Separation of variables is realized in the basis of Schrodinger–Pauli type, description of angular dependence in electromagnetic complex 3-vectors is given in terms of Wigner D-functions. In the case of compact Riemann model a discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three discrete parameters is found. In the case of hyperbolic Lobachevsky model no discrete spectrum for frequencies of electromagnetic modes arises.

Modified dispersion relations in Hořava–Lifshitz gravity and Finsler brane models

We study possible links between quantum gravity phenomenology encoding Lorentz violations as nonlinear dispersions, the Einstein–Finsler gravity models, EFG, and nonholonomic (non-integrable) deformations to Hořava–Lifshitz, HL, and/or Einstein’s general relativity, GR, theories. EFG and its scaling anisotropic versions formulated as Hořava–Finsler models, HF, are constructed as covariant metric compatible theories on (co) tangent bundle to Lorentz manifolds and respective anisotropic deformations. Such theories are integrable in general form and can be quantized following standard methods of deformation quantization, A-brane formalism and/or (perturbatively) as a nonholonomic gauge like model with bi-connection structure. There are natural warping/trapping mechanisms, defined by the maximal velocity of light and locally anisotropic gravitational interactions in a (pseudo) Finsler bulk spacetime, to four dimensional (pseudo) Riemannian spacetimes. In this approach, the HL theory and scenarios of recovering GR at large distances are generated by imposing nonholonomic constraints on the dynamics of HF, or EFG, fields.

TWO-CONNECTION RENORMALIZATION AND NON-HOLONOMIC GAUGE MODELS OF EINSTEIN GRAVITY

A new framework for perturbative quantum gravity is proposed following the geometry of non-holonomic distributions on (pseudo)Riemannian manifolds. There are considered such distributions and adap-ted connections, also completely defined by a metric structure, when gravitational models with infinite many couplings reduce to two-loop renormalizable effective actions. We use a key result from our partner work arXiv: 0902.0911 that the classical Einstein gravity theory can be reformulated equivalently as a non-holonomic gauge model in the bundle of affine/de Sitter frames on pseudo-Riemannian space–time. It is proven that (for a class of non-holonomic constraints and splitting of the Levi–Civita connection into a "renormalizable" distinguished connection, on a base background manifold, and a gauge-like distortion tensor, in total space) a non-holonomic differential renormalization procedure for quantum gravitational fields can be elaborated. Calculation labor is reduced to one- and two-loop levels and renormalization group equations for non-holonomic configurations.

A possible mathematics for the unification of quantum mechanics and general relativity

This paper summarizes and generalizes a recently proposed mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time variables x,t, furnished with an additional indefinite inner product invariant under Poincaré transformations, and isomorphisms of these spaces that preserve the indefinite metric. The indefinite metric is responsible for breaking the symmetry between space and time variables and for selecting a family of Hilbert subspaces that are preserved under Galileo transformations. Within these subspaces the usual quantum mechanics with Schrödinger evolution and t as the evolution parameter is derived. Simultaneously, the Minkowski space-time is isometrically embedded into H, Poincaré transformations have unique extensions to isomorphisms of H and the embedding commutes with Poincaré transformations. The main new result is a proof that the framework accommodates arbitrary pseudo-Riemannian space-times furnished with the action of the diffeomorphism group.

Maxwell equations in complex form of Majorana–Oppenheimer, solutions with cylindric symmetry in Riemann S 3 and Lobachevsky H 3 spaces

Complex formalism of Riemann–Silberstein–Majorana–Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space-time in accordance with the tetrad recipe of Tetrode–Weyl–Fock–Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of static cosmological Einstein model, parameterized by special cylindrical coordinates and realized as a Riemann space of constant positive curvature. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three parameters is found, and corresponding basis electromagnetic solutions have been constructed explicitly. In the case of elliptical model a part of the constructed solutions should be rejected by continuity considerations. Similar treatment is given for the Maxwell equations in hyperbolic Lobachevsky model, the complete basis of electromagnetic solutions in corresponding cylindrical coordinates has been constructed as well, no quantization of frequencies of electromagnetic modes arises.

Geometry of the Unification of Quantum Mechanics and Relativity of a Single Particle

The paper summarizes, generalizes and reveals the physical content of a recently proposed framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time variables x,t, furnished with an additional indefinite inner product invariant under Poincare transformations. The indefinite metric is responsible for breaking the symmetry between space and time variables and for selecting a family of Hilbert subspaces that are preserved under Galileo transformations. Within these subspaces the usual quantum mechanics with Shrodinger evolution and t as the evolution parameter is derived. Simultaneously, the Minkowski space-time is embedded into H as a set of point-localized states, Poincare transformations obtain unique extensions to H and the embedding commutes with Poincare transformations. Furthermore, the framework accommodates arbitrary pseudo-Riemannian space-times furnished with the action of the diffeomorphism group.

BRANES AND QUANTIZATION FOR AN A-MODEL COMPLEXIFICATION OF EINSTEIN GRAVITY IN ALMOST KÄHLER VARIABLES

The general relativity theory is redefined equivalently in almost Kähler variables: symplectic form, θ[g], and canonical symplectic connection, [Formula: see text] (distorted from the Levi–Civita connection by a tensor constructed only from metric coefficients and their derivatives). The fundamental geometric and physical objects are uniquely determined in metric compatible form by a (pseudo) Riemannian metric g on a manifold V enabled with a necessary type nonholonomic 2 + 2 distribution. Such nonholonomic symplectic variables allow us to formulate the problem of quantizing Einstein gravity in terms of the A-model complexification of almost complex structures on V, generalizing the Gukov–Witten method [1]. Quantizing [Formula: see text], we derive a Hilbert space as a space of strings with two A-branes which for the Einstein gravity theory are nonholonomic because of induced nonlinear connection structures. Finally, we speculate on relation of such a method of quantization to curve flows and solitonic hierarchies defined by Einstein metrics on (pseudo) Riemannian spacetimes.

Dynamics of multiparticle systems on the symplectic extension of the Schwarzschild space-time manifold

Methods of extension of the Schwarzschild metrics to the 8D symplectic phase manifold are considered. The method used is based on Sasaki’s metric of the tangent bundle over the pseudo-Riemannian 4D space-time manifold.