General Conformal Transformation Research Articles

Fully nonlinear transformations of the Weyl-Bondi-Metzner-Sachs asymptotic symmetry group

The asymptotic symmetry group of general relativity in asymptotically flat spacetimes can be extended from the Bondi-Metzner-Sachs (BMS) group to the generalized BMS (GMBS) group suggested by Campiglia and Laddha, which includes arbitrary diffeomorphisms of the celestial two-sphere. It can be further extended to the Weyl BMS (BMSW) group suggested by Freidel, Oliveri, Pranzetti and Speziale, which includes general conformal transformations. We compute the action of fully nonlinear BMSW transformations on the leading order Bondi-gauge metric functions: specifically, the induced metric, Bondi mass aspect, angular momentum aspect, and shear. These results generalize previous linearized results in the BMSW context by Freidel et al., and also nonlinear results in the BMS context by Chen, Wang, Wang and Yau. The transformation laws will be useful for exploring implications of the BMSW group.

Conformal gravitational theories in Barthel–Kropina-type Finslerian geometry, and their cosmological implications

We consider dark energy models obtained from the general conformal transformation of the Kropina metric, representing an (alpha , beta )-type Finslerian geometry, constructed as the ratio of the square of a Riemannian metric alpha and the one-form beta . Conformal symmetries appear in many fields of physics, and they may play a fundamental role in our understanding of the Universe. We investigate the possibility of obtaining conformal theories of gravity in the osculating Barthel–Kropina geometric framework, where gravitation is described by an extended Finslerian-type model, with the metric tensor depending on both the base space coordinates and a vector field. We show that it is possible to formulate a family of conformal Barthel–Kropina theories in an osculating geometry with second-order field equations, depending on the properties of the conformal factor, whose presence leads to the appearance of an effective scalar field of geometric origin in the gravitational field equations. The cosmological implications of the theory are investigated in detail by assuming a specific relation between the component of the one-form of the Kropina metric and the conformal factor. The cosmological evolution is thus determined by the initial conditions of the scalar field and a free parameter of the model. We analyze in detail three cosmological models corresponding to different values of the theory parameters. Our results show that the conformal Barthel–Kropina model can provide an acceptable description of the observational data, and may represent a theoretically attractive alternative to the standard Lambda CDM cosmology.

Coupled dark energy model inspired from general conformal transformation

We study the coupled dark energy model constructed from the general conformal transformation in which the coefficient of the conformal transformation depends on both the scalar field and its kinetic term. Under this conformal transformation, the action for subclass of Degenerate Higher-Order Scalar-Tensor (DHOST) theories is related to the Einstein-Hilbert action. The evolution of the background universe has the scaling fixed point which corresponds to acceleration of the universe at late time. For the choices of parameters which make the late-time scaling point stable, the fixed point corresponding to $\phi$-matter-dominated-era ($\phi$MDE) is a saddle point, and the universe can evolve from radiation dominated epoch through $\phi$MDE before reaching the scaling point at late time with the cosmological parameters which satisfy the observational bound. During the $\phi$MDE, the effective equation of state parameter is slightly positive, so that one of possible mechanisms for alleviating the $H_0$ tension can be achieved. In this coupled dark energy model, the effective gravitational coupling for dark matter perturbations on small scales can be smaller than that in the $\Lambda$CDM model. Therefore a growth rate of the dark matter perturbations is suppressed compare with the $\Lambda$CDM model, which implies that the $\sigma_8$ tension could be alleviated.

Generalized conformal transformation and inflationary attractors

We investigate the inflationary attractors in models of inflation inspired from general conformal transformation of general scalar-tensor theories to the Einstein frame. The coefficient of the conformal transformation in our study depends on both the scalar field and its kinetic term. Therefore the relevant scalar-tensor theories display the subset of the class I of the degenerate higher-order scalar-tensor theories in which both the scalar field and its kinetic term can non-minimally couple to gravity. We find that if the conformal coefficient $\Omega$ takes a multiplicative form such that $\Omega \equiv w(\phi)W(X)$ where $X$ is the kinetic term of the field $\phi$, the theoretical predictions of the proposed model can have usual universal attractor independent of any functions of $W(X)$. For the case where $\Omega$ takes an additive form, such that $\Omega \equiv w(\phi) + k(\phi) \Xi(X)$, we find that there are new $\xi$ attractors in addition to the universal ones. We analyze the inflationary observables of these models and compare them to the latest constraints from the Planck collaboration. We find that the observable quantities associated to these new $\xi$ attractors do not satisfy the constraints from Planck data at a strong coupling limit.

Determination of pressure properties of superconducting systems based on characteristic ratios 2\u0394(0)/Tc and \u0394C(Tc)/CN(Tc)

A simplified analytical model of the effect of high pressure on the critical temperature and other thermodynamic properties of superconducting systems is developed using the general conformal transformation method and group-theoretical arguments. Relationships between the characteristic ratios { {mathcal R} }_{1}equiv 2{rm{Delta }}(0)/{T}_{{rm{c}}} and { {mathcal R} }_{2}equiv {rm{Delta }}C({T}_{{rm{c}}})/{C}_{{rm{N}}}({T}_{{rm{c}}}) and the stability of the superconducting state is discussed. Including a single two-parameter fluctuation in the density of states, placed away from the Fermi level, stable solutions determined by { {mathcal R} }_{1} are found. It is shown that the critical temperature Tc(p), as a function of high external pressure, can be predicted from experimental data, based on the values of the two characteristic ratios, the critical temperature, and a pressure coefficient measured at zero pressure. The model can be applied to s-wave low-temperature and high-temperature superconductors, as well as to some novel superconducting systems of the new generation. The problem of emergence of superconductivity under high pressure is explained as well. The discussion is illustrated by using experimental data for superconducting elements available in the literature. A criterion for compatibility of experimental data is formulated, allowing one to identify incompatible measurement data for superconducting systems for which the maximum or the minimum critical temperature is achieved under high pressure.

Scalar-torsion theories of gravity. I. General formalism and conformal transformations

We discuss the most general class of teleparallel scalar-torsion theories of gravity in their covariant formulation. The only restrictions we impose are the invariance of the action under diffeomorphisms and local Lorentz transformations, as well as vanishing direct coupling of the matter fields to the teleparallel spin connection. In this general setting we discuss the implications of local Lorentz invariance and diffeomorphism invariance and derive the general structure of the field equations. Further, we show how different theories of this class are related to each other by conformal transformations of the tetrad and redefinitions of the scalar field. We finally show how the formalism can be generalized to an arbitrary number of scalar fields, and provide a few examples.

Simple analytical model of the effect of high pressure on the critical temperature and other thermodynamic properties of superconductors

Within the general conformal transformation method a simplified analytical model is proposed to study the effect of external hydrostatic pressure on low- and high-temperature superconducting systems. A single fluctuation in the density of states, placed away from the Fermi level, as well as external pressure are included in the model to derive equations for the superconducting gap, free energy difference, and specific heat difference. The zero- and sub-critical temperature limits are discussed by the method of successive approximations. The critical temperature is found as a function of high external pressure. It is shown that there are four universal types of the response of the system, in terms of dependence of the critical temperature on increasing external pressure. Some effects, which should be possible to be observed experimentally in s-wave superconductors, the cuprates (i.e. high-Tc superconductors) and other superconducting materials of the new generation such as two-gap superconductors, are revealed and discussed. An equation for the ratio {{boldsymbol{ {mathcal R} }}}_{{bf{1}}} ≡ 2Δ(0)/Tc, as a function of the introduced parameters, is derived and solved numerically. Analysis of other thermodynamic quantities and the characteristic ratio {{boldsymbol{ {mathcal R} }}}_{{bf{2}}} ≡ ΔC(Tc)/CN(Tc) is performed numerically, and mutual relations between the discussed quantities are investigated. The simple analytical model presented in the paper may turn out to be helpful in searching for novel superconducting components with higher critical temperatures induced by pressure effects.

Shape dynamical loop gravity from a conformal Immirzi parameter

The Immirzi parameter of loop quantum gravity is a one-parameter ambiguity of the theory whose precise interpretation is not universally agreed upon. It is an inherent characteristic of the quantum theory as it appears in the spectra of geometric operators, despite being irrelevant at the classical level. The parameter’s appearance in the area and volume spectra to the same power as the Planck area suggest that it plays a role in determining the fundamental length scale of space. In fact, a consistent interpretation is that it represents a constant rescaling of the kinematical spatial geometry. An interesting realization is that promoting the Immirzi parameter to be a general conformal transformation leads to a system which can be identified as analogous to the linking theory of shape dynamics. A three-dimensional gravitational gauge connection is then constructed within the linking theory in a manner analogous to loop quantum gravity, thereby facilitating the application of the established procedure of loop quantization.

Scattering of out-plane wave by a circular cavity near the right-angle interface in the exponentially inhomogeneous media

By applying the theory of complex function and the image principle, this paper investigate the dynamic stress of wave propagation problems in an exponentially inhomogeneous right-angle plane with a circular cavity. The density is assumed to be an exponential change along x-axis, and the shear modulus is constant. Due to the inhomogeneity, the coefficient of the Helmholtz equation is a variable. Based on the theory of complex function, Helmholtz equation is transformed with a general conformal transformation method. By using the image principle, the right-angle plane is first mirrored as the half-space in order to find the expressions of the incident wave and the reflected wave. Then, the half-space is mapped into the finite one for solving the scattering problem of the wave. In order to verify the validity of this method, wave equation in inhomogeneous media is degenerated to the equation with constant coefficients. By truncating a set of infinite algebraic equations, the coefficient of the series is determined. The expression of scattering wave is given by the traction-free boundary condition. The displacements and stress fields are proposed. Dynamic stress concentration factor (DSCF) around circular cavity is calculated in exponentially inhomogeneous media.

Dynamic Stress Analysis Around a Circular Cavity in Two-Dimensional Inhomogeneous Medium with Density Variation

AbstractBased on the complex function theory and the homogenization principle, an universal approach of solving the dynamic stress concentration around a circular cavity in two-dimensional (2D) inhomogeneous medium is developed. The Helmholtz equation with variable coefficient is converted to the standard Helmholtz equation by means of the general conformal transformation method (GCTM) analytically. As an example, the inhomogeneous medium with density varying as a function of two spatial coordinates and the constant elastic modulus is studied. The dynamic stress concentration factors (DSCF) are calculated numerically. It shows that medium inhomogeneous parameters and wave numbers have significant influence on the dynamic stress concentration by the circular cavity in two-dimensional inhomogeneous medium.

Scattering by circular cavity in radially inhomogeneous medium with wave velocity variation

Based on the theory of complex function and the principle of homogenization, harmonic dynamics stress of a radially infinite inhomogeneous medium with a circular cavity is investigated. Due to the symmetry, wave velocity is assumed to have power-law variation in the radial direction only, and the shear modulus is constant. The Helmholtz equation with a variable coefficient is equivalently transformed into a standard Helmholtz equation with a general conformal transformation method (GCTM). The displacements and stress fields are proposed. Numerical results show that the wave number and the inhomogeneity parameter of the medium have significant effects on the dynamic stress concentration around the circular cavity. The dynamic stress concentration factor (DSCF) becomes singular when the inhomogeneity parameter of medium is close to zero.

Modelling and analysis of the dynamic behavior of inhomogeneous continuum containing a circular inclusion

Based on the complex function theory, an universal approach of solving the dynamic stress concentration around a circular inclusion in two-dimensional (2D) inhomogeneous medium is presented. The inhomogeneous Helmholtz equation with variable coefficient is converted to the standard Helmholtz equation by using the general conformal transformation technique analytically. An inhomogeneity with the density varying as a smooth function of two spatial coordinates and the constant elastic modulus is established to verify the accuracy of analytical results. As a typical example, an exponential variation of the density is introduced for analyzing the dynamic stress concentration factor (DSCF) around the inclusion. Numerical results show the efficiency of the method and the effects of the medium inhomogeneity, reference wave number of the background medium, the reference wave number and shear modulus ratios between the background medium and inclusion.

Canonical-type connection on almost contact manifolds with B-metric

The canonical-type connection on the almost contact manifolds with B-metric is constructed. It is proved that its torsion is invariant with respect to a subgroup of the general conformal transformations of the almost contact B-metric structure. The basic classes of the considered manifolds are characterized in terms of the torsion of the canonical-type connection.

Testing the generalized second law in1+1dimensional conformal vacua: An argument for the causal horizon

The anomalous conformal transformation law of the generalized entropy is found for dilaton gravity coupled to a 1+1 conformal matter sector with central charges $c = \tilde{c}$. (When $c \ne \tilde{c}$ the generalized entropy is not invariant under local Lorentz boosts.) It is shown that a certain second null derivative of the entropy, $S_\text{gen}" + (6/c)(S_\text{out}')^2$, is primary, and therefore retains its sign under a general conformal transformation. Consequently all conformal vacua have increasing entropy on causal horizons. Alternative definitions of the horizon, including apparent or dynamical horizons, can have decreasing entropy in any dimension $D \ge 2$. This indicates that the generalized second law should be defined using the causal horizon.

General conformal transformation method based on Schwarz-Christoffel approach

A general conformal transformation method (CTM) is proposed to construct the conformal mapping between two irregular geometries. In order to find the material parameters corresponding to the conformal transformation between two irregular geometries, two polygons are utilized to approximate the two irregular geometries, and an intermediate geometry is used to connect the mapping relations between the two polygons. Based on these manipulations, the approximate material parameters for TE and TM waves are finally obtained by calculating the Schwarz-Christoffel (SC) mappings. To demonstrate the validity of the method, a phase modulator and a plane focal surface Luneburg lens are designed and simulated by the finite element method. The results show that the conformal transformation can be expanded to the cases that the transformed objects are with irregular geometries.

Brane gravity at low energy

Four dimensional gravity in the low energy limit of a higher dimensional theory has been expected to be a (generalized) Brans-Dicke theory. A subtle point in brane world scenarios is that the system of four dimensional effective gravitational equations is not closed due to bulk gravitational waves and bulk scalars. Nonetheless, weak gravity on the brane can be analyzed completely. We reexamine the theory of weak brane gravity using gauge-invariant gravitational and scalar perturbations around a background warped geometry with a bulk scalar between two flat branes. We obtain a simple condition for the radion stabilization in terms of the scalar field potentials. We show that for general potentials of the scalar field which provides radion stabilization and a general conformal transformation to a frame in which matter on the branes are minimally coupled to the metric, 4-dimensional Einstein gravity, not BD gravity, is restored at low energies on either brane. In contrast, in RS brane world scenario without a bulk scalar, low energy gravity is a BD one. We conjecture that in general brane world scenarios with more than one scalar field one will again encounter the situation that low energy gravity is not described by the Einstein theory. Equipped with the weak gravity results, we discuss the properties of 4D brane gravitational equations, in particular, the value and sign of 4D Newton gravitational coupling.

Brane gravity, higher derivative terms, and nonlocality

In brane world scenarios with a bulk scalar field between two branes it is known that $4$-dimensional Einstein gravity is restored at low energies on either brane. By using a gauge-invariant gravitational and scalar perturbation formalism we extend the theory of weak gravity in the brane world scenarios to higher energies, or shorter distances. We argue that weak gravity on either brane is indistinguishable from $4$-dimensional higher derivative gravity, provided that the interbrane distance (radion) is stabilized, that the background bulk scalar field is changing near the branes and that the background bulk geometry near the branes is warped. This argument holds for a general conformal transformation to a frame in which matter on the branes is minimally coupled to the metric. In particular, Newton's constant and the coefficients of curvature-squared terms in the $4$-dimensional effective action are determined up to an ambiguity of adding a Gauss-Bonnet topological term. In other words, we provide the brane-world realization of the so called ${R}^{2}$ model without utilizing a quantum theory. We discuss the appearance of composite spin-$2$ and spin-$0$ fields in addition to the graviton on the brane and point out a possibility that the spin-$0$ field may play the role of an effective inflaton to drive brane-world inflation. Finally, we conjecture that the sequence of higher derivative terms is an infinite series and, thus, indicates nonlocality in the brane world scenarios.

Stress and Displacement around an Elastic Artificial Rectangular Hole

In engineering practice, rectangular holes (openings) are often encountered. The presence of rectangular holes increases the complexity of the analysis of engineering media (e.g., soils around a diaphragm wall panel) and structural members (e.g., floor slabs). However, there is currently no simple but sufficiently accurate solution readily available for engineers to obtain stress and displacement distributions around an artificial rectangular hole in an elastic plate. This paper presents a simple approximate solution to the problem for an infinite elastic and isotropic plate subjected to uniaxial stress at infinity. A linear elastic plane stress analysis is carried out using complex variables. By introducing a correction factor, simplified and explicitly expressed, a general conformal transformation function that transforms the area outside an arbitrary rectangular hole into the region inside a unit circle is created and used to obtain a solution. The solution is verified by comparing the calculated results using a finite-element method. Displacements caused by the formation of a rectangular hole are investigated. The ease of implementation of the solution is illustrated.

(Super)conformal algebra on the (super)torus

A generalization of the Virasoro algebra has recently been introduced by Krichever and Novikov (KN). The KN algebra describes the algebra of general conformal transformations in a basis appropriate to a genus- g Riemann surface. We examine in detail the genus-one KN algebra, and find explicit expressions for the central extension. We, further, construct explicitly the superconformal algebra of the supertorus, which yields supersymmetric generalizations of the genus-one KN algebra. A novel feature of the odd-spin-structure case is that the algebra includes a central element which is anticommuting. We comment on possible applications to string theory.

Dimensional reduction of symmetries

It has been shown that ordinary ( d−2)-dimensional quantum field theories are equivalent to corresponding quantum field theories defined on a ( d+2)-dimensional superspace with two anticommuting variables. This dimensional reduction is a consequence of superrotation invariance in the superspace. In this letter we study general conformal transformations in the superspace and the properties of field theories which are invariant under such transformations. We show that the symmetries are dimensionally reduced.