Riemann Tensor Research Articles

On-Shell Covariance of Quantum Field Theory Amplitudes

Scattering amplitudes in quantum field theory are independent of the field parametrization, which has a natural geometric interpretation as a form of "coordinate invariance." Amplitudes can be expressed in terms of Riemannian curvature tensors, which makes the covariance of amplitudes under nonderivative field redefinitions manifest. We present a generalized geometric framework that extends this manifest covariance to all allowed field redefinitions. Amplitudes satisfy a recursion relation to all orders in perturbation theory that closely resembles the application of covariant derivatives to increase the rank of a tensor. This allows us to argue that tree-level amplitudes possess a notion of "on-shell covariance," in that they transform as a tensor under any allowed field redefinition up to a set of terms that vanish when the equations of motion and on-shell momentum constraints are imposed. We highlight a variety of immediate applications to effective field theories.

On Einstein-reversible mth root Finsler metrics

The theory of mth root Finsler metrics has been applied to Biology, Ecology, Gravitation, Seismic ray theory, etc. It is regarded as a direct generalization of Riemannian metric in a sense, namely, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann–Finsler metric. The reversibility of Riemannian and Ricci curvatures of Finsler metrics is an essential concept in Finsler geometry. Here, we study the Riemannian curvature of the class of third and fourth root [Formula: see text]-metrics. Then, we find the necessary and sufficient condition under which a cubic and fourth root [Formula: see text]-metric be Einstein-reversible.

Cluster of stars in f(, T) gravity

This paper is devoted to analyze the dynamical impacts of gravity model on the cluster of stars. For this motive, we consider the spherically symmetric interior geometry with anisotropic fluid as analogous to cluster of stars distributions. We express the modified field equations by taking a particular model of , i.e. . In order to explore the evolutionary behavior of cluster of stars, the observational data of a compact star 4U1820 − 30 is used. We construct the modified scalar functions by orthogonal splitting of Riemann tensor in theory of gravity and find the factors causing density irregularities in the framework. We calculate the evolution parameters by using these scalar functions. Moreover, we also investigate the structure scalars for dust ball. The dynamical effects on cluster of stars are examined via structure scalars in the presence of Gauss-Bonnet gravity. It is found that Gauss-Bonnet parameter representing exotic material in the cluster plays a vital role in governing the dynamics of cluster of stars.

Generalized volume-complexity for two-sided hyperscaling violating black branes

In this paper, we investigate generalized volume-complexity mathcal{C} gen for a two-sided uncharged HV black brane in d + 2 dimensions. This quantity which was recently introduced in [48], is an extension of volume in the Complexity=Volume (CV) proposal, by adding higher curvature corrections with a coupling constant λ to the volume functional. We numerically calculate the growth rate of mathcal{C} gen for different values of the hyperscaling violation exponent θ and dynamical exponent z. It is observed that mathcal{C} gen always grows linearly at late times provided that we choose λ properly. Moreover, it approaches its late time value from below. For the case λ = 0, we find an analytic expression for the late time growth rate for arbitrary values of θ and z. However, for λ ≠ 0, the late time growth rate can only be calculated analytically for some specific values of θ and z. We also examine the dependence of the growth rate on d, θ, z and λ. Furthermore, we calculate the complexity of formation obtained from volume-complexity and show that it is not UV divergent. We also examine its dependence on the thermal entropy and temperature of the black brane. At the end, we also numerically calculate the growth rate of mathcal{C} gen for the case where the higher curvature corrections are a linear combination of the Ricci scalar, square of the Ricci tensor and square of the Riemann tensor. We show that for appropriate values of the coupling constants, the late time growth rate is again linear.

Influence of EMSG on complex systems: Spherically symmetric, static case

Through this study, we intend to present a comprehensive summary of the most recent research on complex systems or complexity factor [Formula: see text] in the background of modified gravity theories (MGT). In this way, we examined [Formula: see text] for the novel MGT named [Formula: see text] gravity under the anisotropic static fluid configuration. [Formula: see text] is defined as one of the structure scalars [Formula: see text] which is trace-free component obtained from the orthogonal splitting of the Riemann tensor. Two mass approaches are used which have immense influence on [Formula: see text]. Furthermore, vanishing complexity condition is used with an eye towards finding the solution of the nonlinear system. Two extra models are explored which ensure one more condition for the purpose of comprehensive results.

Sequential warped product manifolds with a semi-symmetric metric connection

In the present paper, we study a new generalization of warped product manifolds, called sequential warped product manifolds, with respect to a semi-symmetric metric connection. We obtain relations for covariant derivatives, Riemannian curvature, Ricci curvature and scalar curvature of the sequential warped product manifolds with respect to the semi-symmetric connection, respectively, and demonstrate the relationship between them and curvatures with respect to the Levi-Civita connection. Also, we consider sequential warped product space-time models, namely sequential generalized Robertson- Walker space-times and sequential standard static space-times, with semi-symmetric metric connections and obtain conditions for such space-times to be Einstein.

Riemannian submersions endowed with a new type of semi-symmetric non-metric connection

In this paper we study relations for the covariant derivative of O?Neill?s tensor fields, Riemannian curvature, Ricci curvature and scalar curvature of the Riemannian submersion from a Riemannian manifold with respect to a new type of semi-symmetric non-metric connection to a Riemannian manifold, respectively, and demonstrate the relationship between them.

К геометрии субримановых многообразий, оснащенных канонической четверть-симметрической связностью

In this article, a sub-Riemannian manifold of contact type is under­stood as a Riemannian manifold equipped with a regular distribution of codimension-one and by a unit structure vector field orthogonal to this distribution. This vector field is called a structural. On a sub-Riemannian manifold of contact type, a quarter-symmetric connection is defined, which is associated with an endomorphism that preserves the distribution of the sub-Riemannian manifold. It is proved that if the connection under study is metric, then the endomorphism associated to it is uniquely de­fined. The structure of the associated endomorphism is found. In the case when the structure vector field is a field of infinitesimal isometries, the quarter-symmetric connection is called the canonical N-connection. An expression is found for the curvature tensor of the canonical N-connection in terms of the Riemann curvature tensor. The properties of the Schouten curvature tensor are investigated, which provide, in particular, the neces­sary symmetries of the curvature tensor of an N-connection for its sec­tional curvature to be well-defined. A relation between the sectional cur­vature of the canonical N-connection and the sectional curvature of the Levi-Civita connection is found. Necessary and sufficient conditions are found under which the sectional curvature of the N-connection and the sectional curvature of the Levi-Civita connection coincide.

Locally product-like statistical submersions

In this paper, the main identities on locally product-like statistical submersions are obtained with the aid of statistical structures and their Riemannian curvature tensors. Some examples of locally product-like statistical submersions are presented. Some results on $F$-invariant, $F^{\ast}$-invariant and antiinvariant locally product-like statistical submersions are given.

Electromagnetic effects on the complexity of static cylindrical object in f(G, T) gravity

In this paper, we investigate complexity of anisotropic cylindrical object under the influence of electromagnetic field in f(G, T) theory, where G and T indicate the Gauss–Bonnet term and trace of the stress–energy tensor, respectively. For this purpose, we calculate the modified field equations, non-conservation equation and mass distributions that assist in comprehending the structure of astrophysical objects. The Riemann tensor is divided into different structure scalars, among which one is called the complexity factor. This factor is used to measure complexity of the system due to the involvement of inhomogeneous energy density, anisotropic pressure and charge. The vanishing of the complexity factor is employed as a constraint to formulate charged static solutions for the Gokhroo–Mehra model and polytropic equation of state. We conclude that the presence of charge reduces the complexity of the anisotropic system.

Higher-derivative couplings and torsional Riemann curvature

Using the most general higher-derivative field redefinitions for the closed spacetime manifolds, we show that the tree-level couplings of the metric, B-field and dilaton at orders α′2 and α′3 that have been recently found by the T-duality, can be written in a particular scheme in terms of the torsional Riemann curvature mathcal{R} and the torsion tensor H. The couplings at order α′2 have structures mathcal{R} 3, H2 mathcal{R} 2, H6, and the couplings at order α′3 have only structures mathcal{R} 4, H2 mathcal{R} 3. Replacing mathcal{R} with the ordinary Riemann curvature, the couplings in the structure H2 mathcal{R} 3 reproduce the couplings found in the literature by the S-matrix method.

Vacua by derivative corrections in mathcal {N} = 1 supergravity with matter multiplets

We study the vacuum structures of four-dimen-sional mathcal {N} = 1 old minimal supergravity with higher derivative corrections. We find that mathcal {N} = 1 supergravity with Riemann curvature square corrections and higher derivative matter chiral multiplets induces a non-trivial de Sitter vacuum, even in the absence of superpotentials. This vacuum generically breaks supersymmetry. We show that the auxiliary fields in the gravity and the chiral multiples play important roles to generate a potential in supersymmetric higher derivative theories.

Decomposition of Riemannian Recurrent Curvature Tensor Manifolds of First Order

Takano [6] premeditated decomposition of curvature tensor in a recurrent Riemannian space. After that, Negi and Bisht [3] defined and deliberated decomposition of recurrent curvature tensor fields in a Kaehlerian manifolds of first order. We have calculated the decomposition of Riemannian recurrent curvature tensor manifolds of first order and some theorems established using the decomposition tensor field.

Functional renormalization for f(Rμνρσ) quantum gravity

We derive new functional renormalisation group flows for quantum gravity, in any dimension. The key new achievement is that the equations apply for any theory of gravity whose underlying Lagrangian $\sim f(R_{\mu\nu\rho\sigma})$ is a function of the Riemann tensor and the inverse metric. The results centrally exploit the benefits of maximally symmetric spaces for the evaluation of operator traces. The framework is highly versatile and offers a wide range of new applications to study quantum gravitational effects in extensions of Einstein gravity, many of which have hitherto been out of reach. The phase diagram and sample flows for Einstein-Hilbert gravity, Gauss-Bonnet, and selected higher-order theories of gravity are given. We also provide an algorithm to find the flow for general polynomial Riemann curvature interactions. The setup vastly enhances the reach of fixed point searches, enabling novel types of search strategies including across the operator space spanned by polynomial curvature invariants, and in extensions of general relativity relevant for cosmology. Further implications, and links with unimodular versions of gravity are indicated.

Self-gravitating spherically symmetric systems with Q h condition in chameleonic Brans–Dicke gravity

Abstract This paper is asserted to explore the self-gravitating spherically symmetric anisotropic fluids in Chameleonic Brans–Dicke theory as dark energy matter. The dissipative and non-dissipative cases for the evolution of the system are discussed evidently satisfying the quasi-homologous condition with vanishing complexity (Y TF ) factor, which is identified in the trace free part of the electric Riemann tensor in splitting of the curvature tensor. We formulate different equations through conformal tensor, mass function, shear stress tensor, scalar field to govern self-gravitating systems. A few models describe center filled fluid distribution whereas some of them have cavities surrounding the center by means of matching conditions on the boundary as well as on inner surfaces. The temperature of the respective models is also discussed here. Finally, we conclude the work by comparing it with GR.

Tachyonic cosmology with cubic contractions of the Riemann tensor

A new cosmological theory is proposed in the theoretical framework of modified gravity theories which is based on a tachyonic field non-minimally coupled with a specific topological invariant constructed with third order contractions of the Riemann tensor. After proposing the action of the cosmological model, the modified Friedmann relations and the Klein–Gordon equations are obtained, describing the corresponding geometrical corrections to the Einstein–Hilbert action. The physical features of the cosmological system are investigated by adopting the dynamical system analysis in the case of an exponential function for the geometrical coupling term. The investigation revealed that the cosmological system can explain the current accelerated expansion of the Universe and the matter dominated epoch, showing a high compatibility to the recent history of our Universe for various values of the coupling coefficients.

Modified Hawking temperatures of black holes in Lorentz violation theory

In this paper, the tunneling of scalar particles near the event horizons of Riemann space time, BTZ black hole and Schwarzschild–de Sitter black hole are investigated by applying Hamilton–Jacobi equation with Lorentz violation theory in curved space time. The modified Hamilton–Jacobi equation is derived from Klein–Gordon equation of scalar particles induced by Lorentz violation theory in curved space time. The Hawking temperatures of Riemann space time and the BTZ black hole are modified due to the effect of Lorentz violation theory. Moreover, the Bekenstein–Hawking entropy near the event horizon of Schwarzschild–de Sitter black hole is also modified due to Lorentz violation theory. It is observed that the modified values of Hawking temperatures and change in Bekenstein–Hawking entropy depend upon the ether-like vectors [Formula: see text].

Are there Einsteinian gravities involving covariant derivatives of the Riemann tensor?

We study the particle content of higher derivative theories of gravity built with contractions of the Riemann tensor and its covariant derivatives. In the absence of the latter, there is a family of theories exhibiting an Einsteinian spectrum known as generalized quasi-topological gravities. In turn, we present a no-go result for the construction of Einsteinian gravities involving covariant derivatives of the Riemann tensor. We find evidences suggesting that (truncated series) finite order Lagrangians with covariant derivatives of the Riemann tensor generically present ghosts in their spectrum. This might be interpreted as a hint of non-locality in any healthy UV completion of General Relativity.

Tidal effect in charged black hole enclosed by thin accretion disc in Rastall gravity

We analyze the tidal effects in a charged black hole enclosed by a thin accretion disc in Rastall gravity (RG). We study the radial geodesic motion in charged black hole under the influence of perfect fluid matter. The radial geodesic motion sharply depends upon the perfect fluid matter. The components of tidal force are obtained by Riemann tensor on tetrad basis and geodesic deviation equation. We solve the geodesic deviation equations analytically for the radially free-falling geodesic around a charged black hole enclosed by a thin accretion disc in RG. The behavior of the geodesic deviation vector for such a geodesic under the influence of the tidal forces is probed. The radial and angular tidal forces disappear at a certain point. The angular tidal forces increases with the increase of Rastall parameter and radius horizon of black hole.

Broken Global Symmetries and Defect Conformal Manifolds.

Just as exactly marginal operators allow one to deform a conformal field theory along the space of theories known as the conformal manifold, appropriate operators on conformal defects allow for deformations of the defects. When a defect breaks a global symmetry, there is a contact term in the conservation equation with an exactly marginal defect operator. The resulting defect conformal manifold is the symmetry breaking coset, and its Zamolodchikov metric is expressed as the two-point function of the exactly marginal operator. As the Riemann tensor on the conformal manifold can be expressed as an integrated four-point function of the marginal operators, we find an exact relation to the curvature of the coset space. We confirm this relation against previously obtained four-point functions for insertions into the 1/2 BPS Wilson loop in N=4 SYM and 3D N=6 theory and the 1/2 BPS surface operator of the 6D N=(2,0) theory.