## What is/are Curved Space?

Curved Space - Finally, we show that on curved surfaces, the motion of vortices and that of fractons agree; thereby opening a route to experimental study of the interplay between fracton physics and curved space.^{[1]}The generalized Duffin–Kemmer–Petiau (DKP) oscillator with electromagnetic interactions in the curved spacetimes is investigated.

^{[2]}We consider separately a case of Abelian extension, for which the RiemannHilbert equations of the dressing scheme are explicitly solvable and give an analogue of Penrose formula in the curved space.

^{[3]}We analyse the response of laser interferometric gravitational wave detectors using the full Maxwell equations in curved spacetime in the presence of weak gravitational waves.

^{[4]}The dynamics of low-energy electrons in general static strained graphene surface is modelled mathematically by the Dirac equation in curved space-time.

^{[5]}We consider the conformal transformation of electrodynamics in curved space.

^{[6]}We extend complete complementarity relations to curved spacetimes by considering a succession of infinitesimal local Lorentz transformations, which implies that complementarity remains valid as the quanton travels through its world line and the complementarity aspects in different points of spacetime are connected.

^{[7]}The speed of the graviton and photon are calculated in a curved spacetime modified by GUP, assuming that these particles have a small mass.

^{[8]}The article describes how thinking about black holes as the most efficient kind of computers in the physical universe also paves the way for developing new ideas about such issues as black hole information paradox, the possibility of emulating the properties of curved space-time with the collective quantum behaviors of certain kind of condensate fluids, ideas of holographic spacetime, gravitational thermodynamics and entropic gravity, the role of quantum entanglements and non-locality in the construction of spacetime, spacetime geometry and the nature of gravitation and dark energy and dark matter, etc.

^{[9]}In this study, we extended the Brownian motion theory in a curved space-time come from a strong gravitational field on the Schwarzschild black hole.

^{[10]}Even close to 10 years after the intervention, the participants remembered several key concepts (such as curved space-time).

^{[11]}We investigate the generalized form of Duffin–Kemmer–Petiau (DKP) equation in the presence of both a position-dependent electrical field and curved spacetime for the 2-dimensional anti-de Sitter sp.

^{[12]}Neutrino trajectories in curved spacetime as well as the particle spin evolution in dense matter of an accretion disk and in the magnetic field are accounted for exactly.

^{[13]}Our theory provides a convenient and powerful tool for investigations of radiation in curved space.

^{[14]}In this paper the Feynman Green function for Maxwell’s theory in curved space-time is studied by using the Fock–Schwinger–DeWitt asymptotic expansion; the point-splitting method is then applied, since it is a valuable tool for regularizing divergent observables.

^{[15]}In this paper we solve the Dirac equation with the Coulomb scalar U(r) and vector V(r) potentials and type-Mie tensor potential in curved space-time whose metric is of type ds2=(1+α2U(r))2(dt2−dr2)−r2dθ2−r2sin2θdϕ2 with spherical symmetry.

^{[16]}The new perturbed metric is determined over the multiplication of isolated background metric of curved space-time for two different massive sources in post-Newtonian theory.

^{[17]}In curved spacetime, nonrelativistic string theory is defined by a renormalizable quantum nonlinear sigma model in background fields, following certain symmetry principles that disallow any deformation towards relativistic string theory.

^{[18]}The partial differential equation in curved space whose solution is the electrostatic potential A0 is obtained following Whittaker (Proc.

^{[19]}We consider scalar fields in curved space-time with local couplings and masses.

^{[20]}This would help us to better understand some of possible applications in astrophysics and cosmology, such as the natural phenomenon of a distinctive gravitational wave signature of merging two black holes associated with curved space–time in general relativity.

^{[21]}In the presence of suitable vortex flows, the density fluctuations are governed by the massive Klein-Gordon equation on a (2+1) curved spacetime, possessing an ergoregion and an event horizon.

^{[22]}However, little work has been completed on the utilisation of morphing composites in the design of curved space structures, e.

^{[23]}The most common methods are to use Noether's first theorem with the 4-parameter Poincare translation, or to write the action in a curved spacetime and perform variation with respect to the metric tensor, then return to a Minkowski spacetime.

^{[24]}The refractive index and curved space relation is formulated using the Riemann-Christoffel curvature tensor.

^{[25]}Atomic structure is described with discrete coordinates in Minkowski space, while the atom itself resides in the curved space-time continuum of the gravitational field, the background space of quantum gravity.

^{[26]}The scheme is a theory describing the dynamics of quantum fields propagating in a curved spacetime background, described by a Lorentzian manifold with a general classical metric gμν.

^{[27]}In this chapter, which forms the central chapter of Part II, the effective action of quantum matter fields in curved spacetime is formulated in terms of functional integrals.

^{[28]}We find exact multi-instanton solutions to the self-dual Yang–Mills equation on a large class of curved spaces with [Formula: see text] isometry, generalizing the results previously found on [Formula: see text].

^{[29]}We investigate how pure-state Einstein-Podolsky-Rosen correlations in the internal degrees of freedom of massive particles are affected by a curved spacetime background described by extended theories of gravity.

^{[30]}These findings greatly extend the potential of using unitarity to bootstrap cosmological observables and to restrict the space of consistent effective field theories on curved spacetimes.

^{[31]}Quantum effects in non-inertial frames and curved space-time have been studied for decades.

^{[32]}We show that the equation of states computed in the curved spacetime of these stars depend explicitly on the metric function.

^{[33]}We establish and generalize this relation to curved spaces by coupling the undeformed theory to 2d gravity.

^{[34]}They are spacetime structures that can potentially exist in curved spacetimes, so in a very wide class of gravity models.

^{[35]}The image polarization is determined by effects including the orientation of the magnetic field in the emitting region, relativistic motion of the gas, strong gravitational lensing by the black hole, and parallel transport in the curved spacetime.

^{[36]}As is well-known, it is very difficult to solve wave equations in curved space-time.

^{[37]}In the target space, this sigma model describes classical strings propagating in a curved spacetime background, whose geometry is described by two distinct metric fields.

^{[38]}Physicists possess an intuitive awareness of Euclidian space and time and Galilean transformation, and are then challenged with Minkowski space-time and Einstein’s curved space-time.

^{[39]}String theory’s holographic QCD duality makes predictions for hadron physics by building models that live in five-dimensional (5D) curved space.

^{[40]}The first one allows us to consider refractivity as spacetime curvature while the second one is used to determine the time/frequency transfers occurring in a curved spacetime.

^{[41]}We study the self-gravitating Abrikosov vortex in curved space with and with-out a (negative) cosmological constant, considering both singular and non-singular solutions with an eye to hairy black holes.

^{[42]}First, a generalized formalism of neutrino (spinor field) interaction with a classical scalar field in curved space-time is presented.

^{[43]}In practice one has to calculate the contribution of the transverse-traceless component of the metric perturbation on a curved spacetime background.

^{[44]}First, the Klein-Gordon equation and Hamilton-Jacobi equation of bosons that are corrected by Lorentz symmetry violation theory in curved space-time are obtained.

^{[45]}Einstein’s general relativity theory including the gravitational field can be expressed under a condensed tensor formulation as E R − Rg = T where E defines the geometry via a curved space-time structure (R) over the gravity field (1/2Rg), embedded in a matter distribution T The fundamental (ten non-linear partial differential) equations of the gravitational field are a kind of the space-time machine using the curvature of a four-dimensional space-time to engender the gravity field carrying away material structures.

^{[46]}First, the minimal subtraction renormalization group in curved space is formulated.

^{[47]}This points are applied in the forms of open and flexible spaces for exploration, easy access to green open spaces, spaces that suits the ergonomics of children, special corners that invoke children’s sense of place, the use of natural colours and curved spaces, and the use of soft materials and gardening to stimulate children’s sense of touch.

^{[48]}This condition is introduced according to the definition of the position operator in curved space.

^{[49]}For our predictions, we trace curvilinear rays to the line of sight using the full set of equations from Hamiltonian optics for a dispersive medium in curved spacetime.

^{[50]}

## quantum field theory

We theoretically investigate superradiance effects in quantum field theories in curved space-times by proposing an analogue model based on Bose--Einstein condensates subject to a synthetic vector potential.^{[1]}Solomon claims that this model violates both the classical energy conditions of special relativity and the quantum energy conditions of quantum field theory in curved spacetime.

^{[2]}It is hoped that probability theories, such as quantum field theories in curved space-time, might be adaptable to the general relativity isoframe concept introduced herein.

^{[3]}Even in the framework of the usual perturbative quantum field theory, there are several approaches leading to theoretically satisfactory models of quantum gravitational effects, starting from quantum field theory in curved spacetime.

^{[4]}A detailed calculation is given by using Boltzmann equations and decay rates obtained using formalism of quantum field theory in curved spacetime.

^{[5]}Success of S-matrix in quantum field theory in Minkowski spacetime naturally leads to the question of construction of S-matrix in a general curved spacetime.

^{[6]}Part II discusses basic aspects of quantum field theory in curved spacetime and perturbative quantum gravity.

^{[7]}Thereby, we establish a local counterpart, and microscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamical laws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black hole dynamics originally shown for stationary black holes by Bardeen, Carter and Hawking.

^{[8]}Violation of the null energy condition plays an important role both in the general theory of relativity and quantum field theory in curved spacetimes.

^{[9]}This is a model example of quantum field theory in curved space–time.

^{[10]}In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes.

^{[11]}Readers will discover the fantastical realm of dark matter, quantum field theory, and curved spacetimes that modern physics has revealed, while also confronting uncomfortable truths about the social dynamics that have led to these discoveries.

^{[12]}The needed formalism of quantum field theory in curved spacetime will be summarized, and applied to the example of scalar particle creation in a spatially flat universe.

^{[13]}We obtain the emitted power using the framework of Quantum Field Theory in Curved Spacetimes at tree level.

^{[14]}We study the thermalization of smeared particle detectors that couple locally to any operator in a quantum field theory in curved spacetimes.

^{[15]}We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics.

^{[16]}These results are interpreted in the language of quantum field theories on curved space-times as a breakdown of the standard semiclassical theory of the backreaction.

^{[17]}We unify and generalize the notions of vacuum and amplitude in linear quantum field theory in curved spacetime.

^{[18]}Matter is described by quantum field theory on curved space-time, whereas gravity is described by a space-time metric which satisfies Einstein's field equations.

^{[19]}We propose and develop a measurement scheme for quantum field theory (QFT) in curved spacetimes, in which the QFT of interest, the “system”, is dynamically coupled to another, the “probe”, in a compact spacetime region.

^{[20]}This chapter describes the quantization of field theories on curved spacetimes, highlighting the differences between quantum field theory on Minkowski space.

^{[21]}The Entropic Dynamics reconstruction of quantum mechanics is extended to quantum field theory in curved space-time.

^{[22]}I consider the problem of reconciling "tunneling" approaches to black-hole radiation with the treatment by quantum field theory in curved space-time.

^{[23]}These lecture notes give an exposition of microlocal analysis methods in the study of Quantum Field Theory on curved spacetimes.

^{[24]}Such first-order equation contains, in the imaginary part of its complex solutions, the complicated second-order field equation that typically arises for the time-dependent frequency of the perturbations in the context of quantum field theory in curved spacetimes.

^{[25]}Inflation is most often described using quantum field theory (QFT) on a fixed, curved spacetime background.

^{[26]}It is particularly useful in the study of Quantum Field Theory, especially in external classical background, typically in curved spacetime, and in statistical mechanics.

^{[27]}Quantization of fields on backgrounds provided by General Relativity is the primary cornerstone of Quantum field theory in curved spaces.

^{[28]}Special relativity -- Riemannian geometry -- Introduction to general relativity -- General relativity -- Einstein's equations -- Black holes -- Kruskal-Szekeres coordinates and geodesics of the Schwarzschild black hole -- Conformal compactifications and Penrose diagrams -- Penrose diagrams of charged & rotating black holes -- Rotating black holes and black hole mechanics -- Black hole mechanics and thermodynamics -- Black hole thermodynamics -- Black holes and entropy -- Hawking and Unruh radiation -- Quantum field theory in curved space-time backgrounds -- Unruh und Hawking effect -- Information loss paradox -- Solitons in String Theory -- Brane solutions -- Dimensional reduction and black holes -- Black holes in string theory from p/D-branes -- Black hole microstate counting -- Asymptotic symmetries in general relativity and black hole hair -- Asymptotic symmetries of 4D space-time geometries -- BMS charges -- The gravitational memory effect -- Current research on BMS-like transformations and charges of black holes -- Quantum hair and quantum black hole vacua.

^{[29]}In the framework of open quantum systems, and combining with the quantum field theory in curved space–time, we study the geometric phase for a static two-level atom immersed in a thermal bath of a.

^{[30]}The functional Schrodinger representation of a nonlinear scalar quantum field theory in curved space-time is shown to emerge as a singular limit from the formulation based on precanonical quantization.

^{[31]}We use the quantum field theory in curved spacetime framework to obtain the emitted power of radiation by computing the one-particle emission amplitude of scalar particles in the curved background.

^{[32]}Keywords: quantum field theory on curved space, physics of the early universe, inflation.

^{[33]}Hawking radiation is one of the most intriguing and elusive predictions of quantum field theory in curved spacetime.

^{[34]}Analogue gravity can be used to reproduce the phenomenology of quantum field theory in curved spacetime and in particular phenomena such as cosmological particle creation and Hawking radiation.

^{[35]}In the studies of quantum field theory in curved spacetime, the ambiguous concept of vacuum state and the particle content is a long-standing debatable aspect.

^{[36]}The aim of this manuscript is to provide a complete and precise formulation of the renormalization picture for perturbative Quantum Field Theory (pQFT) on general curved spacetimes introduced by R.

^{[37]}

## generalized uncertainty principle

In this paper, we solve the Dirac Equation in curved space–time, modified by the generalized uncertainty principle, in the presence of an electromagnetic field.^{[1]}Considering that curved space-time in coordinate or momentum representation should be curved, this paper proposes a new high-order generalized uncertainty principle by modifying the coordinate and momentum operator simultaneously, which could give a self-consistent phenomenological explanation for the existence of the minimum observable length.

^{[2]}

## General Curved Space

Success of S-matrix in quantum field theory in Minkowski spacetime naturally leads to the question of construction of S-matrix in a general curved spacetime.^{[1]}Due to the absence of symmetries under time and spatial translations in a general curved spacetime, the energy and momentum of matter is not conserved as it is in flat space.

^{[2]}Then we consider quantum gravity effects in the vacuum energy of a scalar field in $\mathbb{M}_3 \times S^1$ where $\mathbb{M}_3$ is a general curved spacetime.

^{[3]}We investigate the particle-antiparticle symmetry of the gravitationally coupled Dirac equation, both on the basis of the gravitational central-field problem and in general curved space-time backgrounds.

^{[4]}We elaborate in detail on the advantages of the Hadamard method for the general definition of composite operators in general curved spacetimes, and speculate on possible causes for the appearance of the Pontryagin density in other calculations.

^{[5]}In this work we extend the relativistic hydrodynamics to third order in the gradient expansion for neutral fluids in a general curved spacetime of d dimensions.

^{[6]}A non linear spin 1 field equation is proposed in a general curved space time.

^{[7]}The aim of this manuscript is to provide a complete and precise formulation of the renormalization picture for perturbative Quantum Field Theory (pQFT) on general curved spacetimes introduced by R.

^{[8]}

## Dimensional Curved Space

First, we study the Weyl anomaly for a non-conformal free scalar in a four-dimensional curved spacetime.^{[1]}We consider the Weyl–Yang gauge theory of gravitation in a (4 + 3)-dimensional curved space-time within the scenario of the non-Abelian Kaluza–Klein theory for the source and torsion-free limits.

^{[2]}The equation set is general field equations in three-dimensional flat space, instead of that in four-dimensional curved spacetime.

^{[3]}By exploiting this duality between changes in geometry, and changes in local unitaries, we show that the spacetime-dependent QW simulates the Dirac equation in (2 + 1)–dimensional curved spacetime.

^{[4]}A bstractWe discuss consistency at the quantum level in the rigid N$$ \mathcal{N} $$ = 1 supersymmetric field theories with a U(1)R symmetry in four-dimensional curved space which are formulated via coupling to the new-minimal supergravity background fields.

^{[5]}Relativistic geodesy: Science representing the Earth (or any planet), including the measurement of its gravitational field, in a four-dimensional curved spacetime using differential-geometric methods in the framework of Einstein's theory of gravitation (General Relativity).

^{[6]}

## Negatively Curved Space

LST is non-local so it is not obvious which spaces it can be defined on; we show that holography implies that the theory cannot be put on negatively curved spaces, but only on spaces with zero or positive curvature.^{[1]}The main new tools which allow us to obtain such representations are various higher-point propagator identities, which can be interpreted as generalizations of the well-known flat space star-triangle identity, and which compute integrals over products of three bulk-to-bulk and/or bulk-to-boundary propagators in negatively curved spacetime.

^{[2]}In this book, we study equidistribution and counting problems concerning locally geodesic arcs in negatively curved spaces endowed with potentials, and we deduce, from the special case of tree quotients, various arithmetic applications to equidistribution and counting problems in non-Archimedean local fields.

^{[3]}First we analyze the behavior of chiral fermionic matter on negatively curved space which typically exhibit an enhanced tendency towards chiral symmetry breaking due to the background curvature.

^{[4]}Specifically, Coulomb explosions mimic the non-singular open cosmologies in negatively curved spaces, while breathing modes in conductors model oscillatory universes including the anti-de Sitter space.

^{[5]}

## Euclidean Curved Space

In this article, we propose a Bayesian embedded spherical topic model (ESTM) that combines both knowledge graph and word embeddings in a non-Euclidean curved space, the hypersphere, for better topic interpretability and discriminative text representations.^{[1]}When it comes to aether, a subject rarely mentioned today, it appears to be Isaac Newton’s absolute time and space, modified to fit the Lorentz transformations and the non-Euclidean curved space of Einstein’s General Relativity.

^{[2]}

## Stationary Curved Space

We study the analogy between propagation of light rays in a stationary curved spacetime and in a toroidal (meta-)material.^{[1]}We derive the Mandelstam-Tamm time-energy uncertainty relation for neutrino oscillations in a generic stationary curved spacetime.

^{[2]}

## Two Curved Space

In the second part we present a study first of the Kepler problem and then of the Kepler problem with additional nonlinear terms in these two curved spaces, S 3 (κ > 0) and H 3 (κ < 0).^{[1]}At the same time, many data analysis and classification problems eventually reduce to an optimization, in which the criteria being minimized can be interpreted as the distortion associated with a mapping between two curved spaces.

^{[2]}

## Highly Curved Space

X-ray quasi-periodic oscillations (QPOs) in AGN allow us to probe and understand the nature of accretion in highly curved space-time, yet the most robust form of detection (i.^{[1]}These effects strongly affect the destiny of observables in highly curved space– times.

^{[2]}

## curved space time

What was supposed to have happened was that matter ‘curved and warped space-time’ and when physical masses of this matter were ‘falling’ or were in orbit, they were actually following a straight line in this curved space time which created stretched holes in the ‘fabric of space-time’.^{[1]}In this article we initiate the study of 1+ 2 dimensional wave maps on a curved space time in the low regularity setting.

^{[2]}A non linear spin 1 field equation is proposed in a general curved space time.

^{[3]}

## curved space effect

We consider the initial times of $\chi\chi\,\rightarrow\,\phi\phi$ and observe that the curved space effects promote formation of Bose-Einstein condensate of $\phi$ particles.^{[1]}Even though a naive extrapolation of the linear Regge trajectory on flat space implies a violation of the Higuchi bound (a unitarity bound on the mass of higher-spin particles in de Sitter space), the curved space effects turn out to modify the trajectory to respect the bound.

^{[2]}

## curved space whose

The partial differential equation in curved space whose solution is the electrostatic potential A0 is obtained following Whittaker (Proc.^{[1]}We establish a local monotonicity formula for mean curvature flow into a curved space whose metric is also permitted to evolve simultaneously with the flow, extending the work of Ecker (Ann Math (2) 154(2):503–525, 2001), Huisken (J Differ Geom 31(1):285–299, 1990), Lott (Commun Math Phys 313(2):517–533, 2012), Magni, Mantegazza and Tsatis (J Evol Equ 13(3):561–576, 2013) and Ecker et al.

^{[2]}