This document summarizes research on thin-shell wormholes and gravastars. It first provides background on wormholes and motivations for studying thin-shell constructions. It then describes the thin-shell method using Israel junction conditions to minimize exotic matter at the wormhole throat. Specific thin-shell wormhole models are constructed using the Hayward, linear gas, Chaplygin gas, generalized Chaplygin gas, and modified generalized Chaplygin gas equations of state. Stability analyses of these thin-shell wormholes under perturbations are presented through plots of the parameter spaces allowing for stable configurations.
This document summarizes Ali Övgün's PhD thesis defense on studies of thin-shell wormholes and thin-shell theories. The defense covered 4 of Övgün's 19 publications on the topic. Key points included:
1) Hawking radiation of traversable wormholes was calculated, finding that wormhole throats radiate "phantom energy" or dark energy, reducing the throat size and entropy over time.
2) Thin-shell wormhole construction was discussed, using the Darmois-Israel formalism to match interior and exterior spacetimes across a thin shell or throat, and calculate the surface energy and pressure from extrinsic curvature.
3) One publication presented a particular thin-shell
Hidden gates in universe: Wormholes UCEN 2017 by Dr. Ali Ovgun
Gravity at UCEN 2017: Black holes and Cosmology, November 22, 23 and 24, 2017
The meeting take place at Universidad Central de Chile.
http://www2.udec.cl/~juoliva/gravatucen2017.html
This document contains lecture notes on mechanics of solids and structures from the University of Manchester. It covers topics related to centroids, moments of area, beams, and bending theory. Specifically, it provides definitions and examples of centroids, first and second moments of area, and introduces beam supports and equilibrium, beam shear forces and bending moments, and bending theory. The contact information for the lecturer, Dr. D.A. Bond, is also provided at the top.
M. Dimitrijević, Noncommutative models of gauge and gravity theoriesSEENET-MTP
- The document describes a talk on noncommutative geometry and gravity theories given at a workshop in Serbia.
- Noncommutative geometry arises in string theory and could help address problems in quantum gravity and the standard model. The talk presents an approach using a star product to represent noncommutative algebras.
- Actions for noncommutative gauge theory and gravity are discussed. For gravity, a deformation of the MacDowell-Mansouri action is proposed based on the Seiberg-Witten map. This leads to modified field equations and corrections to the Einstein-Hilbert and cosmological constant terms.
The document discusses methods for calculating the volumes of solids obtained by revolving a region about an axis. It covers the disk method, washer method, and examples of revolving regions about the x-axis and y-axis. Regions are revolved between curves and axes over intervals to obtain solids, and formulas are provided for calculating volumes based on disks, washers, or cylindrical shells. Examples demonstrate setting up and solving problems to find volumes of solids using these methods.
The document discusses methods for calculating the volumes of solids obtained by revolving a region about an axis. It covers the disk method, washer method, and examples of revolving regions about the x-axis and y-axis. Regions are revolved between curves and axes over intervals to obtain solids, and formulas are provided for calculating volumes based on disks, washers, or cylindrical shells. Examples demonstrate setting up and solving problems by identifying graphs, writing formulas, and calculating volumes of revolution.
This document discusses different methods for calculating the volume of solids of revolution:
- The disk method is used when the axis of revolution is part of the boundary and revolves perpendicular strips into disks. Volume is calculated by integrating the area of disks.
- The washer/ring method is used when the strips revolve perpendicular but not touching the axis, forming rings. Volume subtracts the inner radius area from the outer radius area.
- The shell method is used when strips revolve parallel to the axis, forming cylindrical shells. Volume multiplies the shell area by its thickness.
The document provides examples and homework problems for students to practice calculating volumes of solids of revolution using these different methods.
This document summarizes Ali Övgün's PhD thesis defense on studies of thin-shell wormholes and thin-shell theories. The defense covered 4 of Övgün's 19 publications on the topic. Key points included:
1) Hawking radiation of traversable wormholes was calculated, finding that wormhole throats radiate "phantom energy" or dark energy, reducing the throat size and entropy over time.
2) Thin-shell wormhole construction was discussed, using the Darmois-Israel formalism to match interior and exterior spacetimes across a thin shell or throat, and calculate the surface energy and pressure from extrinsic curvature.
3) One publication presented a particular thin-shell
Hidden gates in universe: Wormholes UCEN 2017 by Dr. Ali Ovgun
Gravity at UCEN 2017: Black holes and Cosmology, November 22, 23 and 24, 2017
The meeting take place at Universidad Central de Chile.
http://www2.udec.cl/~juoliva/gravatucen2017.html
This document contains lecture notes on mechanics of solids and structures from the University of Manchester. It covers topics related to centroids, moments of area, beams, and bending theory. Specifically, it provides definitions and examples of centroids, first and second moments of area, and introduces beam supports and equilibrium, beam shear forces and bending moments, and bending theory. The contact information for the lecturer, Dr. D.A. Bond, is also provided at the top.
M. Dimitrijević, Noncommutative models of gauge and gravity theoriesSEENET-MTP
- The document describes a talk on noncommutative geometry and gravity theories given at a workshop in Serbia.
- Noncommutative geometry arises in string theory and could help address problems in quantum gravity and the standard model. The talk presents an approach using a star product to represent noncommutative algebras.
- Actions for noncommutative gauge theory and gravity are discussed. For gravity, a deformation of the MacDowell-Mansouri action is proposed based on the Seiberg-Witten map. This leads to modified field equations and corrections to the Einstein-Hilbert and cosmological constant terms.
The document discusses methods for calculating the volumes of solids obtained by revolving a region about an axis. It covers the disk method, washer method, and examples of revolving regions about the x-axis and y-axis. Regions are revolved between curves and axes over intervals to obtain solids, and formulas are provided for calculating volumes based on disks, washers, or cylindrical shells. Examples demonstrate setting up and solving problems to find volumes of solids using these methods.
The document discusses methods for calculating the volumes of solids obtained by revolving a region about an axis. It covers the disk method, washer method, and examples of revolving regions about the x-axis and y-axis. Regions are revolved between curves and axes over intervals to obtain solids, and formulas are provided for calculating volumes based on disks, washers, or cylindrical shells. Examples demonstrate setting up and solving problems by identifying graphs, writing formulas, and calculating volumes of revolution.
This document discusses different methods for calculating the volume of solids of revolution:
- The disk method is used when the axis of revolution is part of the boundary and revolves perpendicular strips into disks. Volume is calculated by integrating the area of disks.
- The washer/ring method is used when the strips revolve perpendicular but not touching the axis, forming rings. Volume subtracts the inner radius area from the outer radius area.
- The shell method is used when strips revolve parallel to the axis, forming cylindrical shells. Volume multiplies the shell area by its thickness.
The document provides examples and homework problems for students to practice calculating volumes of solids of revolution using these different methods.
The document discusses calculating volumes of revolution by rotating an area about the x-axis or y-axis. It provides the formulas for finding these volumes using integration, with examples of setting up the integrals to calculate specific volumes. It also covers cases where the curve needs to be rearranged in order to substitute it into the integral when rotating about the y-axis.
This document discusses using the method of cylindrical shells to calculate volumes of solids of revolution. It provides an example calculating the volume of the solid obtained by rotating the region between the curves y=2x^2 - x^3 and y=0 about the y-axis. The method involves imagining the solid as being composed of cylindrical shells and using the formula V=2π∫_{a}^{b} x*f(x) dx to calculate the volume, where f(x) is the height of each shell.
This document discusses methods for calculating the volume of solids of revolution generated by rotating plane regions about lines. It covers the disk and washer methods for rotation about axes in the plane of the region. Formulas are provided for calculating volumes of revolution in Cartesian, parametric, and polar coordinate systems by integrating the area of thin cross-sectional disks/washers. Several examples demonstrate applying these formulas to find the volumes of solids generated by rotating curves like ellipses, lines, and cardioids.
The document discusses the concepts of centroid and centre of gravity. The centroid, also known as the center of gravity, is the point location of an object's average position of weight. For symmetrical objects, the centroid will be at the exact center, but for irregularly shaped objects it depends on the weight distribution. Various methods are described for calculating the centroid of standard shapes, composite objects, and through integration. The centroid is important for balancing objects and determining their stability.
1. The document discusses different methods for calculating the volume of solids obtained by revolving a region about an axis, including the disk method, washer method, and examples applying each method.
2. The disk method is used when the region is revolved about the x-axis, and the washer method is used when the region is revolved about the y-axis.
3. Examples are provided to demonstrate calculating the volume of solids generated by revolving regions between curves over intervals about the x-axis and y-axis.
The document provides an overview of functions of a complex variable. Some key points:
1) Functions of a complex variable provide powerful tools in theoretical physics for quantities that are complex variables, evaluating integrals, obtaining asymptotic solutions, and performing integral transforms.
2) The Cauchy-Riemann equations are a necessary condition for a function f(z) = u(x,y) + iv(x,y) to be differentiable at a point. If the equations are satisfied, the function is analytic.
3) Cauchy's integral theorem states that if a function f(z) is analytic in a simply connected region R, the contour integral of f(z) around any closed path in
The document discusses using the shell method to find the volume of solids of revolution. The shell method uses cylindrical shells to calculate volume, whereas the disk method uses disks. It is best to use the shell method when the representative rectangle forming each shell layer is parallel to the axis of rotation, as this allows identifying the radius using the correct variable in the integral. The disk method is best when the rectangle is perpendicular to the axis of rotation. Examples are provided to illustrate setting up integrals for both methods.
This document contains conceptual problems and questions about static equilibrium and elasticity. It includes the following:
1) True/false questions about the conditions for static equilibrium.
2) A question about the tension in different parts of a wire made of aluminum and steel.
3) Derivations of the expression for Young's modulus based on an atomic model and an estimate of the atomic force constant.
4) Questions involving calculating tensions, normal forces, and torque in situations involving objects in equilibrium, such as masts on sailboats and cylinders on steps.
5) Questions involving static equilibrium conditions to solve for quantities like the location of a person's center of gravity and the height a ladder can
ICOMASEF 2013: Influence of the shape on the roughness-induced transitionJean-Christophe Loiseau
- The document discusses how the shape of three-dimensional wall roughness elements can influence transition to turbulence in a boundary layer.
- Direct numerical simulations and stability analyses are used to compare the effects of a cylinder and bump shaped roughness element. The bump induces transition at a higher Reynolds number than the cylinder.
- For both shapes, the most unstable modes are localized over downstream low-speed streaks and extract energy from spanwise shear, resembling localized streak instabilities. However, the bump produces weaker, more localized streaks, resulting in an isolated unstable mode rather than a branch.
(1) The document discusses fluid dynamics concepts including density, pressure, buoyancy, continuity, and Bernoulli's equation. It provides conceptual explanations and sample calculations for comparing volumes and densities of objects, calculating buoyant forces, and relating pressure, velocity, and flow rates in pipes.
(2) One example calculates the force needed to lift a car using a hydraulic lift, where the pressure applied must be the same at both the large and small pistons.
(3) Another example uses continuity and Bernoulli's equations to relate the velocity and pressure of water flowing at different diameters in a pipe. It finds the velocity and pressure change when the pipe narrows.
Dynamical t/U expansion for the doped Hubbard modelWenxingDing1
This document presents a dynamical mean field theory approach to studying the doped Hubbard model. It introduces a new slave spin representation that separates the spin and charge degrees of freedom. Within this representation, the Hubbard model is mapped to an effective spin Hamiltonian and fermion Hamiltonian. A static mean field theory is then developed by decoupling these Hamiltonians. Expressions are derived for the physical Green's functions in terms of the slave spin and fermion Green's functions within the mean field approximation. Finally, the slave spin Green's function is obtained by solving its equation of motion within the atomic limit.
this is the ppt on application of integrals, which includes-area between the two curves , volume by slicing , disk method , washer method, and volume by cylindrical shells,.
this is made by dhrumil patel and harshid panchal.
The document summarizes research on the imbalanced antiferromagnet in an optical lattice. Key points:
1) A mean-field analysis of the Fermi-Hubbard model at half filling predicts a Mott insulator phase transition and Néel antiferromagnetic ordering below a critical temperature.
2) Introducing spin imbalance splits the spin-wave dispersion and leads to three phases in the mean-field phase diagram: Néel, canted, and Ising phases.
3) Topological excitations called merons are predicted at low temperatures, whose unbinding drives a Kosterlitz-Thouless transition lowering the critical temperature compared to mean-field theory.
The document discusses computational aspects of stochastic phase-field models. It begins by motivating the inclusion of thermal noise in phase-field simulations through examples of dendrite formation. It then provides background on the deterministic phase-field and Allen-Cahn models before introducing the stochastic Allen-Cahn equation with additive white noise. The remainder of the document discusses the importance of studying this problem both theoretically and computationally, as well as outlining the topics to be covered in more depth.
This document contains 21 problems solving for the moment of inertia of various shapes. The shapes include rectangles, triangles, semicircles, and composite shapes. For each problem, the relevant dimensions are given, a calculation is shown, and the numerical value of the moment of inertia about the specified axis is provided. Formulas for the moment of inertia of common shapes like rectangles and triangles are used.
The document summarizes key equations in linear elasticity, including:
1. Strain-displacement relations, compatibility relations, and equilibrium equations form the general field equation system with 15 equations and 15 unknowns (displacements, strains, stresses).
2. Hooke's law relates stresses and strains.
3. Boundary conditions include traction, displacement, and mixed conditions and are specified on surfaces.
4. Fundamental problem classifications are the traction problem, displacement problem, and mixed problem.
5. The stress and displacement formulations eliminate unknowns to reduce the field equations to equations involving only stresses or only displacements.
The document discusses various diffraction techniques used to determine crystal structures, including X-ray diffraction (XRD), electron diffraction, and neutron diffraction. It provides details on each technique, such as the wavelength used, how the beams interact with materials, and common applications. Neutron diffraction is described as penetrating deeply into materials and providing information on magnetic structures. Examples of common crystal structures are also presented, such as face-centered cubic and hexagonal close-packed structures. Interstitial sites and packing fractions are discussed for relating structure to properties of materials.
Surveillance refers to the task of observing a scene, often for lengthy periods in search of particular objects or particular behaviour. This task has many applications, foremost among them is security (monitoring for undesirable behaviour such as theft or vandalism), but increasing numbers of others in areas such as agriculture also exist. Historically, closed circuit TV (CCTV) surveillance has been mundane and labour Intensive, involving personnel scanning multiple screens, but the advent of reasonably priced fast hardware means that automatic surveillance is becoming a realistic task to attempt in real time. Several attempts at this are underway.
This document summarizes an analysis of the elastic behavior of multiply coated fiber-reinforced composites. It presents:
1) A model of an n-layered transversely isotropic cylindrical inclusion surrounded by a transversely isotropic matrix, with the goal of estimating overall elastic moduli.
2) A method using "transfer matrices" to solve for displacement and stress fields in each layer subjected to uniform boundary conditions.
3) Derivations of average strains and stresses in each layer and inclusion for various loading modes, including longitudinal shear, normal tension, and in-plane hydrostatic.
4) Recursive algorithms to predict the longitudinal shear modulus, Young's modulus, Poisson's ratio, and plane-
1) The document derives the Bogoliubov-de Gennes (BdG) equation to solve for the eigenstates and eigenenergies of a superconductor with a single static magnetic impurity.
2) Using the BdG equation and Nambu spinor formalism, analytic expressions are obtained for the subgap Shiba states induced by the magnetic impurity. The Shiba state energies are given by E0 = ∆(1 - α2)/(1 + α2), where α is the dimensionless impurity strength.
3) The BdG spinor eigenstates for the Shiba states are derived. They consist of spin-up electrons and spin-down holes for one state, and spin
The document discusses minimal surfaces and the Willmore conjecture. It defines minimal surfaces as surfaces with zero mean curvature and provides examples such as the catenoid, helicoid, and surfaces discovered by Scherk, Enneper, and Costa. The Willmore conjecture proposes a lower bound of 2π2 for the Willmore energy of any smooth immersed torus in R3, with equality achieved by the Clifford torus. The conjecture was recently proved after many partial results over decades of work by various mathematicians.
El 7 de noviembre de 2016, la Fundación Ramón Areces organizó el Simposio Internacional 'Solitón: un concepto con extraordinaria diversidad de aplicaciones inter, trans, y multidisciplinares. Desde el mundo macroscópico al nanoscópico'.
The document discusses calculating volumes of revolution by rotating an area about the x-axis or y-axis. It provides the formulas for finding these volumes using integration, with examples of setting up the integrals to calculate specific volumes. It also covers cases where the curve needs to be rearranged in order to substitute it into the integral when rotating about the y-axis.
This document discusses using the method of cylindrical shells to calculate volumes of solids of revolution. It provides an example calculating the volume of the solid obtained by rotating the region between the curves y=2x^2 - x^3 and y=0 about the y-axis. The method involves imagining the solid as being composed of cylindrical shells and using the formula V=2π∫_{a}^{b} x*f(x) dx to calculate the volume, where f(x) is the height of each shell.
This document discusses methods for calculating the volume of solids of revolution generated by rotating plane regions about lines. It covers the disk and washer methods for rotation about axes in the plane of the region. Formulas are provided for calculating volumes of revolution in Cartesian, parametric, and polar coordinate systems by integrating the area of thin cross-sectional disks/washers. Several examples demonstrate applying these formulas to find the volumes of solids generated by rotating curves like ellipses, lines, and cardioids.
The document discusses the concepts of centroid and centre of gravity. The centroid, also known as the center of gravity, is the point location of an object's average position of weight. For symmetrical objects, the centroid will be at the exact center, but for irregularly shaped objects it depends on the weight distribution. Various methods are described for calculating the centroid of standard shapes, composite objects, and through integration. The centroid is important for balancing objects and determining their stability.
1. The document discusses different methods for calculating the volume of solids obtained by revolving a region about an axis, including the disk method, washer method, and examples applying each method.
2. The disk method is used when the region is revolved about the x-axis, and the washer method is used when the region is revolved about the y-axis.
3. Examples are provided to demonstrate calculating the volume of solids generated by revolving regions between curves over intervals about the x-axis and y-axis.
The document provides an overview of functions of a complex variable. Some key points:
1) Functions of a complex variable provide powerful tools in theoretical physics for quantities that are complex variables, evaluating integrals, obtaining asymptotic solutions, and performing integral transforms.
2) The Cauchy-Riemann equations are a necessary condition for a function f(z) = u(x,y) + iv(x,y) to be differentiable at a point. If the equations are satisfied, the function is analytic.
3) Cauchy's integral theorem states that if a function f(z) is analytic in a simply connected region R, the contour integral of f(z) around any closed path in
The document discusses using the shell method to find the volume of solids of revolution. The shell method uses cylindrical shells to calculate volume, whereas the disk method uses disks. It is best to use the shell method when the representative rectangle forming each shell layer is parallel to the axis of rotation, as this allows identifying the radius using the correct variable in the integral. The disk method is best when the rectangle is perpendicular to the axis of rotation. Examples are provided to illustrate setting up integrals for both methods.
This document contains conceptual problems and questions about static equilibrium and elasticity. It includes the following:
1) True/false questions about the conditions for static equilibrium.
2) A question about the tension in different parts of a wire made of aluminum and steel.
3) Derivations of the expression for Young's modulus based on an atomic model and an estimate of the atomic force constant.
4) Questions involving calculating tensions, normal forces, and torque in situations involving objects in equilibrium, such as masts on sailboats and cylinders on steps.
5) Questions involving static equilibrium conditions to solve for quantities like the location of a person's center of gravity and the height a ladder can
ICOMASEF 2013: Influence of the shape on the roughness-induced transitionJean-Christophe Loiseau
- The document discusses how the shape of three-dimensional wall roughness elements can influence transition to turbulence in a boundary layer.
- Direct numerical simulations and stability analyses are used to compare the effects of a cylinder and bump shaped roughness element. The bump induces transition at a higher Reynolds number than the cylinder.
- For both shapes, the most unstable modes are localized over downstream low-speed streaks and extract energy from spanwise shear, resembling localized streak instabilities. However, the bump produces weaker, more localized streaks, resulting in an isolated unstable mode rather than a branch.
(1) The document discusses fluid dynamics concepts including density, pressure, buoyancy, continuity, and Bernoulli's equation. It provides conceptual explanations and sample calculations for comparing volumes and densities of objects, calculating buoyant forces, and relating pressure, velocity, and flow rates in pipes.
(2) One example calculates the force needed to lift a car using a hydraulic lift, where the pressure applied must be the same at both the large and small pistons.
(3) Another example uses continuity and Bernoulli's equations to relate the velocity and pressure of water flowing at different diameters in a pipe. It finds the velocity and pressure change when the pipe narrows.
Dynamical t/U expansion for the doped Hubbard modelWenxingDing1
This document presents a dynamical mean field theory approach to studying the doped Hubbard model. It introduces a new slave spin representation that separates the spin and charge degrees of freedom. Within this representation, the Hubbard model is mapped to an effective spin Hamiltonian and fermion Hamiltonian. A static mean field theory is then developed by decoupling these Hamiltonians. Expressions are derived for the physical Green's functions in terms of the slave spin and fermion Green's functions within the mean field approximation. Finally, the slave spin Green's function is obtained by solving its equation of motion within the atomic limit.
this is the ppt on application of integrals, which includes-area between the two curves , volume by slicing , disk method , washer method, and volume by cylindrical shells,.
this is made by dhrumil patel and harshid panchal.
The document summarizes research on the imbalanced antiferromagnet in an optical lattice. Key points:
1) A mean-field analysis of the Fermi-Hubbard model at half filling predicts a Mott insulator phase transition and Néel antiferromagnetic ordering below a critical temperature.
2) Introducing spin imbalance splits the spin-wave dispersion and leads to three phases in the mean-field phase diagram: Néel, canted, and Ising phases.
3) Topological excitations called merons are predicted at low temperatures, whose unbinding drives a Kosterlitz-Thouless transition lowering the critical temperature compared to mean-field theory.
The document discusses computational aspects of stochastic phase-field models. It begins by motivating the inclusion of thermal noise in phase-field simulations through examples of dendrite formation. It then provides background on the deterministic phase-field and Allen-Cahn models before introducing the stochastic Allen-Cahn equation with additive white noise. The remainder of the document discusses the importance of studying this problem both theoretically and computationally, as well as outlining the topics to be covered in more depth.
This document contains 21 problems solving for the moment of inertia of various shapes. The shapes include rectangles, triangles, semicircles, and composite shapes. For each problem, the relevant dimensions are given, a calculation is shown, and the numerical value of the moment of inertia about the specified axis is provided. Formulas for the moment of inertia of common shapes like rectangles and triangles are used.
The document summarizes key equations in linear elasticity, including:
1. Strain-displacement relations, compatibility relations, and equilibrium equations form the general field equation system with 15 equations and 15 unknowns (displacements, strains, stresses).
2. Hooke's law relates stresses and strains.
3. Boundary conditions include traction, displacement, and mixed conditions and are specified on surfaces.
4. Fundamental problem classifications are the traction problem, displacement problem, and mixed problem.
5. The stress and displacement formulations eliminate unknowns to reduce the field equations to equations involving only stresses or only displacements.
The document discusses various diffraction techniques used to determine crystal structures, including X-ray diffraction (XRD), electron diffraction, and neutron diffraction. It provides details on each technique, such as the wavelength used, how the beams interact with materials, and common applications. Neutron diffraction is described as penetrating deeply into materials and providing information on magnetic structures. Examples of common crystal structures are also presented, such as face-centered cubic and hexagonal close-packed structures. Interstitial sites and packing fractions are discussed for relating structure to properties of materials.
Surveillance refers to the task of observing a scene, often for lengthy periods in search of particular objects or particular behaviour. This task has many applications, foremost among them is security (monitoring for undesirable behaviour such as theft or vandalism), but increasing numbers of others in areas such as agriculture also exist. Historically, closed circuit TV (CCTV) surveillance has been mundane and labour Intensive, involving personnel scanning multiple screens, but the advent of reasonably priced fast hardware means that automatic surveillance is becoming a realistic task to attempt in real time. Several attempts at this are underway.
This document summarizes an analysis of the elastic behavior of multiply coated fiber-reinforced composites. It presents:
1) A model of an n-layered transversely isotropic cylindrical inclusion surrounded by a transversely isotropic matrix, with the goal of estimating overall elastic moduli.
2) A method using "transfer matrices" to solve for displacement and stress fields in each layer subjected to uniform boundary conditions.
3) Derivations of average strains and stresses in each layer and inclusion for various loading modes, including longitudinal shear, normal tension, and in-plane hydrostatic.
4) Recursive algorithms to predict the longitudinal shear modulus, Young's modulus, Poisson's ratio, and plane-
1) The document derives the Bogoliubov-de Gennes (BdG) equation to solve for the eigenstates and eigenenergies of a superconductor with a single static magnetic impurity.
2) Using the BdG equation and Nambu spinor formalism, analytic expressions are obtained for the subgap Shiba states induced by the magnetic impurity. The Shiba state energies are given by E0 = ∆(1 - α2)/(1 + α2), where α is the dimensionless impurity strength.
3) The BdG spinor eigenstates for the Shiba states are derived. They consist of spin-up electrons and spin-down holes for one state, and spin
The document discusses minimal surfaces and the Willmore conjecture. It defines minimal surfaces as surfaces with zero mean curvature and provides examples such as the catenoid, helicoid, and surfaces discovered by Scherk, Enneper, and Costa. The Willmore conjecture proposes a lower bound of 2π2 for the Willmore energy of any smooth immersed torus in R3, with equality achieved by the Clifford torus. The conjecture was recently proved after many partial results over decades of work by various mathematicians.
El 7 de noviembre de 2016, la Fundación Ramón Areces organizó el Simposio Internacional 'Solitón: un concepto con extraordinaria diversidad de aplicaciones inter, trans, y multidisciplinares. Desde el mundo macroscópico al nanoscópico'.
Yurri Sitenko "Boundary effects for magnetized quantum matter in particle and...SEENET-MTP
This document discusses boundary conditions for quantized spinor matter fields and their impact on physical systems. It proposes a general boundary condition for spinor fields that ensures the self-adjointness of the Dirac Hamiltonian operator. This boundary condition confines the spinor matter inside spatial boundaries. The condition reduces to the MIT bag boundary condition in a specific case. Quantized spinor fields obeying this boundary condition can be used to study phenomena in hot dense magnetized matter found in particle physics and astrophysics.
Hidden Symmetries and Their Consequences in the Hubbard Model of t2g ElectronsABDERRAHMANE REGGAD
(1) The Hubbard model for t2g electrons in transition metal oxides possesses novel hidden symmetries that have significant consequences.
(2) These symmetries prevent long-range spin order at non-zero temperatures and lead to an extraordinary simplification in exact diagonalization studies.
(3) Even with spin-orbit interactions included, the excitation spectrum remains gapless due to a continuous symmetry arising from the hidden symmetries.
Backreaction of hawking_radiation_on_a_gravitationally_collapsing_star_1_blac...Sérgio Sacani
The document summarizes a study investigating the backreaction of Hawking radiation on the interior of a gravitationally collapsing star. It finds that due to the negative energy Hawking radiation inside the star, the collapse stops at a finite radius before a black hole singularity or event horizon can form, meaning the star bounces instead of collapsing fully. The interior metric of a collapsing star is equivalent to that of a closed Friedmann-Robertson-Walker universe. The Oppenheimer-Snyder model of stellar collapse is described, which provides context for analyzing the dynamics involving Hawking radiation.
Quantum gravitational corrections to particle creation by black holesSérgio Sacani
We calculate quantum gravitational corrections to the amplitude for the emission of a Hawking particle
by a black hole. We show explicitly how the amplitudes depend on quantum corrections to the exterior
metric (quantum hair). This reveals the mechanism by which information escapes the black hole. The
quantum state of the black hole is reflected in the quantum state of the exterior metric, which in turn
influences the emission of Hawking quanta.
Riemannian Laplacian Formulation in Oblate Spheroidal Coordinate System Using...iosrjce
IOSR Journal of Applied Physics (IOSR-JAP) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The document discusses thin-shell wormholes constructed from the Hayward regular black hole spacetime. It first introduces wormholes and describes the Hayward metric, which removes the singularity at the center of the black hole. It then constructs a thin-shell wormhole by cutting and pasting two Hayward spacetimes, with the shell supported by various exotic matter models. Stability analyses are performed for different exotic matter equations of state.
Question-no.docx
Chapter7
Question no’s: 2,3,4,5,6,8,10,13,14,15,17,18,19,20,21,27,28,29,31,32,33,36
Chapter 8
Question no’s: 1,2,3,4,6,7,9,13,14,15,19,20,21,22,24,26,28,29,30
ch7.pdf
Chapter 7 Laplace’s Equation: The Potential Produced by Surface Charge
7.13 Problems
7.1 Finding Charge From Potential
The potential in a spherical region r < R is '(x, y, z) = '0(z/R)
3. Find a volume charge density
Ω(r, µ) in the region r < R and a surface charge density æ(µ) on the surface r = R which together
produce this potential. Express your answers in terms of elementary trigonometric functions.
7.2 A Periodic Array of Charged Rings
Let the z-axis be the symmetry axis for an infinite number of identical rings, each with charge
Q and radius R. There is one ring in each of the planes z = 0, z = ±b, z = ±2b, etc. Exploit
the Fourier expansion in Example 1.6 to find the potential everywhere in space. Check that your
solution makes sense in the limit that the cylindrical variable Ω ¿ R, b. Hint: If IÆ(y) and KÆ are
modified Bessel functions,
I
0
Æ(y)KÆ(y) ° IÆ(y)K0Æ(y) = 1/y.
7.3 Two Electrostatic Theorems
Use the orthogonality properties of the spherical harmonics to prove the following for a function
'(r) which satisfies Laplace’s equation in and on an origin-centered spherical surface S of radius
R:
(a)
R
S
dS '(r) = 4ºR2'(0)
(b)
Z
S
dSz'(r) =
4º
3
R
4 @'
@z
ØØØØ
r=0
7.4 Make a Field Inside a Sphere
Find the volume charge density Ω and surface charge density æ which much be placed in and on a
sphere of radius R to produce a field inside the sphere of
E = °2V0
xy
R3
x̂ +
V0
R3
(y
2 ° x2)ŷ ° V0
R
ẑ.
There is no other charge anywhere. Express your answer in terms of trigonometric functions of µ
and ¡.
7.5 Green’s Formula
Let n̂ be the normal to an equipotential surface at a point P . If R1 and R2 are the principal
radii of curvature of the surface at P . A formula due to George Green relates normal derivatives
(@/@n ¥ n̂ · r) of the potential '(r) (which satisfies Laplace’s equation) at the equipotential surface
to the mean curvature of that equipotential surface ∑ = 1
2
(R°11 + R
°1
2 ):
@2'
@n2
+ 2∑
@'
@n
= 0.
Derive Green’s equation by direct manipulation of Laplace’s equation.
7.6 The Channeltron
c∞2009 Andrew Zangwill 278
Chapter 7 Laplace’s Equation: The Potential Produced by Surface Charge
The parallel plates of a channeltron are segmented into conducting strips of width b so the po-
tential can be fixed on the strips at staggered values. We model this using infinite-area plates, a
finite portion of which is shown below. Find the potential '(x, y) between the plates and sketch
representative field lines and equipotentials. Note the orientation of the x and y axes.
1 1
02 2
x
y
d
b
7.7 The Calculable Capacitor
The figure below shows a circle which has been divided into two pairs of segments with equal
arc length by a horizontal bisector and a vertical line. The positive x-axis bisects the segment
labe ...
Classical and Quasi-Classical Consideration of Charged Particles in Coulomb F...ijrap
On the basis of the theory of bound charges the calculation of the motion of the charged particle at the Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
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On the basis of the theory of bound charges the calculation of the motion of the charged particle at the
Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
the Riemanniam space-time. The comparison with the general relativity theory (GRT) and special relativity
theory (SRT) results in the Schwarzshil'd field when the particle falls on the Schwarzshil'd and Coulomb
centres is carried out. It is shown that the proton and electron can to create a stable connection with the
dimensions of the order of the classic electron radius. The perihelion shift of the electron orbit in the
proton field is calculated. This shift is five times greater than in SRT and when corrsponding substitution of
the constants it is 5/6 from GRT. By means of the quantization of adiabatic invariants in accordance with
the method closed to the Bohr and Sommerfeld one without the Dirac equation the addition to the energy
for the fine level splitting is obtained. It is shown that the Caplan's stable orbits in the hydrogen atom
coincide with the Born orbits.
The document discusses the Robertson-Walker metric and its application to cosmological redshifts. It contains the following key points:
1. The Robertson-Walker metric describes a homogeneous and isotropic universe and can account for the observed redshift of distant galaxies by considering the expansion of the universe over time.
2. The metric allows for spaces with positive, negative, or zero curvature, corresponding to closed, open, or flat geometries for the universe.
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1) The document discusses the problem of a particle sliding off a moving hemisphere, with the goal of finding the angle at which the particle's horizontal velocity is maximum.
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1) The document discusses the problem of a particle sliding off a moving hemisphere, using conservation of momentum and energy equations to derive an expression for the particle's horizontal velocity vx as a function of the angle θ.
2) Setting the derivative of vx equal to zero yields a cubic equation that determines the angle θ at which vx is maximized, corresponding to the particle losing contact with the hemisphere.
3) For the special case where the particle and hemisphere masses are equal (ratio r = 1), the cubic equation can be solved to find θ ≈ 42.9 degrees.
The Gravity Probe B experiment tested two predictions of general relativity using gyroscopes in a satellite orbiting Earth. It measured the geodetic precession and frame-dragging precession predicted by Einstein to within 0.2% and 18.4% accuracy, respectively, confirming his theory of gravitation. Technical challenges arose from the gyroscopes not being perfectly spherical, leading to greater errors than anticipated. The experiment was a decades-long effort involving NASA, Stanford University, and collaboration with other institutions.
The document presents an analytical method called the dynamic stiffness matrix approach to analyze the torsional vibrations and buckling of thin-walled beams of open section that are resting on an elastic foundation. The method is used to study a thin-walled beam that is clamped at one end and simply supported at the other. Numerical results for the natural frequencies and buckling loads are obtained for different values of warping and elastic foundation parameters.
Probing Extreme Physics With Compact ObjctsSérgio Sacani
This document discusses compact objects like white dwarfs, neutron stars, and black holes, which exist under extreme conditions of density, temperature, gravity, and magnetism. It summarizes key findings and areas of ongoing research regarding these objects. In particular, it highlights how studying the cooling of neutron stars can help constrain their interior physics, and how quantum electrodynamics effects in strong magnetic fields can be probed by observing the spectra and polarization of neutron star atmospheres.
An analysis of the symmetries of Cosmological Billiards;
Talk presented at
Fourteenth Marcel Grossmann Meeting - MG14, University of Rome "La Sapienza" , Rome, July 12-18, 2015,
Parallel Session Exact Solutions (Physical Aspects) on 14 July 2015
Simulating the universe requires accounting for gravitational interactions, setting initial conditions based on cosmological models, and evolving systems using numerical simulations. Initial conditions are generated to match observed power spectra, with particles placed according to density fluctuations. Simulations now include dark matter, gas and model galaxy formation through various techniques to account for baryonic physics. Current simulations can reproduce realistic galaxy disks, but further work is still needed to fully simulate the observed universe.
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Weak Gravitational Lensing and Gauss-Bonnet Theorem discusses using the Gauss-Bonnet theorem to calculate light deflection angles in various spacetime geometries, including:
1. Calculating the deflection angle of light near a Schwarzschild black hole using the optical geometry and Gaussian curvature.
2. Computing the deflection of light by dyonic wormholes in Einstein-Maxwell-Dilaton theory and showing the deflection depends on the electric, magnetic, and dilaton charges.
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1) The document analyzes particle collisions near the horizon of 1+1 dimensional Hořava-Lifshitz black holes.
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EMU M.Sc. Thesis Presentation
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The document discusses dark matter and provides evidence for its existence from various astronomical observations. It notes that while ordinary matter makes up only about 4% of the universe, dark matter accounts for about 23%. Various properties of dark matter are described, including that it interacts gravitationally but does not emit or absorb light. Possible candidates for dark matter are discussed, including WIMPs (Weakly Interacting Massive Particles), which are favored from both astronomical data and particle physics models. The document outlines how WIMPs could have been thermally produced in the early universe to account for the observed dark matter abundance.
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Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
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Context. The observation of several L-band emission sources in the S cluster has led to a rich discussion of their nature. However, a definitive answer to the classification of the dusty objects requires an explanation for the detection of compact Doppler-shifted Brγ emission. The ionized hydrogen in combination with the observation of mid-infrared L-band continuum emission suggests that most of these sources are embedded in a dusty envelope. These embedded sources are part of the S-cluster, and their relationship to the S-stars is still under debate. To date, the question of the origin of these two populations has been vague, although all explanations favor migration processes for the individual cluster members. Aims. This work revisits the S-cluster and its dusty members orbiting the supermassive black hole SgrA* on bound Keplerian orbits from a kinematic perspective. The aim is to explore the Keplerian parameters for patterns that might imply a nonrandom distribution of the sample. Additionally, various analytical aspects are considered to address the nature of the dusty sources. Methods. Based on the photometric analysis, we estimated the individual H−K and K−L colors for the source sample and compared the results to known cluster members. The classification revealed a noticeable contrast between the S-stars and the dusty sources. To fit the flux-density distribution, we utilized the radiative transfer code HYPERION and implemented a young stellar object Class I model. We obtained the position angle from the Keplerian fit results; additionally, we analyzed the distribution of the inclinations and the longitudes of the ascending node. Results. The colors of the dusty sources suggest a stellar nature consistent with the spectral energy distribution in the near and midinfrared domains. Furthermore, the evaporation timescales of dusty and gaseous clumps in the vicinity of SgrA* are much shorter ( 2yr) than the epochs covered by the observations (≈15yr). In addition to the strong evidence for the stellar classification of the D-sources, we also find a clear disk-like pattern following the arrangements of S-stars proposed in the literature. Furthermore, we find a global intrinsic inclination for all dusty sources of 60 ± 20◦, implying a common formation process. Conclusions. The pattern of the dusty sources manifested in the distribution of the position angles, inclinations, and longitudes of the ascending node strongly suggests two different scenarios: the main-sequence stars and the dusty stellar S-cluster sources share a common formation history or migrated with a similar formation channel in the vicinity of SgrA*. Alternatively, the gravitational influence of SgrA* in combination with a massive perturber, such as a putative intermediate mass black hole in the IRS 13 cluster, forces the dusty objects and S-stars to follow a particular orbital arrangement. Key words. stars: black holes– stars: formation– Galaxy: center– galaxies: star formation
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THIN-SHELL WORMHOLES AND GRAVASTARS PUCV SEMINAR 2017
1. THIN-SHELL WORMHOLES AND
GRAVASTARS
Are wormholes or ’gravastars’ mimicking gravitational-wave sig-
nals from black holes?
Dr. Ali Övgün
Ph.D in Physics,
Eastern Mediterranean University, Northern Cyprus with Prof. Mustafa Halilsoy
21 September 2017
Now FONDECYT Postdoc. Researcher at Instituto de Física,
Pontificia Universidad Católica de Valparaíso with Prof. Joel Saavedra
2.
3. My Research Areas
1. COSMOLOGY: Inflation and acceleration of universe with Nonlinear
Electrodynamics 1 PAPER
2. WORMHOLES: Thinshell Wormholes and Wormholes 9 PAPER
3. GRAVITATIONAL LENSING 6 PAPER
4. COMPACT STARS, BLACK HOLES, GRAVASTARS 2 PAPERS
5. WAVE PROPERTIES OF BHs or WHs: Hawking Radiation,
Greybody factors, Quasinormal Modes of Black Holes, Quantum
Singularities 20 PAPER
6. PARTICLE PROPERTIES OF BHs or WHs: Geodesics, particle
collision near BHs and BSW effect of BHs 2 PAPER
4. Table of contents
1. INTRODUCTION
2. WHAT IS A WORMHOLE?
3. MOTIVATION
4. THIN-SHELL WORMHOLES
5. HAYWARD THIN-SHELL WH IN 3+1-D
6. ROTATING THIN-SHELL WORMHOLE
7. THIN-SHELL GRAVASTARS
19. Figure 8: Wormhole
-We do not know how to open the throat without exotic matter.
-Thin-shell methods with Israel junction conditions can be used to
minimize the exotic matter needed.
However, the stability must be saved.
21. History of Wormholes
Figure 10: Firstly , Flamm’s work on the WH physics using the Schwarzschild
solution (1916).
22. Figure 11: Einstein and Rosen (ER) (1935), ER bridges connecting two
identicalsheets.
23. Figure 12: J.Wheeler used ”geons” (self-gravitating bundles of electromagnetic
fields) by giving the first diagram of a doubly-connected-space (1955).
Wheeler added the term ”wormhole” to the physics literature at the
quantum scale.
28. Traversable Wormhole Construction Criteria
• Obey the Einstein field equations.
• Must have a throat that connects two asymptotically flat regions of
spacetime.
• No horizon, since a horizon will prevent two-way travel through the
wormhole.
• Tidal gravitational forces experienced by a traveler must be bearably
small.
• Traveler must be able to cross through the wormhole in a finite and
reasonably small proper time.
• Physically reasonable stress-energy tensor generated by the matter
and fields.
• Solution must be stable under small perturbation.
• Should be possible to assemble the wormhole. ie. assembly should
require much less than the age of the universe.
30. Traversable Lorentzian Wormholes
The first defined traversable WH is Morris Thorne WH with a the
red-shift function f(r) and a shape function b(r) :
ds2
= −e2f(r)
dt2
+
1
1 − b(r)
r
dr2
+ r2
(dθ2
+ sin2
θdϕ2
). (1)
• Spherically symmetric and static
• Radial coordinate r such that circumference of circle centered
around throat given by 2πr
• r decreases from +∞ to b = b0 (minimum radius) at throat, then
increases from b0 to +∞
• At throat exists coordinate singularity where r component diverges
• Proper radial distance l(r) runs from −∞ to +∞ and vice versa
31.
32.
33.
34. Figure 18: This hypothesis was originally put forward by Mazur and Mottola in
2004.
• Gravastar literally means Gravitational Vacuum Condensate Star
• Black holes havea very large entropy value.
• Gravastars, on the other hand, have quite a low entropy.
• As a star collapses further the particles fall into a Bose-Einstein state
where the entire star nears absolute zero and get very compact.
• Acts as a giant atom composed of bosons.
• The interior of Gravastars is a de Sitter Spacetime, a positive
vacuum energy, an internal negative pressure.
35. • Mazur and Mottola think that event horizon is actually the outer
shell of the Bose-Einstein matter, similar to neutron star
41. • Constructing WHs with non-exotic (normal matter) source is a
difficult issue in General Relativity.
• First, Visser use the thin-shell method to construct WHs by
minimizing the exotic matter on the throat of the WHs.
42.
43. Input: Two space-times
-Use the Darmois –Israel formalism and match an interior spacetime to
an exterior spacetime
-Use the Lanczos equations to find a surface energy density σ and a
surface pressure p.
-Use the energy conservation to find the equation of motion of particle
on the throat of the thin-shell wormhole
-Check the stability by using different EoS equations.
Outputs Thin-shell wormhole
σ < 0 with extrinsic curvature K > 0 of the throat means it required
exotic matter.
45. A. Ovgun et.al Eur. Phys. J. C 74 (2014): 2796
• The metric of the Hayward BH is given by
ds2
= −f(r)dt2
+ f(r)−1
dr2
+ r2
dΩ2
. (2)
with the metric function
f (r) =
(
1 −
2mr2
r3 + 2ml2
)
(3)
and
dΩ2
= dθ2
+ sin2
θdϕ2
. (4)
• It is noted that m and l are free parameters.
• At large r, the metric function behaves as a Schwarzchild BH
lim
r→∞
f (r) → 1 −
2m
r
+ O
(
1
r4
)
, (5)
whereas at small r becomes a de Sitter BH
lim
r→0
f (r) → 1 −
r2
2
+ O
(
r5
)
. (6)
46. • It is noted that the singularity located at r = 0.
• Thin-shell is located at r = a .
• The throat must be outside of the horizon (a > rh).
• Then we paste two copies of it at the point of r = a.
• For this reason the thin-shell metric is taken as
ds2
= −dτ2
+ a (τ)
2 (
dθ2
+ sin2
θdϕ2
)
(7)
where τ is the proper time on the shell.
• The Einstein equations on the shell are
[
Kj
i
]
− [K] δj
i = −Sj
i (8)
where [Kij] = K+
ij − K−
ij .
• It is noted that the extrinsic curvature tensor is Kj
i.
• Moreover, K stands for its trace.
• Setting coordinates ξi
= (τ, θ), the extrinsic curvature formula
connecting the two sides of the shell is simply given by
K±
ij = −n±
γ
(
∂2
xγ
∂ξi∂ξj
+ Γγ
αβ
∂xα
∂ξi
∂xβ
∂ξj
)
, (9)
47. where the unit normals (nγ
nγ = 1) are
n±
γ = ± gαβ ∂H
∂xα
∂H
∂xβ
−1/2
∂H
∂xγ
, (10)
with H(r) = r − a(τ).
• The surface stresses, i.e., surface energy density σ and surface
pressures Sθ
θ = p = Sϕ
ϕ , are determined by the surface stress-energy
tensor Sj
i.
• The energy and pressure densities are obtained as
σ = −
4
a
√
f (a) + ˙a2 (11)
p = 2
(√
f (a) + ˙a2
a
+
¨a + f′
(a) /2
√
f (a) + ˙a2
)
. (12)
• Then they reduce to simple form in a static configuration (a = a0)
σ0 = −
4
a0
√
f (a0) (13)
and
p0 = 2
(√
f (a0)
a0
+
f′
(a0) /2
√
f (a0)
)
. (14)
48. Once σ ≥ 0 and σ + p ≥ 0 hold, then WEC is satisfied.
• It is obvioust hat negative energy density violates the WEC, and
consequently we are in need of the exotic matter for constructing
thin-shell WH.
• Stability of such a WH is investigated by applying a linear
perturbation with the following EoS
p = ψ (σ) (15)
• Moreover the energy conservation is
Sij
;j = 0 (16)
which in closed form it equals to
Sij
,j + Skj
Γiµ
kj + Sik
Γj
kj = 0 (17)
after the line element in Eq.(7) is used, it opens to
∂ (
σa2
)
+ p
∂ (
a2
)
= 0. (18)
49. • The 1-D equation of motion is
˙a2
+ V (a) = 0, (19)
in which V (a) is the potential,
V (a) = f −
(aσ
4
)4
. (20)
• The equilibrium point at a = a0 means V′
(a0) = 0 and V′′
(a0) ≥ 0.
• Then it is considered that f1 (a0) = f2 (a0), one finds V0 = V′
0 = 0.
• To obtain V′′
(a0) ≥ 0 we use the given p = ψ (σ) and it is found as
follows
σ′
(
=
dσ
da
)
= −
2
a
(σ + ψ) (21)
and
σ′′
=
2
a2
(σ + ψ) (3 + 2ψ′
) , (22)
where ψ′
= dψ
dσ . After we use ψ0 = p0, finally it is found that
V′′
(a0) = f′′
0 −
1 [
(σ0 + 2p0)
2
+ 2σ0 (σ0 + p0) (1 + 2ψ′
(σ0))
]
(23)
50. • The equation of motion of the throat, for a small perturbation
becomes
˙a2
+
V′′
(a0)
2
(a − a0)2
= 0.
• Note that for the condition of V′′
(a0) ≥ 0 , TSW is stable where the
motion of the throat is oscillatory with angular frequency
ω =
√
V′′(a0)
2 .
Some models of EoS
Now we use some models of matter to analyze the effect of the
parameter of Hayward in the stability of the constructed thin-shell WH.
Linear gas
For a LG, EoS is choosen as
ψ = η0 (σ − σ0) + p0 (24)
in which η0 is a constant and ψ′
(σ0) = η0.
51. Figure 19: Stability of Thin-Shell WH supported by LG.
Fig. shows the stability regions in terms of η0 and a0 with different
Hayward’s parameter. It is noted that the S shows the stable regions.
52. Chaplygin gas
For CG, we choose the EoS as follows
ψ = η0
(
1
σ
−
1
σ0
)
+ p0 (25)
where η0 is a constant and ψ′
(σ0) = − η0
σ2
0
.
53. Figure 20: Stability of Thin-Shell WH supported by CG.
In Fig., the stability regions are shown in terms of η0 and a0 for different
values of ℓ. The effect of Hayward’s constant is to increase the stability
of the Thin-Shell WH.
54. Generalized Chaplygin gas
The EoS of the GCG is taken as
ψ (σ) = η0
(
1
σν
−
1
σν
0
)
+ p0 (26)
where ν and η0 are constants. We check the effect of parameter ν in the
stability and ψ becomes
ψ (σ) = p0
(σ0
σ
)ν
. (27)
55. Figure 21: Stability of Thin-Shell WH supported by GCG.
We find ψ′
(σ0) = − p0
σ0
ν. In Fig., the stability regions are shown in terms
of ν and a0 with various values of ℓ.
56. Modified Generalized Chaplygin gas
In this case, the MGCG is
ψ (σ) = ξ0 (σ − σ0) − η0
(
1
σν
−
1
σν
0
)
+ p0 (28)
in which ξ0, η0 and ν are free parameters. Therefore,
ψ′
(σ0) = ξ0 + η0
η0ν
σν+1
0
. (29)
57. Figure 22: Stability of Thin-Shell WH supported by MGCG.
To go further we set ξ0 = 1 and ν = 1. In Fig., the stability regions are
plotted in terms of η0 and a0 with various values of ℓ. The effect of
Hayward’s constant is to increase the stability of the Thin-Shell WH.
58. Logarithmic gas
Lastly LogG is choosen by follows
ψ (σ) = η0 ln
σ
σ0
+ p0 (30)
in which η0 is a constant. For LogG, we find that
ψ′
(σ0) =
η0
σ0
. (31)
59. Figure 23: Stability of Thin-Shell WH supported by LogG.
In Fig., the stability regions are plotted to show the effect of Hayward’s
parameter clearly. The effect of Hayward’s constant is to increase the
stability of the Thin-Shell WH.
60. Conclusions
• On the thin-shell we use the different type of EoS with the form
p = ψ (σ) and plot possible stable regions.
• We show the stable and unstable regions on the plots.
• Stability simply depends on the condition of V′′
(a0) > 0.
• We show that the parameter ℓ, which is known as Hayward parameter
has a important role.
• Moreover, for higher ℓ value the stable regions are increased.
• It is checked the small velocity perturbations for the throat.
• It is found that throat of the thin-shell WH is not stable against such
kind of perturbations.
• Hence, energy density of the WH is found negative so that we need
exotic matter.
62. A. Ovgun, Eur.Phys.J.Plus 131 (2016) no.11, 389
Constructing The Rotating Thin-Shell Wormhole
The 5-d rotating Myers-Perry (5DRMP) black hole solution, which is the
generalization of the Kerr solution to higher dimensions, is given by the
following space-time metric :
ds2
= −F(r)2
dt2
+G(r)2
dr2
+r2
gabdxa
dxb
+ H(r)2
[dψ + Badxa
− K(r)dt]
2
,
(32)
in which
G(r)2
=
(
1 +
r2
ℓ2
−
2MΞ
r2
+
2Ma2
r4
)−1
, (33)
H(r)2
= r2
(
1 +
2Ma2
r4
)
, K(r) =
2Ma
r2H(r)2
, (34)
F(r) =
r
G(r)H(r)
, Ξ = 1 −
a2
ℓ2
, (35)
where B = Badxa
and
gabdxa
dxb
=
1 (
dθ2
+ sin2
θ dϕ2
)
, B =
1
cos θ dϕ . (36)
63. Note that taking the limit of the Anti-de-Sitter (AdS) length ℓ → ∞ ,
the asymptotically flat case can be recovered. One writes the mass M
and angular momentum J of the spacetime as
M =
πM
4
(
3 +
a2
ℓ2
)
, J = πMa . (37)
For convenience we move to a comoving frame to eliminate cross terms
in the induced metrics by introducing
dψ −→ dψ′
+ K±(R(t))dt . (38)
We choose a radius R(t), which is the throat of the wormhole, and take
two copies of this manifold ˜M± for the interior and exterior regions with
r ≥ R to paste them at an identical hypersurface
Σ = {xµ
: t = T (τ), r = R(τ)}, which is parameterized by coordinates
yi
= {τ, ψ, θ, ϕ} on the 4-d surface.
The line element in the interior and the exterior sides become
ds2
± = −F±(r)2
dt2
+ G±(r)2
dr2
+ r2
dΩ + H±(r)2
{dψ′
+ Badxa
+ [K±(R(t)) −
For simplicity in the comoving frame, we drop the prime on ψ′
. The
geodesically complete manifold is satisfied as ˜M = ˜M+ U ˜M−. We use
64. the Darmois-Israel formalism to construct the rotating thin-shell
wormhole. Using the Israel junction conditions, the Einstein’s equations
for the wormhole produce
ρ = −
β(R2
H)′
4π R3
, φ = −
J (RH)′
2π2R4H
, (40)
P =
H
4πR3
[
R2
β
]′
, ∆P =
β
4π
[
H
R
]′
, (41)
where primes stand for d/dR and
β ≡ F(R)
√
1 + G(R)2 ˙R2 . (42)
Without rotation or in the case of a corotating frame, the momentum φ
and the anisotropic pressure term ∆P are equal to zero. Consequently,
the static energy and pressure densities at the throat of wormhole R =
65. R0 are given by
ρ0 = −
F(R2
0H)′
4π R3
0
, φ0 = −
J (R0H)′
2π2R4
0H
, (43)
P0 =
H
4πR3
0
[
R2
0F
]′
, ∆P0 =
F
4π
[
H
R0
]′
. (44)
To check the stability of the wormhole, we use the linear equation of
state (EoS):
P = ωρ. (45)
The stability of the wormhole solution depends upon the conditions of
V′′
eff (R0) > 0 and V′
eff (R0) = Veff (R0) = 0 as
Veff ∼
1
2
V′′
eff (R0) (R − R0)
2
. (46)
Let us then introduce x = R − R0 and write the equation of motion
again:
˙x2
+
1
2
V′′
eff (R0) x2
= 0 (47)
which after a derivative with respect to time reduces to
¨x +
1
V′′
eff (R0) x = 0. (48)
66. We show the conditions for a positive stability value. Our main aim is to
discover the behavior of V′′
eff (R0) as
V′′
eff =
1
R0
10
[−40 (R0
√
R0
4
+ 2 a2
R0
4 )−2 ω
R0
−4 ω
a2
+ 2 R0
10
+
(
12 a2
− 12
)
R0
6
+40 a2
R0
4
].
67. Note that a = ω = 0 corresponds to a non-rotating case. It is easy to see
that a has a crucial role in this stability analysis; the stability regions are
shown in the following Figs.:
猀
猀
Figure 24: Stability of wormhole supported by linear gas in terms of ω and R0
for a = 0.1.
70. Discussions
• We have studied a TSWH with rotation in 5-D constructed using a
Myers-Perry black hole with cosmological constants using a cut-and-paste
procedure.
• The standard stability approach has been applied by considering a linear
gas model at the TSWH throat.
• A key feature of the current analysis is the inclusion of rotation in the
form of non-zero values of angular momentum.
• Another key aspect of the current analysis is the focus on different values
of parameter (a), which plays a crucial role in making the TSWH more
stable in five dimensions.
• We observe that the stability of the wormhole is fundamentally linked to
the behavior of the constant (a).
• The amount of exotic matter required to support the TSWH is always a
crucial issue; unfortunately, we are not able to completely eliminate
exotic matter during the constructing of the stable rotating TSWH.
72. A. Ovgun et. al Eur. Phys. J. C (2017) 77:566
Exterior Of Gravastars: Noncommutative geometry inspired
Charged BHs
• The metric of a noncommutative charged black hole is described by the
metric given in S. Ansoldi et al.Phys.Lett.B645,261(2007):
ds2
= −f(r)dt2
+ f(r)−1
dr2
+ r2
dΩ2
, (51)
with f(r) =
(
1 − 2Mθ
r +
Q2
θ
r2
)
.
73. STRUCTURE EQUATIONS OF CHARGED GRAVASTARS
• The metrics of interior is the nonsingular de Sitter spacetimes:
ds2
= −(1 −
r2
−
α2
)dt2
− + (1 −
r2
−
α2
)−1
dr2
− + r2
−dΩ2
− (52)
and exterior of noncommutative geometry inspired charged spacetimes:
ds2
= −f(r)+dt2
+ + f(r)−1
+ dr2
+ + r2
+dΩ2
+. (53)
• Then using the relation of Si
j = diag(−σ, P, P), one can find the surface
energy density, σ, and the surface pressure, P, as follows:
σ = −
1
4πa
√
1 −
2Mθ+
a
+
Q2
θ+
a2
+ ˙a2 −
√(
1 −
a2
α2
)
+ ˙a2
,(54)
P =
1
8πa
1 + ˙a2
+ a¨a − Mθ+
a
√
1 − 2Mθ+
a +
Q2
θ+
a2 + ˙a2
−
(
1 + a¨a + ˙a2
− 2a2
α2
)
√(
1 − a2
α2
)
+ ˙a2
.
(55)
74. • To calculate the surface mass of the thin-shell, one can use this equation
Ms(a) = 4πa2
σ. To find stable solution, we consider a static case
[a0 ∈ (r−, r+)].
Then the surface charge and pressure at static case reduce to
σ(a0) = −
1
4πa0
√
1 −
2Mθ+
a0
+
Q2
θ+
a2
0
−
√(
1 −
a2
0
α2
)
, (56)
P(a0) =
1
8πa0
1 − Mθ+
a0
√
1 − Mθ+
a0
+
Q2
θ+
a2
0
−
(
1 −
2a2
0
α2
)
√(
1 −
a2
0
α2
)
.
(57)
75. Figure 27: We plot σ0 + p0 as a function of Mθ and a0. We choose Qθ = 1
and α = 0.4. Note that in this region the NEC is satisfied.
76. Stability of the Charged Thin-shell Gravastars in Noncommutative
Geometry
• We use the surface energy density σ(a) on the thin-shell of the gravastars
to check stability:
1
2
˙a2
+ V(a) = 0, (58)
with the potential,
V(a) =
1
2
{
1 −
B(a)
a
−
[
Ms(a)
2a
]2
−
[
D(a)
Ms(a)
]2
}
. (59)
It is noted that B(a) and D(a) are
B(a) =
[(
2Mθ+ −
Q2
θ+
a
)
+ ( a3
α2 )
]
2
, D(a) =
[(
2Mθ+ −
Q2
θ+
a
)
− ( a3
α2 )
]
2
. (60
77. • One can also easily obtain the surface mass as a function of the potential:
Ms(a) = −a
√
1 −
2Mθ+
a
+
Q2
θ+
a2
− 2V(a) −
√
(1 −
a2
α2
) − 2V(a)
.
(61)
• Then the surface charge and the pressure are rewritten in terms of
potential as follows:
σ = −
1
4πa
√
1 −
2Mθ+
a
+
Q2
θ+
a2
− 2V −
√
(1 −
a2
α2
) − 2V
, (62)
P =
1
8πa
1 − 2V − aV′
− Mθ+
a
√
1 − 2Mθ+
a +
Q2
θ+
a2 − 2V
−
1 − 2V − aV′
−
( 2a
α2
)
√
(1 − a2
α2 ) − 2V
. (63)
• To find the stable solution, we linearize it using the Taylor expansion
around the a0 to second order as follows:
V(a) =
1
V′′
(a0)(a − a0)2
+ O[(a − a0)3
] . (64)
78. • Note that for stability, the conditions are V(a0) = V′
(a0) = 0,
˙a0 = ¨a0 = 0 and V′′
(a0) > 0.
• Using the relation Ms(a) = 4πσ(a)a2
, we use the M′′
s (a0) instead of
V′′
(a0) ≥ 0 as following:
M′′
s (a0) ≥
1
4a3
0
[
2(Mθ++Q2
θ+)
a0
]2
[1 − 2Mθ+
a0
+
Q2
θ+
a2
0
]3/2
−
[
−a3
0
α2 ]2
[(1 −
a2
0
α2 )]3/2
+
1
2
2Q2
θ+
a3
0
√
1 − 2Mθ+
a0
+
Q2
θ+
a2
0
−
4a0
α2
√
(1 −
a2
0
α2 )
, (65)
• so for the stable solution
V′′
(a0) = −
3M2
s (a0)
4a4
0
+
[M′
s(a0)
a3
0
−
M′′
s (a0)
4a2
0
]
Ms(a0)
−
B′′
(a0)
2a0
+
B′
(a0)
a2
−
B(a0)
a3
−
M′
s(a0)2
4a2
79. −
D′2
(a0) + D(a0)D′′
(a0)
M2
s (a0)
+
4D(a0)D′
(a0)M′
s(a0) + D2
(a0)M′′
s (a0)
M3
s (a0)
−
3D2
(a0)(M′
s)2
(a0)
M4
s (a0)
, (66)
where
M′
s(a0) = 8πa0σ0 − 8πa0(σ0 + p0), (67)
and
M′′
s (a0) = 8πσ0 − 32π(σ0 + p0)
+ 4π [2(σ0 + p0) + 4(σ0 + p0)(1 + η)] . (68)
• Moreover, we have also introduced η(a) = P′
(a)/σ′
(a)|a0 , as a parameter
which will play a fundamental role in determining the stability regions of
the respective solutions.
• Generally, η interpreted as the speed of sound, so that one would expect
the range of 0 < η ≤ 1, that the speed of sound should not exceed the
speed of light.
80. • But the range of η may be lying outside the range of 0 < η ≤ 1, on the
surface layer.
• Therefore, in this work the range of η will be relaxed and we use
graphical reputation to determine the stability regions given by the Eq.
(34), due to the complexity of the expression.
81. Figure 28: Stability regions of the charged gravastar in terms of η = P′
/σ′
as
a function of a0. We choose Mθ = 2, Qθ = 1.5, α = 0.4.
Figure 29: Stability regions of the charged gravastar in terms of η = P′
/σ′
as
a function of a0. We choose Mθ = 1.5, Qθ = 1, α = 0.2.
82. Figure 30: Stability regions of the charged gravastar in terms of η = P′
/σ′
as
a function of a0. We choose Mθ = 3, Qθ = 2.5, α = 0.5.
83. Conclusions
• We have studied the stability of a particular class of thin-shell gravastar
solutions, in the context of charged noncommutative geometry.
• We have considered the de Sitter geometry in the interior of the
gravastar by matching an exterior charged noncommutative solution at a
junction interface situated outside the event horizon.
• We have showed that gravastar’s shell satisfies the null energy conditions.
• We further explored the gravastar solution by the dynamical stability of
the transition layer, which is sufficient close to the event horizon.
• We have found that for specific choices of mass Mθ, charge Qθ and the
values of α, the stable configurations of the surface layer do exists which
is sufficiently close to where the event horizon is expected to form.