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
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.
- The document discusses the 4/3 problem as it relates to the gravitational field of a uniform massive ball moving at constant velocity.
- It derives expressions for the gravitational field potentials both inside and outside the moving ball using the superposition principle and Lorentz transformations.
- Calculations show that the effective mass of the gravitational field found from the field energy does not equal the effective mass found from the field momentum, with a ratio of approximately 4/3, demonstrating that the 4/3 problem exists for gravitational fields as it does for electromagnetic fields.
Differential geometry three dimensional spaceSolo Hermelin
This presentation describes the mathematics of curves and surfaces in a 3 dimensional (Euclidean) space.
The presentation is at an Undergraduate in Science (Math, Physics, Engineering) level.
Plee send comments and suggestions to improvements to solo.hermelin@gmail.com. Thanks/
More presentations can be found at my website http://www.solohermelin.com.
Natural frequencies of magnetoelastic longitudinal wave propagationAlexander Decker
This document summarizes research on the natural frequencies of longitudinal wave propagation in an orthotropic circular cylinder permeated by a magnetic field. It begins by introducing the topic and providing background. It then presents the basic equations that govern magnetoelastic wave propagation. Specifically, it outlines the equations of motion for an elastic solid in a magnetic field, including the Lorentz force term. It also presents Maxwell's equations in the quasi-static approximation. The document derives a frequency equation for this problem and describes numerically solving for the natural frequencies of the first three modes for different material properties and cylinder geometries.
This document summarizes the key concepts around harmonic excitation of undamped and damped systems from Chapter 2. It introduces the important concept of resonance that occurs when the driving frequency matches the natural frequency of the system. For an undamped system under harmonic excitation, the response is the sum of the homogeneous and particular solutions. The particular solution assumes the form of the driving force. When the driving frequency approaches the natural frequency, the amplitude of the response increases dramatically. For a damped system, the particular solution includes a phase shift.
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.
This document discusses methods for determining areas, volumes, centroids, and moments of inertia of basic geometric shapes. It begins by introducing the method of integration for calculating areas and volumes. Standard formulas are provided for areas of rectangles, triangles, circles, sectors, and parabolic spandrels. Formulas are also provided for volumes of parallelepipeds, cones, spheres, and solids of revolution. The concepts of center of gravity, centroid, and center of mass are defined. Equations are given for calculating the centroids of uniform bodies, plates, wires, and line segments. Methods for finding centroids of straight lines, arcs, semicircles, and quarter circles are illustrated.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
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.
- The document discusses the 4/3 problem as it relates to the gravitational field of a uniform massive ball moving at constant velocity.
- It derives expressions for the gravitational field potentials both inside and outside the moving ball using the superposition principle and Lorentz transformations.
- Calculations show that the effective mass of the gravitational field found from the field energy does not equal the effective mass found from the field momentum, with a ratio of approximately 4/3, demonstrating that the 4/3 problem exists for gravitational fields as it does for electromagnetic fields.
Differential geometry three dimensional spaceSolo Hermelin
This presentation describes the mathematics of curves and surfaces in a 3 dimensional (Euclidean) space.
The presentation is at an Undergraduate in Science (Math, Physics, Engineering) level.
Plee send comments and suggestions to improvements to solo.hermelin@gmail.com. Thanks/
More presentations can be found at my website http://www.solohermelin.com.
Natural frequencies of magnetoelastic longitudinal wave propagationAlexander Decker
This document summarizes research on the natural frequencies of longitudinal wave propagation in an orthotropic circular cylinder permeated by a magnetic field. It begins by introducing the topic and providing background. It then presents the basic equations that govern magnetoelastic wave propagation. Specifically, it outlines the equations of motion for an elastic solid in a magnetic field, including the Lorentz force term. It also presents Maxwell's equations in the quasi-static approximation. The document derives a frequency equation for this problem and describes numerically solving for the natural frequencies of the first three modes for different material properties and cylinder geometries.
This document summarizes the key concepts around harmonic excitation of undamped and damped systems from Chapter 2. It introduces the important concept of resonance that occurs when the driving frequency matches the natural frequency of the system. For an undamped system under harmonic excitation, the response is the sum of the homogeneous and particular solutions. The particular solution assumes the form of the driving force. When the driving frequency approaches the natural frequency, the amplitude of the response increases dramatically. For a damped system, the particular solution includes a phase shift.
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.
This document discusses methods for determining areas, volumes, centroids, and moments of inertia of basic geometric shapes. It begins by introducing the method of integration for calculating areas and volumes. Standard formulas are provided for areas of rectangles, triangles, circles, sectors, and parabolic spandrels. Formulas are also provided for volumes of parallelepipeds, cones, spheres, and solids of revolution. The concepts of center of gravity, centroid, and center of mass are defined. Equations are given for calculating the centroids of uniform bodies, plates, wires, and line segments. Methods for finding centroids of straight lines, arcs, semicircles, and quarter circles are illustrated.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
Optical Aberration is the phenomenon of Image Distortion due to Optics Imperfection.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations visit my website at http://www.solohermelin.com.
This presentation is in the Optics Folder. Since some of the Figures were not downloaded I recommend to see the presentation on my website.
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
Dr. Arpan Bhattacharyya (Indian Institute Of Science, Bangalore)Rene Kotze
1. The document discusses entanglement entropy functionals for higher derivative gravity theories. It proposes new area functionals for computing entanglement entropy in higher derivative theories containing polynomials of curvature tensors.
2. These functionals are derived using the Lewkowycz-Maldacena interpretation of generalized entropy. However, attempting to derive the extremal surface equations from these functionals using bulk equations of motion leads to inconsistencies and ambiguities in some higher derivative theories like Gauss-Bonnet gravity.
3. The document suggests that the source of ambiguity lies in the limiting procedure used to extract the divergences near the conical singularity. Different limiting paths can lead to different extremal surface equations, indicating no unique prescription
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
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
This document summarizes Vishnu Jejjala's talk "The Geometry of Generations" given at the University of the witwatersrand on 23 September 2014. It discusses using the geometry of vacuum moduli spaces of supersymmetric gauge theories to understand phenomenological aspects of the Standard Model, such as the number of generations. It provides examples of calculating the vacuum moduli space for simple supersymmetric gauge theories and outlines a process for obtaining the moduli space from the F-term and D-term equations.
Hello, I am Subhajit Pramanick. I and my friend, Sougata Dandapathak, both presented this ppt in our college seminar. It is basically based on the origin of calculus of variation. It consists of several topics like the history of it, the origin of it, who developed it, application of it, advantages and disadvantages etc. The main aim of this presentation is to increase our mathematical as well as physical conception on advanced classical mechanics. We hope you will all enjoy by reading this presentation. Thank you.
- This document discusses modeling and energy methods for determining equations of motion and natural frequencies of systems. It provides alternative approaches to calculating these values when forces or torques are difficult to determine directly.
- Energy methods are useful for more complicated multi-degree of freedom and distributed mass systems that will be discussed later. Potential and kinetic energy equations are presented for springs and various mass configurations.
- The conservation of energy principle and Lagrange's equations can be used to derive equations of motion from the kinetic and potential energy of a system, providing alternative ways to model dynamic behavior. Examples are worked through for simple spring-mass and pendulum systems.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
This document discusses the application of dynamical groups and coherent states in quantum optics and molecular spectroscopy. It provides an introduction to using Lie groups and algebras to describe quantum systems and defines coherent states. Specific applications discussed include using dynamical symmetries to calculate energy levels of systems like the harmonic oscillator and hydrogen atom. Coherent states are used to derive classical equations of motion and represent open quantum systems. Examples of coherent state dynamics are shown for two-level and three-level atoms interacting with laser fields.
N. Bilic - "Hamiltonian Method in the Braneworld" 2/3SEENET-MTP
This document discusses braneworld models and the Randall-Sundrum model. It begins by introducing the relativistic particle and string actions used to describe dynamics in higher dimensions. It then summarizes the two Randall-Sundrum models: RS I contains two branes separated in a fifth dimension to address the hierarchy problem, while RS II has the negative tension brane sent to infinity and observers on a single positive tension brane. Finally, it derives the RS II model solution, using Gaussian normal coordinates and imposing junction conditions at the brane.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
This document provides information about the dynamics of machinery course for several mechanical engineering students. It includes the learning objectives, symbols and definitions, response of a damped system under harmonic motion, an example problem, and key concepts about magnification factor, phase angle, and total response of a system. The example calculates the total response of a single-degree-of-freedom system subjected to an external harmonic force and free vibration.
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
Estimate the hidden States, Parameters, Signals of a Linear Dynamic Stochastic System from Noisy Measurements. It requires knowledge of probability theory. Presentation at graduate level in math., engineering
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com. Since a few Figure were not downloaded I recommend to see the presentation on my website at RADAR Folder, Tracking subfolder.
Describes the Static and Dynamic Equation of Gears.
In the download process a few figures are missing.
I recommend to visit my website, in the Simulation Folder, for a better view of this presentation.
For graduate students in engineering. Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects, visit my website at http://www.solohermelin.com
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
Establishment of New Special Deductions from Gauss Divergence Theorem in a Ve...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Optical Aberration is the phenomenon of Image Distortion due to Optics Imperfection.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations visit my website at http://www.solohermelin.com.
This presentation is in the Optics Folder. Since some of the Figures were not downloaded I recommend to see the presentation on my website.
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
Dr. Arpan Bhattacharyya (Indian Institute Of Science, Bangalore)Rene Kotze
1. The document discusses entanglement entropy functionals for higher derivative gravity theories. It proposes new area functionals for computing entanglement entropy in higher derivative theories containing polynomials of curvature tensors.
2. These functionals are derived using the Lewkowycz-Maldacena interpretation of generalized entropy. However, attempting to derive the extremal surface equations from these functionals using bulk equations of motion leads to inconsistencies and ambiguities in some higher derivative theories like Gauss-Bonnet gravity.
3. The document suggests that the source of ambiguity lies in the limiting procedure used to extract the divergences near the conical singularity. Different limiting paths can lead to different extremal surface equations, indicating no unique prescription
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
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
This document summarizes Vishnu Jejjala's talk "The Geometry of Generations" given at the University of the witwatersrand on 23 September 2014. It discusses using the geometry of vacuum moduli spaces of supersymmetric gauge theories to understand phenomenological aspects of the Standard Model, such as the number of generations. It provides examples of calculating the vacuum moduli space for simple supersymmetric gauge theories and outlines a process for obtaining the moduli space from the F-term and D-term equations.
Hello, I am Subhajit Pramanick. I and my friend, Sougata Dandapathak, both presented this ppt in our college seminar. It is basically based on the origin of calculus of variation. It consists of several topics like the history of it, the origin of it, who developed it, application of it, advantages and disadvantages etc. The main aim of this presentation is to increase our mathematical as well as physical conception on advanced classical mechanics. We hope you will all enjoy by reading this presentation. Thank you.
- This document discusses modeling and energy methods for determining equations of motion and natural frequencies of systems. It provides alternative approaches to calculating these values when forces or torques are difficult to determine directly.
- Energy methods are useful for more complicated multi-degree of freedom and distributed mass systems that will be discussed later. Potential and kinetic energy equations are presented for springs and various mass configurations.
- The conservation of energy principle and Lagrange's equations can be used to derive equations of motion from the kinetic and potential energy of a system, providing alternative ways to model dynamic behavior. Examples are worked through for simple spring-mass and pendulum systems.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
This document discusses the application of dynamical groups and coherent states in quantum optics and molecular spectroscopy. It provides an introduction to using Lie groups and algebras to describe quantum systems and defines coherent states. Specific applications discussed include using dynamical symmetries to calculate energy levels of systems like the harmonic oscillator and hydrogen atom. Coherent states are used to derive classical equations of motion and represent open quantum systems. Examples of coherent state dynamics are shown for two-level and three-level atoms interacting with laser fields.
N. Bilic - "Hamiltonian Method in the Braneworld" 2/3SEENET-MTP
This document discusses braneworld models and the Randall-Sundrum model. It begins by introducing the relativistic particle and string actions used to describe dynamics in higher dimensions. It then summarizes the two Randall-Sundrum models: RS I contains two branes separated in a fifth dimension to address the hierarchy problem, while RS II has the negative tension brane sent to infinity and observers on a single positive tension brane. Finally, it derives the RS II model solution, using Gaussian normal coordinates and imposing junction conditions at the brane.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
This document provides information about the dynamics of machinery course for several mechanical engineering students. It includes the learning objectives, symbols and definitions, response of a damped system under harmonic motion, an example problem, and key concepts about magnification factor, phase angle, and total response of a system. The example calculates the total response of a single-degree-of-freedom system subjected to an external harmonic force and free vibration.
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
Estimate the hidden States, Parameters, Signals of a Linear Dynamic Stochastic System from Noisy Measurements. It requires knowledge of probability theory. Presentation at graduate level in math., engineering
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com. Since a few Figure were not downloaded I recommend to see the presentation on my website at RADAR Folder, Tracking subfolder.
Describes the Static and Dynamic Equation of Gears.
In the download process a few figures are missing.
I recommend to visit my website, in the Simulation Folder, for a better view of this presentation.
For graduate students in engineering. Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects, visit my website at http://www.solohermelin.com
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
Establishment of New Special Deductions from Gauss Divergence Theorem in a Ve...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document discusses how St. Francis of Assisi Parish in Centerville, Ohio was awarded the Pentecost Award for their outstanding evangelization efforts through the JustFaith program. JustFaith is a 30-week process that focuses on conversion and value classification, leading parishioners at St. Francis to start various social ministry programs focused on issues like agriculture, prison ministry, and helping unemployed workers. As a result of JustFaith, St. Francis has seen a doubling in size of their St. Vincent de Paul Society, an increase in Lenten Rice Bowl collections, and numerous parishioners become actively involved in social justice causes.
MRP, MRP II, ERP, Malzeme ihtiyaç planlaması, materials requirement planning, üretim kaynakları planlaması, manufacturing resource planning, kurumsal kaynak planlaması, enterprise resource planning
The document discusses two discoveries related to repair mechanisms in the body. The first is a discovery using polymer nano-shells that deliver molecules to damaged bone to stimulate the body's own bone cells to repair itself, reducing risks from foreign cells. The second is a Nobel Prize-winning discovery of a DNA repair mechanism failure link to cancer susceptibility, which could help identify high-risk patients and fuel cancer immunotherapy research working with drug companies. Both discoveries are seen as important progress for diseases like cancer and bone injuries that could improve patients' quality of life worldwide.
Grafico diario del dax perfomance index para el 10 08-2012Experiencia Trading
Este documento presenta un análisis técnico del índice Dax Performance del 10 de agosto de 2012. Incluye gráficos que representan varias medias simples como líneas de soporte y resistencia potenciales. Analiza posibles escenarios si el índice mantiene o pierde ciertos niveles clave, y ofrece objetivos alcistas y resistencias. También incluye una explicación de cómo se construyeron los gráficos y cómo interpretarlos.
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.
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'.
A Pedagogical Discussion on Neutrino Wave Packet EvolutionCheng-Hsien Li
This document discusses the time evolution of a neutrino wave packet. It presents a method to calculate higher-order corrections to the wave packet solution by expanding the momentum distribution and energy terms. The results show that including higher-order terms causes the wave packet to evolve into a spherical shape as expected by relativity. While the solution is limited to early times, it demonstrates that higher-order terms are needed to accurately model the wave packet's evolution into a spherical wavefront.
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.
Study of Parametric Standing Waves in Fluid filled Tibetan Singing bowlSandra B
I completed a Summer Project (May-July 2015) in Physics entitled "Study of
Parametric Standing Waves in Fluid filled Tibetan Singing bowl" under the
guidance of Dr. S. Shankaranarayanan at Indian Institute of Science Education
and Research (IISER-TVM). The project was to theoretically analyze and solve the
non-linear equations for the patterns of wave formed on the surface of Tibet
Singing Bowl for ideal and viscous fluid.
Slides of my talk at IISc Bangalore on nanomechanics and finite element analysis for statics and dynamics of nanoscale structures such as carbon nanotube, graphene, ZnO nanotube and BN nano sheet.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
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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Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
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1. Studies on Thin-shells and Thin-shell
Wormholes
PhD THESIS DEFENSE, 2016
Ali Övgün
Supervisor: Mustafa Halilsoy
Co-supervisor: S. Habib Mazhariomusavi
29 June 2016
Eastern Mediterranean University
2. Publications
4/19 of my publications which I will present today.
1. Existence of traversable wormholes in the spherical stellar
systems
A. Ovgun, M. Halilsoy. Astrophys.Space Sci. 361 (2016) 214
2. On a Particular Thin-shell Wormhole
A. Ovgun, I. Sakalli. arXiv:1507.03949 (accepted to publish in
TMPh)
3. Thin-shell wormholes from the regular Hayward black hole
M. Halilsoy, A. Ovgun, S.H. Mazharimousavi Eur.Phys.J. C74 (2014)
2796
4. Tunnelling of vector particles from Lorentzian wormholes in 3+1
dimensions
I. Sakalli, A. Ovgun. Eur.Phys.J.Plus 130 (2015) no.6, 110
1
3. Table of contents
1. INTRODUCTION
2. WHAT IS A WORMHOLE?
3. MOTIVATION
4. HAWKING RADIATION OF THE TRAVERSABLE WORMHOLES
5. THIN-SHELL WORMHOLES
6. ON A PARTICULAR THIN-SHELL WH
7. HAYWARD THIN-SHELL WH IN 3+1-D
2
18. 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.
2
20. History of Wormholes
Figure 10: Firstly , Flamm’s work on the WH physics using the Schwarzschild
solution (1916).
2
21. Figure 11: Einstein and Rosen (ER) (1935), ER bridges connecting two
identicalsheets.
2
22. 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.
2
24. Figure 14: Then Morris, Thorne, and Yurtsever investigated the requirements
of the energy condition for traversable WHs.
2
25. Figure 15: A technical way to make thin-shell WHs by Visser (1988).
2
26. Traversable Wormhole Construction Criteria
• Spherically symmetric and static metric
• 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.
2
27. 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
+ sin
2
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
32. Figure 19: The radiation is due to the black hole capturing one of a
particle-antiparticle pair created spontaneously near to the event horizon.
2
33. • For studying the HR of traversable WHs, we consider a general
spherically symmetric and dynamic WH with a past outer
trapping horizon.
• The traversable WH metric can be transformed into the
generalized retarded Eddington-Finkelstein coordinates as
following
ds2
= −Cdu2
− 2dudr + r2
(
dθ2
+ Bdφ2
)
, (2)
where C = 1 − 2M/r and B = sin
2
θ.
• Proca equation in a curved space-time :
1
√
−g
∂µ
(√
−gψν;µ
)
+
m2
ℏ2
ψν
= 0, (3)
in which the wave functions are defined as ψν = (ψ0, ψ1, ψ2, ψ3).
• Use WKB approximation, the following HJ ansätz is substituted
into Eq. (3)
ψν = (c0, c1, c2, c3) e
i
ℏ S(u,r,θ,ϕ)
, (4)
2
34. • Furthermore, we define the action S(u, r, θ, ϕ) as following
S(u, r, θ, ϕ) = S0(u, r, θ, ϕ)+ℏS1(u, r, θ, ϕ)+ℏ2
S2(u, r, θ, ϕ)+.... (5)
• Then one can use the separation of variables method to the
action S0(u, r, θ, ϕ):
S0 = Eu − W(r) − jθ − kϕ, (6)
• It is noted that E and (j, k) are energy and real angular
constants, respectively.
• After inserting Eqs. (4), (5), and (6) into Eq. (3), a matrix equation
2
35. ∆ (c0,c1, c2, c3)
T
= 0 (to the leading order in ℏ) is obtained by
∆11 = 2B [∂rW(r)]
2
r2
,
∆12 = ∆21 = 2m2
r2
B + 2B∂rW(r)Er2
+ 2Bj2
+ 2k2
,
∆13 = −
2∆31
r2
= −2Bj∂rW(r),
∆14 =
∆41
Br2
= −2k∂rW(r),
∆22 = −2BCm2
r2
+ 2E2
r2
B − 2j2
BC − 2k2
C, (7)
∆23 =
−2∆32
r2
= 2jBC∂rW(r) + 2EjB,
∆24 =
∆42
Br2
= 2kC∂rW(r) + 2kE,
∆33 = m2
r2
B + 2BEr2
∂rW(r) + r2
BC [∂rW(r)]
2
+ k2
,
∆34 =
−∆43
2B
= −kj,
∆44 = −2r2
BC [∂rW(r)]
2
− 4BEr2
∂rW(r) − 2B(m2
r2
+ j2
).
2
36. The determinant of the ∆-matrix (det∆ = 0) is used to get
det∆ = 64Bm2
r2
{
1
2
r2
BC [∂rW(r)]
2
+ BEr2
∂rW(r) +
B
2
(
m2
r2
+ j2
)
+
k2
2
}3
=
(8)
• Then the Eq. (8) is solved for W(r)
W±(r) =
∫ (
−E
C
±
√
E2
C2
−
m2
C
−
j2
CB2r2
−
k2
Cr2
)
dr. (9)
• The above integral near the horizon (r → r0) reduces to
W±(r) ≃
∫ (
−E
C
±
E
C
)
dr. (10)
• The probability rate of the ingoing/outgoing particles only
depend on the imaginary part of the action.
• Eq. (10) has a pole at C = 0 on the horizon.
• Using the contour integration in the upper r half-plane, one
obtains
W± = iπ
(
−E
2κ|H
±
E
2κ|H
)
. (11)
2
37. From which
ImS = ImW±, (12)
that the κ|H = ∂rC/2 is the surface gravity.
• Note that the κ|H is positive quantity because the throat is an
outer trapping horizon.
• When we define the probability of incoming particles W+ to
100% such as Γabsorption ≈ e−2ImW
≈ 1.
• Consequently W− stands for the outgoing particles.
• Then we calculate the tunneling rate of the vector particles as
Γ =
Γemission
Γabsorption
= Γemission ≈ e−2ImW−
= e
2πE
κ|H . (13)
• The Boltzmann factor Γ ≈ e−βE
where β is the inverse
temperature is compared with the Eq. (13) to obtain the Hawking
temperature T|H of the traversable WH as
T|H = −
κ|H
2π
, (14)
2
38. • Surprisingly, we derive the the negative T|H that past outer
trapping horizon of the traversable WH radiate thermal
phantom energy (i.e. dark energy)
• Additionally, the radiation of phantom energy has an effect of
reduction of the size of the WH’s throat and its entropy.
• The main reason of this negativeness is the phantom energy
which is located at the throat of WH.
• Moreover, as a result of the phantom energy, the ordinary
matter can travel backward in time.
• Nonetheless, this does not create a trouble. The total entropy of
universe always increases, hence it prevents the violation of the
second law of thermodynamics.
• Moreover, in our different work, we show that the gravitino also
tunnels through WH and we calculate the tunneling rate of the
emitted gravitino particles from traversable WH.
40. • Constructing WHs with non-exotic (normal matter) source is a
difficult issue in General Relativity.
• On this purpose, firstly , Visser use the thin-shell method to
construct WHs by minimizing the exotic matter on the throat of
the WHs.
2
41. • We need to introduce some conditions on the
energy-momentum tensor.
2
42. 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.
-Check Stability by using the
Outputs Thin-shell wormhole
σ < 0 with extrinsic curvature K > 0 of the throat means it required
exotic matter.
2
44. • The line element of a Scalar Hairy Black Hole (SHBH)
investigated by Mazharimousavi and Halilsoy is
ds2
= −f(r)dt2
+
4r2
dr2
f(r)
+ r2
dθ2
, (15)
where
f(r) =
r2
l2
− ur. (16)
• Here u and l are constants.
• Event horizon of the BH is located at rh = uℓ2
.
• It is noted that the singularity located at r = 0.
• Firstly we take two identical copies of the SHBHs with (r ≥ a):
M±
= (x|r ≥ 0),
• The manifolds are bounded by hypersurfaces M+
and M−
, to get
the single manifold M = M+
+ M−
,
2
45. • We glue them together at the surface of the junction
Σ±
= (x|r = a).
where the boundaries Σ are given.
• The spacetime on the shell is
ds2
= −dτ2
+ a(τ)2
dθ2
, (17)
where τ represents the proper time .
• 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
)
, (18)
where the unit normals (nγ
nγ = 1) are
n±
γ = ± gαβ ∂H
∂xα
∂H
∂xβ
−1/2
∂H
∂xγ
, (19)
with H(r) = r − a(τ).
2
46. • The non zero components of n±
γ are calculated as
nt = ∓2a˙a, (20)
nr = ±2
√
al2(4˙a2l2a − l2u + a)
(l2u − a)
, (21)
where the dot over a quantity denotes the derivative with
respect to τ.
• Then, the non-zero extrinsic curvature components yield
K±
ττ = ∓
√
−al2(8˙a2
l2
a + 8¨al2
a2
− l2
u + 2a)
4a2l2
√
−4˙a2l2a − l2u + a
, (22)
K±
θθ = ±
1
2a
3
2 l
√
4˙a2l2a − l2u + a. (23)
• Since Kij is not continuous around the shell, we use the Lanczos
equation:
Sij = −
1
8π
(
[Kij] − [K]gij
)
. (24)
where K is the trace of Kij, [Kij] = K+
ij − K−
ij .
2
47. • Firstly, K+
= −K−
= [Kij] while [Kij] = 0.
• For the conservation of the surface stress–energy Sij
j = 0.
• Sij is stress energy-momentum tensor at the junction which is
given in general by
Si
j = diag(σ, −p), (25)
with the surface pressure p and the surface energy density σ.
• Due to the circular symmetry, we have
Ki
j =
Kτ
τ 0
0 Kθ
θ
. (26)
Thus, from Eq.s (25) and (24) one obtains the surface pressure
and surface energy density .
• Using the cut and paste technique, we can demount the interior
regions r < a of the geometry, and links its exterior parts.
2
48. • The energy density and pressure are
σ = −
1
8πa
3
2 l
√
4˙a2l2a − l2u + a, (27)
p =
1
16πa
3
2 l
(
8˙a2
l2
a + 8¨al2
a2
− l2
u + 2a
)
√
4˙a2l2a − l2u + a
. (28)
Then for the static case (a = a0), the energy and pressure
quantities reduce to
σ0 = −
1
8πa
3
2
0 l
√
−l2u + a0, (29)
p0 =
1
16πa
3
2
0 l
(
−l2
u + 2a0
)
√
−l2u + a0
. (30)
Once σ ≥ 0 and σ + p ≥ 0 hold, then WEC is satisfied.
• It is obvious from Eq. (24) that negative energy density violates
the WEC, and consequently we are in need of the exotic matter
for constructing thin-shell WH.
2
49. • We note that the total matter supporting the WH is given by
Ωσ =
∫ 2π
0
[ρ
√
−g] r=a0
dϕ = 2πa0σ(a0) = −
1
4a
1
2
0 |l|
√
−l2u + a0.
(31)
• Stability of the WH is investigated using the linear perturbation
so that the EoS is
p = ψ(σ), (32)
where ψ(σ) is an arbitrary function of σ.
• It can be written in terms of the pressure and energy density:
d
dτ
(σa) + ψ
da
dτ
= −˙aσ. (33)
• From above equation, one reads
σ′
= −
1
a
(2σ + ψ), (34)
and its second derivative yields
σ′′
=
2
a2
( ˜ψ + 3)(σ +
ψ
2
). (35)
2
50. where prime and tilde symbols denote derivative with respect to
a and σ, respectively.
• The conservation of energy for the shell is in general given by
˙a2
+ V = 0, (36)
where the effective potential V is found from Eq. (27)
V =
1
4l2
−
u
4a
− 16a2
σ2
π2
. (37)
• In fact, Eq. (36) is nothing but the equation of the oscillatory
motion in which the stability around the equilibrium point
a = a0 is conditional on V′
(a0) = 0 and V′′
(a0) ≥ 0.
• We finally obtain
V′′
= −
1
2a3
[
64π2
a5
(
(σσ′
)
′
+ 4σ′ σ
a
+
σ2
a2
)
+ u
]
a=a0
, (38)
2
51. or equivalently,
V′′
=
1
2a3
{−64π2
a3
[
(2ψ′
+ 3)σ2
+ ψ(ψ′
+ 3)σ + ψ2
]
− u}
a=a0
.
(39)
• 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 .
2
52. Some Models of EoS Supporting Thin-Shell WH
In this section, we use particular gas models (linear barotropic gas
(LBG) , chaplygin gas (CG) , generalized chaplygin gas (GCG) and
logarithmic gas (LogG) ) to explore the stability of TSW.
Stability analysis of Thin-Shell WH via the LBG
The equation of state of LBG is given by
ψ = ε0σ, (40)
and hence
ψ′
(σ0) = ε0, (41)
where ε0 is a constant parameter. By changing the values of l and u
in Eq. (35), we illustrate the stability regions for TSW, in terms of ε0
and a0.
2
53. Figure 20: Stability Regions via the LBG
Stability analysis of Thin-Shell WH via CG
The equation of state of CG that we considered is given by
ψ = ε0(
1
σ
−
1
σ0
) + p0, (42)
2
54. and one naturally finds
ψ′
(σ0) =
−ε0
σ2
0
. (43)
After inserting Eq. (39) into Eq. (35), The stability regions for
thin-shell WH supported by CG is plotted in Fig.
2
56. Stability analysis of Thin-Shell WH via GCG
By using the equation of state of GCG
ψ = p0
(σ0
σ
)ε0
, (44)
and whence
ψ′
(σ0) = −ε0
p0
σ0
, (45)
Substituting Eq. (41) in Eq. (35), one can illustrate the stability
regions of thin-shell WH supported by GCG as seen in Fig.
2
58. Stability analysis of Thin-Shell WH via LogG
• In our final example, the equation of state for LogG is selected
as follows (ε0, σ0, p0 are constants)
ψ = ε0 ln(
σ
σ0
) + p0, (46)
which leads to
ψ′
(σ0) =
ε0
σ0
. (47)
• After inserting the above expression into Eq. (35), we show the
stability regions of thin-shell WH supported by LogG in Fig.
2
60. • In summary, we have constructed thin-shell WH by gluing two
copies of SHBH via the cut and paste procedure.
• To this end, we have used the fact that the radius of throat must
be greater than the event horizon of the metric given: (a0 > rh).
• We have used LBG, CG, GCG, and LogG EoS to the exotic matter.
• Then, the stability analysis (V′′
(a0) ≥ 0) is plotted.
• We show the stability regions in terms a0 andε0
2
62. • The metric of the Hayward BH is given by
ds2
= −f(r)dt2
+ f(r)−1
dr2
+ r2
dΩ2
. (48)
with the metric function
f (r) =
(
1 −
2mr2
r3 + 2ml2
)
(49)
and
dΩ2
= dθ2
+ sin
2
θdϕ2
. (50)
• 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
)
, (51)
whereas at small r becomes a de Sitter BH
lim
r→0
f (r) → 1 −
r2
l2
+ O
(
r5
)
. (52)
2
63. • 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
+ sin
2
θdϕ2
)
(53)
where τ is the proper time on the shell.
• The Einstein equations on the shell are
[
Kj
i
]
− [K] δj
i = −Sj
i (54)
where [X] = X2 − X1,.
• It is noted that the extrinsic curvature tensor is Kj
i.
• Moreover, K stands for its trace.
• 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.
2
64. • The energy and pressure densities are obtained as
σ = −
4
a
√
f (a) + ˙a2 (55)
p = 2
(√
f (a) + ˙a2
a
+
¨a + f′
(a) /2
√
f (a) + ˙a2
)
. (56)
• Then they reduce to simple form in a static configuration
(a = a0)
σ0 = −
4
a0
√
f (a0) (57)
and
p0 = 2
(√
f (a0)
a0
+
f′
(a0) /2
√
f (a0)
)
. (58)
• Stability of such a WH is investigated by applying a linear
perturbation with the following EoS
p = ψ (σ) (59)
2
65. • Moreover the energy conservation is
Sij
;j = 0 (60)
which in closed form it equals to
Sij
,j + Skj
Γiµ
kj + Sik
Γj
kj = 0 (61)
after the line element in Eq.(53) is used, it opens to
∂
∂τ
(
σa2
)
+ p
∂
∂τ
(
a2
)
= 0. (62)
• The 1-D equation of motion is
˙a2
+ V (a) = 0, (63)
in which V (a) is the potential,
V (a) = f −
(aσ
4
)4
. (64)
• The equilibrium point at a = a0 means V′
(a0) = 0 and
V′′
(a0) ≥ 0.
2
66. • 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) (65)
and
σ′′
=
2
a2
(σ + ψ) (3 + 2ψ′
) , (66)
where ψ′
= dψ
dσ . After we use ψ0 = p0, finally it is found that
V′′
(a0) = f′′
0 −
1
8
[
(σ0 + 2p0)
2
+ 2σ0 (σ0 + p0) (1 + 2ψ′
(σ0))
]
(67)
2
67. 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 (68)
in which η0 is a constant and ψ′
(σ0) = η0.
2
68. Figure 24: 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.
2
69. Chaplygin gas
For CG, we choose the EoS as follows
ψ = η0
(
1
σ
−
1
σ0
)
+ p0 (69)
where η0 is a constant and ψ′
(σ0) = −η0
σ2
0
.
2
70. Figure 25: 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.
2
71. Generalized Chaplygin gas
The EoS of the GCG is taken as
ψ (σ) = η0
(
1
σν
−
1
σν
0
)
+ p0 (70)
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
. (71)
2
72. Figure 26: 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.
2
73. Modified Generalized Chaplygin gas
In this case, the MGCG is
ψ (σ) = ξ0 (σ − σ0) − η0
(
1
σν
−
1
σν
0
)
+ p0 (72)
in which ξ0, η0 and ν are free parameters. Therefore,
ψ′
(σ0) = ξ0 + η0
η0ν
σν+1
0
. (73)
2
74. Figure 27: 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.
2
75. Logarithmic gas
Lastly LogG is choosen by follows
ψ (σ) = η0 ln
σ
σ0
+ p0 (74)
in which η0 is a constant. For LogG, we find that
ψ′
(σ0) =
η0
σ0
. (75)
2
76. Figure 28: 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.
2
77. • In this section we construct thin-shell WHs from the Hayward
BH.
• 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.
2