This paper is my work on Einstein's Relativity theory for speed greater than light.The paper mainly focus on to generalize the Lorentz factor , equation of negative energy and negative mass thus proving the existence of wormholes and the derivation of new space metric.
Errata of Seismic analysis of structures by T.K. Dattatushardatta
This document provides corrections to errors found in textbook chapters 1 through 5 on structural dynamics and earthquake engineering. Over 100 errors are noted, including incorrect equation numbers, variables, values, figure references, and text. The corrections range from single character or number changes to entire rewritten paragraphs. The high-level issues corrected include equation forms, variable definitions, figure contents, example solutions, and references to other parts of the text.
This document discusses response analysis for structures subjected to specified ground motions. It begins by introducing time history analysis and the different methods that can be used, including direct integration and Fourier transform techniques. It then reviews concepts for modeling single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems subjected to ground motions. Examples are provided to demonstrate how to derive the mass, stiffness and forcing vectors/matrices for different structural models. Methods for specifying multi-support excitation and deriving transformation matrices (r-matrices) relating responses at different supports are also described through examples.
Me307 machine elements formula sheet Erdi Karaçal Mechanical Engineer Univers...Erdi Karaçal
1. The document discusses various topics related to stress analysis including moment of inertias, stresses from different load cases, principal stresses, stress states, stresses in cylinders, and deflection analysis using Castigliano's theorem.
2. Design considerations for static strength are covered for both ductile and brittle materials using theories such as maximum normal stress and distortion energy.
3. Fatigue strength design includes determining the endurance limit based on material properties and adjusting it using factors for surface finish, size, and loading conditions.
This document describes a damped oscillation graph of a spring-mass system experiencing dry frictional force. The system performs damped harmonic oscillation as the mass oscillates back and forth along the horizontal surface. The document provides the equation of motion, solution, and a Maple code to plot the decreasing amplitude oscillation graph over multiple periods as a function of time.
Formul me-3074683 Erdi Karaçal Mechanical Engineer University of GaziantepErdi Karaçal
1. The document discusses various topics related to stress analysis and design including moment of inertias, stresses, deflection analysis, design for static strength, fatigue design, tolerances and fits, power screws, and bolted joints.
2. Formulas are provided for calculating stresses and strains under different loading conditions as well as determining critical loads, deflections, endurance limits, and stresses in various mechanical elements.
3. Design considerations for different materials, loading types, and failure theories are outlined for static and fatigue strength analysis. Guidelines for screw thread stresses, efficiency, and joint stiffness are also summarized.
1) Four positive charges are located at the corners of a square in the xy-plane. A fifth positive charge is located 8cm from the others. The total force on the fifth charge is calculated to be 4.0x10-4 N directed along the z-axis.
2) Two charges of Q1 coulombs are located at z=±1. For a third charge Q2 to produce zero total electric field at (0,1,0), Q2 must lie along the y-axis at y=1±21/4|Q2|/Q1, where the sign depends on the sign of Q2.
3) The total force on a 50nC charge
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Errata of Seismic analysis of structures by T.K. Dattatushardatta
This document provides corrections to errors found in textbook chapters 1 through 5 on structural dynamics and earthquake engineering. Over 100 errors are noted, including incorrect equation numbers, variables, values, figure references, and text. The corrections range from single character or number changes to entire rewritten paragraphs. The high-level issues corrected include equation forms, variable definitions, figure contents, example solutions, and references to other parts of the text.
This document discusses response analysis for structures subjected to specified ground motions. It begins by introducing time history analysis and the different methods that can be used, including direct integration and Fourier transform techniques. It then reviews concepts for modeling single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems subjected to ground motions. Examples are provided to demonstrate how to derive the mass, stiffness and forcing vectors/matrices for different structural models. Methods for specifying multi-support excitation and deriving transformation matrices (r-matrices) relating responses at different supports are also described through examples.
Me307 machine elements formula sheet Erdi Karaçal Mechanical Engineer Univers...Erdi Karaçal
1. The document discusses various topics related to stress analysis including moment of inertias, stresses from different load cases, principal stresses, stress states, stresses in cylinders, and deflection analysis using Castigliano's theorem.
2. Design considerations for static strength are covered for both ductile and brittle materials using theories such as maximum normal stress and distortion energy.
3. Fatigue strength design includes determining the endurance limit based on material properties and adjusting it using factors for surface finish, size, and loading conditions.
This document describes a damped oscillation graph of a spring-mass system experiencing dry frictional force. The system performs damped harmonic oscillation as the mass oscillates back and forth along the horizontal surface. The document provides the equation of motion, solution, and a Maple code to plot the decreasing amplitude oscillation graph over multiple periods as a function of time.
Formul me-3074683 Erdi Karaçal Mechanical Engineer University of GaziantepErdi Karaçal
1. The document discusses various topics related to stress analysis and design including moment of inertias, stresses, deflection analysis, design for static strength, fatigue design, tolerances and fits, power screws, and bolted joints.
2. Formulas are provided for calculating stresses and strains under different loading conditions as well as determining critical loads, deflections, endurance limits, and stresses in various mechanical elements.
3. Design considerations for different materials, loading types, and failure theories are outlined for static and fatigue strength analysis. Guidelines for screw thread stresses, efficiency, and joint stiffness are also summarized.
1) Four positive charges are located at the corners of a square in the xy-plane. A fifth positive charge is located 8cm from the others. The total force on the fifth charge is calculated to be 4.0x10-4 N directed along the z-axis.
2) Two charges of Q1 coulombs are located at z=±1. For a third charge Q2 to produce zero total electric field at (0,1,0), Q2 must lie along the y-axis at y=1±21/4|Q2|/Q1, where the sign depends on the sign of Q2.
3) The total force on a 50nC charge
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Design of a Lift Mechanism for Disabled PeopleSamet Baykul
DATE: 2019.01
In this project, a lift mechanism for especially disabled people has been designed. It is known as home lift, platform lift, vertical lift or through floor lift. These products operate by moving up and down. The lift mechanism consists of powertrain, linkage system, and a raising platform.
- Design of a a shaft, connecting rods, pins and weldings
- Static force analysis
- Building shear and moment diagrams
- Calculation of mechanical design parameters
1) The document contains examples calculating various vector operations such as finding unit vectors, magnitudes, dot and cross products, and vector components.
2) It also contains examples finding vector fields, surfaces where vector field components are equal to scalars, and demonstrating properties of vector fields such as being everywhere parallel.
3) The document tests understanding of vector concepts through multiple practice problems.
A Note on the Derivation of the Variational Inference Updates for DILNTomonari Masada
This document describes the derivation of the variational inference updates for the Dirichlet-logistic normal model. It begins by defining the joint distribution and obtaining a lower bound on the log evidence using Jensen's inequality. It then examines each term in the lower bound, applying integrals and derivations. This results in an expression for the lower bound involving parameters of the model distributions. The document concludes by stating assumptions made about some of the distributions and providing the update equations for variational distributions q(Cmn) and q(Zmk).
The document discusses modeling dynamic systems and earthquake response. It covers basic concepts like Fourier transforms, single and multi-degree of freedom systems, modal analysis, and elastic response spectra. Numerical methods are presented for dynamic analysis in the frequency and time domains, including the finite element method and method of complex response. Examples of earthquake records and harmonic motion are shown.
2d beam element with combined loading bending axial and torsionrro7560
The document discusses beam theory and finite element modeling of beams and frames. It provides information on modeling beams using one-dimensional beam elements with cubic shape functions. The formulation describes defining the element stiffness matrix and calculating the element's contribution to the global structural stiffness matrix and force vector based on applied loads. Boundary conditions and sample problems are presented to demonstrate the element modeling approach.
The document discusses the longest common subsequence (LCS) problem and presents a dynamic programming approach to solve it. It defines key terms like subsequence and common subsequence. It then presents a theorem that characterizes an LCS and shows it has optimal substructure. A recursive solution and algorithm to compute the length of an LCS are provided, with a running time of O(mn). The b table constructed enables constructing an LCS in O(m+n) time.
This document summarizes key points from a lecture on steady-state heat conduction in multiple dimensions:
1) It introduces the Laplace equation that governs two-dimensional steady-state heat conduction problems. 2) Analytical solutions can be obtained for some simple geometries using separation of variables, and the document works through examples of this approach. 3) Numerical and graphical techniques can also be used to analyze more complex multi-dimensional heat conduction problems. 4) The concept of a shape factor is introduced to simplify calculating heat flow through objects of various shapes.
This document contains a marking scheme for a Physics sample question paper from 2018. It includes 5 sections - Section A with 5 one-mark questions, Section B with 6 questions ranging from 1/2 to 3 marks, Section C with 4 questions ranging from 1 to 5 marks, Section D with 3 one-mark questions and Section E with 7 questions ranging from 1 to 5 marks. The marking scheme provides details about the expected answers for each question such as concepts, principles, diagrams, calculations and numerical values involved. It also specifies the marks allocated for different parts of each question.
Convergence methods for approximated reciprocal and reciprocal-square-rootKeigo Nitadori
This document discusses convergence methods for approximating reciprocal (1/x) and reciprocal-square-root (1/√x) values using polynomials expanded in a Taylor series. It presents the general forms for reciprocal and reciprocal-square-root approximations up to eighth order. For reciprocal-square-root, the well-known Newton-Raphson method and its optimized form for fused multiply-add hardware are described. Examples for other functions like reciprocal-cube-root are also provided. Higher order methods are noted to require more registers for coefficient storage.
1) The document uses dimensional analysis to derive relationships between fundamental physical constants like the Planck constant h, Boltzmann constant k, speed of light c, gravitational constant G, and Stefan-Boltzmann constant σ.
2) It then applies these relationships and principles of thermodynamics to analyze properties of black holes, deriving expressions for the area of the event horizon, entropy, temperature, and evaporation time of a black hole in terms of its mass.
3) Finally, it considers a black hole approaching thermal equilibrium with a background radiation bath and shows that as it evaporates and loses mass over time, it will move away from equilibrium.
The document contains 10 problems involving electromagnetic induction and Maxwell's equations. Problem 10.1 involves calculating the voltage and current in a circuit with a changing magnetic flux. Problem 10.2 replaces a voltmeter with a resistor and calculates the resulting current. Problem 10.3 calculates the emf induced in closed paths with changing magnetic fluxes.
Physics Notes: Solved numerical of Physics first yearRam Chand
1. The document is a physics textbook covering solved numerical problems for the Sindh Textbook Board.
2. It was written by Dr. Ram Chand Raguel and covers topics like scalars and vectors, motion, statics, gravitation, and optics.
3. The author has visited research institutions in the US, Malaysia, Italy, and China and is a member of the American Association of Physics Teachers.
El documento presenta el plan de una clase sobre sumas de dos y tres cifras que será dictada a estudiantes de matemáticas. La clase utilizará una situación problema para introducir el tema y luego explicará los conceptos básicos de la suma. Seguidamente, el profesor realizará ejercicios en el tablero y supervisará a los estudiantes mientras resuelven ejercicios. Al final, los estudiantes tendrán una tarea.
This document is a dissertation submitted by Abdul Kasukari Athumani in partial fulfillment of the requirements for a Bachelor's degree in Rural and Regional Development Planning from the Institute of Rural Development Planning in Dodoma, Tanzania in August 2014. The dissertation assesses anti-corruption clubs in two secondary schools in Dodoma municipality. It collected data through interviews, observations, and focus group discussions with 67 respondents including club members, teachers, and a municipal anti-corruption officer. The study aimed to evaluate club membership, activities, and challenges. It found that clubs face issues like lack of training, few organized events, and management problems. The document recommends that stakeholders support anti-corruption clubs programs to address these challenges.
Several students at Freed-Hardeman University shaved their heads to support fellow student Caleb Sams, who had been diagnosed with a rare form of cancer called sarcoma. Twenty-five young men participated in the fundraiser, which raised over $2,500 for Sams's medical expenses. One participant, Jake Wilbanks, said he was happy to shave his head if it helped comfort Sams during his battle with cancer. The event showed the support from students and social clubs on campus for Sams as he undergoes chemotherapy treatments.
Shashikant S.H. has over 25 years of experience in operations, production planning, and troubleshooting in industries including cast steel, alloy steel, and sinter making. He currently serves as Deputy General Manager and Head of Plant at Jindal SAW Ltd, where he is responsible for day-to-day production, planning, and inter-departmental administration. Previously, he held production roles at Aqua Alloys Pvt Ltd and Chandra Alloys Pvt Ltd. He has a Bachelor's degree in Metallurgy and seeks to utilize his expertise to maximize efficiency and accomplish organizational goals.
This document is a research report submitted by Himanshi Singh, a student in the 6th semester of the Bachelor of Business Administration program at RSMT in Varanasi, India, with a roll number of 118134045, to her professor Brijesh Yadav in partial fulfillment of the requirements for her BBA degree.
Auditing and marketing discussion 7 for profstan onlyallhomeworktutors
ACL software can be used by auditors to find irregularities or patterns in transactions by analyzing relevant data fields, helping identify control weaknesses. Three types of database update anomalies are insertion, deletion, and update anomalies, which can be minimized by ensuring data is complete, only authorized users can delete data, and data is reviewed before being saved. Auditors should understand data normalization principles and have IT skills to analyze system processes and data flows using audit tools.
Design of a Lift Mechanism for Disabled PeopleSamet Baykul
DATE: 2019.01
In this project, a lift mechanism for especially disabled people has been designed. It is known as home lift, platform lift, vertical lift or through floor lift. These products operate by moving up and down. The lift mechanism consists of powertrain, linkage system, and a raising platform.
- Design of a a shaft, connecting rods, pins and weldings
- Static force analysis
- Building shear and moment diagrams
- Calculation of mechanical design parameters
1) The document contains examples calculating various vector operations such as finding unit vectors, magnitudes, dot and cross products, and vector components.
2) It also contains examples finding vector fields, surfaces where vector field components are equal to scalars, and demonstrating properties of vector fields such as being everywhere parallel.
3) The document tests understanding of vector concepts through multiple practice problems.
A Note on the Derivation of the Variational Inference Updates for DILNTomonari Masada
This document describes the derivation of the variational inference updates for the Dirichlet-logistic normal model. It begins by defining the joint distribution and obtaining a lower bound on the log evidence using Jensen's inequality. It then examines each term in the lower bound, applying integrals and derivations. This results in an expression for the lower bound involving parameters of the model distributions. The document concludes by stating assumptions made about some of the distributions and providing the update equations for variational distributions q(Cmn) and q(Zmk).
The document discusses modeling dynamic systems and earthquake response. It covers basic concepts like Fourier transforms, single and multi-degree of freedom systems, modal analysis, and elastic response spectra. Numerical methods are presented for dynamic analysis in the frequency and time domains, including the finite element method and method of complex response. Examples of earthquake records and harmonic motion are shown.
2d beam element with combined loading bending axial and torsionrro7560
The document discusses beam theory and finite element modeling of beams and frames. It provides information on modeling beams using one-dimensional beam elements with cubic shape functions. The formulation describes defining the element stiffness matrix and calculating the element's contribution to the global structural stiffness matrix and force vector based on applied loads. Boundary conditions and sample problems are presented to demonstrate the element modeling approach.
The document discusses the longest common subsequence (LCS) problem and presents a dynamic programming approach to solve it. It defines key terms like subsequence and common subsequence. It then presents a theorem that characterizes an LCS and shows it has optimal substructure. A recursive solution and algorithm to compute the length of an LCS are provided, with a running time of O(mn). The b table constructed enables constructing an LCS in O(m+n) time.
This document summarizes key points from a lecture on steady-state heat conduction in multiple dimensions:
1) It introduces the Laplace equation that governs two-dimensional steady-state heat conduction problems. 2) Analytical solutions can be obtained for some simple geometries using separation of variables, and the document works through examples of this approach. 3) Numerical and graphical techniques can also be used to analyze more complex multi-dimensional heat conduction problems. 4) The concept of a shape factor is introduced to simplify calculating heat flow through objects of various shapes.
This document contains a marking scheme for a Physics sample question paper from 2018. It includes 5 sections - Section A with 5 one-mark questions, Section B with 6 questions ranging from 1/2 to 3 marks, Section C with 4 questions ranging from 1 to 5 marks, Section D with 3 one-mark questions and Section E with 7 questions ranging from 1 to 5 marks. The marking scheme provides details about the expected answers for each question such as concepts, principles, diagrams, calculations and numerical values involved. It also specifies the marks allocated for different parts of each question.
Convergence methods for approximated reciprocal and reciprocal-square-rootKeigo Nitadori
This document discusses convergence methods for approximating reciprocal (1/x) and reciprocal-square-root (1/√x) values using polynomials expanded in a Taylor series. It presents the general forms for reciprocal and reciprocal-square-root approximations up to eighth order. For reciprocal-square-root, the well-known Newton-Raphson method and its optimized form for fused multiply-add hardware are described. Examples for other functions like reciprocal-cube-root are also provided. Higher order methods are noted to require more registers for coefficient storage.
1) The document uses dimensional analysis to derive relationships between fundamental physical constants like the Planck constant h, Boltzmann constant k, speed of light c, gravitational constant G, and Stefan-Boltzmann constant σ.
2) It then applies these relationships and principles of thermodynamics to analyze properties of black holes, deriving expressions for the area of the event horizon, entropy, temperature, and evaporation time of a black hole in terms of its mass.
3) Finally, it considers a black hole approaching thermal equilibrium with a background radiation bath and shows that as it evaporates and loses mass over time, it will move away from equilibrium.
The document contains 10 problems involving electromagnetic induction and Maxwell's equations. Problem 10.1 involves calculating the voltage and current in a circuit with a changing magnetic flux. Problem 10.2 replaces a voltmeter with a resistor and calculates the resulting current. Problem 10.3 calculates the emf induced in closed paths with changing magnetic fluxes.
Physics Notes: Solved numerical of Physics first yearRam Chand
1. The document is a physics textbook covering solved numerical problems for the Sindh Textbook Board.
2. It was written by Dr. Ram Chand Raguel and covers topics like scalars and vectors, motion, statics, gravitation, and optics.
3. The author has visited research institutions in the US, Malaysia, Italy, and China and is a member of the American Association of Physics Teachers.
El documento presenta el plan de una clase sobre sumas de dos y tres cifras que será dictada a estudiantes de matemáticas. La clase utilizará una situación problema para introducir el tema y luego explicará los conceptos básicos de la suma. Seguidamente, el profesor realizará ejercicios en el tablero y supervisará a los estudiantes mientras resuelven ejercicios. Al final, los estudiantes tendrán una tarea.
This document is a dissertation submitted by Abdul Kasukari Athumani in partial fulfillment of the requirements for a Bachelor's degree in Rural and Regional Development Planning from the Institute of Rural Development Planning in Dodoma, Tanzania in August 2014. The dissertation assesses anti-corruption clubs in two secondary schools in Dodoma municipality. It collected data through interviews, observations, and focus group discussions with 67 respondents including club members, teachers, and a municipal anti-corruption officer. The study aimed to evaluate club membership, activities, and challenges. It found that clubs face issues like lack of training, few organized events, and management problems. The document recommends that stakeholders support anti-corruption clubs programs to address these challenges.
Several students at Freed-Hardeman University shaved their heads to support fellow student Caleb Sams, who had been diagnosed with a rare form of cancer called sarcoma. Twenty-five young men participated in the fundraiser, which raised over $2,500 for Sams's medical expenses. One participant, Jake Wilbanks, said he was happy to shave his head if it helped comfort Sams during his battle with cancer. The event showed the support from students and social clubs on campus for Sams as he undergoes chemotherapy treatments.
Shashikant S.H. has over 25 years of experience in operations, production planning, and troubleshooting in industries including cast steel, alloy steel, and sinter making. He currently serves as Deputy General Manager and Head of Plant at Jindal SAW Ltd, where he is responsible for day-to-day production, planning, and inter-departmental administration. Previously, he held production roles at Aqua Alloys Pvt Ltd and Chandra Alloys Pvt Ltd. He has a Bachelor's degree in Metallurgy and seeks to utilize his expertise to maximize efficiency and accomplish organizational goals.
This document is a research report submitted by Himanshi Singh, a student in the 6th semester of the Bachelor of Business Administration program at RSMT in Varanasi, India, with a roll number of 118134045, to her professor Brijesh Yadav in partial fulfillment of the requirements for her BBA degree.
Auditing and marketing discussion 7 for profstan onlyallhomeworktutors
ACL software can be used by auditors to find irregularities or patterns in transactions by analyzing relevant data fields, helping identify control weaknesses. Three types of database update anomalies are insertion, deletion, and update anomalies, which can be minimized by ensuring data is complete, only authorized users can delete data, and data is reviewed before being saved. Auditors should understand data normalization principles and have IT skills to analyze system processes and data flows using audit tools.
This certificate certifies that a research report titled "Recruitment and Selection Process in Reliance Communication. LTD, Varanasi" submitted by Himanshi Singh for her Bachelor of Business Administration degree fulfills the requirements and standards for the degree program. The report was conducted during the 2014-2015 academic session under the guidance of Dr. P.N. Singh and Mr. Bijendra Yadav. The certificate recommends that the report be sent for evaluation.
This Man Had Biryani Thrice A Day For 2 Whole Months And Managed To Lose WeightNishanth Appari
Nishanth Appari, a 23-year-old man from Hyderabad, ate chicken biryani three times a day for 50 days and lost weight instead of gaining weight. He ate approximately 50 packets of biryani over the 60 day period, spending Rs. 5,000, and lost close to 3 kgs and 2 inches overall. The purpose of his experiment was to show that eating certain foods occasionally is okay and that weight is determined more by quantity than food quality alone.
Esta presentación se corresponde sobre la Violencia contra las Mujeres y sus consecuencias, sus luchas por el reconocimiento de derechos y de nuevos derechos en pro de llevar una vida digna sin violencia, en condiciones equitativas y de equidad de género. Saber que existen delitos, que están contemplados en una norma y que puede ser exigible ante los órganos de administración de justicia, es un gran logro. Conquistar nuevos derechos depende de nuestra conciencia de toda la sociedad.
El documento describe una actividad lúdica de aprendizaje en la que los estudiantes fueron divididos en equipos para responder preguntas. Los resultados mostraron que los estudiantes estuvieron atentos, motivados y participativos, compartiendo opiniones y apoyándose mutuamente. La actividad creó un ambiente favorable para el aprendizaje y la interacción entre los estudiantes.
Reliance Communications is the flagship company of Anil Dhirubhai Ambani Group. It was founded by Dhirubhai Ambani in 1999 with a vision to make communication tools accessible to all. Reliance Communications is now India's largest integrated telecom operator with over 40 million subscribers. It offers both wireless and wireline services including voice, data, broadband, national and international long distance. The company has established a pan-India fiber optic network and owns the world's largest private submarine cable system. Reliance Communications aims to further Dhirubhai Ambani's dream of a digitally empowered India.
- Kimchi originated in Korea in the 7th century as a way to preserve vegetables during winter using fermentation. It has since spread globally but retains significance as a symbol of Korean culture.
- Different regions of Korea have developed their own varieties of kimchi using local vegetables and seasonings. As Koreans immigrated abroad, they adapted kimchi to appeal more to local tastes.
- Though adapted, kimchi remains a staple of Korean cuisine and an expression of cultural identity both in Korea and Korean diaspora communities worldwide.
This document introduces complex integration and provides examples of evaluating integrals along paths in the complex plane. It expresses integrals in terms of real and imaginary parts involving line integrals of functions. Key points made include:
- Complex integrals can be interpreted as line integrals over paths in the complex plane.
- Integrals of analytic functions over closed paths, like the unit circle, may yield simple results like 2πi or 0.
- Blasius' theorem relates forces and moments on a cylinder in fluid flow to complex integrals around the cylinder boundary.
1) The document derives an equation to calculate the velocity of an accelerating spaceship over time by considering infinitesimal changes in velocity and using the velocity addition formula.
2) This yields an equation showing velocity is equal to the speed of light multiplied by the hyperbolic tangent of acceleration times time over the speed of light.
3) The document then introduces the concept of rapidity, defined as the time integral of acceleration, and shows that rapidities add through simple addition while velocities add through a more complex formula.
1) The document derives an equation to calculate the velocity of an accelerating spaceship over time by considering infinitesimal changes in velocity and using the velocity addition formula.
2) This yields an equation showing velocity is equal to the speed of light multiplied by the hyperbolic tangent of acceleration times time over the speed of light.
3) The document then introduces the concept of rapidity, defined as the time integral of acceleration, and shows that rapidity provides a useful way to write expressions in relativity, with velocities simply being the speed of light multiplied by the hyperbolic tangent of rapidity.
This document provides solutions to problems from a numerical methods textbook. It includes MATLAB code and step-by-step workings for problems related to numerical integration, differential equations, curve fitting, and other numerical methods topics. Solutions are provided for single-step and multi-step problems across 10 chapters. MATLAB code demonstrates the numerical methods and allows plotting of model outputs and errors. Equations, tables, and plots are used to show the results of applying numerical techniques to example problems.
This document discusses base excitation in vibration analysis and provides examples. It begins by introducing base excitation as an important class of vibration analysis that involves preventing vibrations from passing through a vibrating base into a structure. Examples of base excitation include vibrations in cars, satellites, and buildings during earthquakes. The document then provides mathematical models and equations to analyze single degree of freedom base excitation systems. Graphs of transmissibility ratios are presented and examples are worked through, such as calculating car vibration amplitude at different speeds. Rotating unbalance is also covered as another source of vibration excitation.
Neutrino mass and neutron decay are analyzed. The maximum energy of an electron emitted in neutron decay is calculated to be approximately 0.782 MeV. The speed of the anti-neutrino when the electron reaches maximum energy is calculated to be approximately 1.265 x 10-3 c.
Light levitation of a glass hemisphere is considered. Snell's law is used to derive an expression for the minimum laser power required to levitate the hemisphere that balances the optical force and the weight of the hemisphere.
Modern Control Engineering Problems Ch 3.pdfMahamad Jawhar
This document provides solutions to example problems involving block diagram simplification and obtaining state-space representations and transfer functions of systems. The solutions walk through simplifying block diagrams by moving branch points and combining blocks. They also show how to define state variables and derive state-space models and transfer functions for various mechanical, electrical, and control systems. Key steps include redrawing diagrams, defining state variables as outputs of integrators, and setting up state and output equations in matrix form.
This document describes an experiment to determine the damping constant of a spring and hook system. A spring with an attached hook was stretched and released, and its motion was recorded. The effective mass and spring constant were calculated. The position over time was plotted and showed an exponentially decaying oscillation. The damping ratio was estimated by taking the logarithm of the amplitude ratios between peaks and finding the slope. This gave a damping ratio of 0.006 and a calculated damping constant of approximately 0.059.
EMF ELECTROSTATICS:
Coulomb’s Law, Electric Field of Different Charge Configurations using Coulomb’s Law, Electric Flux, Field Lines, Gauss’s Law in terms of E (Integral Form and Point Form), Applications of Gauss’s Law, Curl of the Electric Field, Electric Potential, Calculation of Electric Field Through Electric Potential for given Charge Configuration, Potential Gradient, The Dipole, Energy density in the Electric field.
This document provides an introduction to electrostatics and discusses electric charges, atomic structure, early theories of electricity from Thales and Franklin, conduction and insulation, Coulomb's law for electrostatic force between two charges, and examples of calculating electrostatic force and potential. It defines key terms like electrostatics, charge, proton, electron, conductor, insulator, and Coulomb's law. Diagrams and examples throughout illustrate electrostatic concepts and their applications.
I am Rachael W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Massachusetts Institute of Technology, USA
I have been helping students with their homework for the past 6 years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
This document contains partial solutions to homework problems from dynamics courses taught between 2002-2003. It was compiled by the author from PDF files to help recipients with their studies. It includes solutions for chapters 13-17 and past exams, but is not a complete set of answers. The mass, pulley, and rod problems solved here provide example solutions that could aid readers in learning concepts in engineering dynamics.
This document provides the mathematical modeling and transfer function of an overhead crane. It defines the variables used such as position, cable length, and angle. It derives the kinetic and potential energies of the crane platform and pendulum and determines the Lagrangian. From this, it obtains the equations of motion for the crane position and cable angle. Small angle approximations yield a transfer function that relates the crane position to the applied force as a function of system parameters like masses and cable length.
This document discusses solving cubic equations using the cubic formula. It provides the steps to:
1. Calculate coefficients a, b, and c from the equation parameters.
2. Determine the nature of the roots based on the discriminant.
3. Use the cubic formula or quadratic formula to find the exact roots.
4. Several examples are provided to demonstrate solving cubic equations.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
Hyperon and charmed baryon masses and axial charges from Lattice QCDChristos Kallidonis
Poster presented at the Electromagnetic Interactions on Nucleons and Nuclei 2013 (EINN2013) Conference, held in Paphos, Cyprus. We present results on the masses and axial charges of all forty light, strange and charm baryons, obtained from Lattice QCD simulations
Numerical Methods in Mechanical Engineering - Final ProjectStasik Nemirovsky
Final Project for the class of "Numerical Methods in Mechanical Engineering" - MECH 309.
In this project, various engineering problems were analyzed and solved using advanced numerical approximation methods and MATLAB software.
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...AbdullaAlAsif1
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
1. International Refereed Journal of Engineering & Technology (IRJET) – Volume I Issue I
27
Advanced Theory of Relativity
Mihir Kumar Jha , EEE Department ,Global academy of technology , Bangalore , India
ABSTRACT- The motive of this paper is to derive the
generalized equation for relativistic mass and rest mass
equation, equation for negative mass , relativistic kinetic
energy followed by the derivation of new metric of space -
time for speed greater than light.
INTRODUCTION
At the dawn of 20th
century, physicist believed that they
possessed enough knowledge to deduce the mystery of the
universe. During this golden era, a patent clerk in Switzerland
named Albert Einstein proposed two theories of relativity
including the famous equation E=MC². He used the relativistic
mass and moving mass equation given as follows
m = m₀/ (√ (1-v²/c²))--------1
Above equation was used to derive the energy mass equation.
If a close look is taken on the equation, there are two major
drawbacks described as below:
If speed is equal to light speed C, the equation is undefined.
And, If
speed is greater than light speed C, the mass becomes complex.
In order to overcome these drawbacks, the equation has to be
generalized which is valid for greater then equal to light speed
and thus determining the equation for negative energy and
negative mass.
DELAY FACTOR (Ф)
Consider a particle of mass m, initially moving with light speed
c. Let the time taken by the ball to cover a distance L₀ is Δt.
Now, let us consider the ball speed increases from c to (c+v)
then the time required to cover the same distance is Δt’.
Mathematically,
L₀= c*Δt -------------- 2 at v=c
And, L₀ =(c+v)* Δt’ ---------3 at v= (v+c)
From the above two equations, we can say
c*Δt = (c+v)* Δt’ --------------------4
Δt’ = Δt / (1+v/c)
Δt’ = Δt*Ф -----------------------5
Where, Ф is delay factor given by,
Ф = 1/(1+v/c)
LENGTH CONTRACTION
Consider the above explanation for delay factor, in which the
ball travels the distance L₀ at time Δt with speed c and covers
distance L at time Δt’ with the same speed. Then from equation
4 we have
c*Δt = (c+v)*Δt’ ----------------------6
c*Δt = c(1+v/c)*Δt’
c*Δt= (c*Δt’)*(1+v/c)
L₀ = L(1+v/c)
Hence, L = L₀/(1+v/c) ---------------7
Or, L= L₀*Ф
RELATEVISTIC MASS AND REST MASS
Consider a particle having a rest mass m₀ and moving mass m,
which is moving initially with a velocity v₁ and attains a
velocity v after time t. The initial and final momentum are
given by the following equations
P(i) = m₁v₁ for (v<c)------------------------8
P(f) = m₂v for(v>= c)------------------------ 9
The total momentum is the sum of initial momentum and final
momentum given by
P(t) = P(i) + P(f) -------------- 10
The total kinetic energy of the body is given by the equation
E= ∫ 𝐹. 𝑑𝑠 ----------------------------11
= ∫(dP/dt). ds
= ∫(ds/dt). dP
= ∫ 𝑣. 𝑑𝑃
= ∫ 𝑣. [𝑚₂𝑑𝑣 + 𝑣𝑑𝑚₂]
= ∫ 𝑚₂𝑣. 𝑑𝑣 + ∫ 𝑣². 𝑑𝑚₂
= ∫ 𝑚₂𝑣. 𝑑𝑣 + ∫(𝑣². 𝑑𝑚₂/dv ) .dv
Differentiating the above equation with respect to v, we get
P(t) = m₂v + v²dm₂/dv
P(t) = P(f) + v²dm₂/dv
P(t) – P(f) = v²dm₂/dv
P(i) = v²dm₂/dv
P(i) ∫ (
1
v2) dv
𝑐
0
= ∫ 𝑑𝑚
𝑚₂
𝑚₁
₂
-P(i)[1/c – 1/0] = m₁ - m₂ ------------A
Since [1/v] for v=0 is undefined value we need to calculate the
value through limit
I₁ = lim
𝑥→0
1/𝑥
= lim
𝑥→0
(𝑠𝑖𝑛²x + cos²x)/𝑥
= lim
𝑥→0
(𝑠𝑖𝑛2
x)/x + lim
𝑥→0
(cos²x)/𝑥 =0 (B)
Now, let I = lim
𝑥→0
(cos²x)/𝑥
Multiplying and dividing with x we have
I = lim
𝑥→0
(xcos²x)/𝑥²
Applying L- Hospital rule we get
I= lim
𝑥→0
(−2x(cosx ∗ sinx) + cos²x)/2𝑥
= lim(
𝑥→0
− cosx ∗ sinx) +1/2 lim
𝑥→0
cos²x/𝑥
= 0 + ( ½) I
I – ½ I = 0
Or, I = 0 ----------------C
Substituting the value of lim
𝑥→0
(cos²x)/𝑥 in equation B we have
I₁ = lim
𝑥→0
𝑠𝑖𝑛2x
x
+ 0
Again applying L- Hospital rule we get
I₁ = lim
𝑥→0
2𝑠𝑖𝑛x∗cosx
1
= 0
Hence the value of lim
𝑣→0
1/𝑣 is zero.
2. International Refereed Journal of Engineering & Technology (IRJET) – Volume I Issue I
28
Coming back to our equation A we have
-P(i)*[1/c – 1/0] = m₁ - m₂
- P(i)*1/c = m₁ - m₂ ----------D
Now Substituting for P(i) in above equation D we have
-m₁v₁/c = m₁ - m₂
Or, m₂ = m₁*(1-v₁/c) --------------13
In general,
m = m₀*(1+v/c) ------------------14
m = m₀/Ф [-v represents velocity in opposite direction]
RELATEVISTIC FORCE
Consider a mass kept on a surface of the earth. When the mass
is stationary (i:e v = 0) the only acceleration acting on the
particle is gravity (g). When particle starts gaining velocity the
force on the particle is given by
F = dp/dt ----------------------------15
= d(m.v)/dt
= d(m₀.v)*(1+v/c))/dt
= (m₀*a₀)*(1+v/c)²
= (m₀*(1+v/c))*(a₀*(1+v/c))
=(m₀*(1+v/c))*(g*(1+v/c))------16
[at v= 0 ; a₀ = g ]
F = F’/ Ф²
NEGATIVE ENERGY AND NEGATIVE MASS
Negative mass is a concept of matter whose mass is of opposite
sign to the mass of normal matter and the energy produced by
these matter is called negative energy.
In order to find the equation for negative mass let us consider
the total energy of a particle, given by the equation
E= ∫ 𝐹. 𝑑𝑠
𝑐
0
= ∫(𝑑𝑝/𝑑𝑡). 𝑑𝑠 = ∫ 𝑣. 𝑑𝑝
= ∫ 𝑑𝑝𝑣 – ∫ 𝑝. 𝑑𝑣
= mc² - ∫ mvdv
𝑐
0
= mc² - m₀ ∫ v ∗ (1 +
v
c
)dv
𝑐
0
= mc² - m₀[
v2
2
+
v³
3c
]0
𝑐
= mc² - m [c²/2 +c²/3 ]
= mc² - 5m₀c²/6
= mc² - 5mc²/6(1+v/c)
= m [c²- 5c²/6(1+v/c)]
= mc² [1-5c/6(c-v)]
Now we consider the value of v as v₁ such that [5c/6(c-v)] is
equal to [5c/6(c-v₁)].Thus
E = mc²[1-c/(c-v₁)]
= mc² [(c-v₁+c)/(c-v)]
= mc²(-v₁/(c-v)) = -mv₁c²/(c-v) ----------------------17
Hence ,
E = - mvc/(1-v/c)
E = 𝑀−
𝑣𝑐 ∗ Ф
Were M¯ is negative mass and E is the negative energy’
The above equation clearly states that the negative energy and
negative mass is generated at velocities equal to or greater than
light speed.
RELATEVISTIC ENERGY
From equation 17 we have
E = mc²[1-c/(c-v)]
= m₀c²(1-v/c)[1-1/(1-v/c)]
= m₀c²[(1-v/c)-1 ]
E = m₀c²[(1/ Ф)-1] -------------------------------19
SPACE-TIME INTERVAL
Space time is defined as any mathematical model that
combines space and time into a single interwoven continuum.
The mathematical form is a series of differential equation for
speed less than light is given by
Δs² = - (cΔt)² + Δx² + Δy² + Δz²--------20
And, the mathematical form of space time for speed greater or
equal to light speed is given by
Δs = cΔt + Δx + Δy + Δz ----------------21
In order to derive the above differential equation, we need to
find the equations for energy dependence on distance (s) and
time (t).
The energy of a particle having a velocity v is given by
E = - ∫ F. ds
𝑣
0
------------------------22
Substituting for force F in above equation
E = ∫ [m₀g/(1 + v/c)²] ds
𝑣
0
= m₀g ∫ [(1 + v/c)²] ds
𝑣
0
-------------23
= m₀g∫ [(1 + v/c)²]v dt
𝑡
0
--------------24
ENERGY DEPENDENCE ON DISTANCE
Since, S = S₀/(1+v/c)
Or, S₀/S = (1+v/c)
Substituting the value of Ф in equation 23
E = m₀g ∫ [S₀²/S²] ds
𝑆
0
= m₀gS₀²∫ (1/S² )ds
𝑆
0
= -[m₀g*(S₀/S)]
= -[m₀g*(S₀/S²)]*S
Substituting for S₀/S in above equation we get
E = -m*a*S -----------------------25
ENERGY DEPENDENCE ON TIME
Since t = t₀/(1+v/c)
Or, t₀/t = (1+v/c) and
v = c*(t₀/t -1)
Substituting t’/t and v in equation 24 we get
E = m₀g∫ [c ∗ (
t₀
t
– 1) ∗
t₀2
t²
] dt
𝑡
0
------26
=-[m₀gc(
t0
t
)²(-t/2) + (t₀/t²)t] ---27
= m₀a*c(t/2)
Let t/2 = T
Then the above equation can be written as
E = m₀a*c*T---------------28
Comparing both the energy equations we get
S = cT---------------------29
3. International Refereed Journal of Engineering & Technology (IRJET) – Volume I Issue I
29
The above equation says that the mechanics formulas hold
good for quantum level also. Since in mechanics the speed
distance and time is related as s = v*t
FOR VELOCITY LESS THAN LIGHT SPEED
From equation 29 if the speed decreases by c to (c-v) then the
Above equation is written as
S= (c-v)*T
Differentiating the above equation we get
dS = c*dT - d(V*T) ----------------------30
dS=c*dT-dS’ [ S = V*T ]
dS = c*dT - √(dx² + dy² + dz²) ---------31
dS – c*dT = -√(dx² + dy² + dz²)
squaring both sides we get
dS² + (c*dT)² - 2*dS*c*dT = (dx² + dy² + dz²)
------32
dS²+ (c*dT)² -dS*c*dt = (dx² + dy² + dz²)
-------------33 [since T=t/2]
since,
-2(dS*c*dT) = -dS²+dS√(dx² + dy² +dz²)**
----------------34 [from eq 31]
-2(dS*c*dt)/2 =-dS²- dS√(dx²+ dy²+ dz²)
-(dS*c*dt) = - dS² - dS√(dx² + dy² + dz²) ------------34
Substituting equation 31 in equation 29 we get
dS² +(c*dT)² -dS² - dS*√(dx² + dy² + dz²)
= (dx² + dy² + dz²) ---------------35
-dS*√(dx² + dy² + dz²) = - (c*dT)² +(dx² + dy² + dz²)
-dS*dS’ = - (c*dT)² +(dx² + dy² + dz²)
[ dS = √(dx² + dy² + dz²)]
dS₁² = - (c*dT)² +(dx² + dy² + dz²) ------36
Therefore we can write the above equation in general,
ΔS² = - (c*Δt)² + Δx² + Δy² + Δz² ------37
**[dS = c*dT + √(dx² + dy² + dz²)
Multiplying both side by dS
dS² = c*dT*dS + dS*√(dx² + dy² + dz²) -c*dT*dS = - ds² +
dS*√(dx² + dy² + dz²)]
FOR VELOCITY GREATER THAN LIGHT SPEED
From equation 29 we have
S = cT*(1+ V/c)
Differentiating the above equation with respect to t we get
dS/dt = c* dT/dt + d(VT)
dS = c*dT + dS
dS = c*dT + √( dx² + dy² + dz²)] ----- 38
Now consider two reference frame R(a) and R(b) where R(a) is
in three dimensional reference frame and R(b) is in four
dimensional reference frame given that both reference frame
coincide at t=0. Then the co-ordinate of three dimension given
by √( dx² + dy² + dz²)] can be written as dx’ + dy’ + dz’ in four
dimension.
For example if the co-ordinate in 3-D is s(x,y,z)=(3i,4j,0k) then
s = √ (3² + 4² + 0²) = 5 then in 4-D co-ordinate it can be
written as simply the algebraic sum of 4-D co-ordinate i.e. s =
(1i ,4y,0z,0t) where 4+1=5 or (1i,2j,1k,1t) where 1+2+1+1=5
and any other value which satisfies the above condition. Thus
using this concept in equation 35, it can be written in 4-D as
the simple algebraic sum of the 4-D co-ordinate whose sum is
equal to the value of 3-D co-ordinates
Thus,
dS = c*dT + dx’ + dy’ + dz’ -------------39
in general equation 36 can be written as
dS = c*dt + dx + dy + dz -------------40
The interesting thing about the above equation is the
Pythagoras theorem ceases as velocity increases greater than
the light speed and the particle can be seen in multiple co-
ordinates at the same time in four dimension in it’s line of
motion. The above equation is called mihir space time
equation.
MIHIR METRIC
In this section, we will find a solution to Einstein’s Field
equations that describes a
gravitational field exterior to an isolated sphere of mass M
assumed to be at rest.
Let us place this sphere at the origin of our coordinate system.
For simplicity, we will use spherical coordinates ρ, φ, θ:
x = ρ sin φ cos θ
y = ρ sin φ sin θ
z = ρ cos φ
Starting from the flat Mihir space time of advanced relativity
and changing to
spherical coordinates, we have the invariant interval from
equation 40
dS = c*dt + dx + dy + dz
= dt + dρ + ρ dφ + ρ sin φdθ ------41
The mihir space time tensor mμν is given by
Mμν= (
g₀₀ g₀₁ g₀₂ g₀₃
g₁₀ g₁₁ g₁₂ g₁₃
g₂₀ g₂₁ g₂₂ g₂₃
g₃₀ g₃₁ g₃₂ g₃₃
)
= (
𝑔𝑡𝑡 𝑔𝑡𝑟 𝑔𝑡𝜑 𝑔𝑡𝜃
𝑔𝑟𝑡 𝑔𝑟𝑟 𝑔𝑟𝜑 𝑔𝑟𝜃
𝑔𝜑𝑡 𝑔𝜑𝑟 𝑔𝜑𝜑 𝑔𝜑𝜃
𝑔𝜃𝑡 𝑔𝜃𝑟 𝑔𝜃𝜑 𝑔𝜃𝜃
)
In addition to having a spherically symmetric solution to
Einstein’s field equations
we also want it to be static. That is, the gravitational field is
unchanging with time
and independent of φ and θ. If it were not independent of φ and
θ then we would be
able to define a preferred direction in the space, but since this
is not so
grθ = grφ = gθr = gφr = gθφ = gφθ = 0
Similarly, to prevent a preferred direction in space time, we can
further restrict the
Coefficients
gtθ = gtφ = gθt = gφt = 0
Also, with a static and unchanging gravitational field, all of the
metric coefficients must be independent of t and the metric
should remain unchanged if we were to reverse time, that is,
apply the transformation t → −t. With that constraint, only dt
leaves ds unchanged and thus implies 𝑔𝑡𝑟 =𝑔 𝑟𝑡 = 0. At this
point we have reduced mμν to the following
4. International Refereed Journal of Engineering & Technology (IRJET) – Volume I Issue I
30
mμν = (
𝑔𝑡𝑡 0 0 0
0 𝑔𝑟𝑟 0 0
0 0 𝑔𝜑𝜑 0
0 0 0 𝑔𝜃𝜃
)
We may then write the general form of the static spherically
symmetric space time as
𝑔𝑡𝑡 dt + 𝑔𝑟𝑟 dρ + 𝑔𝜑𝜑 dφ + 𝑔𝜃𝜃 sin φdθ
Now let us solve for the gμν coefficients. We start with a
generalization of the invariant interval in flat space time from
equation
.ds = U(ρ) dt + V (ρ) dρ +W(ρ) (ρ dφ + ρ sin φdθ)
where U, V,W are functions of ρ only. Since we can redefine
our radial coordinate, so let r = ρW(ρ), and then we can define
some A(r) and B(r) so that the above becomes
ds = A(r) dt + B(r) dr + r dφ + r sin φ---42
We next define functions m = m(r) and n = n(r) so that
A(r) = 𝑒 𝑚(𝑟)
= 𝑒2𝑚
and B(r) = 𝑒 𝑛(𝑟)
= 𝑒2𝑛
We set A(r) and B(r) equal to exponentials because we know
that they must be strictly positive, and with a little foresight, it
will make calculations easier later on. Then substituting into
equation (13) we have
ds = 𝑒2𝑚
dt + 𝑒2𝑛
dr + r dφ + r sin φdθ -----43
Thus mihir space time can be written as
Mμν = (
𝑒2𝑚
0 0 0
0 𝑒2𝑛
0 0
0 0 𝑟 0
0 0 0 𝑟 sin 𝜑
) -----44
Since Mμν is a diagonal matrix, g = det(𝑔𝑖𝑗) = 𝑒2𝑚+2𝑛
r² sinφ.In
order to solve for a static, spherically symmetric solution we
must find m(r) and n(r). However, to solve for them we must
use the Ricci Tensor given by
which relies on the Christoffel symbols given by
EVALUATION OF CHRISTOFFEL SYMBOLS
From equation 44 , we see that 𝑔 𝜇𝜈 = 0 for μ ≠ ν, and so 𝑔 𝜇𝜇 =
1/𝑔 𝜇𝜇 and 𝑔 𝜇𝜈 = 0
if μ ≠ ν. Thus, the coefficient 𝑔 𝛽𝜆 is 0 unless β = λ and
substituting this into the above we have
Notice that Γλ
𝜇𝜈 = Γλ
𝜈𝜇 so we have three cases: λ = ν, μ = ν
≠ λ, and μ, ν, λ distinct.
Recall that 𝑔 𝜇𝜈 = 0 for μ ≠ ν as we will use that to simplify
several terms.
Case 1. For λ = ν
Case 2. For μ = ν ≠ λ:
Case 3. For μ, ν, λ distinct:
Using the values for 𝑔 𝜇𝜈 from equation (44) we can calculate
the nonzero Christoffel symbols (in terms of m, n, r, and φ )
where ‘ = d/dr
Γ°₁₀ = Γ°₀₁ = m’ Γ¹₀₀ = -m’𝑒2𝑚−2𝑛
Γ¹₁₁ = n’
Γ¹₂₂ = -(1/2) 𝑒−2𝑛
Γ²₁₂ = Γ²₂₁ = 1/2r
Γ¹₃₃ = (1/2) 𝑒−2𝑛
sin 𝜑
Γ³₁₃ = Γ³₃₁ = 1/2r
Γ³₂₃ = Γ³₃₂ = (½) cot 𝜑
Γ²₃₃ = -(cos 𝜑)/2
And We know that ,
½ log 𝑔 = ½ log[𝑒2𝑚+2𝑛
𝑟² sin𝜑] = m + n + log(r) + ½ log
[sin𝜑] ----------------45
RICCI TENSOR COMPONENT
However, 𝑅 𝜇𝜈 for some values of μ and ν will be in terms of m,
n, r, and φ which we
must set equal to zero. Let us calculate those terms now and
show that the other
terms reduce to zero afterwards. Recall x° = t, x¹ = r, x² = φ, x³
= θ, equation (22),
and the nonzero values of the Christoffel symbols presented
previously
= 𝑒2𝑚−2𝑛
[ 2m’² + m’’ - m’n’ + m’/r ]
= [ m’’ + m’² - m’n’ –n’/r – 1/2r² ]
= [ -
½- ½ cot² 𝜑 + 3/2 (n’𝑒−2𝑛
) + ½(𝑒−2𝑛
) ]
And,
5. International Refereed Journal of Engineering & Technology (IRJET) – Volume I Issue I
31
= sin 𝜑 [1/2𝑛′(𝑒−2𝑛
) + ½ -½ (m’𝑒−2𝑛
)
+½r(𝑒−2𝑛
) ]
And the value of ricci component at 𝜇, 𝜈 = 1,2 is (0,0,-(cot
φ)/4r , (cot φ)/4r)
Thus, 𝑅 𝜇𝜈= {
0 + 0 +
cot 𝜑
4𝑟
−
cot 𝜑
4𝑟
= 0 , 𝑓𝑜𝑟 𝜇, 𝜈 = {1,2}
0 + 0 + 0 + 0 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜇 𝜈
----------46
[For detailed explanation of above value refer the derivation of
Schwarzschild Metric]
Now, we have
R₀₀ = 𝑒2𝑚−2𝑛
[ 2m’² + m’’ - m’n’ + m’/r ]
---------47
R₁₁ = [ m’’ + m’² - m’n’ –n’/r – 1/2r² ]
----------48
R₂₂ = [ -½- ½ cot² 𝜑 + 3/2 (n’𝑒−2𝑛
) + ½(𝑒−2𝑛
) ] --------------
-------------49
R₃₃ = sin 𝜑 [1/2𝑛′(𝑒−2𝑛
) + ½ -½ (m’𝑒−2𝑛
) + ½r(𝑒−2𝑛
) ] ------
-----------------50
SOLVING FOR THE COEFFICIENTS
Since we are finding the solution for an isolated sphere of
mass, M, the Field Equations from equation (46) imply that
outside this mass, all components of the Ricci tensor are zero.
We can then set the above equations equal to zero and solve the
system for m and n so we can find an exact metric that
describes the gravitational field of a static spherically
symmetric space-time. Hence, we need
2m’² + m’’ - m’n’ + m’/r = 0 ------------51
m’’ + m’² - m’n’ –n’/r – 1/2r² = 0 ------52
-½- ½ cot² 𝜑 + 3/2 (n’𝑒−2𝑛
) + ½(𝑒−2𝑛
) = 0
-----------------------53 1/2𝑛′(𝑒−2𝑛
) + ½ -½
(m’𝑒−2𝑛
) + ½r(𝑒−2𝑛
) = 0
-------------------54
Subtracting equation 51 from equation 52 we get
m’² + m’/r + n’/r + 1/2r² = 0 ----------55
Now partially differentiating the above equation with respect to
m’ we get
2m’ + 1/r = 0
Or, m’ = -1/(2r) ----------------------56
Substituting m’ in equation we get
n’ = 1/(2r) ---------------------57
Therefore from equation 56 and 57 we have
m’ + n’ = 0 ----------------58
or, m + n = constant
and m = -n -----------------59
Now from equation 54 we have
½𝑛′(𝑒−2𝑛
) + ½ -½ (m’𝑒−2𝑛
) + ½r(𝑒−2𝑛
) = 0
Or, (2𝑒−2𝑛
)/r = -1
= 𝑒2𝑚
= -r /2 -----60 [ since , m = -n ]
We will need to solve for r in terms of M to find the exact
metric. With this in mind, suppose we release a ‘test’ particle
with so little mass that it does not disturb the space-time
metric. Also, suppose we release it from rest so that initially
𝑑𝑥 𝜇
= 0 for μ = 1, 2, 3
Now from equation we have
ds = 𝑒2𝑚
dt + 𝑒2𝑛
dr + r dφ + r sin φdθ
= 𝑒2𝑚
dt + 0 + 0 +0
However, for time like intervals, we can relate proper time τ
between two events as
dτ = ds/c
The second equality follows from our use of geometrized units;
we set c = 1 in the
formulation of Mihir space time. We then have
dτ = ds =𝑒2𝑚
𝑑𝑡
dt/dτ = 1/𝑒2𝑚
--------------------61
and, (dt/dτ) ² = (1/𝑒2𝑚
)² ----------62
From the geodesic equation we have
Notice that it relies on 𝑑𝑥 𝜇
/ dτ which motivates our previous
change from ds to dτ.
Recall that we are attempting to solve for r and that we want
our metric to reduce
to Newtonian gravitation at the weak field limit. Towards that
end, and having the
benefit of a little foresight, let us find the geodesic equation for
𝑥 𝜆
= 𝑥1
= r at the instant that we release our ‘test’ particle.
Remember that we are releasing this particle from rest and so
𝑑𝑥 𝜆
/𝑑𝜏= 0 for λ ≠ 0 and so we are left with only the following
in our summation
Now we see more of the motivation for switching to proper
time τ as we can easily substitute in from equation (61) and
move the right hand term to the other side to have
d²r/ dτ² = -m’(𝑒2𝑚−2𝑛
) 𝑒4𝑚
= - m’ 𝑒(2𝑚+2𝑚)−4𝑚
[as m= -n]
= -m’
= - ( -1/2r) = 1/2r ---------63
[since m’ = -1/2r]
Conveniently, we know that this must reduce to the predictions
of Newtonian gravity for the limit of a weak gravitational field
and so
d²r/ dτ² = −GM/r² --------------------64
Thus from equation 63 and 64 we have
1/2r = −GM/r²
or, r = -2MG = -2M --------------------65
Substituting for r in equation 60 we get
𝑒2𝑚
= -r/2 = 2MG/2 = M
And, 𝑒2𝑚
= 𝑒−2𝑛
= -2/r = 2/2MG = 1/M
[since G=1 for geometrized units ]
6. International Refereed Journal of Engineering & Technology (IRJET) – Volume I Issue I
32
Using this to substitute back into equation (60) and recalling
that m = −n we can
find an exact solution for equation (43)
ds = (-r/2) dt + (-2/r) dr +rdφ + r sin φ dθ
------------------66
And since from equation 65 we have r= - 2M substituting in
equation 66 we get
ds = Mdt + (1/M)d(-2M) + (-2M) dφ +
(- 2M) sin φ dθ -----------------67
The above equation is called MIHIR metric.