Number-Flux Vector and Stress-Energy Tensor (6 pp.) Search. These lectures represent an introductory graduate course in general relativity, both its foun-dations and applications. This course, which concentrates on special relativity, is normally taken by physics majors in their sophomore year. An introduction to Einstein's theory of gravitation. Welcome to this question, learners. OpenRelativity is an open-source toolkit to simulate effects of special relativity by varying the speed of light. Spherical compact sources I.21. So, we must provide an instance of a case. So linearized general relativity is a situation in which we are only going to consider space times that are nearly at. (It really passes more slowly; everything slows down relative to . 1. In Newtonian gravity the potential due to two point masses is simply the sum of the potentials for each mass, but clearly this does not carry over to general relativity (outside the weak-field limit). Research Areas Astrophysics Theory (617) 258-8523 sahughes@mit.edu Office: Ronald McNair Building, 37-602A Lab (s): Hughes Group - Astrophysical General Relativity @ MIT To qualify for the gift, the existing solution to the problem below must be either (1) missing, (2) handwritten, or (3) demonstrably incorrect (with the demonstrating to be done by you). Many well-known exact solutions belong to one of several types, depending upon the intended physical interpretation of the stressenergy tensor: In addition to such well established phenomena as fluids or electromagnetic waves, one can contemplate models in which the gravitational field is produced entirely by the field energy of various exotic hypothetical fields: One possibility which has received little attention (perhaps because the mathematics is so challenging) is the problem of modeling an elastic solid. That's why we created RelativityOne, a powerful and easy-to-use SaaS solution for big-time legal challenges. Below we have compiled exams and solution sets for as many back years as we can find. (The speed of light: 3.00 x 108 m/s) 2. Science . Collections. A few pedagogical gems are hidden here (e.g., the nicest proof of the Bianchi identity I've ever seen). This result, known as the positive energy theorem was finally proven by Richard Schoen and Shing-Tung Yau in 1979, who made an additional technical assumption about the nature of the stressenergy tensor. Aspects: EXOTIC SOLUTIONS: Exotic solutions in General Relativity by Francisco S. N. Lobo [2007/10] Aspects: FLRW METRIC: The Friedman--Lematre--Robertson--Walker Metric: A Centennial Review by R. Barnes [2022/01] Aspects: FRAME DRAGGING: Frame dragging: Meaning, myths, and . For a physical object in general relativity, when it moves, its motion will cause "ripples" in the space-time caused by gravitational backreaction of its own presence. physics mit opencourseware string theory max planck institut fr physik lectures on string theory lmu string theory and m theory cambridge core david tong string theory. van Holten - arXiv , 1997. The course web site includes the syllabus, reading, and assignment problem sets. The theory of relativity is also known as Albert Einstein's theory of relativity. On the other, they may be far too restrictive: the most popular energy conditions are apparently violated by the Casimir effect. Even after such symmetry reductions, the reduced system of equations is often difficult to solve. As you can see, not all exams have solution sets, and some of the posted solutions are messy or incomplete. This information was graciously provided by the NASA-sponsored "Ask the Space Scientist" web page and its author, Dr. Sten Odenwald. This naive approach usually works best if one uses a frame field rather than a coordinate basis. //-->. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications: gravitational radiation, black holes, and cosmology. Home. Cosmology II.20. Below we have compiled exams and solution sets for as many back years as we can find. Black holes I.23. This second kind of symmetry approach has often been used with the NewmanPenrose formalism, which uses spinorial quantities for more efficient bookkeeping. Dates: June 11, 12, 14, 15. teaching Physics 8.962, the graduate course in GR at MIT, during the Spring of 1996. Browse Course Material. In addition to the warm, fuzzy sensation of knowing you helped out your fellow exam-takers, and the fame and prestige of having your insightful solution admired by future generations of Course 8 students, the really important thing is that every neatly typeset and correct solution to a complete written exam problem will be gratefully rewarded with a shiny physREFS water bottle. 8.962: General relativity by Professor Scott A. Hughes. These are often stated in terms of the Petrov classification of the possible symmetries of the Weyl tensor, or the Segre classification of the possible symmetries of the Ricci tensor. Al- . General relativity is a beautiful theory, our standard theory of gravity, and an essential component of the working knowledge of the theoretical physicist, cosmologist, and astrophysicist. Equations of Motion for a General Orbit. As you can see, not all exams have solution sets, and some of the posted solutions are messy or incomplete. in Physics in 1993. II. The simplest involves imposing symmetry conditions on the metric tensor, such as stationarity (symmetry under time translation) or axisymmetry (symmetry under rotation about some symmetry axis). With sufficiently clever assumptions of this sort, it is often possible to reduce the Einstein field equation to a much simpler system of equations, even a single partial differential equation (as happens in the case of stationary axisymmetric vacuum solutions, which are characterized by the Ernst equation) or a system of ordinary differential equations (as happens in the case of the Schwarzschild vacuum). Ask questions, read articles, submit tickets, and more. It can study stellar stability, the orbits (with radiation . Just another site mit general relativity solutions You will have one week to complete each assignment. Product Description. On Problems and Solutions of General Relativity-151120-RG-200927.pdf On Problems and Solutions of General Relativity -151120-2005-1.pdf On Problems and Solutions of General Relativity -151120-1908.pdf General Relativity Notes by Edmund Bertschinger [MIT] [1999] General Relativity Today by Thibault Damour [2006/10] . Australia: 0011-800-99094949. Written Exams are administered and results are available prior to the fall and spring semesters each year. Collections. google_color_url = "CC0000"; These symmetry groups are often infinite dimensional, but this is not always a useful feature. Relativity is just a concept that states objects in motion are relative to each other. We welcome simlilar compilations for other divisions! Adopting the assumption that the scalar field has a constant kinetic term, we derive in a fully covariant manner a set of conditions under which the Euler-Lagrange equations allow General Relativity solutions as exact solutions in the presence of a . 8.962: General relativity by Professor Scott A. Hughes MIT has a one semester course in general relativity, which I have taught several times. MIT OpenCourseWare is a web-based publication of virtually all MIT course content. ciples of relativity had been proposed, but Einstein was the rst to state it clearly and hammer out all the coun-terintuitive consequences. This is a supplementary problem book or student's manual, consisting of 150 problems in each of special and general relativity. A free, confidential service for discussing your options. This course will introduce you to the concepts behind special relativity including, but not limited to, length contraction, time dilation, the Lorentz transformation, relativistic kinematics, Doppler shifts, and even so-called paradoxes.. Wald, Robert M. General Relativity. Spherical compact sources II.22. via MIT OpenCourseWare This show combines the Masters thesis work of Zachary Sherin 15 with the teaching skills of Dr. Gerd Kortemeyer from Michigan State University, both of whom worked on A Slower Speed of Light in 2012. Geometry for General Relativity, Notes written by Sam Johnson, 2016. This is a textbook on general relativity for upper-division undergraduates majoring in physics, at roughly the same level as Rindler's Essential Relativity or Hartle's Gravity. The General Relativity Tutorial John Baez This is bunch of interconnected web pages that serve as an informal introduction to that beautiful and amazingly accurate theory of gravity called general relativity.The goal is to explain the basic equation in this theory - Einstein's equation - with a minimum of fuss and muss. The full field equation is highly nonlinear, so we really want to prove that the Minkowski vacuum is stable under small perturbations which are treated using the fully nonlinear field equation. My personal spin on this topic is to spend half of the course focused on the formal mathematical framework (not getting too sophisticated, just making sure to carefully develop Cosmology. Contribute to chinapedia/wikipedia.en development by creating an account on GitHub. On this Wikipedia the language links are at the top of the page across from the article title. The collection concentrates on papers with a direct bearing on classical general relativity, from conceptual breakthroughs and experimental tests of the theory to . And two main ideas w ere found to solve all paradoxes. This course also provides a brief introduction to some concepts of general relativity, including the principle . Solutions can also be organized using the Segre classification of the possible algebraic symmetries of the Ricci tensor: The remaining Segre types have no particular physical interpretation and most of them cannot correspond to any known type of contribution to the stressenergy tensor. This approach is essentially the idea behind the post-Newtonian approximations used in constructing models of a gravitating system such as a binary pulsar. General Relativity theory, developed by Einstein in 1907-1915, states that being at rest in the gravitational field and accelerating are identical physically. Physicists made many exotic predictions using general relativity. General relativity is a theory of gravitation that Albert Einstein developed between 1907 and 1915, with contributions from many others after 1915. The e ects that we attribute to the force of gravity are due to the bending and warping of spacetime, from falling cats, to orbiting spinning planets, to the motion of the cosmos on the grandest scale. Date issued 2006-06 URI https://hdl.handle.net/1721.1/127941 Other identifiers 8.962-Spring2006 Other identifiers 8.962 These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. google_ad_client = "pub-2137895145944627"; Topicsinclude manifolds, Riemannian geometry, Einstein's equations, andthreeapplications: grav- . A key feature of this universe is that there is a . Class Central aggregates courses from many providers to help you find the best courses on almost any subject, wherever they exist. Act on it. Solutions for Various Combinations of Omega m, Omega Lambda and Omega k. ( PDF) 20. A Slower Speed of Light combines accessible gameplay and a fantasy setting with theoretical and computational physics research to deliver an engaging and pedagogically rich experience. These solutions reflect assignments made by Professor Igor Klebanov at Princeton University during his semester course on General Relativity during the fall of 2006. paper)I.eneral relativity (Physics)1 G I. Einstein's General Relativity and Gravitation (English) Course Information This course is listed as Physics 255: General Relativity in the UCI course catalogue. A concise, direct examination of general relativity and black holes, Exploring Black Holes provides tools that motivate tools that motivate readers to become active participants in carrying out their own investigations about curved spacetime near earth and black holes. General relativity is physicist Albert Einstein 's understanding of how gravity affects the fabric of space-time. E. Taylor and J. Wheeler, Spacetime Physics(Freeman, 1992) Introduction and the geometric viewpoint on physics.. 2. The toolkit was developed during the creation of the game A Slower Speed of Light. Instructor: Prof. Scott Hughes Course Number: 8.962 Departments: Physics As Taught In: Spring 2020 Level: Graduate Topics. Poisson, Eric. In the above field equations, Presently, it seems that no exact solutions for this specific type are known. Emmy Noether showed that a slight but profound generalization of Lie's notion of symmetry can result in an even more powerful method of attack. Nonetheless, several effective techniques for obtaining exact solutions have been established. Share to Twitter Share to Facebook Share to Pinterest. Hence as one commenter has pointed out, it presupposes curved spacetime. Quite remarkably, both the Ernst equation (which arises several ways in the studies of exact solutions) and the NLS turn out to be completely integrable. It has the reputation of being dif cult but Bernard Schutz, with his Learn Vector Algebra and Calculus (For Beginners) Both special and general relativity heavily rely on vectors and vector calculus. blockbuster store still open near haarlem. Black holes II. The function a(t) is known as the scale factor, and it tells us "how big" the spacelike slice is at the moment t. 10:30 am-11:30 am. Expand the Effective Potential in the Weak-Field Limit. It covers advanced material, but is designed to be understandable for students who haven't had a first course in the subject. 1.1 What is a principle of relativity? If, in your preparation for the general exams, you work out a solution to a problem, please consider writing it up and sending us a copy. 4 A General Relativity Workbook-- Online Student Manual Chapter 5 5.1.1 Hint: Equation 5.18 predicts a vector pointing to the upper right, . 8.962 is MIT's graduate course in general relativity, which covers the basic principles of Einstein's general theory of relativity, differential geometry, experimental tests of general relativity, black holes, and cosmology. Albert Einstein settled on his 'general' theory in 1915, a decade after he came up with a 'special' theory of relativity that applied a universal speed of light to the assumption that the laws of physics stay the same inside any given frame of reference.. Browse. Report this file. If you are looking to have a career in these, these General Relativity books for beginners may be the perfect spot for you. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. general relativity an extension of special relativity to a curved spacetime. Topics include Einstein's postulates, the Lorentz transformation, relativistic effects and paradoxes, and applications involving electromagnetism and particle physics. The concepts involved in special relativity seem remote and inaccessible to us in our daily lives. The theory explained the anomalous orbit of Mercury, but the first major triumph came in 1919 when Arthur Eddington and his colleagues measured the influence of the Sun's gravity on light from stars during a total solar eclipse. All of . The bending of light around the Sun is small, but . Your use of the MIT OpenCourseWare site and course materials is subject to the . Introduction to Special Relativity. This is due to the rocket's acceleration, which equals 9.8 m/s 2. 8.962 is MIT's graduate course in general relativity, which covers the basic principles of Einstein's general theory of relativity, differential geometry, experimental tests of general relativity, black holes, and cosmology. General Exam Preparation Overview The general exam consists of two parts: written and oral. For example, in a truck, two kids are playing catch the ball game and the truck is moving with a constant speed i.e., 50 mph and the kids are also moving with the speed of 50 . google_color_bg = "FFFFFF"; Please do email me if you find any typos or mistakes. This is a generalization of linearized theory. Email This BlogThis! The notes as a whole are available as gr-qc/9712019. G . Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensor unspecified (see the Ricci decomposition), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or non-gravitational fields, in the sense that the immediate presence "here and now" of non-gravitational energymomentum causes a proportional amount of Ricci curvature "here and now". Professor Hughes' research is in astrophysical general relativity, focusing in particular upon black holes and gravitational-wave sources. Taking this limit on the Eq. Abstract: I will talk about the status of the stability of Kerr conjecture in General Relativity based on recent results obtained in collaboration with Jeremie Szeftel and Elena Giorgi. The work presented in this paper is the initial form of . For example, an observer can see the ball fall the same way on the rocket and on Earth. These solutions reflect assignments made by Professor Igor Klebanov at Princeton University during his semester course on General Relativity during the fall of 2006. The number flux 4-vector, and its use in defining a conservation law. Then, one can prove that solutions exist at least locally, using ideas not terribly dissimilar from those encountered in studying other differential equations. Download Course. Unfortunately, no such characterization is known. Given the difficulty of constructing explicit small families of solutions, much less presenting something like a "general" solution to the Einstein field equation, or even a "general" solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions. This is illustrated by the following procedure: This shows that there are two complementary ways to use general relativity: Within the first approach the alleged stressenergy tensor must arise in the standard way from a "reasonable" matter distribution or non-gravitational field. Classical Theory of Fields, by Landau and Lifschitz. The main course textbook was Bernard Schutz' A First Course in General Relativity and a few of the homework problems came from the text. 1916 was the year of the publication of the final form of general relativity. Below we have sketched a classification by physical interpretation. Intended for game developers, educators, and anyone interested in physics, OpenRelativity can help people create, test, and share experiments to explore the effects of special relativity. GitHub export from English Wikipedia. Menu. The principle of equivalence..7. The fall 2015 is the first exam that was administered in the current format. solutions to almost half of Schutz s exercises, and includes 125 brand-new supplementary . Local means in space and time: the local Minkowski frame is a freely-falling observer. MIT OpenCourseWare is a web-based publication of virtually all MIT course content. Hence, Einstein Kerry seems more reliable and can make more accurate predictions. Once again, the creative tension between elegance and convenience, respectively, has proven difficult to resolve satisfactorily. Look at weak gravity, also known as linearized general relativity. Instructor: Scott Hughes View the complete course: https://ocw.mit.edu/8-962S20 8.962 is MIT's graduate course in general relativity, which covers the basic . This course, which concentrates on special relativity, is normally taken by physics majors in their sophomore year. Stewart, Advanced General Relativity, Cambridge University Press, Cambridge, 1993. Introduction to Tensor Calculus for General Relativity (34 pp.) {\displaystyle T^{\alpha \beta }} If I am in this situation, then I can choose coordinates, such that my space time metric is the metric of . Subscribe to the OCW Newsletter General Relativity Autumn 2017 Lecture Notes, Exercises, Solutions, and more . Post-doctoral researcher, Chip-based platform for exploring the interface between quantum mechanics and general relativity Leiden University Leiden, South Holland, Netherlands Apply on company website When and Where. Monday 13h15 -- 15h, Tuesday 12h15 -- 14h, room 119 In order to formulate his theory, Einstein had to reinterpret fundamental concepts of our experience (such as time, space . Einstein's general theory of relativity, Einstein, relativity, differential geometry, general relativity, black holes, cosmology, Hamiltonian Dynamics, Curvature, Acceleration, Hilbert action, Orthonormal bases, White dwarfs, neutron stars, Kruskal coordinates, Wormholes, Hawking radiation, Kerr solution . 3 comments: Chun_zzj 17 November 2020 at 14:31. A related idea involves imposing algebraic symmetry conditions on the Weyl tensor, Ricci tensor, or Riemann tensor. For example, an observer can see the ball fall the same way on the rocket and on Earth. Lecture Notes 1: Wed 02/07/2018, Review of Special Relativity I. Lecture 10 Slides, Mon 03/12/2018: Mathematical Concepts and Definitions These lecture notes are a lightly edited version of the ones I handed out while teaching Physics 8.962, the graduate course in General Relativity at MIT, during Spring 1996. There are introductory GR courses in Part II (Mathematics or Natural Sciences) so, although self-contained, this course does not cover topics usually covered in a rst course, e.g., the Schwarzschild solution, the solar system tests, and cosmological solutions. Quantum Field Theory General Relativity is the classical theory that describes the evolution of systems under the e ect of gravity. Physics (8) - Archived; Search DSpace. Dynamical Equations for the Scale Factor a - Including Ordinary Matter, Dark Matter, and Dark Energy. We present the classical tests that support the theory and discuss the solutions of Einstein's equations that are relevant to . 10.J. Solutions Read Pdf Free string theory solution manual pdf pdf string theory scribd solutions for string theory 101 lmu answers to the problems in a first . The graph illustrates the two situations. Live visuals from our OpenRelativity game engine were projected onto the dome of the Charles Hayden Planetarium to demonstrate the theory of Special Relativity. masters level) students. general relativity, part of the wide-ranging physical theory of relativity formed by the German-born physicist Albert Einstein. In practice, this notion is pretty clear, especially if we restrict the admissible non-gravitational fields to the only one known in 1916, the electromagnetic field. Course Info Instructor Prof. Scott Hughes Departments Physics Topics Science Physics Relativity Learning Resource Types Quite mathematically sophisticated, and rather terse. The GR berbuch; typically the final arbiter of right and wrong in this subject. Our hope is that software created with OpenRelativity will allow a greater number of people access to a field that is often regarded as difficult to comprehend. Geodesics..10. Only a few parts, including the treatment of the stress-energy tensor are adapted in accordance with later reformulations of the theory, and contravariant coordinates are consistently labeled by superscripts. In fact, even today, more than 100 years after General Relativity was first put forth, there are still only about ~20 exact solutions known in relativity, and a spacetime with two point masses in . This is no coincidence, since this phenomenon is also related to the notions of Noether and Lie regarding symmetry. If you were on a spaceship travelling at 0.50c away from a star, what speed would the starlight pass you? (Graph by Prof. Saul Rappaport.) This course also provides a brief introduction to some concepts of general relativity, including the principle . These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations). They are intended for academic use only. However, there are some notable exact solutions which contain two or three contributions, including: The Einstein field equations are a system of coupled, nonlinear partial differential equations. Complete Lecture Notes from Spring 2017, by Andrew Turner. The Einstein field equation (variant derivation)..14. lakeshore high school sports calendar. areas of GR, with fully worked solutions, making it all the more dicult for . Syllabus 1. Spherical Solutions to the Einstein Equations: 19: White Dwarfs and Neutron Stars: 20: Schwarzschild and its Geodesics: 21: 4 Classical Tests of General Relativity . 8.033 Relativity, Fall 2003. Date issued 2006-06 URI https://hdl.handle.net/1721.1/127941 Other identifiers 8.962-Spring2006 Other identifiers In the Ether Theory, light moves with speed c relative to the ether. General Relativity by Wald Solutions. Research Question Previously, I have reported problems on MIT open course, Phy. See the department webpage for more details. About. Brief introduction to Newtonian cosmology. 8.224 Exploring Black Holes: General Relativity and Astrophysics Prereq: 8.033 or 8.20 Acad Year 2022-2023: Not offered Acad Year 2023-2024: U (Fall) 3-0-9 units Study of physical effects in the vicinity of a black hole as a basis for understanding general relativity, astrophysics, and elements of cosmology. General relativity is outlined as the classical field theory of gravity, emphasizing physical phenomena rather than mathematical formalism. Critical Closure Density; Open, Closed, Flat Universes. 8.962 is MITs graduate course in general relativity, which covers the basic principles of Einsteins general theory of relativity, differential geometry, experimental tests of general relativity, black holes, and cosmology. General Relativity Kevin Zhou This course will introduce you to the concepts behind special relativity including, but not limited to, length contraction, time dilation, the Lorentz , The theory of special relativity, originally proposed by Albert Einstein in his famous 1905 paper, has had profound consequences on our view of physics, space, and time. LIGO scientific collaboration / T. Pyle / Caltech / MIT. OCW is open and available to the world and is a permanent MIT activity . Extra meetings can be held by appointment. Massachusetts Institute of Technology blockbuster store still open near haarlem. We can ask: what happens as the incoming radiation interacts with the ambient field? The problems, which have been developed, tested and refined by the authors over the past two decades, are a mixture of . 5.0 rating, based on 1 Class Central review, Start your review of General Relativity (Spring 2020). General Relativity: Solutions to exercises in Lecture I January 22, 2018 Exercise 1 Consider a binary system of gravitating objects of masses Mand m. First consider the case in which mM and where the small-mass object is in quasi-circular orbit around the more massive object. . Solution manual of the one of the best books on Einstein's General Relativity by J B Hartle Solution manual of the one of the best books on Einstein's General Relativity by J B Hartle (PDF) James B. Hartle - Gravity_ An Introduction to Einstein's General Relativity_ Solutions.pdf | Akshay SB - Academia.edu Hong Kong: 001-800-99094949. Linearized gravity II: Dynamic sources.16. Science . ISBN: 9780226870335. Cosmology I.19. 20012023 Massachusetts Institute of Technology, Related Subjects; Brief History of Physics, Galilean Transformation, Inertial Reference Frames, Classical Wave Equations; Transformation to Other Frames, First Discussion of Minkowski Diagrams, World Lines, Derivation of Lorentz-Einstein Transformations, Alternative Looks at Time Dilation and Length Contraction, Astrophysical Examples; Relativistic and Superluminal Jets, Doppler Effect and Angle Transformation via Transformation of Phase of Plane Waves, Twin Paradox with Constant Velocity Plus a Reversal, Short Discourse on the Calculus of Variations, The Euler-Lagrange Equations and Constants of the Motion, Extremal Aging for Inertially Moving Clocks, Optional Problems in the Use of the Calculus of Variations as Applied to Lagragian Mechanics and Other Problems in the Extremization of Path Integrals, Relativistic Momentum Inferred from Gedanken Experiment with Inelastic Collisions, Relativistic Relations between Force and Acceleration, Relativistic Version of Work-Energy Theorem, Kinetic Energy, Rest Energy, Equivalence of Mass-Energy, Atomic Mass Excesses, Semi-Empirical Binding Energy Equation, Two Photons Producing an Electron/Positron Pair, Formal Transformation of E and P as a Four-Vector, Magnetic Force due to Current-Bearing Wire, Strong and Weak Principles of Equivalence, Local Equivalence of Gravity and Acceleration, Relative Acceleration of Test Particles in Falling Elevator of Finite Size, Analogy between the Metric Tensor and the Ordinary Potential, and between Einsteins Field Equations and Poissons Equation, Cosmological Redshifts and the Hubble Law, Dynamical Equations for the Scale Factor a - Including Ordinary Matter, Dark Matter, and Dark Energy, Critical Closure Density; Open, Closed, Flat Universes, Solutions for Various Combinations of Omega, Relation between Scale Factor and Z from the Doppler Shift, Lookback Age as a Function of Z for Various Values of Omega, Acceleration Parameter as a Function of Scale Factor, Current S Status of Cosmology, Unsolved Puzzles, Handout Defining Einstein Field Equations, Einstein Tensor, Stress-Energy Tensor, Curvature Scalar, Ricci Tensor, Christoffel Symbols, Riemann Curvature Tensor, Symmetry Arguments by Which 6 Schwarzschild Metric Tensor Components Vanish, Symmetry Arguments for Why the Non-zero Components are Functions of Radius Only, The Differential Equations for G00 and G11, Shell Radius vs. Bookkeepers Radial Coordinate, Use Euler Equations (for External Aging) in Connection with the Schwarzschild Metric to find Constants of the Motion E and L, Derive the Full Expression for the Effective Potential, Derive Analytic Results for Radial Motion, Compare Speeds and Energies for Bookkeeper and Shell Observers, Explain How these can be Numerically Integrated, Expand the Effective Potential in the Weak-Field Limit, Keplers Third Law in the Schwarzschild Metric, Relativistic Precession in the Weak-Field Limit, Derivation of the Last Stable Circular Orbit at 6M, Derive Differential Equation for the Trajectories, Derive Expression for Light Bending in the Weak-Field Limit.
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