3 Classical Symmetries and Conservation Laws
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A Mathematical Derivation of the General Relativistic Schwarzschild
A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components ................. -
Quantum Phase Space in Relativistic Theory: the Case of Charge-Invariant Observables
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 3, 1448–1453 Quantum Phase Space in Relativistic Theory: the Case of Charge-Invariant Observables A.A. SEMENOV †, B.I. LEV † and C.V. USENKO ‡ † Institute of Physics of NAS of Ukraine, 46 Nauky Ave., 03028 Kyiv, Ukraine E-mail: [email protected], [email protected] ‡ Physics Department, Taras Shevchenko Kyiv University, 6 Academician Glushkov Ave., 03127 Kyiv, Ukraine E-mail: [email protected] Mathematical method of quantum phase space is very useful in physical applications like quantum optics and non-relativistic quantum mechanics. However, attempts to generalize it for the relativistic case lead to some difficulties. One of problems is band structure of energy spectrum for a relativistic particle. This corresponds to an internal degree of freedom, so- called charge variable. In physical problems we often deal with such dynamical variables that do not depend on this degree of freedom. These are position, momentum, and any combination of them. Restricting our consideration to this kind of observables we propose the relativistic Weyl–Wigner–Moyal formalism that contains some surprising differences from its non-relativistic counterpart. This paper is devoted to the phase space formalism that is specific representation of quan- tum mechanics. This representation is very close to classical mechanics and its basic idea is a description of quantum observables by means of functions in phase space (symbols) instead of operators in the Hilbert space of states. The first idea about this representation has been proposed in the early days of quantum mechanics in the well-known Weyl work [1]. -
26-2 Spacetime and the Spacetime Interval We Usually Think of Time and Space As Being Quite Different from One Another
Answer to Essential Question 26.1: (a) The obvious answer is that you are at rest. However, the question really only makes sense when we ask what the speed is measured with respect to. Typically, we measure our speed with respect to the Earth’s surface. If you answer this question while traveling on a plane, for instance, you might say that your speed is 500 km/h. Even then, however, you would be justified in saying that your speed is zero, because you are probably at rest with respect to the plane. (b) Your speed with respect to a point on the Earth’s axis depends on your latitude. At the latitude of New York City (40.8° north) , for instance, you travel in a circular path of radius equal to the radius of the Earth (6380 km) multiplied by the cosine of the latitude, which is 4830 km. You travel once around this circle in 24 hours, for a speed of 350 m/s (at a latitude of 40.8° north, at least). (c) The radius of the Earth’s orbit is 150 million km. The Earth travels once around this orbit in a year, corresponding to an orbital speed of 3 ! 104 m/s. This sounds like a high speed, but it is too small to see an appreciable effect from relativity. 26-2 Spacetime and the Spacetime Interval We usually think of time and space as being quite different from one another. In relativity, however, we link time and space by giving them the same units, drawing what are called spacetime diagrams, and plotting trajectories of objects through spacetime. -
Spacetime Diagrams(1D in Space)
PH300 Modern Physics SP11 Last time: • Time dilation and length contraction Today: • Spacetime • Addition of velocities • Lorentz transformations Thursday: • Relativistic momentum and energy “The only reason for time is so that HW03 due, beginning of class; HW04 assigned everything doesn’t happen at once.” 2/1 Day 6: Next week: - Albert Einstein Questions? Intro to quantum Spacetime Thursday: Exam I (in class) Addition of Velocities Relativistic Momentum & Energy Lorentz Transformations 1 2 Spacetime Diagrams (1D in space) Spacetime Diagrams (1D in space) c · t In PHYS I: v In PH300: x x x x Δx Δx v = /Δt Δt t t Recall: Lucy plays with a fire cracker in the train. (1D in space) Spacetime Diagrams Ricky watches the scene from the track. c· t In PH300: object moving with 0<v<c. ‘Worldline’ of the object L R -2 -1 0 1 2 x object moving with 0>v>-c v c·t c·t Lucy object at rest object moving with v = -c. at x=1 x=0 at time t=0 -2 -1 0 1 2 x -2 -1 0 1 2 x Ricky 1 Example: Ricky on the tracks Example: Lucy in the train ct ct Light reaches both walls at the same time. Light travels to both walls Ricky concludes: Light reaches left side first. x x L R L R Lucy concludes: Light reaches both sides at the same time In Ricky’s frame: Walls are in motion In Lucy’s frame: Walls are at rest S Frame S’ as viewed from S ... -3 -2 -1 0 1 2 3 .. -
Symmetries and Pre-Metric Electromagnetism
Symmetries and pre-metric electromagnetism ∗ D.H. Delphenich ∗∗ Physics Department, Bethany College, Lindsborg, KS 67456, USA Received 27 April 2005, revised 14 July 2005, accepted 14 July 2005 by F. W. Hehl Key words Pre-metric electromagnetism, exterior differential systems, symmetries of differential equations, electromagnetic constitutive laws, projective relativity. PACS 02.40.k, 03.50.De, 11.30-j, 11.10-Lm The equations of pre-metric electromagnetism are formulated as an exterior differential system on the bundle of exterior differential 2-forms over the spacetime manifold. The general form for the symmetry equations of the system is computed and then specialized to various possible forms for an electromagnetic constitutive law, namely, uniform linear, non-uniform linear, and uniform nonlinear. It is shown that in the uniform linear case, one has four possible ways of prolonging the symmetry Lie algebra, including prolongation to a Lie algebra of infinitesimal projective transformations of a real four-dimensional projective space. In the most general non-uniform linear case, the effect of non-uniformity on symmetry seems inconclusive in the absence of further specifics, and in the uniform nonlinear case, the overall difference from the uniform linear case amounts to a deformation of the electromagnetic constitutive tensor by the electromagnetic field strengths, which induces a corresponding deformation of the symmetry Lie algebra that was obtained in the linear uniform case. Contents 1 Introduction 2 2 Exterior differential systems 4 2.1 Basic concepts. ………………………………………………………………………….. 4 2.2 Exterior differential systems on Λ2(M). …………………………………………………. 6 2.3 Canonical forms on Λ2(M). ………………………………………………………………. 7 3. Symmetries of exterior differential systems 10 3.1 Basic concepts. -
1 Lifts of Polytopes
Lecture 5: Lifts of polytopes and non-negative rank CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Lifts of polytopes 1.1 Polytopes and inequalities Recall that the convex hull of a subset X n is defined by ⊆ conv X λx + 1 λ x0 : x; x0 X; λ 0; 1 : ( ) f ( − ) 2 2 [ ]g A d-dimensional convex polytope P d is the convex hull of a finite set of points in d: ⊆ P conv x1;:::; xk (f g) d for some x1;:::; xk . 2 Every polytope has a dual representation: It is a closed and bounded set defined by a family of linear inequalities P x d : Ax 6 b f 2 g for some matrix A m d. 2 × Let us define a measure of complexity for P: Define γ P to be the smallest number m such that for some C s d ; y s ; A m d ; b m, we have ( ) 2 × 2 2 × 2 P x d : Cx y and Ax 6 b : f 2 g In other words, this is the minimum number of inequalities needed to describe P. If P is full- dimensional, then this is precisely the number of facets of P (a facet is a maximal proper face of P). Thinking of γ P as a measure of complexity makes sense from the point of view of optimization: Interior point( methods) can efficiently optimize linear functions over P (to arbitrary accuracy) in time that is polynomial in γ P . ( ) 1.2 Lifts of polytopes Many simple polytopes require a large number of inequalities to describe. -
THE HOMOGENEITY SCALE of the UNIVERSE 1 Introduction On
THE HOMOGENEITY SCALE OF THE UNIVERSE P. NTELIS Astroparticle and Cosmology Group, University Paris-Diderot 7 , 10 rue A. Domon & Duquet, Paris , France In this study, we probe the cosmic homogeneity with the BOSS CMASS galaxy sample in the redshift region of 0.43 <z<0.7. We use the normalised counts-in-spheres estimator N (<r) and the fractal correlation dimension D2(r) to assess the homogeneity scale of the universe. We verify that the universe becomes homogenous on scales greater than RH 64.3±1.6 h−1Mpc, consolidating the Cosmological Principle with a consistency test of ΛCDM model at the percentage level. Finally, we explore the evolution of the homogeneity scale in redshift. 1 Introduction on Cosmic Homogeneity The standard model of Cosmology, known as the ΛCDM model is based on solutions of the equations of General Relativity for isotropic and homogeneous universes where the matter is mainly composed of Cold Dark Matter (CDM) and the Λ corresponds to a cosmological con- stant. This model shows excellent agreement with current data, be it from Type Ia supernovae, temperature and polarisation anisotropies in the Cosmic Microwave Background or Large Scale Structure. The main assumption of these models is the Cosmological Principle which states that the universe is homogeneous and isotropic on large scales [1]. Isotropy is well tested through various probes in different redshifts such as the Cosmic Microwave Background temperature anisotropies at z ≈ 1100, corresponding to density fluctuations of the order 10−5 [2]. In later cosmic epochs, the distribution of source in surveys the hypothesis of isotropy is strongly supported by in X- ray [3] and radio [4], while the large spectroscopic galaxy surveys such show no evidence for anisotropies in volumes of a few Gpc3 [5].As a result, it is strongly motivated to probe the homogeneity property of our universe. -
Quantum Mechanics and Relativity): 21St Century Analysis
Noname manuscript No. (will be inserted by the editor) Joule's 19th Century Energy Conservation Meta-Law and the 20th Century Physics (Quantum Mechanics and Relativity): 21st Century Analysis Vladik Kreinovich · Olga Kosheleva Received: December 22, 2019 / Accepted: date Abstract Joule's Energy Conservation Law was the first \meta-law": a gen- eral principle that all physical equations must satisfy. It has led to many important and useful physical discoveries. However, a recent analysis seems to indicate that this meta-law is inconsistent with other principles { such as the existence of free will. We show that this conclusion about inconsistency is based on a seemingly reasonable { but simplified { analysis of the situation. We also show that a more detailed mathematical and physical analysis of the situation reveals that not only Joule's principle remains true { it is actually strengthened: it is no longer a principle that all physical theories should satisfy { it is a principle that all physical theories do satisfy. Keywords Joule · Energy Conservation Law · Free will · General Relativity · Planck's constant 1 Introduction Joule's Energy Conservation Law: historically the first meta-law. Throughout the centuries, physicists have been trying to come up with equa- tions and laws that describe different physical phenomenon. Before Joule, how- ever, there were no general principles that restricted such equations. James Joule showed, in [13{15], that different physical phenomena are inter-related, and that there is a general principle covering all the phenom- This work was supported in part by the US National Science Foundation grants 1623190 (A Model of Change for Preparing a New Generation for Professional Practice in Computer Science) and HRD-1242122 (Cyber-ShARE Center of Excellence). -
Noether's Theorems and Gauge Symmetries
Noether’s Theorems and Gauge Symmetries Katherine Brading St. Hugh’s College, Oxford, OX2 6LE [email protected] and Harvey R. Brown Sub-faculty of Philosophy, University of Oxford, 10 Merton Street, Oxford OX1 4JJ [email protected] August 2000 Consideration of the Noether variational problem for any theory whose action is invariant under global and/or local gauge transformations leads to three distinct the- orems. These include the familiar Noether theorem, but also two equally important but much less well-known results. We present, in a general form, all the main results relating to the Noether variational problem for gauge theories, and we show the rela- tionships between them. These results hold for both Abelian and non-Abelian gauge theories. 1 Introduction There is widespread confusion over the role of Noether’s theorem in the case of local gauge symmetries,1 as pointed out in this journal by Karatas and Kowalski (1990), and Al-Kuwari and Taha (1991).2 In our opinion, the main reason for the confusion is failure to appreciate that Noether offered two theorems in her 1918 work. One theorem applies to symmetries associated with finite dimensional arXiv:hep-th/0009058v1 8 Sep 2000 Lie groups (global symmetries), and the other to symmetries associated with infinite dimensional Lie groups (local symmetries); the latter theorem has been widely forgotten. Knowledge of Noether’s ‘second theorem’ helps to clarify the significance of the results offered by Al-Kuwari and Taha for local gauge sym- metries, along with other important and related work such as that of Bergmann (1949), Trautman (1962), Utiyama (1956, 1959), and Weyl (1918, 1928/9). -
Simplicial Complexes
46 III Complexes III.1 Simplicial Complexes There are many ways to represent a topological space, one being a collection of simplices that are glued to each other in a structured manner. Such a collection can easily grow large but all its elements are simple. This is not so convenient for hand-calculations but close to ideal for computer implementations. In this book, we use simplicial complexes as the primary representation of topology. Rd k Simplices. Let u0; u1; : : : ; uk be points in . A point x = i=0 λiui is an affine combination of the ui if the λi sum to 1. The affine hull is the set of affine combinations. It is a k-plane if the k + 1 points are affinely Pindependent by which we mean that any two affine combinations, x = λiui and y = µiui, are the same iff λi = µi for all i. The k + 1 points are affinely independent iff P d P the k vectors ui − u0, for 1 ≤ i ≤ k, are linearly independent. In R we can have at most d linearly independent vectors and therefore at most d+1 affinely independent points. An affine combination x = λiui is a convex combination if all λi are non- negative. The convex hull is the set of convex combinations. A k-simplex is the P convex hull of k + 1 affinely independent points, σ = conv fu0; u1; : : : ; ukg. We sometimes say the ui span σ. Its dimension is dim σ = k. We use special names of the first few dimensions, vertex for 0-simplex, edge for 1-simplex, triangle for 2-simplex, and tetrahedron for 3-simplex; see Figure III.1. -
Chapter 5 the Relativistic Point Particle
Chapter 5 The Relativistic Point Particle To formulate the dynamics of a system we can write either the equations of motion, or alternatively, an action. In the case of the relativistic point par- ticle, it is rather easy to write the equations of motion. But the action is so physical and geometrical that it is worth pursuing in its own right. More importantly, while it is difficult to guess the equations of motion for the rela- tivistic string, the action is a natural generalization of the relativistic particle action that we will study in this chapter. We conclude with a discussion of the charged relativistic particle. 5.1 Action for a relativistic point particle How can we find the action S that governs the dynamics of a free relativis- tic particle? To get started we first think about units. The action is the Lagrangian integrated over time, so the units of action are just the units of the Lagrangian multiplied by the units of time. The Lagrangian has units of energy, so the units of action are L2 ML2 [S]=M T = . (5.1.1) T 2 T Recall that the action Snr for a free non-relativistic particle is given by the time integral of the kinetic energy: 1 dx S = mv2(t) dt , v2 ≡ v · v, v = . (5.1.2) nr 2 dt 105 106 CHAPTER 5. THE RELATIVISTIC POINT PARTICLE The equation of motion following by Hamilton’s principle is dv =0. (5.1.3) dt The free particle moves with constant velocity and that is the end of the story. -
1 the Basic Set-Up 2 Poisson Brackets
MATHEMATICS 7302 (Analytical Dynamics) YEAR 2016–2017, TERM 2 HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS These notes are intended to be read as a supplement to the handout from Gregory, Classical Mechanics, Chapter 14. 1 The basic set-up I assume that you have already studied Gregory, Sections 14.1–14.4. The following is intended only as a succinct summary. We are considering a system whose equations of motion are written in Hamiltonian form. This means that: 1. The phase space of the system is parametrized by canonical coordinates q =(q1,...,qn) and p =(p1,...,pn). 2. We are given a Hamiltonian function H(q, p, t). 3. The dynamics of the system is given by Hamilton’s equations of motion ∂H q˙i = (1a) ∂pi ∂H p˙i = − (1b) ∂qi for i =1,...,n. In these notes we will consider some deeper aspects of Hamiltonian dynamics. 2 Poisson brackets Let us start by considering an arbitrary function f(q, p, t). Then its time evolution is given by n df ∂f ∂f ∂f = q˙ + p˙ + (2a) dt ∂q i ∂p i ∂t i=1 i i X n ∂f ∂H ∂f ∂H ∂f = − + (2b) ∂q ∂p ∂p ∂q ∂t i=1 i i i i X 1 where the first equality used the definition of total time derivative together with the chain rule, and the second equality used Hamilton’s equations of motion. The formula (2b) suggests that we make a more general definition. Let f(q, p, t) and g(q, p, t) be any two functions; we then define their Poisson bracket {f,g} to be n def ∂f ∂g ∂f ∂g {f,g} = − .