DOCSLIB.ORG

The Equation of Motion in Rindler Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Equation of Motion in Rindler Space Open Access Journal of Physics Volume 2, Issue 4, 2018, PP 24-28 ISSN 2637-5826 The Equation of Motion in Rindler Space Sanchita Das1, Somenath Chakrabarty2 1Department of physics,visva bharati santiniketan 731235,india *Corresponding Author: Somenath Chakrabarty, Department of physics,visva bharati santiniketan 731235,india ABSTRACT The equation of motion of a massive particle in Rindler space or equivalently in an uniform gravitational field has been studied in both classical and quantum mechanical scenarios. The classical geodesics of motion of the particle and the wave function for the quantum mechanical equation satisfied by the particle are obtained. It has been observed that it is equivalent to the problem of free fall of the particle at the center. Unlike the conventional scenario where the fall occurs at the origin, which is a singularity, here it takes place at the point which is at a finite distance from the origin (singularity). Further, the classical trajectories are not a perihelion type. These are closed type. The massive particle returns to its original position if disturbed from the equilibrium position (having minimum energy). PACS number: 03.65.Ge,03.65.Pm,03.30.+p,04.20.-q INTRODUCTION local acceleration or local uniform gravitational field. Hence the line element in Rindler space in It is well known that the Lorentz transformations 1+1 –dimension is given by of space time coordinates are between the inertial frame of references [1], whereas the 푑푠2 = −푑휏2 = − 1 + 훼푥 2푑푡2 + 푑푥2 (3) Rindler transformations are between an inertial Where 훕 is the proper time. Then one can define frame and the frame undergoing uniform the classical action integral in the form [1] accelerated motion [2-10]. The space here is 퐵 퐵 called the Rindler space, exactly like the 푑휏 = 퐿 푑흀 (4) Minkowski space in the Lorentz transformation 퐴 퐴 scenario. Further, the Rindler space is locally Where 휆 is an affine parameter which is flat, whereas the flatness of Minkowski space is changing along the trajectory and having the 푑휏 global in nature. Now according to the principle dimension of time. Hence 퐿 = , the 푑휆 of equivalence a frame undergoing accelerated Lagrangian of the particle, explicitly given motion in absence of gravity is equivalent to a by[11-13] frame at rest in presence of gravitational field. 1 The strength of the gravitational field is exactly 푑푡 2 푑푥 2 2 퐿 = 1 + 훼푥 2 + − (5) equal to the magnitude of the acceleration of the 푑휆 푑휆 moving frame. Therefore the Rindler space may be considered to be a space associated with a The aim of the present article is to investigate frame at rest in presence of a uniform the classical motion as well as the quantum gravitational field. The Rindler space-time mechanical motion of the non-zero mass particle transformations in natural unit (푐 = ћ = 1) and in Rindler space. Apparently from the title of in 1+1 dimension are given by [7-10] this article it may sound that it is a very old problem, however so far our knowledge is 1 푡 = + 푥′ sinh 훼푡′ (1) concerned it has not been studied before. 훼 Therefore we expect that this work should be 1 푥 = + 푥′ cosh 훼푡′ (2) reported in some reputed journal. In this article 훼 we have obtained the classical geodesics of Where primed coordinates are in the non-inertial motion and the wave functions for the quantum frame, whereas the unprimed are in the inertial mechanical equation satisfied by the particle. It one. The Rindler space-time transformations are has been observed that the problem is equivalent therefore exactly like the Lorentz transformations to the problem of free fall. We have also but in uniformly accelerated frame. Here α is the investigated this free fall problem of the Open Access Journal of Physics V2● 14 ● 2018 24 The Equation of Motion in Rindler Space particle. Although the gravitational field is (see Appendix for some outline to obtain this considered to be uniform throughout, however expression). Now changing the variable from 푥 the energy of the particle is minimum at the 1 to 푢 = 1 + 훼푥 and redefining 휏 → 퐶2훼흉 , we origin. As a consequence the particle will fall at have from the above equation, the point which is at a finite distance from the 푑2푢 1 origin 푥 = 0 .Now in the Rindler coordinate + = 0 (10) system, the portion 푥 > |푡| of the Minkowski 푑휏 2 푢3 space is called the Rindler wedge. The second This is the classical equation of motion for the wedge 푥 < −|푡|can be obtained by reflection. particle in Rindler space. Integrating both sides We call the first one as right wedge and the with respect to 푢 , this equation may be second one the left wedge of Rindler space. The transformed to the form null rays act as the event horizons for Rindler 1 푑푢 2 1 observers. An observer in the right wedge − = 푐표푛푠푡푎푛푡 (11) 2 푑휏 2푢2 cannot see any eventsin the left wedge. These two regions are casually disjoint two Re-writing 훕 in its original form and defining 푑푢 universes.However, exactly like the Minkowski 푝 = as the momentum of the particle of unit 푑휏 space the past and the present can be defined mass, we have from the above equation and are casually connected. Therefore the actual 2 2 1 푝 퐶훼 singularity 푥 = − is not an accessible to an − 2 = 푐표푛푠푡푎푛푡 = 퐸 (12) 훼 2 2푢 observer in right wedge. The origin (which is Where the first term on the left hand side is like not a singularity) is set at 푥 = 0. To the best of the kinetic energy for a unit mass and the second knowledge this problem has not been done term is an attractive inverse square potential, before.Various sections of the article are whereas the constant on the right hand side may organized in the following manner. In the next be treated as the total energy of the particle. We section we shall develop the formalism of the further assume that the energy of the particle is classical motion of the particle in Rindler space. negative in nature, i.e., we are considering In section-3 we have discussed the quantum bounded motion of the classical particle. Now mechanical motion in Rindler space and finally expressing the above equation in the usual form, in section-4 we have given the conclusion of our given by work. 1 푝2 + 푉 푢 = 퐸 (13) CLASSICAL MOTION IN RINDLER SPACE 2 퐶훼2 Now the well-known Euler-Lagrange equation Where 푉 푢 = − , the inverse square 2푢2 is given by attractive potential. Then depending on the 2 휕 휕퐿 휕퐿 strength 0.5퐶훼 of the attractive potential 푉 푢 , 휕푞 − = 0 (6) the particle may fall at the center( 푥 = 0 or 휕휆 휕 푖 휕푞 푖 휕휆 푢 = 1 ), where the momentum becomes zero Where 푞푖 = 푡 or 푥. Let us first consider 푞푖 = 푡, (푝 = 0, i.e., particle is in the rest condition) and the universal time coordinate. Since t is cyclic in this situation 퐸 = 푉 푢 = 1 = −0.5퐶훼2, the 휕퐿 coordinate, = 0 and therefore maximum depth of the potential, which is also 휕푡 the minimum possible value of the total energy. 휕푡 퐶 = (7) Since it is negative it gives the maximum 휕휏 1+훼푥 2 binding at 푥 = 0 or 푢 = 1 . Since α is the Where 퐶 is a constant and to obtain this uniform acceleration, the parameter 퐶 may be equation we have used the expression for the treated as the strength of the potential 푉 푢 . Lagrangian as given by equation (5). We next One can calculate the two classical turning consider 푞푖 = 푥, then with some little algebra, points in case the motion is oscillatory between we have from the Euler-Lagrange equation two extreme points, where the particle momentum becomes zero. These are at 푢 = 휕푡 2 푑2푥 1 + 훼푥 훼 + 2 = 0 (8) 푢0 = 1 , the central point in the transformed 휕휏 푑휏 1 휕푡 special coordinate and 푢 = 푢 = 0.5퐶|퐸| 2훼 On substituting the expression for from 1 휕휏 for 퐸 negative = −|퐸|. In this case, of course equation (7) we have the motion is periodic. However, if because of 훼퐶 푑2푥 strong attractive gravitational potential at + = 0 (9) 1+훼푥 3 푑휏 2 푢 = 푢0 (for the large value of 퐶) the particle 25 Open Access Journal of Physics V2● 14 ● 2018 The Equation of Motion in Rindler Space falls at the center, i.e., at 푥 = 0 or 푢 = 푢0 = 1 Using the changed variable 푢(= 1 + 훼푥) , we then the motion will no longer remain have, oscillatory. In fig.(1) we have shown this ퟏ ퟏ ퟐ ퟐ ퟏ potential well. The coordinate 푥 is plotted along 퐂ퟐ ퟏ+훂퐱 ퟏ α훕 = − ퟏ − − ퟏ − ퟐ (16) 푥 -axis, where the magnitude of depth of 훂 ퟏ 퐂 퐂ퟐ potential |푉 푢 | is plotted along 푦 -axis. Therefor in actual scenario this is not the Since the uniform acceleration α of the frame is potential hill but is potential well. But unlike the completely arbitrary, we shall study the conventional scenario of fall at the center variation of 훼푥 with α훕, keeping in problem, there is no singularity at the origin consideration that α remains constant within this (푥 = 0) , or at the center of the gravitating 2 푥 range and is static in nature. The above object[14-16]. The value of 푉(푢) is −0.5퐶훼 at equation can also be expressed in the following the origin and → −1 for large 푢 values.
Load more

Recommended publications
  • Line Element in Noncommutative Geometry
    Line element in noncommutative geometry P. Martinetti G¨ottingenUniversit¨at Wroclaw, July 2009 . ? ? - & ? !? The line element p µ ν ds = gµν dx dx is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - . ? !? The line element p µ ν ds = gµν dx dx & ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - !? The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance ? -Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry).
    [Show full text]
  • ASSIGNMENTS Week 4 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof
    ASSIGNMENTS Week 4 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof. A. Achterberg, Dr. S. Larsen and Dr. F. Saueressig Exercise 1: Flat space in spherical coordinates For constant time t the line element of special relativity reduces to the flat three-dimensional Euclidean metric Euclidean 2 SR 2 2 2 2 (ds ) = (ds ) =const = dx + dy + dz . (1) − |t Use the coordinate transformation x = r sin θ cos φ, y = r sin θ sin φ , z = r cos θ , (2) with θ [0,π] and φ [0, √2 π[ to express the line element in spherical coordinates. ∈ ∈ Exercise 2: Distances, areas and volumes For constant time the metric describing the spatial part of a homogeneous closed Friedmann- Robertson Walker (FRW) universe is given by 2 2 dr 2 2 2 2 ds = + r dθ + sin θ dφ , r [ 0 , a ] . (3) 1 (r/a)2 ∈ − The scale factor a(t) depends on time only so that, for t = const, it can be considered as a fixed number. Compute from this line element a) The proper circumference of the sphere around the equator. b) The proper distance from the center of the sphere up to the coordinate radius a. c) The proper area of the two-surface at r = a. d) The proper volume inside r = a. Compare your results to the ones expected from flat Euclidean space. Which of the quantities are modified by the property that the line element (3) is curved? Exercise 3: The metric is covariantly constant In the lecture we derived a formula for the Christoffel symbol Γα 1 gαβ g + g g .
    [Show full text]
  • Appendix a Tensor Mathematics
    Appendix A Tensor Mathematics A.l Introduction In physics we are accustomed to relating quantities to other quantities by means of mathematical expressions. The quantities that represent physical properties of a point or of an infinitesimal volume in space may be of a single numerical value without direction, such as temperature, mass, speed, dist­ ance, specific gravity, energy, etc.; they are defined by a single magnitude and we call them scalars. Other physical quantities have direction as well and their magnitude is represented by an array of numerical values, their number corresponding to the number of dimensions of the space, three numerical values in a three-dimensional space; such quantities are called vectors, for example velocity, acceleration, force, etc., and the three numerical values are the components of the vector in the direction of the coordinates of the space. Still other physical quantities such as moment of inertia, stress, strain, permeability coefficients, electric flux, electromagnetic flux, etc., are repre­ sented in a three-dimensional space by nine numerical quantities, called components, and are known as tensors. We will introduce a slightly different definition. We shall call the scalars tensors of order zero, vectors - tensors of order one and tensors with nine components - tensors of order two. There are, of course, in physics and mathematics, tensors of higher order. Tensors of order three have an array of 27 components and tensors of order four have 81 components and so on. Couple stresses that arise in materials with polar forces are examples of tensors of order three, and the Riemann curvature tensor that appears in geodesics is an example of a tensor of order four.
    [Show full text]
  • 1 Special Relativity
    1 Special relativity 1.1 Minkowski space Inertial frames and the principle of relativity Newton’s Lex Prima (or the Galilean law of inertia) states: Each force-less mass point stays at rest or moves on a straight line at constant speed. In a Cartesian inertial coordinate system, Newton’s lex prima becomes d2x d2y d2z = = = 0 . (1.1) dt2 dt2 dt2 Most often, we call such a coordinate system just an inertial frame. Newton’s first law is not just a trivial consequence of its second one, but may be seen as practical definition of those reference frames for which his following laws are valid. Since the symmetries of Euclidean space, translations a and rotations R, leave Eq. (1.1) invariant, all frames connected by x′ = Rx + a to an inertial frame are inertial frames too. Additionally, proper Galilean transformations x′ = x + vt connect inertial frames moving with relative speed v. The principle of relativity states that the physics in all inertial frames is the same. If we consider two frames with relative velocity along the x direction, then the most general linear transformation between the two frames is t′ At + Bx At + Bx ′ x Dt + Ex A(x vt) ′ = = − . (1.2) y y y ′ z z z Newtonian physics assumes the existence of an absolute time, t = t′, and thus A = 1 and B = 0. This leads to the “common sense” addition law of velocities. Time differences ∆t and space differences 2 2 2 2 ∆x12 = (x1 x2) + (y1 y2) + (z1 z2) (1.3) − − − are separately invariant. Lorentz transformations In special relativity, we replace the Galilean transformations as symmetry group of space and time by Lorentz transformations Λ.
    [Show full text]
  • Metric Tensor & Chriostofell Symbols
    MATHEMATICAL PHYSICS UNIT – 8 Metric Tensor & Chriostofell Symbols PRESENTED BY: DR. RAJESH MATHPAL ACADEMIC CONSULTANT SCHOOL OF SCIENCES U.O.U. TEENPANI, HALDWANI UTTRAKHAND MOB:9758417736,7983713112 STRUCTURE OF UNIT 8.1. INTRODUCTION 8.2. RIEMANNIAN SPACE: METRIC TENSOR jk 풋 8.3. FUNDAMENTAL TENSORS gjk, g AND 휹풌 8.4. CHRISTOFELL’S 3-INDEX SYMBOLS 8.5. GEODESICS 8.1. INTRODUCTION In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. A metric tensor is called positive-definite if it assigns a positive value g(v, v) > 0 to every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p, q) whose value at a pair of points p and q is the distance from p to q.
    [Show full text]
  • PHY103A: Lecture # 3 (Text Book: Intro to Electrodynamics by Griffiths, 3Rd Ed.)
    Semester II, 2017-18 Department of Physics, IIT Kanpur PHY103A: Lecture # 3 (Text Book: Intro to Electrodynamics by Griffiths, 3rd Ed.) Anand Kumar Jha 08-Jan-2018 Notes • The first tutorial is tomorrow (Tuesday). • Updated lecture notes will be uploaded right after the class. • Office Hour – Friday 2:30-3:30 pm • Phone: 7014(Off); 962-142-3993(Mobile) [email protected]; [email protected] • Tutorial Sections have been finalized and put up on the webpage. • Course Webpage: http://home.iitk.ac.in/~akjha/PHY103.htm 2 3 Summary of Lecture # 2: 2 Gradient of a scalar 1 0 1 + + 2 ≡ � � � 3 3 2 1 0 1 2 3 Divergence of a vector 2 1 ⋅ = + + 0 ⋅ 1 Curl of a vector × 2 2 2 1 0 1 2 1 × = � � � 0 1 = + + 2 2 1 0 1 2 3 − � − � − � Integral Calculus: The ordinary integral that we know of is of the form: In vector calculus we encounter many other types of integrals. ∫ Line Integral: � ⋅ Vector field Infinitesimal Displacement vector Or Line element = + + If the path is a closed loop � then the� line �integral is written as • When do� we ⋅ need line integrals? Work done by a force along a given path involves line integral. 4 Example (G: Ex. 1.6) Q: = + ( + 1) ? What is the line integral from A to2 B along path (1) and (2)? � 2 � Along path (1) We have = + . (i) = ; =1; �= � = 1 (ii) = ; =2; = 4( 2 + 1) =10 � ∫ ⋅ ∫ 2 � ∫ ⋅ ∫1 Along path (2): = + ; = ; = = ( + 2 + ) =10 � � 2 2 2 ∫ ⋅ ∫1 2 =11-10=1 This means that if represented the force vector, ⋅ ∮ it would be a non-conservative force 5 Surface Integral: or � ⋅ � ⋅ Vector field Infinitesimal area vector Closed loop • For a closed surface, the area vector points outwards.
    [Show full text]
  • AST1100 Lecture Notes
    AST1100 Lecture Notes 9–10 The special theory of relativity: Four vectors and relativistic dynamics 1 Worldlines In the spacetime diagram in figure 1 we see the path of a particle (or any object) through spacetime. We see the different positions (x, t) in space and time that the particle has passed through. Such a path showing the points in spacetime that an object passed is called a worldline. We will now study two events A and B (on the worldline of a particle) which are separated by a small spacetime interval ∆s. These events could be the particle emitting two flashes of light or the particle passing through two specific points in space. The corresponding space and time intervals between these two events in the laboratory frame are called ∆t and ∆x. From the figure you see that ∆t > ∆x. You can see that this also holds for every small spacetime interval along the path. This has to be this way: The speed of the particle at a given instant is v = ∆x/∆t. If ∆x = ∆t then v = 1 and the particle travels at the speed of light. That ∆t > ∆x simply means that the particle travels at a speed v < c which it must. The worldline of a photon would thus be a line at 45◦ with the coordinate axes. The worldline of any material particle will therefore always make less than 45◦ with the time axis. Events which are separated by spacetime distances such that ∆t > ∆x are called timelike events. Timelike events may be causally connected since a particle with velocity v < c would have the possibility to travel from one of the events to the other event.
    [Show full text]
  • Riemann Extension of Minkowski Line Element in the Rindler Chart
    Gen. Math. Notes, Vol. 30, No. 1, September 2015, pp.21-27 ISSN 2219-7184; Copyright c ICSRS Publication, 2015 www.i-csrs.org Available free online at http://www.geman.in Riemann Extension of Minkowski Line Element in the Rindler Chart H.G. Nagaraja1 and D. Harish2 1;2Department of Mathematics, Bangalore University Central College Campus, Bengaluru-560001, India 1E-mail: [email protected] 2E-mail: [email protected] (Received: 16-7-15 / Accepted: 29-8-15) Abstract In this paper, we discuss Riemannian extension of Minkowski metric in Rindler coordinates and its geodesics. Keywords: Riemannian curvature, Riemann extension, Flat metric, geodesic equations, constant positive curvature. 1 Introduction Patterson and Walker[7] have defined Riemann extensions and showed how a Riemannian structure can be given to the 2n dimensional tangent bundle of an n− dimensional manifold with given non-Riemannian structure. This shows Riemann extension provides a solution of the general problem of embedding a manifold M carrying a given structure in a manifold M 0 carrying another structure, the embedding being carried out in such a way that the structure on M 0 induces in a natural way the given structure on M. The Riemann ex- tension of Riemannian or non-Riemannian spaces can be constructed with the i help of the Christoffel coefficients Γjk of corresponding Riemann space or with i connection coefficients Πjk in the case of the space of affine connection[5]. The theory of Riemann extensions has been extensively studied by Afifi[1] and Dryuma[2],[3], [4] [5]. 22 H.G. Nagaraja et al.
    [Show full text]
  • 2 Basics of General Relativity 15
    2 BASICS OF GENERAL RELATIVITY 15 2 Basics of general relativity 2.1 Principles behind general relativity The theory of general relativity completed by Albert Einstein in 1915 (and nearly simultaneously by David Hilbert) is the current theory of gravity. General relativity replaced the previous theory of gravity, Newtonian gravity, which can be understood as a limit of general relativity in the case of isolated systems, slow motions and weak fields. General relativity has been extensively tested during the past century, and no deviations have been found, with the possible exception of the accelerated expansion of the universe, which is however usually explained by introducing new matter rather than changing the laws of gravity [1]. We will not go through the details of general relativity, but we try to give some rough idea of what the theory is like, and introduce a few concepts and definitions that we are going to need. The principle behind special relativity is that space and time together form four-dimensional spacetime. The essence of general relativity is that gravity is a manifestation of the curvature of spacetime. While in Newton's theory gravity acts directly as a force between two bodies1, in Einstein's theory the gravitational interac- tion is mediated by spacetime. In other words, gravity is an aspect of the geometry of spacetime. Matter curves spacetime, and this curvature affects the motion of other matter (as well as the motion of the matter generating the curvature). This can be summarised as the dictum \matter tells spacetime how to curve, spacetime tells matter how to move" [2].
    [Show full text]
  • ASSIGNMENTS Week 3 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof
    ASSIGNMENTS Week 3 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof. A. Achterberg, Dr. S. Larsen and Dr. F. Saueressig This exercise explicitly keeps track of the speed of light. Use c = 3 × 108 m/s for explicit compu- tations. For solving exercises 1 and 2 the following compilation of masses and distances may be used: Earth: 27 Mass M⊕ = 5.97 × 10 g 2 −3 GM⊕/c = 4.43 × 10 m 6 Equatorial radius R⊕ = 6.378 × 10 m Rotation period 8.62 × 104 s Sun: 33 Mass M⊙ = 1.99 × 10 g 2 3 GM⊙/c = 1.48 × 10 m 8 Radius R⊙ = 6.96 × 10 m Exercise 1: the global positioning system (GPS) (Hand-in assignment) The GPS system consists of 24 satellites which orbit the earth on 6 equally spaced orbital planes. Each satellite emits a signal encoding its time of emission te and the location of the satellite. An observer who receives the signal at time tr, that is an interval ∆t = tr − te later, knows that she is located somewhere on a sphere of radius c∆t centered on the satellite. Signals from two satellites narrow the location down to the intersection of two spheres. Since GPS receivers typically don’t include clocks with sufficient accuracy to precisely pinpoint the arrival time of the satellite signals, it requires the signals from four satellites to locate a position in four-dimensional spacetime. In order to facilitate the computation, assume that the GPS satellite is in a 12-hour circular equatorial orbit of radius Rs from the Earth’s center.
    [Show full text]
  • Equations of Motion of Schwarzschild, Reissner-Nordstrom and Kerr
    Equations of Motion of Schwarzschild, Reissner-Nordstr¨om and Kerr Particles Peter A. Hogan,∗ School of Physics University College Dublin Belfield, Dublin 4, Ireland March 13, 2018 Abstract A technique for extracting from the appropriate field equations the relativistic motion of Schwarzschild, Reissner-Nordstr¨om and Kerr par- ticles moving in external fields is motivated and illustrated. The key assumptions are that (a) the particles are isolated and (b) near the particles the wave fronts of the radiation generated by their motion are smoothly deformed spheres. No divergent integrals arise in this approach. The particles are not test particles. The formalism is used, however, to derive the Mathisson–Papapetrou equations of motion of spinning test particles, neglecting spin-spin terms. 1 Introduction We describe a method for modeling the relativistic motion of (uncharged arXiv:1508.02564v1 [gr-qc] 11 Aug 2015 or charged) particles moving in external gravitational and electromagnetic fields in general relativity. The particles modify the fields in which they move and thus are not test particles. The modified fields near the particles are predominantly the Schwarschild, Reissner–Nordstr¨om or Kerr fields and thus the particles will be referred to as Schwarschild, Reissner–Nordstr¨om or Kerr particles respectively. The origin of the approach described here is the seminal and challenging paper by Robinson and Robinson [1], with the disconcertingly modest title of “Equations of Motion in the Linear Approxi- mation”, published in the festschrift in honor of Professor J. L. Synge. This ∗Email: [email protected] 1 paper inspired early work involving space–times which are more special than those required here and which can be found in a series of papers by Hogan and Imaeda [2], [3], [4] (see also [5]).
    [Show full text]
  • Special Relativity2 1.1 Introduction
    General relativity Spring 2021 Syksy R¨as¨anen CONTENTS 1 Contents 1 Special relativity2 1.1 Introduction................................2 1.2 Newtonian spacetime...........................3 1.2.1 Illustrating the structure of Newtonian spacetime.......3 1.2.2 Symmetries of Newtonian spacetime..............3 1.3 Minkowski space.............................5 1.3.1 The Minkowski metric......................5 1.3.2 Natural units...........................8 1.3.3 Sets of points at constant distance...............8 1.3.4 The line element.........................9 1.3.5 Poincar´etransformations.................... 11 1.3.6 Dot product and raising and lowering indices......... 14 1.3.7 Lorentz boosts and velocity................... 14 1.4 Dynamics in SR............................. 16 1.4.1 Generalising Newton's second law............... 16 1.5 Electrodynamics............................. 18 1.5.1 Maxwell equations........................ 18 1.5.2 The road in reverse........................ 20 Preface These are the lecture notes of the course General relativity at the University of Helsinki. They owe a heavy debt to the lecture notes of Hannu Kurki-Suonio, which were in turn influenced by Sean Carroll's lecture notes and his book Spacetime and Geometry. Unless otherwise noted, figures are by Hannu Kurki-Suonio. I thank Fernando Bracho Blok for LaTeXing parts of these lecture notes. Any errors are my responsibility; let me know when you catch them.1 General relativity is one of the two fundamental theories we currently have. The other is quantum field theory, in particular its application to the Standard Model of particle physics. Both of these branches grew from addressing inadequacies of Newtonian mechanics in the 20th century. Quantum physics was developed in bits and pieces in strong correspondence with observations as the classical description of matter was found to be inadequate.
    [Show full text]