DOCSLIB.ORG

ASSIGNMENTS Week 4 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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 . (4) µν ≡ 2 h µβ,ν νβ,µ − µν,βi Here the “, α” indicates a derivative with respect to the coordinate xα, i.e., g ∂ g ∂gµν . µν,α ≡ α µν ≡ ∂xα Note that an “upper index” in the denominator corresponds to a “lower index” in the numerator. Figure 1: The embedding of the (r, φ)-slice of the wormhole geometry (7) into three-dimensional flat space. Based on the Christoffel symbol, we can define a covariant derivative Dµ by its action on (the components of) a tensor by D T β1...βm ∂ T β1...βm µ α1...αn ≡ µ α1...αn ith position n m (5) ν β1...βm βi β1... ν ...βm Γ αiµ Tα1... ν ...αn + Γ νµ Tα1...αn . − X X z}|{ i=1 ith position i=1 |{z} Use this definition to show that the covariant derivative is compatible with the metric, i.e. Dµ gαβ = 0 . (6) β β This implies that the covariant derivative commutes with the metric, e.g., Dµ gαβ u = gαβ Dµ u . Exercise 4: Transversing a wormhole (hand-in exercise) The wormhole geometry depicted in Figure 1 allows to connect either two different (potentially far away) points of the universe or even different universes. The line element of the geometry is 2 2 2 2 2 2 2 2 ds = dt dr (r + b )(dθ + sin θdφ ) . (7) − − The factor b is coordinate independent parameter that determines the size of the wormhole’s throat. a) Use the fact that geodesics extremize proper time to derive the geodesic equation for this particular geometry from the variational principle. Hint: Write down the Lagrangian = 1 2 L (ds/dσ)2 / for the line element (7) and compute the Lagrange equation for the four components of the position vector xα = t,r,θ,φ . { } b) Verify your result by explicitly computing the Christoffel Symbols with upper index t and t r 0 1 r, i.e., Γ αβ and Γ αβ. Write down the geodesic equation for x = t and x = r. Does your answer agree with part a)? 2 c) Consider a spaceship that starts at coordinate position r = R and falls freely and radially through the wormhole throat. For a given initial velocity ur = U, how much time does it take on the spaceships own clock to fall through the wormhole and reach the corresponding point r = R? − Exercise 5: The warp-drive spacetime (optional) The Alcubierre warp-drive solution uses coordinates (t,x,y,z) and a spaceship moving along the curve x = xs(t),y = 0, z = 0, lying in the t-x-plane and moving through the origin. The line element specifying the metric is 2 2 2 2 2 ds = dt [dx V (t) f(r ) dt] dy dz . (8) − − s s − − Here V (t) dx (t)/dt is the velocity of the spaceship and r (x x (t))2 + y2 + z2 marks s ≡ s s ≡ − s its position. The function f(rs) is any smooth function interpolating between f(0) = 1 and f(R) = 0. If necessary, you can use f(r ) = 1 (r /R)4 for concreteness. s − s To see what is interesting about this geometry, consider two stationary space stations at xA and xB, separated by the coordinate distance D along the x-axis, and the spaceship connecting the two space stations in an elapsed coordinate time T < D. This looks like the ship has traveled faster than the speed of light! Indeed the curve xs(t) must have Vs > 1 for some time. Establish that, owed to the special curvature of the warp-drive spacetime, the spaceship is always moving at less than the local velocity of light, even if the coordinate velocity Vs is greater than 1. a) The light cones at a point in the t-x plane are the curves emerging from the point with ds2 = 0. Give a qualitative sketch of the light cone structure entailed by the geometry in the t-x-plane. Distinguish between the regions where f(rs) is vanishing and non-vanishing. Compare the light cone structure to the one of flat space. b) Show that at every point along the curve xs(t) the four-velocity of the ship lies insight the local light cone. Remark: Unfortunately, spacetimes as (8) are excluded in known classical physics. Creating the “warped” spacetime requires matter fields with negative local energy density. All known classical fields have positive energy density, however. Quantum mechanics allows negative energy densities, but it is presently unclear whether this can be used to create a geometry as the one above. 3.
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]
  • 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]
  • Visualizing Curved Spacetime Rickard M
    Visualizing curved spacetime Rickard M. Jonssona) Department of Theoretical Physics, Physics and Engineering Physics, Chalmers University of Technology and Go¨teborg University, 412 96 Gothenburg, Sweden ͑Received 23 April 2003; accepted 8 October 2004͒ I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems ͑Minkowski systems͒ living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying the illustrations is first to specify a field of timelike four-velocities u␮. Then, at every point, one performs a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. The new positive definite metric, called the absolute metric, can be covariantly related to the original Lorentzian metric. For the special case of a two-dimensional original metric, the absolute metric may be embedded in three-dimensional Euclidean space as a curved surface. © 2005 American Association of Physics Teachers. ͓DOI: 10.1119/1.1830500͔ I. INTRODUCTION II. INTRODUCTION TO CURVED SPACETIME Consider a clock moving along a straight line. Special Einstein’s theory of gravity is a geometrical theory and is relativity tells us that the clock will tick more slowly than the well suited to be explained by images. For instance, the way clocks at rest as illustrated in Fig.
    [Show full text]