DOCSLIB.ORG

C:\Book\Booktex\C1s4.DVI

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
C:\Book\Booktex\C1s4.DVI 108 1.4 DERIVATIVE OF A TENSOR § In this section we develop some additional operations associated with tensors. Historically, one of the basic problems of the tensor calculus was to try and find a tensor quantity which is a function of the metric ∂g ∂2g tensor g and some of its derivatives ij , ij , .... A solution of this problem is the fourth order ij ∂xm ∂xm∂xn Riemann Christoffel tensor Rijkl to be developed shortly. In order to understand how this tensor was arrived at, we must first develop some preliminary relationships involving Christoffel symbols. Christoffel Symbols Let us consider the metric tensor gij which we know satisfies the transformation law ∂xa ∂xb g = gab . αβ ∂xα ∂xβ Define the quantity c a b 2 a b a 2 b ∂gαβ ∂gab ∂x ∂x ∂x ∂ x ∂x ∂x ∂ x (α, β, γ)= = + gab + gab ∂xγ ∂xc ∂xγ ∂xα ∂xβ ∂xα∂xγ ∂xβ ∂xα ∂xβ∂xγ 1 and form the combination of terms [(α, β, γ)+(β,γ,α) (γ,α,β)] to obtain the result 2 − a b c b 2 a 1 ∂gαβ ∂gβγ ∂gγα 1 ∂gab ∂gbc ∂gca ∂x ∂x ∂x ∂x ∂ x + = + + gab . (1.4.1) 2 ∂xγ ∂xα − ∂xβ 2 ∂xc ∂xa − ∂xb ∂xα ∂xβ ∂xγ ∂xβ ∂xα∂xγ In this equation the combination of derivatives occurring inside the brackets is called a Christoffel symbol of the first kind and is defined by the notation 1 ∂g ∂g ∂g [ac, b]=[ca, b]= ab + bc ac . (1.4.2) 2 ∂xc ∂xa − ∂xb The equation (1.4.1) defines the transformation for a Christoffel symbol of the first kind and can be expressed as ∂xa ∂xb ∂xc ∂2xa ∂xb [αγ,β]=[ac, b] + gab . (1.4.3) ∂xα ∂xβ ∂xγ ∂xα∂xγ ∂xβ Observe that the Christoffel symbol of the first kind [ac, b] does not transform like a tensor. However, it is symmetric in the indices a and c. At this time it is convenient to use the equation (1.4.3) to develop an expression for the second derivative term which occurs in that equation as this second derivative term arises in some of our future considerations. ∂xβ To solve for this second derivative we can multiply equation (1.4.3) by gde and simplify the result to the ∂xd form ∂2xe ∂xa ∂xc ∂xβ = gde[ac, d] + [αγ,β] gde. (1.4.4) ∂xα∂xγ − ∂xα ∂xγ ∂xd ∂xd ∂xe The transformation gde = gλµ allows us to express the equation (1.4.4) in the form ∂xλ ∂xµ ∂2xe ∂xa ∂xc ∂xe = gde[ac, d] + gβµ[αγ,β] . (1.4.5) ∂xα∂xγ − ∂xα ∂xγ ∂xµ 109 Define the Christoffel symbol of the second kind as i i 1 ∂g ∂g ∂g = = giα[jk,α]= giα kα + jα jk . (1.4.6) jk kj 2 ∂xj ∂xk − ∂xα This Christoffel symbol of the second kind is symmetric in the indices j and k and from equation (1.4.5) we see that it satisfies the transformation law µ ∂xe e ∂xa ∂xc ∂2xe = + . (1.4.7) αγ ∂xµ ac ∂xα ∂xγ ∂xα∂xγ Observe that the Christoffel symbol of the second kind does not transform like a tensor quantity. We can use the relation defined by equation (1.4.7) to express the second derivative of the transformation equations in terms of the Christoffel symbols of the second kind. At times it will be convenient to represent the Christoffel symbols with a subscript to indicate the metric from which they are calculated. Thus, an alternative notation for i is the notation i . jk jk g EXAMPLE 1.4-1. (Christoffel symbols) Solve for the Christoffel symbol of the first kind in terms of the Christoffel symbol of the second kind. Solution: By the definition from equation (1.4.6) we have i = giα[jk,α]. jk We multiply this equation by gβi and find i g = δα[jk,α]=[jk,β] βi jk β and so i 1 N [jk,α]=gαi = gα + + gαN . jk 1 jk ··· jk EXAMPLE 1.4-2. (Christoffel symbols of first kind) Derive formulas to find the Christoffel symbols of the first kind in a generalized orthogonal coordinate system with metric coefficients 2 gij =0 for i = j and g = h ,i=1, 2, 3 6 (i)(i) (i) where i is not summed. Solution: In an orthogonal coordinate system where gij =0fori = j we observe that 6 1 ∂g ∂g ∂g [ab, c]= ac + bc ab . (1.4.8) 2 ∂xb ∂xa − ∂xc Here there are 33 = 27 quantities to calculate. We consider the following cases: 110 CASE I Let a = b = c = i, then the equation (1.4.8) simplifies to 1 ∂g [ab, c]=[ii, i]= ii (no summation on i). (1.4.9) 2 ∂xi From this equation we can calculate any of the Christoffel symbols [11, 1], [22, 2], or [33, 3]. CASE II Let a = b = i = c, then the equation (1.4.8) simplifies to the form 6 1 ∂g [ab, c]=[ii, c]= ii (no summation on i and i = c). (1.4.10) −2 ∂xc 6 since, gic =0fori = c. This equation shows how we may calculate any of the six Christoffel symbols 6 [11, 2], [11, 3], [22, 1], [22, 3], [33, 1], [33, 2]. CASE III Let a = c = i = b, and noting that gib =0fori = b, it can be verified that the equation (1.4.8) 6 6 simplifies to the form 1 ∂g [ab, c]=[ib, i]=[bi, i]= ii (no summation on i and i = b). (1.4.11) 2 ∂xb 6 From this equation we can calculate any of the twelve Christoffel symbols [12, 1] = [21, 1] [31, 3] = [13, 3] [32, 3] = [23, 3] [21, 2] = [12, 2] [13, 1] = [31, 1] [23, 2] = [32, 2] CASE IV Let a = b = c and show that the equation (1.4.8) reduces to 6 6 [ab, c]=0, (a = b = c.) 6 6 This represents the six Christoffel symbols [12, 3] = [21, 3] = [23, 1] = [32, 1] = [31, 2] = [13, 2] = 0. From the Cases I,II,III,IV all twenty seven Christoffel symbols of the first kind can be determined. In practice, only the nonzero Christoffel symbols are listed. EXAMPLE 1.4-3. (Christoffel symbols of the first kind)Find the nonzero Christoffel symbols of the first kind in cylindrical coordinates. Solution: From the results of example 1.4-2 we find that for x1 = r, x2 = θ, x3 = z and 1 2 2 g11 =1,g22 =(x ) = r ,g33 =1 the nonzero Christoffel symbols of the first kind in cylindrical coordinates are: 1 ∂g [22, 1] = 22 = x1 = r −2 ∂x1 − − 1 ∂g [21, 2] = [12, 2] = 22 = x1 = r. 2 ∂x1 111 EXAMPLE 1.4-4. (Christoffel symbols of the second kind) Find formulas for the calculation of the Christoffel symbols of the second kind in a generalized orthogonal coordinate system with metric coefficients 2 gij =0 for i = j and g = h ,i=1, 2, 3 6 (i)(i) (i) where i is not summed. Solution: By definition we have i = gim[jk,m]=gi1[jk,1] + gi2[jk,2] + gi3[jk,3] (1.4.12) jk By hypothesis the coordinate system is orthogonal and so 1 gij =0 for i = j and gii = i not summed. 6 gii The only nonzero term in the equation (1.4.12) occurs when m = i and consequently i [jk,i] = gii[jk,i]= no summation on i.(1.4.13) jk g ii We can now consider the four cases considered in the example 1.4-2. CASE I Let j = k = i and show i [ii, i] 1 ∂g 1 ∂ = = ii = ln g no summation on i.(1.4.14) ii g 2g ∂xi 2 ∂xi ii ii ii CASE II Let k = j = i and show 6 i [jj,i] 1 ∂gjj = = − no summation on i or j.(1.4.15) jj g 2g ∂xi ii ii CASE III Let i = j = k and verify that 6 j j [jk,j] 1 ∂g 1 ∂ = = = jj = ln g no summation on i or j. (1.4.16) jk kj g 2g ∂xk 2 ∂xk jj jj jj CASE IV For the case i = j = k we find 6 6 i [jk,i] = =0,i= j = k no summation on i. jk g 6 6 ii The above cases represent all 27 terms. 112 EXAMPLE 1.4-5. (Notation) In the case of cylindrical coordinates we can use the above relations and find the nonzero Christoffel symbols of the second kind: 1 1 ∂g = 22 = x1 = r 22 −2g ∂x1 − − 11 2 2 1 ∂g 1 1 = = 22 = = 12 21 2g ∂x1 x1 r 22 Note 1: The notation for the above Christoffel symbols are based upon the assumption that x1 = r, x2 = θ and x3 = z. However, in tensor calculus the choice of the coordinates can be arbitrary. We could just as well have defined x1 = z,x2 = r and x3 = θ. In this latter case, the numbering system of the Christoffel symbols changes. To avoid confusion, an alternate method of writing the Christoffel symbols is to use coordinates in place of the integers 1,2 and 3. For example, in cylindrical coordinates we can write θ θ 1 r = = and = r. rθ θr r θθ − If we define x1 = r, x2 = θ, x3 = z, then the nonzero Christoffel symbols are written as 2 2 1 1 = = and = r. 12 21 r 22 − In contrast, if we define x1 = z,x2 = r, x3 = θ, then the nonzero Christoffel symbols are written 3 3 1 2 = = and = r.
Load more

Recommended publications
  • 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 .................
    [Show full text]
  • Laplacians in Geometric Analysis
    LAPLACIANS IN GEOMETRIC ANALYSIS Syafiq Johar syafi[email protected] Contents 1 Trace Laplacian 1 1.1 Connections on Vector Bundles . .1 1.2 Local and Explicit Expressions . .2 1.3 Second Covariant Derivative . .3 1.4 Curvatures on Vector Bundles . .4 1.5 Trace Laplacian . .5 2 Harmonic Functions 6 2.1 Gradient and Divergence Operators . .7 2.2 Laplace-Beltrami Operator . .7 2.3 Harmonic Functions . .8 2.4 Harmonic Maps . .8 3 Hodge Laplacian 9 3.1 Exterior Derivatives . .9 3.2 Hodge Duals . 10 3.3 Hodge Laplacian . 12 4 Hodge Decomposition 13 4.1 De Rham Cohomology . 13 4.2 Hodge Decomposition Theorem . 14 5 Weitzenb¨ock and B¨ochner Formulas 15 5.1 Weitzenb¨ock Formula . 15 5.1.1 0-forms . 15 5.1.2 k-forms . 15 5.2 B¨ochner Formula . 17 1 Trace Laplacian In this section, we are going to present a notion of Laplacian that is regularly used in differential geometry, namely the trace Laplacian (also called the rough Laplacian or connection Laplacian). We recall the definition of connection on vector bundles which allows us to take the directional derivative of vector bundles. 1.1 Connections on Vector Bundles Definition 1.1 (Connection). Let M be a differentiable manifold and E a vector bundle over M. A connection or covariant derivative at a point p 2 M is a map D : Γ(E) ! Γ(T ∗M ⊗ E) 1 with the properties for any V; W 2 TpM; σ; τ 2 Γ(E) and f 2 C (M), we have that DV σ 2 Ep with the following properties: 1.
    [Show full text]
  • Kähler Manifolds, Ricci Curvature, and Hyperkähler Metrics
    K¨ahlermanifolds, Ricci curvature, and hyperk¨ahler metrics Jeff A. Viaclovsky June 25-29, 2018 Contents 1 Lecture 1 3 1.1 The operators @ and @ ..........................4 1.2 Hermitian and K¨ahlermetrics . .5 2 Lecture 2 7 2.1 Complex tensor notation . .7 2.2 The musical isomorphisms . .8 2.3 Trace . 10 2.4 Determinant . 11 3 Lecture 3 11 3.1 Christoffel symbols of a K¨ahler metric . 11 3.2 Curvature of a Riemannian metric . 12 3.3 Curvature of a K¨ahlermetric . 14 3.4 The Ricci form . 15 4 Lecture 4 17 4.1 Line bundles and divisors . 17 4.2 Hermitian metrics on line bundles . 18 5 Lecture 5 21 5.1 Positivity of a line bundle . 21 5.2 The Laplacian on a K¨ahlermanifold . 22 5.3 Vanishing theorems . 25 6 Lecture 6 25 6.1 K¨ahlerclass and @@-Lemma . 25 6.2 Yau's Theorem . 27 6.3 The @ operator on holomorphic vector bundles . 29 1 7 Lecture 7 30 7.1 Holomorphic vector fields . 30 7.2 Serre duality . 32 8 Lecture 8 34 8.1 Kodaira vanishing theorem . 34 8.2 Complex projective space . 36 8.3 Line bundles on complex projective space . 37 8.4 Adjunction formula . 38 8.5 del Pezzo surfaces . 38 9 Lecture 9 40 9.1 Hirzebruch Signature Theorem . 40 9.2 Representations of U(2) . 42 9.3 Examples . 44 2 1 Lecture 1 We will assume a basic familiarity with complex manifolds, and only do a brief review today. Let M be a manifold of real dimension 2n, and an endomorphism J : TM ! TM satisfying J 2 = −Id.
    [Show full text]
  • HARMONIC MAPS Contents 1. Introduction 2 1.1. Notational
    HARMONIC MAPS ANDREW SANDERS Contents 1. Introduction 2 1.1. Notational conventions 2 2. Calculus on vector bundles 2 3. Basic properties of harmonic maps 7 3.1. First variation formula 7 References 10 1 2 ANDREW SANDERS 1. Introduction 1.1. Notational conventions. By a smooth manifold M we mean a second- countable Hausdorff topological space with a smooth maximal atlas. We denote the tangent bundle of M by TM and the cotangent bundle of M by T ∗M: 2. Calculus on vector bundles Given a pair of manifolds M; N and a smooth map f : M ! N; it is advantageous to consider the differential df : TM ! TN as a section df 2 Ω0(M; T ∗M ⊗ f ∗TN) ' Ω1(M; f ∗TN): There is a general for- malism for studying the calculus of differential forms with values in vector bundles equipped with a connection. This formalism allows a fairly efficient, and more coordinate-free, treatment of many calculations in the theory of harmonic maps. While this approach is somewhat abstract and obfuscates the analytic content of many expressions, it takes full advantage of the algebraic symmetries available and therefore simplifies many expressions. We will develop some of this theory now and use it freely throughout the text. The following exposition will closely fol- low [Xin96]. Let M be a smooth manifold and π : E ! M a real vector bundle on M or rank r: Throughout, we denote the space of smooth sections of E by Ω0(M; E): More generally, the space of differential p-forms with values in E is given by Ωp(M; E) := Ω0(M; ΛpT ∗M ⊗ E): Definition 2.1.
    [Show full text]
  • General Relativity Fall 2019 Lecture 11: the Riemann Tensor
    General Relativity Fall 2019 Lecture 11: The Riemann tensor Yacine Ali-Ha¨ımoud October 8th 2019 The Riemann tensor quantifies the curvature of spacetime, as we will see in this lecture and the next. RIEMANN TENSOR: BASIC PROPERTIES α γ Definition { Given any vector field V , r[αrβ]V is a tensor field. Let us compute its components in some coordinate system: σ σ λ σ σ λ r[µrν]V = @[µ(rν]V ) − Γ[µν]rλV + Γλ[µrν]V σ σ λ σ λ λ ρ = @[µ(@ν]V + Γν]λV ) + Γλ[µ @ν]V + Γν]ρV 1 = @ Γσ + Γσ Γρ V λ ≡ Rσ V λ; (1) [µ ν]λ ρ[µ ν]λ 2 λµν where all partial derivatives of V µ cancel out after antisymmetrization. σ Since the left-hand side is a tensor field and V is a vector field, we conclude that R λµν is a tensor field as well { this is the tensor division theorem, which I encourage you to think about on your own. You can also check that explicitly from the transformation law of Christoffel symbols. This is the Riemann tensor, which measures the non-commutation of second derivatives of vector fields { remember that second derivatives of scalar fields do commute, by assumption. It is completely determined by the metric, and is linear in its second derivatives. Expression in LICS { In a LICS the Christoffel symbols vanish but not their derivatives. Let us compute the latter: 1 1 @ Γσ = @ gσδ (@ g + @ g − @ g ) = ησδ (@ @ g + @ @ g − @ @ g ) ; (2) µ νλ 2 µ ν λδ λ νδ δ νλ 2 µ ν λδ µ λ νδ µ δ νλ since the first derivatives of the metric components (thus of its inverse as well) vanish in a LICS.
    [Show full text]
  • Solving the Geodesic Equation
    Solving the Geodesic Equation Jeremy Atkins December 12, 2018 Abstract We find the general form of the geodesic equation and discuss the closed form relation to find Christoffel symbols. We then show how to use metric independence to find Killing vector fields, which allow us to solve the geodesic equation when there are helpful symmetries. We also discuss a more general way to find Killing vector fields, and some of their properties as a Lie algebra. 1 The Variational Method We will exploit the following variational principle to characterize motion in general relativity: The world line of a free test particle between two timelike separated points extremizes the proper time between them. where a test particle is one that is not a significant source of spacetime cur- vature, and a free particles is one that is only under the influence of curved spacetime. Similarly to classical Lagrangian mechanics, we can use this to de- duce the equations of motion for a metric. The proper time along a timeline worldline between point A and point B for the metric gµν is given by Z B Z B µ ν 1=2 τAB = dτ = (−gµν (x)dx dx ) (1) A A using the Einstein summation notation, and µ, ν = 0; 1; 2; 3. We can parame- terize the four coordinates with the parameter σ where σ = 0 at A and σ = 1 at B. This gives us the following equation for the proper time: Z 1 dxµ dxν 1=2 τAB = dσ −gµν (x) (2) 0 dσ dσ We can treat the integrand as a Lagrangian, dxµ dxν 1=2 L = −gµν (x) (3) dσ dσ and it's clear that the world lines extremizing proper time are those that satisfy the Euler-Lagrange equation: @L d @L − = 0 (4) @xµ dσ @(dxµ/dσ) 1 These four equations together give the equation for the worldline extremizing the proper time.
    [Show full text]
  • 3. Introducing Riemannian Geometry
    3. Introducing Riemannian Geometry We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric. The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry.Infact,strictlyspeakingwewillneeda slightly di↵erent kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the di↵erences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons. 3.1 The Metric In Section 1, we informally introduced the metric as a way to measure distances between points. It does, indeed, provide this service but it is not its initial purpose. Instead, the metric is an inner product on each vector space Tp(M). Definition:Ametric g is a (0, 2) tensor field that is: Symmetric: g(X, Y )=g(Y,X). • Non-Degenerate: If, for any p M, g(X, Y ) =0forallY T (M)thenX =0. • 2 p 2 p p With a choice of coordinates, we can write the metric as g = g (x) dxµ dx⌫ µ⌫ ⌦ The object g is often written as a line element ds2 and this expression is abbreviated as 2 µ ⌫ ds = gµ⌫(x) dx dx This is the form that we saw previously in (1.4).
    [Show full text]
  • The Riemann Curvature Tensor
    The Riemann Curvature Tensor Jennifer Cox May 6, 2019 Project Advisor: Dr. Jonathan Walters Abstract A tensor is a mathematical object that has applications in areas including physics, psychology, and artificial intelligence. The Riemann curvature tensor is a tool used to describe the curvature of n-dimensional spaces such as Riemannian manifolds in the field of differential geometry. The Riemann tensor plays an important role in the theories of general relativity and gravity as well as the curvature of spacetime. This paper will provide an overview of tensors and tensor operations. In particular, properties of the Riemann tensor will be examined. Calculations of the Riemann tensor for several two and three dimensional surfaces such as that of the sphere and torus will be demonstrated. The relationship between the Riemann tensor for the 2-sphere and 3-sphere will be studied, and it will be shown that these tensors satisfy the general equation of the Riemann tensor for an n-dimensional sphere. The connection between the Gaussian curvature and the Riemann curvature tensor will also be shown using Gauss's Theorem Egregium. Keywords: tensor, tensors, Riemann tensor, Riemann curvature tensor, curvature 1 Introduction Coordinate systems are the basis of analytic geometry and are necessary to solve geomet- ric problems using algebraic methods. The introduction of coordinate systems allowed for the blending of algebraic and geometric methods that eventually led to the development of calculus. Reliance on coordinate systems, however, can result in a loss of geometric insight and an unnecessary increase in the complexity of relevant expressions. Tensor calculus is an effective framework that will avoid the cons of relying on coordinate systems.
    [Show full text]
  • Derivation of the Geodesic Equation and Defining the Christoffel Symbols
    Derivation of the Geodesic Equation and De¯ning the Christo®el Symbols Dr. Russell L. Herman March 13, 2008 We begin with the line element 2 ® ¯ ds = g®¯dx dx (1) where g®¯ is the metric with ®; ¯ = 0; 1; 2; 3. Also, we are using the Einstein summation convention in which we sum over repeated indices which occur as a subscript and superscript pair. In order to ¯nd the geodesic equation, we use the variational principle which states that freely falling test particles follow a path between two ¯xed points in spacetime which extremizes the proper time, ¿. The proper time is de¯ned by d¿ 2 = ¡ds2: (We are assuming that c = 1.) So, formally, we have Z B p Z B q 2 ® ¯ ¿AB = ¡ds = ¡g®¯dx dx : A A In order to write this as an integral that we can compute, we consider a parametrized worldline, x® = x®(σ); where the parameter σ = 0 at point A and σ = 1 at point B. Then, we write Z 1 · ® ¯ ¸1=2 Z 1 · ® ¸ dx dx dx ® ¿AB = ¡g®¯ dσ ´ L ; x dσ: (2) 0 dσ dσ 0 dσ £ ¤ dx® ® Here we have introduced the Lagrangian, L dσ ; x : We note also that d¿ L = : dσ Therefore, for functions f = f(¿(σ)), we have df df d¿ df = = L : dσ d¿ dσ d¿ We will use this later to change derivatives with respect to our arbitrary pa- rameter σ to derivatives with respect to the proper time, ¿: 1 Using variational methods as seen in classical dynamics, we obtain the Euler- Lagrange equations in the form µ ¶ d @L @L ¡ + = 0: (3) dσ @(dxγ /dσ) @xγ We carefully compute these derivatives for the general metric.
    [Show full text]
  • Cartan Connection and Curvature Forms
    CARTAN CONNECTION AND CURVATURE FORMS Syafiq Johar syafi[email protected] Contents 1 Connections and Riemannian Curvature 1 1.1 Affine Connection . .1 1.2 Levi-Civita Connection and Riemannian Curvature . .2 1.3 Riemannian Curvature . .3 2 Covariant External Derivative 4 2.1 Connection Form . .4 2.2 Curvature Form . .5 2.3 Explicit Formula for E = TM ..................................5 2.4 Levi-Civita Connection and Riemannian Curvature Revisited . .6 3 An Application: Uniformisation Theorem via PDEs 6 1 Connections and Riemannian Curvature Cartan formula is a way of generalising the Riemannian curvature for a Riemannian manifold (M; g) of dimension n to a differentiable manifold. We start by defining the Riemannian manifold and curvatures. We begin by defining the usual notion of connection and the Levi-Civita connection. 1.1 Affine Connection Definition 1.1 (Affine connection). Let M be a differentiable manifold and E a vector bundle over M. A connection or covariant derivative at a point p 2 M is a map D : Γ(E) ! Γ(T ∗M ⊗ E) with the 1 properties for any V; W 2 TpM; σ; τ 2 Γ(E) and f 2 C (M), we have that DV σ 2 Ep with the following properties: 1. D is tensorial over TpM i.e. DfV +W σ = fDV σ + DW σ: 2. D is R-linear in Γ(E) i.e. DV (σ + τ) = DV σ + DV τ: 3. D satisfies the Leibniz rule over Γ(E) i.e. if f 2 C1(M), we have DV (fσ) = df(V ) · σ + fDV σ: Note that we did not require any condition of the metric g on the manifold, nor the bundle metric on E in the abstract definition of a connection.
    [Show full text]
  • Arxiv:1412.2393V4 [Gr-Qc] 27 Feb 2019 2.6 Geodesics and Normal Coordinates
    Riemannian Geometry: Definitions, Pictures, and Results Adam Marsh February 27, 2019 Abstract A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. As much material as possible is developed for manifolds with connection (omitting a metric) to make clear which aspects can be readily generalized to gauge theories. The presentation in most cases does not assume a coordinate frame or zero torsion, and the coordinate-free, tensor, and Cartan formalisms are developed in parallel. Contents 1 Introduction 2 2 Parallel transport 3 2.1 The parallel transporter . 3 2.2 The covariant derivative . 4 2.3 The connection . 5 2.4 The covariant derivative in terms of the connection . 6 2.5 The parallel transporter in terms of the connection . 9 arXiv:1412.2393v4 [gr-qc] 27 Feb 2019 2.6 Geodesics and normal coordinates . 9 2.7 Summary . 10 3 Manifolds with connection 11 3.1 The covariant derivative on the tensor algebra . 12 3.2 The exterior covariant derivative of vector-valued forms . 13 3.3 The exterior covariant derivative of algebra-valued forms . 15 3.4 Torsion . 16 3.5 Curvature .
    [Show full text]
  • Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan
    Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan Structures and Some Applications to Physics∗ Waldyr Alves Rodrigues Jr. Institute of Mathematics Statistics and Scientific Computation IMECC-UNICAMP CP 6065 13083760 Campinas SP Brazil e-mail: [email protected] or [email protected] May 28, 2018 Abstract In this paper after recalling some essential tools concerning the theory of differential forms in the Cartan, Hodge and Clifford bundles over a Riemannian or Riemann-Cartan space or a Lorentzian or Riemann-Cartan spacetime we solve with details several exercises involving different grades of difficult. One of the problems is to show that a recent formula given in [10] for the exterior covariant derivative of the Hodge dual of the torsion 2-forms is simply wrong. We believe that the paper will be useful for students (and eventually for some experts) on applications of differential geometry to some physical problems. A detailed account of the issues discussed in the paper appears in the table of contents. Contents 1 Introduction 3 arXiv:0712.3067v6 [math-ph] 4 Dec 2008 2 Classification of Metric Compatible Structures (M; g;D) 4 2.1 Levi-Civita and Riemann-Cartan Connections . 5 2.2 Spacetime Structures . 5 3 Absolute Differential and Covariant Derivatives 5 ∗published in: Ann. Fond. L. de Broglie 32 (special issue dedicate to torsion), 424-478 (2008). 1 ^ 4 Calculus on the Hodge Bundle ( T ∗M; ·; τg) 8 4.1 Exterior Product . 8 4.2 Scalar Product and Contractions . 9 4.3 Hodge Star Operator ? ........................ 9 4.4 Exterior derivative d and Hodge coderivative δ .
    [Show full text]