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Covariant derivatives and metrics

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

This chapter examines relations between covariant derivatives and metrics. It covers metric compatible covariant derivatives; torsion free covariant derivatives on T*M; the Levi-Civita ... More

This chapter examines relations between covariant derivatives and metrics. It covers metric compatible covariant derivatives; torsion free covariant derivatives on T*M; the Levi-Civita connection/covariant derivative; a formula for the Levi-Civita connection; covariantly constant sections; an example of the Levi-Civita connection; and the curvature of the Levi-Civita connection.Less

Riemannian Geometry

Valeri P. Frolov and Andrei Zelnikov

in Introduction to Black Hole Physics

The key notions of the differential and Riemannian geometry necessary for understanding the General Relativity are introduced here. We provide the reader with the necessary tools for study the ... More

The key notions of the differential and Riemannian geometry necessary for understanding the General Relativity are introduced here. We provide the reader with the necessary tools for study the properties of black holesand their interaction with matter and fields. We tried to omit unnecessary details and to make the presentation simpler. We focus on an explanation of the basic concepts and explicit formulas, which can be used for concrete calculations, and give hints, which might be used to simplify these calculations.Less

Tensor Formalism for General Relativity

Ta-Pei Cheng

in A College Course on Relativity and Cosmology

This chapter introduces the basic tensor formalism needed for a proper formulation of general relativity. In a curved space, one must work with the covariant derivative, which is a combination of the ... More

This chapter introduces the basic tensor formalism needed for a proper formulation of general relativity. In a curved space, one must work with the covariant derivative, which is a combination of the ordinary derivative and the first derivatives of the metric (Christoffel symbols). The covariant derivative of a tensor is itself a tensor. The related concept of parallel transport is introduced, and the Riemann curvature tensor is derived by parallel transport of a vector around a closed path. The equation of geodesic deviation is presented as another method to arrive at the curvature tensor. The symmetries and contraction properties of the Riemann curvature (including the Bianchi identity) are considered in order to find the desired tensors for the Einstein equation. The approach to this general relativity field equation via the principle of least action is also sketched. The relevant mathematics of Schwarzschild solution and the cosmological constant are outlined.Less

Covariant derivatives, connections and curvature

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

This chapter examines the notion of the curvature of a covariant derivative or connection. It begins by describing two notions involving differentiation of differential forms and vector fields that ... More

This chapter examines the notion of the curvature of a covariant derivative or connection. It begins by describing two notions involving differentiation of differential forms and vector fields that require no auxiliary choices. These are used to define curvature when covariant derivatives reappear in the story. It then explains the notion of curvature and gives an example. It also discusses curvature and commutators; connections and curvature; and the horizontal subbundle.Less

Curved Space

Nicholas Manton and Nicholas Mee

in The Physical World: An Inspirational Tour of Fundamental Physics

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that ... More

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that describes general curved spaces of arbitrary dimension. The chapter begins with a comparison of Euclidean geometry and spherical geometry. The concept of the geodesic is introduced. The discovery of hyperbolic geometry is discussed. Gaussian curvature is defined. Tensors are introduced. The metric tensor is defined and simple examples are given. This leads to the use of covariant derivatives, expressed in terms of Christoffel symbols, the Riemann curvature tensor and all machinery of Riemannian geometry, with each step illustrated by simple examples. The geodesic equation and the equation of geodesic deviation are derived. The final section considers some applications of curved geometry: configuration space, mirages and fisheye lenses.Less

The covariant derivative and the curvature

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

This chapter first considers the tangent spaces of a non-connected manifold, in which the tangent t at the set of points p in the manifold is an element of the tangent space at p. Afterward, the ... More

This chapter first considers the tangent spaces of a non-connected manifold, in which the tangent t at the set of points p in the manifold is an element of the tangent space at p. Afterward, the chapter summarizes the elementary introduction to the exterior calculus of Chapter 5 of Book 2. Next, the chapter studies the Lie bracket and Lie derivative, before moving on to the covariant derivative and a connected manifold. The covariant derivative in particular is introduced to ensure the effectiveness of the Lie brackets and the Lie derivative. From here, this chapter considers the torsion of a covariant derivative and finally to the curvature of a covariant derivative.Less

Some elements of general relativity

Rodolfo Gambini and Jorge Pullin

in A First Course in Loop Quantum Gravity

This chapter introduces the idea that the gravitational field is described as a deformation of space-time. It briefly discusses general coordinates, and the definitions of vectors and tensors ... More

This chapter introduces the idea that the gravitational field is described as a deformation of space-time. It briefly discusses general coordinates, and the definitions of vectors and tensors suitable in such context. The chapter explains the ideas of covariant derivative and curvature, and briefly discusses the Riemann curvature tensor and the Ricci and Einstein tensors. It presents the Einstein equations and some of their solutions. The chapter also covers diffeomorphism, the Lie derivative, the 3+1 decomposition of space-time, and the concepts of triads and tensor densities.Less

Derivatives and curvature

Steven Carlip

in General Relativity: A Concise Introduction

This chapter develops tensor calculus: integration on manifolds, Cartan calculus for differential forms, connections and covariant derivatives, and the Levi-Civita connection used in general ... More

This chapter develops tensor calculus: integration on manifolds, Cartan calculus for differential forms, connections and covariant derivatives, and the Levi-Civita connection used in general relativity. It then introduces the Riemann curvature tensor in several different ways, including the most directly physical picture of the curvature as a measure of the convergence of neighboring geodesics. The chapter concludes with a discussion of Cartan’s beautiful formulation of the connection and curvature in the language of differential forms.Less

Curvilinear coordinates

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

This chapter presents a discussion on curvilinear coordinates in line with the introduction on Cartesian coordinates in Chapter 1. First, the chapter introduces a new system C of curvilinear ... More

This chapter presents a discussion on curvilinear coordinates in line with the introduction on Cartesian coordinates in Chapter 1. First, the chapter introduces a new system C of curvilinear coordinates xⁱ = xⁱ(Xj) (also sometimes referred to as Gaussian coordinates), which are nonlinearly related to Cartesian coordinates. It then introduces the components of the covariant derivative, before considering parallel transport in a system of curvilinear coordinates. Next, the chapter shows how connection coefficients of the covariant derivative as well as the Euclidean metric can be related to each other. Finally, this chapter turns to the kinematics of a point particle as well as the divergence and Laplacian of a vector and the Levi-Civita symbol and the volume element.Less

Spacetime symmetries

Michael Kachelriess

in Quantum Fields: From the Hubble to the Planck Scale

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the ... More

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.Less

Differential geometry

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant ... More

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.Less

Two-Dimensional Models and Bosonization Method

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Chapter 31 discussed the generic O(N) non-linear σ-model. We have noticed that the abelian case N = 2 is special because the RG β-function vanishes in two dimensions. The corresponding O(2) invariant ... More

Chapter 31 discussed the generic O(N) non-linear σ-model. We have noticed that the abelian case N = 2 is special because the RG β-function vanishes in two dimensions. The corresponding O(2) invariant spin model is specially interesting: it provides an example of the celebrated Kosterlitz–Thouless phase transition and will be examined in Chapter 33. However, a thorough discussion of this model requires a new technique: we have to establish relations, special to two dimensions, between fermion and boson local field theories, by a method called bosonization. This chapter illustrates the derivation involves several steps with the help of various other two-dimensional models which are physically interesting in their own right. It first studies the free massless boson and fermion fields. It evaluates the determinant of the covariant fermion derivative in the presence of an external gauge field. The result is at the basis of the bosonization technique. The chapter then discusses the sine–Gordon (SG) model. It solves the Schwinger model, QED with massless fermions. Finally, it demonstrates the equivalence between the SG model and two fermion models with current-current interaction: the Thirring model and another model with two species of fermions.Less

Riemannian manifolds

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

This chapter is about Riemannian manifolds. It first discusses the metric manifold and the Levi-Civita connection, determining if the metric is Riemannian or Lorentzian. Next, the chapter turns to ... More

This chapter is about Riemannian manifolds. It first discusses the metric manifold and the Levi-Civita connection, determining if the metric is Riemannian or Lorentzian. Next, the chapter turns to the properties of the curvature tensor. It states without proof the intrinsic versions of the properties of the Riemann–Christoffel tensor of a covariant derivative already given in Chapter 2. This chapter then performs the same derivation as in Chapter 4 by obtaining the Einstein equations of general relativity by varying the Hilbert action. However, this will be done in the intrinsic manner, using the tools developed in the present and the preceding chapters.Less

ε‎-Invariance

Ercüment H. Ortaçgil

in An Alternative Approach to Lie Groups and Geometric Structures

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.