Integration in General Relativity
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The Levi-Civita Tensor Noncovariance and Curvature in the Pseudotensors
The Levi-Civita tensor noncovariance and curvature in the pseudotensors space A. L. Koshkarov∗ University of Petrozavodsk, Physics department, Russia Abstract It is shown that conventional ”covariant” derivative of the Levi-Civita tensor Eαβµν;ξ is not really covariant. Adding compensative terms, it is possible to make it covariant and to be equal to zero. Then one can be introduced a curvature in the pseudotensors space. There appears a curvature tensor which is dissimilar to ordinary one by covariant term including the Levi-Civita density derivatives hence to be equal to zero. This term is a little bit similar to Weylean one in the Weyl curvature tensor. There has been attempted to find a curvature measure in the combined (tensor plus pseudotensor) tensors space. Besides, there has been constructed some vector from the metric and the Levi-Civita density which gives new opportunities in geometry. 1 Introduction Tensor analysis which General Relativity is based on operates basically with true tensors and it is very little spoken of pseudotensors role. Although there are many objects in the world to be bound up with pseudotensors. For example most of particles and bodies in the Universe have angular momentum or spin. Here is a question: could a curvature tensor be a pseudotensor or include that in some way? It’s not clear. Anyway, symmetries of the Riemann-Christopher tensor are not compatible with those of the Levi-Civita pseudotensor. There are examples of using values in Physics to be a sum of a tensor and arXiv:0906.0647v1 [physics.gen-ph] 3 Jun 2009 a pseudotensor. -
Arxiv:0911.0334V2 [Gr-Qc] 4 Jul 2020
Classical Physics: Spacetime and Fields Nikodem Poplawski Department of Mathematics and Physics, University of New Haven, CT, USA Preface We present a self-contained introduction to the classical theory of spacetime and fields. This expo- sition is based on the most general principles: the principle of general covariance (relativity) and the principle of least action. The order of the exposition is: 1. Spacetime (principle of general covariance and tensors, affine connection, curvature, metric, tetrad and spin connection, Lorentz group, spinors); 2. Fields (principle of least action, action for gravitational field, matter, symmetries and conservation laws, gravitational field equations, spinor fields, electromagnetic field, action for particles). In this order, a particle is a special case of a field existing in spacetime, and classical mechanics can be derived from field theory. I dedicate this book to my Parents: Bo_zennaPop lawska and Janusz Pop lawski. I am also grateful to Chris Cox for inspiring this book. The Laws of Physics are simple, beautiful, and universal. arXiv:0911.0334v2 [gr-qc] 4 Jul 2020 1 Contents 1 Spacetime 5 1.1 Principle of general covariance and tensors . 5 1.1.1 Vectors . 5 1.1.2 Tensors . 6 1.1.3 Densities . 7 1.1.4 Contraction . 7 1.1.5 Kronecker and Levi-Civita symbols . 8 1.1.6 Dual densities . 8 1.1.7 Covariant integrals . 9 1.1.8 Antisymmetric derivatives . 9 1.2 Affine connection . 10 1.2.1 Covariant differentiation of tensors . 10 1.2.2 Parallel transport . 11 1.2.3 Torsion tensor . 11 1.2.4 Covariant differentiation of densities . -
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. -
The Language of Differential Forms
Appendix A The Language of Differential Forms This appendix—with the only exception of Sect.A.4.2—does not contain any new physical notions with respect to the previous chapters, but has the purpose of deriving and rewriting some of the previous results using a different language: the language of the so-called differential (or exterior) forms. Thanks to this language we can rewrite all equations in a more compact form, where all tensor indices referred to the diffeomorphisms of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations). The matter of this appendix is not intended to provide a complete nor a rigorous introduction to this formalism: it should be regarded only as a first, intuitive and oper- ational approach to the calculus of differential forms (also called exterior calculus, or “Cartan calculus”). The main purpose is to quickly put the reader in the position of understanding, and also independently performing, various computations typical of a geometric model of gravity. The readers interested in a more rigorous discussion of differential forms are referred, for instance, to the book [22] of the bibliography. Let us finally notice that in this appendix we will follow the conventions introduced in Chap. 12, Sect. 12.1: latin letters a, b, c,...will denote Lorentz indices in the flat tangent space, Greek letters μ, ν, α,... tensor indices in the curved manifold. For the matter fields we will always use natural units = c = 1. Also, unless otherwise stated, in the first three Sects. -
On Certain Identities Involving Basic Spinors and Curvature Spinors
ON CERTAIN IDENTITIES INVOLVING BASIC SPINORS AND CURVATURE SPINORS H. A. BUCHDAHL (received 22 May 1961) 1. Introduction The spinor analysis of Infeld and van der Waerden [1] is particularly well suited to the transcription of given flat space wave equations into forms which'constitute possible generalizations appropriate to Riemann spaces [2]. The basic elements of this calculus are (i) the skew-symmetric spinor y^, (ii) the hermitian tensor-spinor a*** (generalized Pauli matrices), and (iii) M the curvature spinor P ykl. When one deals with wave equations in Riemann spaces F4 one is apt to be confronted with expressions of somewhat be- wildering appearance in so far as they may involve products of a large number of <r-symbols many of the indices of which may be paired in all sorts of ways either with each other or with the indices of the components of the curvature spinors. Such expressions are generally capable of great simplifi- cation, but how the latter may be achieved is often far from obvious. It is the purpose of this paper to present a number of useful relations between basic tensors and spinors, commonly known relations being taken for gran- ted [3], [4], [5]. That some of these new relations appear as more or less trivial consequences of elementary identities is largely the result of a diligent search for their simplest derivation, once they had been obtained in more roundabout ways. Certain relations take a particularly simple form when the Vt is an Einstein space, and some necessary and sufficient conditions relating to Ein- stein spaces and to spaces of constant Riemannian curvature are considered in the last section. -
FROM DIFFERENTIATION in AFFINE SPACES to CONNECTIONS Jovana -Duretic 1. Introduction Definition 1. We Say That a Real Valued
THE TEACHING OF MATHEMATICS 2015, Vol. XVIII, 2, pp. 61–80 FROM DIFFERENTIATION IN AFFINE SPACES TO CONNECTIONS Jovana Dureti´c- Abstract. Connections and covariant derivatives are usually taught as a basic concept of differential geometry, or more precisely, of differential calculus on smooth manifolds. In this article we show that the need for covariant derivatives may arise, or at lest be motivated, even in a linear situation. We show how a generalization of the notion of a derivative of a function to a derivative of a map between affine spaces naturally leads to the notion of a connection. Covariant derivative is defined in the framework of vector bundles and connections in a way which preserves standard properties of derivatives. A special attention is paid on the role played by zero–sets of a first derivative in several contexts. MathEduc Subject Classification: I 95, G 95 MSC Subject Classification: 97 I 99, 97 G 99, 53–01 Key words and phrases: Affine space; second derivative; connection; vector bun- dle. 1. Introduction Definition 1. We say that a real valued function f :(a; b) ! R is differen- tiable at a point x0 2 (a; b) ½ R if a limit f(x) ¡ f(x ) lim 0 x!x0 x ¡ x0 0 exists. We denote this limit by f (x0) and call it a derivative of a function f at a point x0. We can write this limit in a different form, as 0 f(x0 + h) ¡ f(x0) (1) f (x0) = lim : h!0 h This expression makes sense if the codomain of a function is Rn, or more general, if the codomain is a normed vector space. -
Math 396. Covariant Derivative, Parallel Transport, and General Relativity
Math 396. Covariant derivative, parallel transport, and General Relativity 1. Motivation Let M be a smooth manifold with corners, and let (E, ∇) be a C∞ vector bundle with connection over M. Let γ : I → M be a smooth map from a nontrivial interval to M (a “path” in M); keep in mind that γ may not be injective and that its velocity may be zero at a rather arbitrary closed subset of I (so we cannot necessarily extend the standard coordinate on I near each t0 ∈ I to part of a local coordinate system on M near γ(t0)). In pseudo-Riemannian geometry E = TM and ∇ is a specific connection arising from the metric tensor (the Levi-Civita connection; see §4). A very fundamental concept is that of a (smooth) section along γ for a vector bundle on M. Before we give the official definition, we consider an example. Example 1.1. To each t0 ∈ I there is associated a velocity vector 0 ∗ γ (t0) = dγ(t0)(∂t|t0 ) ∈ Tγ(t0)(M) = (γ (TM))(t0). Hence, we get a set-theoretic section of the pullback bundle γ∗(TM) → I by assigning to each time 0 t0 the velocity vector γ (t0) at that time. This is not just a set-theoretic section, but a smooth section. Indeed, this problem is local, so pick t0 ∈ I and an open U ⊆ M containing γ(J) for an open ∞ neighborhood J ⊆ I around t0, with J and U so small that U admits a C coordinate system {x1, . , xn}. Let γi = xi ◦ γ|J ; these are smooth functions on J since γ is a smooth map from I into M. -
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). -
Weyl's Spin Connection
THE SPIN CONNECTION IN WEYL SPACE c William O. Straub, PhD Pasadena, California “The use of general connections means asking for trouble.” —Abraham Pais In addition to his seminal 1929 exposition on quantum mechanical gauge invariance1, Hermann Weyl demonstrated how the concept of a spinor (essentially a flat-space two-component quantity with non-tensor- like transformation properties) could be carried over to the curved space of general relativity. Prior to Weyl’s paper, spinors were recognized primarily as mathematical objects that transformed in the space of SU (2), but in 1928 Dirac showed that spinors were fundamental to the quantum mechanical description of spin—1/2 particles (electrons). However, the spacetime stage that Dirac’s spinors operated in was still Lorentzian. Because spinors are neither scalars nor vectors, at that time it was unclear how spinors behaved in curved spaces. Weyl’s paper provided a means for this description using tetrads (vierbeins) as the necessary link between Lorentzian space and curved Riemannian space. Weyl’selucidation of spinor behavior in curved space and his development of the so-called spin connection a ab ! band the associated spin vector ! = !ab was noteworthy, but his primary purpose was to demonstrate the profound connection between quantum mechanical gauge invariance and the electromagnetic field. Weyl’s 1929 paper served to complete his earlier (1918) theory2 in which Weyl attempted to derive electrodynamics from the geometrical structure of a generalized Riemannian manifold via a scale-invariant transformation of the metric tensor. This attempt failed, but the manifold he discovered (known as Weyl space), is still a subject of interest in theoretical physics. -
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. -
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 . -
Differential Geometry Lecture 18: Curvature
Differential geometry Lecture 18: Curvature David Lindemann University of Hamburg Department of Mathematics Analysis and Differential Geometry & RTG 1670 14. July 2020 David Lindemann DG lecture 18 14. July 2020 1 / 31 1 Riemann curvature tensor 2 Sectional curvature 3 Ricci curvature 4 Scalar curvature David Lindemann DG lecture 18 14. July 2020 2 / 31 Recap of lecture 17: defined geodesics in pseudo-Riemannian manifolds as curves with parallel velocity viewed geodesics as projections of integral curves of a vector field G 2 X(TM) with local flow called geodesic flow obtained uniqueness and existence properties of geodesics constructed the exponential map exp : V ! M, V neighbourhood of the zero-section in TM ! M showed that geodesics with compact domain are precisely the critical points of the energy functional used the exponential map to construct Riemannian normal coordinates, studied local forms of the metric and the Christoffel symbols in such coordinates discussed the Hopf-Rinow Theorem erratum: codomain of (x; v) as local integral curve of G is d'(TU), not TM David Lindemann DG lecture 18 14. July 2020 3 / 31 Riemann curvature tensor Intuitively, a meaningful definition of the term \curvature" for 3 a smooth surface in R , written locally as a graph of a smooth 2 function f : U ⊂ R ! R, should involve the second partial derivatives of f at each point. How can we find a coordinate- 3 free definition of curvature not just for surfaces in R , which are automatically Riemannian manifolds by restricting h·; ·i, but for all pseudo-Riemannian manifolds? Definition Let (M; g) be a pseudo-Riemannian manifold with Levi-Civita connection r.