Manifolds and tensors

2019 ◽  
pp. 25-36
Author(s):  
Steven Carlip
Keyword(s):  
Differential Geometry ◽  
General Relativity ◽  
Differential Forms ◽  
General Covariance ◽  
Mathematical Basis ◽  
Metric Tensor ◽  
Diffeomorphism Invariance ◽  
Starting Point ◽  
Tangent Vectors

The mathematical basis of general relativity is differential geometry. This chapter establishes the starting point of differential geometry: manifolds, tangent vectors, cotangent vectors, tensors, and differential forms. The metric tensor is introduced, and its symmetries (isometries) are described. The importance of diffeomorphism invariance (or “general covariance”) is stressed.

Differential geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Differential Geometry ◽  
Vector Fields ◽  
Differential Forms ◽  
Curvilinear Coordinates ◽  
Moving Frames ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Vector Calculus ◽  
Vector Version ◽  
Definition Of

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.

scholarly journals Gravity as entanglement, entanglement as gravity

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Reference Frame ◽  
Common Sense ◽  
Reference Frames ◽  
General Covariance ◽  
The Universe

A generalized and unifying viewpoint to both general relativity and quantum mechanics and information is investigated. It may be described as a generaliztion of the concept of reference frame from mechanics to thermodynamics, or from a reference frame linked to an element of a system, and thus, within it, to another reference frame linked to the whole of the system or to any of other similar systems, and thus, out of it. Furthermore, the former is the viewpoint of general relativity, the latter is that of quantum mechanics and information.Ciclicity in the manner of Nicolas Cusanus (Nicolas of Cusa) is complemented as a fundamental and definitive property of any totality, e.g. physically, that of the universe. It has to contain its externality within it somehow being namely the totality. This implies a seemingly paradoxical (in fact, only to common sense rather logically and mathematically) viewpoint for the universe to be repesented within it as each one quant of action according to the fundamental Planck constant.That approach implies the unification of gravity and entanglement correspondiing to the former or latter class of reference frames. An invariance, more general than Einstein's general covariance is to be involved as to both classes of reference frames unifying them. Its essence is the unification of the discrete and cotnitinuous (smooth). That idea underlies implicitly quantum mechanics for Bohr's principle that it study the system of quantum microscopic entities and the macroscopic apparatus desribed uniformly by the smmoth equations of classical physics.e

On the existence of metric differential geometries based on the notion of area

1950 ◽  
Vol 46 (1) ◽  
pp. 67-72 ◽  
Author(s):  
F. Brickell
Keyword(s):  
Differential Geometry ◽  
Fourth Order ◽  
Homogeneous Function ◽  
Two Dimensional ◽  
Fundamental Function ◽  
Metric Tensor ◽  
Partial Derivatives ◽  
Dimensional Plane ◽  
Plane Element

The problem of constructing an n-dimensional metric differential geometry based on the idea of a two-dimensional area has given rise to several publications, notably by A. Kawaguchi and S. Hokari (1), E. T. Davies (2), and R. Debever (3). In this geometry the area of a two-dimensional plane element is defined by a fundamental function L(xi, uhk), where the xi are point coordinates and the uhk are the coordinates of the simple bivector representing the plane element. L is supposed to be a positive homogeneous function of the first degree with respect to the variables uij, and to possess continuous partial derivatives up to and including those of the fourth order. With these assumptions the problem of the construction of the metric differential geometry splits into two problems; the first of these is the problem of constructing a metric tensor gij(xr, uhk), and the second is the problem of constructing an affine connexion. We deal with the first problem only in this paper.

scholarly journals ON THE HAMILTONIAN FORMULATION OF THE EINSTEIN–HILBERT ACTION IN TWO DIMENSIONS

2006 ◽  
Vol 21 (11) ◽  
pp. 899-905 ◽  
Author(s):  
N. KIRIUSHCHEVA ◽  
S. V. KUZMIN
Keyword(s):  
Hamiltonian Formulation ◽  
Two Dimensions ◽  
General Covariance ◽  
Sufficient Condition ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Metric Tensor ◽  
Gauge Transformations ◽  
Total Divergence ◽  
Hilbert Action

It is shown that if general covariance is to be preserved (i.e. a coordinate system is not fixed) the well-known triviality of the Einstein field equations in two dimensions is not a sufficient condition for the Einstein–Hilbert action to be a total divergence. Consequently, a Hamiltonian formulation is possible without any modification of the two-dimensional Einstein–Hilbert action. We find the resulting constraints and the corresponding gauge transformations of the metric tensor.

Differential forms on a Kähler manifold

1951 ◽  
Vol 47 (3) ◽  
pp. 504-517 ◽  
Author(s):  
W. V. D. Hodge
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Complex Manifold ◽  
Differential Forms ◽  
Kähler Manifold ◽  
Systematic Account ◽  
General Plan ◽  
Kahler Manifold ◽  
Frequent Use ◽  
Compact Kähler Manifold

While a number of special properties of differential forms on a Kähler manifold have been mentioned in the literature on complex manifolds, no systematic account has yet been given of the theory of differential forms on a compact Kähler manifold. The purpose of this paper is to show how a general theory of these forms can be developed. It follows the general plan of de Rham's paper (2) on differential forms on real manifolds, and frequent use will be made of results contained in that paper. For convenience we begin by giving a brief account of the theory of complex tensors on a complex manifold, and of the differential geometry associated with a Hermitian, and in particular a Kählerian, metric on such a manifold.

scholarly journals Energy–Momentum Pseudotensor and Superpotential for Generally Covariant Theories of Gravity of General Form

Universe ◽  
2020 ◽  
Vol 6 (10) ◽  
pp. 173
Author(s):  
Roman Ilin ◽  
Sergey Paston
Keyword(s):  
General Relativity ◽  
Conservation Laws ◽  
Wide Class ◽  
Equations Of Motion ◽  
General Covariance ◽  
Independent Variables ◽  
Theories Of Gravity ◽  
Generally Covariant ◽  
Derivatives Of ◽  
Mimetic Gravity

The current paper is devoted to the investigation of the general form of the energy–momentum pseudotensor (pEMT) and the corresponding superpotential for the wide class of theories. The only requirement for such a theory is the general covariance of the action without any restrictions on the order of derivatives of the independent variables in it or their transformation laws. As a result of the generalized Noether procedure, we obtain a recurrent chain of the equations, which allows one to express canonical pEMT as a divergence of the superpotential. The explicit expression for this superpotential is also given. We discuss the structure of the obtained expressions and the conditions for the derived pEMT conservation laws to be satisfied independently (fully or partially) by the equations of motion. Deformations of the superpotential form for theories with a change in the independent variables in action are also considered. We apply these results to some interesting particular cases: general relativity and its modifications, particularly mimetic gravity and Regge–Teitelboim embedding gravity.

Sign in / Sign up

Export Citation Format

Share Document