Differential Forms

2021 ◽  
pp. 326-357
Author(s):  
Moataz H. Emam
Keyword(s):  
General Relativity ◽  
Differential Forms ◽  
Modern Theory ◽  
Maxwell Fields ◽  
Classical Fields

In this chapter we present the modern theory of differential forms and see how it applies to the classical fields studied in the previous chapter. We apply the theory to Maxwell fields as well as to Cartan’s formulation of general relativity. A discussion of the generalized Stokes theorem is given.

Least Action and Classical Fields

2021 ◽  
pp. 286-325
Author(s):  
Moataz H. Emam
Keyword(s):  
General Relativity ◽  
Momentum Tensor ◽  
Point Particle ◽  
Kerr Metric ◽  
Field Equations ◽  
Tensor Fields ◽  
Particle Mechanics ◽  
Least Action ◽  
Principle Of Least Action ◽  
Maxwell Fields

We present the principle of least action and see how it is used in non-relativistic point particle mechanics, relativistic point particle mechanics, general relativity, derivation of field equations for scalar, vector and tensor fields as well as the energy momentum tensor. Towards the end we present examples of solutions of Einstein-Maxwell fields: The Reissner-Nordstrom metric, Kerr metric, and Kerr- Newman metric.

scholarly journals DIFFERENTIAL FORMS AND WAVE EQUATIONS FOR GENERAL RELATIVITY

1998 ◽  
Vol 07 (06) ◽  
pp. 857-885 ◽  
Author(s):  
STEPHEN R. LAU
Keyword(s):  
General Relativity ◽  
Wave Equation ◽  
Degrees Of Freedom ◽  
Wave Equations ◽  
Evolution Equations ◽  
Differential Forms ◽  
Extrinsic Curvature ◽  
The Cauchy Problem ◽  
Hyperbolic Form ◽  
Tensor Index

In recent papers, Choquet–Bruhat and York and Abrahams, Anderson, Choquet–Bruhat, and York (we refer to both works jointly as AACY) have cast the 3 + 1 evolution equations of general relativity in gauge-covariant and causal "first-order symmetric hyperbolic form," thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's two-form, which in the "time-gauge" is built linearly from the "extrinsic curvature one-form." The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt–Deser–Misner gravitational momentum.

Primordial quantum perturbations

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Scalar Field ◽  
Energy Momentum Tensor ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
Slow Roll ◽  
Inflationary Models

This chapter shows that fluctuations of quantum origin are generated during inflation and that this process supplies initial conditions compatible with the observations. These fluctuations are therefore an important prediction of inflationary models. The chapter thus begins with a study of perturbations during inflation, proceeding in a similar manner to the previous chapter by finding the perturbation of the energy–momentum tensor of the scalar field. Another method of deriving the equations of motion of the perturbations is to start from the action of general relativity coupled to a scalar field, and expand to second order in the metric and scalar field perturbations. The chapter then proceeds with the determination of the initial conditions and the slow-roll inflation.

scholarly journals Manifest Covariant Hamiltonian Theory of General Relativity

2016 ◽  
Vol 8 (2) ◽  
pp. 60 ◽  
Author(s):  
Claudio Cremaschini ◽  
Massimo Tessarotto
Keyword(s):  
General Relativity ◽  
Einstein Equation ◽  
First Principles ◽  
Hamiltonian Structure ◽  
Invariance Property ◽  
Curved Space ◽  
Covariant Approach ◽  
Hamiltonian Theory ◽  
Classical Fields ◽  
The Continuum

The problem of formulating a manifest covariant Hamiltonian theory of General Relativity in the presence of source fields is addressed, by extending the so-called “DeDonder-Weyl” formalism to the treatment of classical fields in curved space-time. The theory is based on a synchronous variational principle for the Einstein equation, formulated in terms of superabundant variables. The technique permits one to determine the continuum covariant Hamiltonian structure associated with the Einstein equation. The corresponding continuum Poisson bracket representation is also determined. The theory relies on first-principles, in the sense that the conclusions are reached in the framework of a non-perturbative covariant approach, which allows one to preserve both the 4-scalar nature of Lagrangian and Hamiltonian densities as well as the gauge invariance property of the theory.

scholarly journals Hamiltonian of new general relativity using differential forms

2020 ◽  
Vol 35 (02n03) ◽  
pp. 2040014
Author(s):  
Manuel Hohmann
Keyword(s):  
General Relativity ◽  
Recent Work ◽  
Differential Forms ◽  
Teleparallel Gravity ◽  
Alternative Derivation ◽  
Alternative Approach ◽  
Gravity Theories

In a recent work we derived the kinematic Hamiltonian and primary constraints of the new general relativity class of teleparallel gravity theories and showed that these theories can be grouped in 9 classes, based on the presence or absence of primary constraints in their Hamiltonian. Here we demonstrate an alternative approach towards this result, by using differential forms instead of tensor components throughout the calculation. We prove that also this alternative derivation yields the same results and show how they are related to each other.

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.

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