The Maxwell equations

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Electromagnetic Field ◽  
Angular Momentum ◽  
Variational Principle ◽  
Maxwell Equations ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Field Energy ◽  
Total Charge ◽  
Faraday’S Law ◽  
System Energy

This chapter presents Maxwell equations determining the electromagnetic field created by an ensemble of charges. It also derives these equations from the variational principle. The chapter studies the equation’s invariances: gauge invariance and invariance under Poincaré transformations. These allow us to derive the conservation laws for the total charge of the system and also for the system energy, momentum, and angular momentum. To begin, the chapter introduces the first group of Maxwell equations: Gauss’s law of magnetism, and Faraday’s law of induction. It then discusses current and charge conservation, a second set of Maxwell equations, and finally the field–energy momentum tensor.

A note on spinor form of Lovelock differential identity

2019 ◽  
Vol 16 (09) ◽  
pp. 1950145
Author(s):  
Vladimir N. Trishin
Keyword(s):  
General Relativity ◽  
Electromagnetic Field ◽  
Maxwell Equations ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Differential Identity ◽  
Spinor Form ◽  
Order Identity

The paper is devoted to 2-spinor calculus methods in general relativity. New spinor form of the Lovelock differential identity is suggested. This identity is second-order identity for the Riemann curvature tensor. We provide an example that our spinorial treatment of Lovelock identity is effective for the description of solutions of Einstein–Maxwell equations. It is shown that the covariant divergence of Lipkin’s zilch tensor for the free Maxwell field vanishes on the solutions of Einstein–Maxwell equations if and only if the energy–momentum tensor of the electromagnetic field is Weyl-compatible.

scholarly journals Gravitational energy in the framework of embedding and splitting theories

2018 ◽  
Vol 27 (02) ◽  
pp. 1750188 ◽  
Author(s):  
D. A. Grad ◽  
R. V. Ilin ◽  
S. A. Paston ◽  
A. A. Sheykin
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Gravitational Energy ◽  
Einstein Equations ◽  
Field Energy ◽  
Energy Localization ◽  
Embedding Theory ◽  
Isometric Embeddings ◽  
Definition Of ◽  
Noether Current

We study various definitions of the gravitational field energy based on the usage of isometric embeddings in the Regge–Teitelboim approach. For the embedding theory, we consider the coordinate translations on the surface as well as the coordinate translations in the flat bulk. In the latter case, the independent definition of gravitational energy–momentum tensor appears as a Noether current corresponding to global inner symmetry. In the field-theoretic form of this approach (splitting theory), we consider Noether procedure and the alternative method of energy–momentum tensor defining by varying the action of the theory with respect to flat bulk metric. As a result, we obtain energy definition in field-theoretic form of embedding theory which, among the other features, gives a nontrivial result for the solutions of embedding theory which are also solutions of Einstein equations. The question of energy localization is also discussed.

scholarly journals TORSION AND THE ELECTROMAGNETIC FIELD

1999 ◽  
Vol 08 (02) ◽  
pp. 141-151 ◽  
Author(s):  
V. C. DE ANDRADE ◽  
J. G. PEREIRA
Keyword(s):  
General Relativity ◽  
Electromagnetic Field ◽  
Gauge Invariance ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Local Gauge ◽  
Local Gauge Invariance

In the framework of the teleparallel equivalent of general relativity, we study the dynamics of a gravitationally coupled electromagnetic field. It is shown that the electromagnetic field is able not only to couple to torsion, but also, through its energy–momentum tensor, produce torsion. Furthermore, it is shown that the coupling of the electromagnetic field with torsion preserves the local gauge invariance of Maxwell's theory.

scholarly journals Observables for Quarks and Gluons Orbital Angular Momentum Distributions

2015 ◽  
Vol 37 ◽  
pp. 1560039
Author(s):  
Simonetta Liuti ◽  
Aurore Courtoy ◽  
Gary R. Goldstein ◽  
J. Osvaldo Gonzalez Hernandez ◽  
Abha Rajan
Keyword(s):  
Angular Momentum ◽  
Orbital Angular Momentum ◽  
Spin Asymmetry ◽  
Energy Momentum Tensor ◽  
Deeply Virtual Compton Scattering ◽  
Virtual Compton Scattering ◽  
Momentum Tensor ◽  
Transverse Momentum Distribution ◽  
Kinetic Term ◽  
Momentum Distributions

We discuss the observables that have been recently put forth to describe quarks and gluons orbital angular momentum distributions. Starting from a standard parameterization of the energy momentum tensor in QCD one can single out two forms of angular momentum, a so-called kinetic term – Ji decomposition – or a canonical term – Jaffe-Manohar decomposition. Orbital angular momentum has been connected in each decomposition to a different observable, a Generalized Transverse Momentum Distribution (GTMD), for the canonical term, and a twist three Generalized Parton Distribution (GPD) for the kinetic term. While the latter appears as an azimuthal angular modulation in the longitudinal target spin asymmetry in deeply virtual Compton scattering, due to parity constraints, the GTMD associated with canonical angular momentum cannot be measured in a similar set of experiments.

Multivector Fields of Noether Symmetries in the Lagrangian Formalism and Belinfante Tensor

2021 ◽  
Vol 61 ◽  
pp. 53-78
Author(s):  
Halima Loumi-Fergane ◽  
Keyword(s):  
Angular Momentum ◽  
Energy Momentum Tensor ◽  
Internal Symmetry ◽  
Momentum Tensor ◽  
Gauge Fields ◽  
Lagrangian Formalism ◽  
Order Partial Differential Equation ◽  
Spin Angular Momentum ◽  
Relativistic Mechanic ◽  
Invariant Gauge

Elsewhere, we gave the explicit expressions of the multivectors fields associated to infinitesimal symmetries which gave rise to Noether currents for classical field theories and relativistic mechanic using the Second Order Partial Differential Equation SOPDE condition for the Poincar\'e-Cartan form.\\ The main objective of this paper is to reformulate the multivector fields associated to translational and rotational symmetries of the gauge fields in particular those of the electromagnetic field which gave rise to symmetrical and invariant gauge energy-momentum tensor and the orbital angular momentum. The spin angular momentum appears however because of the internal symmetry inside the fiber.

Dynamics of extended bodies in general relativity - II. Moments of the charge-current vector

1970 ◽  
Vol 319 (1539) ◽  
pp. 509-547 ◽  
Keyword(s):  
Tensor Field ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
The Body ◽  
Total Charge ◽  
Subsequent Paper ◽  
Multipole Moments ◽  
Current Vector ◽  
Charge Current ◽  
Infinite Set

The problem is considered of defining multipole moments for a tensor field given on a curved spacetime, with the aim of applying this to the energy-momentum tensor and charge-current vector of an extended body. Consequently, it is assumed that the support of the tensor field is bounded in spacelike directions. A definition is proposed for ‘a set of multipole moments’ of such a tensor field relative to an arbitrary bitensor propagator. This definition is not fully determinate, but any such set of moments completely determines the original tensor field. By imposing additional conditions on the moments in two different ways, two uniquely determined sets of moments are obtained for a vector field J α . The first set, the complete moments , always exists and agrees with moments defined less explicitly by Mathisson. If V α J α = 0, as is the case for the charge-current vector, these moments are interrelated by an infinite set of corresponding restrictions. The second set, the reduced moments , exists if and only if V α J α = 0. These avoid such an infinite set of interrelations, there being instead only one such restriction, the constancy of the total charge of the body. The energy-momentum tensor will be treated in a subsequent paper.

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