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Encyclopedia > Formulation of Maxwell's equations in special relativity

In special relativity, in order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form (cgs units): Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ...

{ 4 pi over c }J^ b = {partial F^{ab} over {partial x^a} } equiv partial_a F^{ab} equiv {F^{ab}}_{,a} ,!,

and

0 = partial_c F_{ab} + partial_b F_{ca} + partial_a F_{bc} equiv {F_{ab}}_{,c} + {F_{ca}}_{,b} +{F_{bc}}_{,a} equiv epsilon_{dabc} {F^{bc}}_{,a}

where , J^a is the 4-current, , F^{ab} is the field strength tensor, , epsilon_{abcd} is the Levi-Civita symbol, and In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, ρ the charge density, and j the conventional current density. ... In electromagnetism, the electromagnetic tensor, or electromagnetic field tensor, F, is defined as: where Ai is the vector potential. ... The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ...

{ partial over { partial x^a } } equiv partial_a equiv {}_{,a} equiv (partial/partial ct, nabla)

is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations. The four-gradient is the four-vector generalization of the gradient: and is sometimes also represented as D. The square of D is the four-Laplacian, which is called the dAlembertian operator: . As it is the dot product of two four-vectors, the dAlembertian is a Lorentz invariant... In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ...


The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and Ampere's law with Maxwell's correction. The second equation is an expression of the homogenous equations, Faraday's law of induction and the absence of magnetic monopoles. In physics, a magnetic monopole is a hypothetical particle that may be loosely described as a magnet with only one pole (see electromagnetic theory for more on magnetic poles). ...

Contents


Charge conservation

The 4-current is a contravariant vector given by: This page does not deal with the statistical concept covariance of random variables, nor with the computer science concepts of covariance and contravariance. ...

where ρ is the charge density and mathbf{J} is the current density.


The 4-current satisfies the continuity equation

Field strength tensor and the 4-potential

The field strength tensor, an antisymmetric tensor, can be written: In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. ...

F^{ab} = partial^b A^a - partial^a A^b ,!

where

is the 4-potential, φ is the scalar potential and mathbf{A} is the vector potential. We have assumed the Lorenz gauge: The four-vector electromagnetic potential is defined in SI units (and gaussian units in parentheses) as in which φ is the electrical potential, and A is the magnetic potential, a vector potential. ... It has been suggested that this article or section be merged with Potential. ... In vector calculus, a vector potential is a vector field whose curl is a given vector field. ... The Lorenz gauge (or Lorenz gauge condition) was published by the Danish physicist Ludwig Lorenz. ...

.

The field strength tensor is written in terms of fields as:

The fact that both electric and magnetic fields are combined into a single tensor expresses the fact that, according to relativity, both of these are different aspects of the same thing—by changing frames of reference, what seemed to be an electric field in one frame can appear as a magnetic field in another frame, and vice versa.


Maxwell's equations, in the absence of sources, reduce to a wave equation in the field strength: Lasers used for visual effects during musical performance. ...

.

Here, is the d'Alembertian operator. In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ...


Different authors sometimes employ different sign conventions for the above tensors and 4-vectors (which does not affect the physical interpretation).


The covariant version of the field strength tensor , F_{ab} is related to to contravariant version , F^{ab} by the Minkowski metric tensor η In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

.

Lorentz force

Fields are detected by their effect on the motion of matter. Electromagnetic fields affect the motion of particles through the Lorentz equation. The Lorentz force equation can be written in terms of the field strength tensor as The electromagnetic field (EMF) is composed of two related vectorial fields, the electric field and the magnetic field. ...

m c { d u^{alpha} over { d tau } } = { {} over {} }F^{alpha beta} q u_{beta}

where m is the particle mass, q is the charge, and Mass is a property of a physical object that quantifies the amount of matter it contains. ... ‹ The template below has been proposed for deletion. ...

u_{beta} = eta_{beta alpha } u^{alpha } = eta_{beta alpha } { d x^{alpha } over {d tau} }

is the 4-velocity of the particle. Here, τ is c times the proper time of the particle. In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector (vector in four-dimensional spacetime) that replaces classical velocity (a three-dimensional vector). ... Proper time is time as measured by the clock for an observer who is traveling through spacetime. ...


Electromagnetic stress-energy tensor

The electromagnetic stress-energy tensor is related to the field strength tensor by: This article is in need of attention from an expert on the subject. ...

See also

Lasers used for visual effects during musical performance. ...

References

[1] Einstein, A. (1961). Relativity: The Special and General Theory, New York: Crown. ISBN 0-517-029618.
[2] Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation, San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.
[3] Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition), Oxford: Pergamon. ISBN 0-08-018176-7.
[4] R. P. Feynman, F. B. Moringo, and W. G. Wagner (1995). Feynman Lectures on Gravitation, Addison-Wesley. ISBN 0-201-62734-5.


General subfields within physics

Atomic, molecular, and optical physics | Classical mechanics | Condensed matter physics | Continuum mechanics | Electromagnetism | Special relativity | General relativity | Particle physics | Quantum field theory | Quantum mechanics | Statistical mechanics | Thermodynamics A Superconductor demonstrating the Meissner Effect. ... Atomic, molecular, and optical physics is the study of matter-matter and light-matter interactions on the scale of single atoms or structures containing a few atoms. ... It has been suggested that this article or section be merged with Newtonian mechanics. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, which exerts a force on those particles that possess the property of electric charge, and is in turn affected by the presence and motion of such particles. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... Particles erupt from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...


 
 

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