 Electromagnetic tensor

Electromagnetism Electricity · Magnetism Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy currentThe electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism. The field tensor was first used after the 4dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows some physical laws to be written in a very concise form.
Contents
Definition
 Mathematical note: In this article, the abstract index notation will be used.
The electromagnetic tensor starts with the Electromagnetic fourpotential:

 and its covariant form is found by multiplying by the Minkowski metric η of signature (+,−,−,−) :
where

 is the vector potential and are its components
 is the scalar potential and
 is the speed of light.
Electric and magnetic fields are derived from the vector potentials and the scalar potential with two formulas:
By definition, the electromagnetic tensor is the exterior derivative of the differential 1form A_{μ}:
F is therefore a differential 2form on spacetime. In an inertial frame, the matrices of F read:
or
Properties
From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties:
 antisymmetry: (hence the name bivector).
 six independent components.
If one forms an inner product of the field strength tensor a Lorentz invariant is formed:
The product of the tensor with its dual tensor gives the pseudoscalar invariant:
where is the completely antisymmetric unit pseudotensor of the fourth rank or LeviCivita symbol. Caution: the sign for the above invariant depends on the convention used for the LeviCivita symbol. The convention used here is = +1.
Notice that:
Significance
Hidden beneath the surface of this complex mathematical equation is an ingenious unification of Maxwell's equations for electromagnetism. Consider the electrostatic equation
which tells us that the divergence of the electric field vector is equal to the charge density, and the electrodynamic equation
that is the change of the electric field with respect to time, minus the curl of the magnetic field vector, is equal to negative 4π times the current density.
These two equations for electricity reduce to
where

 is the 4current.
The same holds for magnetism. If we take the magnetostatic equation
which tells us that there are no "true" magnetic charges, and the magnetodynamics equation
which tells us the change of the magnetic field with respect to time plus the curl of the electric field is equal to zero (or, alternatively, the curl of the electric field is equal to the negative change of the magnetic field with respect to time). With the electromagnetic tensor, the equations for magnetism reduce to
The field tensor and relativity
The field tensor derives its name from the fact that the electromagnetic field is found to obey the tensor transformation law, this general property of (nongravitational) physical laws being recognised after the advent of special relativity. This theory stipulated that all the (nongravitational) laws of physics should take the same form in all coordinate systems  this led to the introduction of tensors. The tensor formalism also leads to a mathematically elegant presentation of physical laws. For example, Maxwell's equations of electromagnetism may be written using the field tensor as:

 and
The second equation implies conservation of charge:
These laws can be generalised to curved spacetime by simply replacing partial with covariant derivatives:

 and
where the semicolon represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime):
Lagrangian formulation of classical electromagnetism without charges and currents
When there are no electric charges (ρ=0) and no electric currents (j=0), Classical electromagnetism and Maxwell's equations can be derived from the action defined:
where

 is over space and time.
This means the Lagrangian density is
The far left and far right terms are the same because μ and ν are just dummy indices after all. The two middle terms are also the same, so the Lagrangian density is
We can then plug this into the EulerLagrange equation of motion for a field:
The second term is zero because the Lagrangian in this case only contains derivatives. So the EulerLagrange equation becomes:
The quantity in parentheses above is just the field tensor, so this finally simplifies to
That equation is just another way of writing the two inhomogeneous Maxwell's equations as long as you make the substitutions:
where and take on the values of 1, 2, and 3.
When there are charges or currents, the Lagrangian needs an extra term to account for the coupling between them and the electromagnetic field. In that case is equal to the 4current instead of zero.
Role in quantum electrodynamics and field theory
The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity, from to incorporate the creation and annihilation of photons (and electrons).
In quantum field theory it is used as the template for the gauge field strength tensor. By being employed in addition to the local interaction Lagrangian it reprises its usual role in QED.
See also
 Application of tensor theory in physics
 Classification of electromagnetic fields
 Covariant formulation of classical electromagnetism
References
 Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 0195146654.
 Jackson, John D. (1999). Classical Electrodynamics. John Wiley & Sons, Inc.. ISBN 047130932X.
 Peskin, Michael E.; Schroeder, Daniel V. (1995). An Introduction to Quantum Field Theory. Perseus Publishing. ISBN 0201503972.
Categories: Electromagnetism
 Minkowski spacetime
 Relativity
 Tensors
 Tensors in general relativity
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