Electromagnetic tensor
Electromagnetism 


In electromagnetism, the 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 in spacetime of a physical system. 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.
Definition
The electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic fourpotential, A, a differential 1form:^{[1]}^{[2]}
Therefore, F is a differential 2form—that is, an antisymmetric rank2 tensor field—on Minkowski space. In component form,
where is the 4gradient and is the 4potential.
SI units for Maxwell's equations and the particle physicist's sign convention for the signature of Minkowski space (+,−,−,−), will be used throughout this article.
Relationship with the classical fields
The electromagnetic tensor is completely isomorphic to the electric and magnetic fields, though the electric and magnetic fields change with the choice of the reference frame, while the electromagnetic tensor does not. In general, the relationship is quite complicated, but in Cartesian coordinates, using the coordinate system's own reference frame, the relationship is very simple.
where c is the speed of light, and
where is the LeviCivita symbol.
In contravariant matrix form,
The covariant form is given by index lowering,
The mixedvariance form appears in the Lorentz force equation when using the contravariant fourvelocity: , where
From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is being assumed, and the electric and magnetic fields are with respect to coordinate system's own reference frame, as in the equations above.
Properties
The matrix form of the field tensor yields the following properties:^{[3]}
 Antisymmetry:
 Six independent components: In Cartesian coordinates, these are simply the three spatial components of the electric field (E_{x}, E_{y}, E_{z}) and magnetic field (B_{x}, B_{y}, B_{z}).
 Inner product: If one forms an inner product of the field strength tensor a Lorentz invariant is formed
 Pseudoscalar invariant: The product of the tensor with its dual tensor gives the Lorentz invariant:
 Determinant:
Significance
This tensor simplifies and reduces Maxwell's equations as four vector calculus equations into two tensor field equations. In electrostatics and electrodynamics, Gauss's law and Ampère's circuital law are respectively:
and reduce to the inhomogeneous Maxwell equation:
where
is the 4current. In magnetostatics and magnetodynamics, Gauss's law for magnetism and Maxwell–Faraday equation are respectively:
which reduce to Bianchi identity:
or using the index notation with square brackets^{[note 1]} for the antisymmetric part of the tensor:
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 simpler presentation of physical laws.
The inhomogeneous Maxwell equation leads to the continuity equation:
implying conservation of charge.
Maxwell's laws above can be generalised to curved spacetime by simply replacing partial derivatives with covariant derivatives:
 and
where the semicolon notation 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
Classical electromagnetism and Maxwell's equations can be derived from the action:
where
 is over space and time.
This means the Lagrangian density is
The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is
Substituting this into the Euler–Lagrange equation of motion for a field:
So the Euler–Lagrange equation becomes:
The quantity in parentheses above is just the field tensor, so this finally simplifies to
That equation is another way of writing the two inhomogeneous Maxwell's equations (namely, Gauss's law and Ampère's circuital law) using the substitutions:
where i, j, k take the values 1, 2, and 3.
Quantum electrodynamics and field theory
The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity to incorporate the creation and annihilation of photons (and electrons):
where the first part in the right hand side, containing the Dirac spinor , represents the Dirac field. 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
 Classification of electromagnetic fields
 Covariant formulation of classical electromagnetism
 Electromagnetic stress–energy tensor
 Gluon field strength tensor
 Ricci calculus
 Riemann–Silberstein vector
Notes
 ^ By definition,
So if
then
References
 ↑ J. A. Wheeler; C. Misner; K. S. Thorne (1973). Gravitation. W.H. Freeman & Co. ISBN 0716703440.
 ↑ D. J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN 8177582933.
 ↑ J. A. Wheeler; C. Misner; K. S. Thorne (1973). Gravitation. W.H. Freeman & Co. ISBN 0716703440.
 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.