Electromagnetic tensor

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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. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely.

Electromagnetism Branch of science concerned with the phenomena of electricity and magnetism

Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force is carried by electromagnetic fields composed of electric fields and magnetic fields, is responsible for electromagnetic radiation such as light, and is one of the four fundamental interactions in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force.

Electromagnetic field Electric and magnetic fields produced by moving charged objects

An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of non-comoving charged objects at any distance of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction. It is one of the four fundamental forces of nature.

Tensor Algebraic object with geometric applications

In mathematics, a tensor is an algebraic object related to a vector space and its dual space that can take several different forms, for example, a scalar, a tangent vector at a point, a cotangent vector at a point, or a multi-linear map between vector spaces. Euclidean vectors and scalars are the simplest tensors. While tensors are defined independent of any basis, the literature on physics often refers to them by their components in a basis related to a particular coordinate system.

Contents

Definition

The electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A, a differential 1-form: [1] [2]

Electromagnetic four-potential Relativistic vector field

An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.

Therefore, F is a differential 2-form—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form,

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

where is the four-gradient and is the four-potential.

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

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.

The signature(v, p, r) of a metric tensor g is the number of positive, zero, and negative eigenvalues of the real symmetric matrix gab of the metric tensor with respect to a basis. In physics, the v represents for the time or virtual dimension, and the p for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers (v, p) implying r = 0 or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signature (1, 3, 0), respectively.

Minkowski space mathematical space setting which eases explanation of special relativity

In mathematical physics, Minkowski space is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity.

Relationship with the classical fields

The electric and magnetic fields can be obtained from the components of the electromagnetic tensor. The relationship is simplest in Cartesian coordinates:

Electric field Vector field representing the Coulomb force per unit charge that would be exerted on a test charge at each point due to other electric charges

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

Magnetic field Spatial distribution of vectors allowing the calculation of the magnetic force on a test particle

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

Cartesian coordinate system Coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

where c is the speed of light, and

where is the Levi-Civita tensor. This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will transform covariantly, and the fields in the new frame will be given by the new components.

In contravariant matrix form,

The covariant form is given by index lowering,

The Faraday tensor's Hodge dual is

From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is assumed, and the electric and magnetic fields are with respect to the coordinate system's reference frame, as in the equations above.

Properties

The matrix form of the field tensor yields the following properties: [3]

  1. Antisymmetry:
  2. Six independent components: In Cartesian coordinates, these are simply the three spatial components of the electric field (Ex, Ey, Ez) and magnetic field (Bx, By, Bz).
  3. Inner product: If one forms an inner product of the field strength tensor a Lorentz invariant is formed
    meaning this number does not change from one frame of reference to another.
  4. Pseudoscalar invariant: The product of the tensor with its Hodge dual gives a Lorentz invariant:
    where is the rank-4 Levi-Civita symbol. The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is .
  5. Determinant:
    which is proportional to the square of the above invariant.

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 four-current. 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 physical laws being recognised after the advent of special relativity. This theory stipulated that all the 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 semi-colon 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.

Hamiltonian form

The Hamiltonian density can be obtained with the usual relation,

.

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

Notes

    1. ^ By definition,

      So if

      then

    1. J. A. Wheeler; C. Misner; K. S. Thorne (1973). Gravitation . W.H. Freeman & Co. ISBN   0-7167-0344-0.
    2. D. J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN   81-7758-293-3.
    3. J. A. Wheeler; C. Misner; K. S. Thorne (1973). Gravitation . W.H. Freeman & Co. ISBN   0-7167-0344-0.

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