Maxwell's equations in curved spacetime

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Induced spacetime curvature

In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields (or energy/matter in general) induce curvature in spacetime,[1] Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

When working in the presence of bulk matter, it is preferable to distinguish between free and bound electric charges. Without that distinction, the vacuum Maxwell's equations are called the "microscopic" Maxwell's equations. When the distinction is made, they are called the macroscopic Maxwell's equations.

The electromagnetic field also admits a coordinate-independent geometric description, and Maxwell's equations expressed in terms of these geometric objects are the same in any spacetime, curved or not. Also, the same modifications are made to the equations of flat Minkowski space when using local coordinates that are not Cartesian. For example, the equations in this article can be used to write Maxwell's equations in spherical coordinates. For these reasons, it may be useful to think of Maxwell's equations in Minkowski space as a special case, rather than Maxwell's equations in curved spacetimes as a generalization.

Summary[edit]

In general relativity, the metric, gαβ, is no longer a constant (like ηαβ as in Examples of metric tensor) but can vary in space and time, and the equations of electromagnetism in a vacuum become:

where fμ is the density of the Lorentz force, gαβ is the reciprocal of the metric tensor gαβ, and g is the determinant of the metric tensor. Notice that Aα and Fαβ are (ordinary) tensors while , Jν, and fμ are tensor densities of weight +1. Despite the use of partial derivatives, these equations are invariant under arbitrary curvilinear coordinate transformations. Thus if one replaced the partial derivatives with covariant derivatives, the extra terms thereby introduced would cancel out. (Cf. manifest covariance#Example.)

The electromagnetic potential[edit]

The electromagnetic potential is a covariant vector, Aα which is the undefined primitive of electromagnetism. As a covariant vector, its rule for transforming from one coordinate system to another is

Electromagnetic field[edit]

The electromagnetic field is a covariant antisymmetric tensor of rank 2 which can be defined in terms of the electromagnetic potential by

To see that this equation is invariant, we transform the coordinates (as described in the classical treatment of tensors)

This definition implies that the electromagnetic field satisfies

which incorporates Faraday's law of induction and Gauss's law for magnetism. This is seen by

Although there appear to be 64 equations in Faraday-Gauss, it actually reduces to just four independent equations. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0 = 0) or render redundant all the equations except for those with λ,μ,ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.

The Faraday-Gauss equation is sometimes written

where the semicolon indicates a covariant derivative, comma indicate a partial derivative, and square brackets indicate anti-symmetrization (see Ricci calculus for the notation). The covariant derivative of the electromagnetic field is

where Γαβ γ is the Christoffel symbol which is symmetric in its lower indices.

Electromagnetic displacement[edit]

The electric displacement field, D and the auxiliary magnetic field, H form an antisymmetric contravariant rank 2 tensor density of weight +1. In a vacuum, this is given by

This equation is the only place where the metric (and thus gravity) enters into the theory of electromagnetism. Furthermore, the equation is invariant under a change of scale, that is, multiplying the metric by a constant has no effect on this equation. Consequently, gravity can only affect electromagnetism by changing the speed of light relative to the global coordinate system being used. Light is only deflected by gravity because it is slower when near to massive bodies. So it is as if gravity increased the index of refraction of space near massive bodies.

More generally, in materials where the magnetizationpolarization tensor is non-zero, we have

The transformation law for electromagnetic displacement is

where the Jacobian determinant is used. If the magnetization-polarization tensor is used, it has the same transformation law as the electromagnetic displacement.

Electric current[edit]

The electric current is the divergence of the electromagnetic displacement. In a vacuum,

If magnetization-polarization is used, then this just gives the free portion of the current

This incorporates Ampere's Law and Gauss's Law.

In either case, the fact that the electromagnetic displacement is antisymmetric implies that the electric current is automatically conserved

because the partial derivatives commute.

The Ampere-Gauss definition of the electric current is not sufficient to determine its value because the electromagnetic potential (from which it was ultimately derived) has not been given a value. Instead, the usual procedure is to equate the electric current to some expression in terms of other fields, mainly the electron and proton, and then solve for the electromagnetic displacement, electromagnetic field, and electromagnetic potential.

The electric current is a contravariant vector density, and as such it transforms as follows

Verification of this transformation law

So all that remains is to show that

which is a version of a known theorem (see Inverse functions and differentiation#Higher derivatives).

Lorentz force density[edit]

The density of the Lorentz force is a covariant vector density given by

The force on a test particle subject only to gravity and electromagnetism is

where pα is the linear 4-momentum of the particle, t is any time coordinate parameterizing the world line of the particle, Γβαγ is the Christoffel symbol (gravitational force field), and q is the electric charge of the particle.

This equation is invariant under a change in the time coordinate; just multiply by and use the chain rule. It is also invariant under a change in the x coordinate system.

Using the transformation law for the Christoffel symbol

we get

Lagrangian[edit]

In a vacuum, the Lagrangian for classical electrodynamics (in joules/meter3) is a scalar density

where

The four-current should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables.

If we separate free currents from bound currents, the Lagrangian becomes

Electromagnetic stress–energy tensor[edit]

As part of the source term in the Einstein field equations, the electromagnetic stress–energy tensor is a covariant symmetric tensor

using a metric of signature (-,+,+,+). If using the metric with signature (+,-,-,-), the expression for will have opposite sign. The stress–energy tensor is trace-free

because electromagnetism propagates at the invariant speed.

In the expression for the conservation of energy and linear momentum, the electromagnetic stress–energy tensor is best represented as a mixed tensor density

From the equations above, one can show that

where the semicolon indicates a covariant derivative.

This can be rewritten as

which says that the decrease in the electromagnetic energy is the same as the work done by the electromagnetic field on the gravitational field plus the work done on matter (via the Lorentz force), and similarly the rate of decrease in the electromagnetic linear momentum is the electromagnetic force exerted on the gravitational field plus the Lorentz force exerted on matter.

Derivation of conservation law

which is zero because it is the negative of itself (see four lines above).

Electromagnetic wave equation[edit]

The nonhomogeneous electromagnetic wave equation in terms of the field tensor is modified from the special relativity form to

where Racbd is the covariant form of the Riemann tensor and is a generalization of the d'Alembertian operator for covariant derivatives. Using

Maxwell's source equations can be written in terms of the 4-potential [ref 2, p. 569] as,

or, assuming the generalization of the Lorenz gauge in curved spacetime

where is the Ricci curvature tensor.

This the same form of the wave equation as in flat spacetime, except that the derivatives are replaced by covariant derivatives and there is an additional term proportional to the curvature. The wave equation in this form also bears some resemblance to the Lorentz force in curved spacetime where Aa plays the role of the 4-position.

Nonlinearity of Maxwell's equations in a dynamic spacetime[edit]

When Maxwell's equations are treated in a background independent manner, that is, when the spacetime metric is taken to be a dynamical variable dependent on the electromagnetic field, then the electromagnetic wave equation and Maxwell's equations are nonlinear. This can be seen by noting that the curvature tensor depends on the stress–energy tensor through the Einstein field equation

where

is the Einstein tensor, G is the gravitational constant, gab is the metric tensor, and R (scalar curvature) is the trace of the Ricci curvature tensor. The stress–energy tensor is composed of the stress-energy from particles, but also stress-energy from the electromagnetic field. This generates the nonlinearity.

Geometric formulation[edit]

In the differential geometric formulation of the electromagnetic field, the antisymmetric Faraday tensor can be considered as the Faraday 2-form F. In this view, one of Maxwell's two equations is dF= 0, where d is the exterior derivative operator. This equation is completely coordinate and metric independent and says that the electro-magnetic flux through a closed two dimensional surface in space time is topological, more precisely, depends only on its homology class (a generalization of the integral form of Gauss law and Maxwell-Faraday equation as the homology class in Minkowski space is automatically 0). By the Poincaré lemma, this equation implies, (at least locally) that there exists a 1-form A satisfying F = d A. The other Maxwell equation is d * F = J. In this context, J is the current 3-form (or even more precise, twisted three form), the asterisk * denotes the Hodge star operator, and d is the exterior derivative operator. The dependence of Maxwell's equation on the metric of spacetime lies in the Hodge star operator * on two forms, which is conformally invariant. Written this way, Maxwell's equation is the same in any space time, manifestly coordinate invariant, and convenient to use (even in Minkowski space or Euclidean space and time especially with curvilinear coordinates).

An even more geometric interpretation is that the Faraday two form F is (up to a factor i) the curvature 2-form of a U(1)-connection on a principal U(1)-bundle whose sections represent charged fields. The connection is much like the vector potential since every connection can be written as for a "base" connection and F = F0 + d A. In this view, the Maxwell "equation", d F= 0, is a mathematical identity known as the Bianchi identity. The equation d * F = J is the only equation with any physical content in this formulation. This point of view is particularly natural when considering charged fields or quantum mechanics. It can be interpreted as saying that, much like gravity can be understood as being the result of the necessity of a connection to parallel transport vectors at different points, electromagnetic phenomena, or more subtle quantum effects like the Aharanov-Bohm effect, can be understood as a result from the necessity of a connection to parallel transport charged fields or wave sections at different points. In fact, just as the Riemann tensor is the holonomy of the Levi Civita connection along an infinitesimal closed curve, the curvature of the connection is the holonomy of the U(1)-connection.

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