 Maxwell's equations in curved spacetime

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 reader is assumed to be familiar with the four dimensional form of electromagnetism in flat spacetime and basic mathematics of curved spacetime.
The electromagnetic field also admits a coordinateindependent 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
In general relativity, the equations of electromagnetism in a vacuum become:
where f_{μ} is the density of Lorentz force, g^{αβ} is the reciprocal of the metric tensor g_{αβ}, and g is the determinant of the metric tensor. Notice that A_{α} and F_{αβ} are (ordinary) tensors while , J^{μ}, and f_{μ} 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.
The electromagnetic potential
The electromagnetic potential is a covariant vector, which is the undefined primitive of electromagnetism. As a covariant vector, its rule for transforming from one coordinate system to another is
Electromagnetic field
The electromagnetic field is a covariant antisymmetric rank 2 tensor 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 FaradayGauss, 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 FaradayGauss equation is sometimes written
where the semicolon indicates a covariant derivative, comma indicate a partial derivative, and square brackets indicate antisymmetrization. The covariant derivative of the electromagnetic field is
where Γ^{α}_{β γ} is the Christoffel symbol which is symmetric in its lower indices.
Electromagnetic displacement
The electric displacement field, and the auxiliary magnetic field, form an antisymmetric contravariant rank 2 tensor density of weight +1. In a vacuum, this is given by
Notice that this equation is the only place where the metric (and thus gravity) enters into the theory of electromagnetism. Furthermore even here, 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 nonzero, we have
The transformation law for electromagnetic displacement is
where the Jacobian determinant is used. If the magnetizationpolarization tensor is used, it has the same transformation law as the electromagnetic displacement.
Electric current
The electric current is the divergence of the electromagnetic displacement. In a vacuum,
If magnetizationpolarization 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 AmpereGauss definition of the electric current is not sufficient to determine its value because the electromagnetic potential (from which is 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
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 p is the linear 4momentum of the particle, t is any time coordinate parameterizing the world line of the particle, Γ 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
In a vacuum, the Lagrangian for classical electrodynamics (in joules/meter^{3}) is a scalar density
where The fourcurrent 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 stressenergy tensor
Main article: Electromagnetic stressenergy tensorAs part of the source term in the Einstein field equations, the electromagnetic stressenergy tensor is a covariant symmmetric tensor
which is tracefree
because electromagnetism propagates at the invariant speed.
In the expression for the conservation of energy and linear momentum, the electromagnetic stressenergy 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
The nonhomogeneous electromagnetic wave equation in terms of the field tensor is modified from the special relativity form to
where R_{acbd} 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 4potential [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 A^{a} plays the role of the 4position.
Nonlinearity of Maxwell's equations in a dynamic spacetime
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 stressenergy tensor through the Einstein field equation
where
is the Einstein tensor, G is the gravitational constant, g_{ab} is the metric tensor, and R (scalar curvature) is the trace of the Ricci curvature tensor. The stressenergy tensor is composed of the stressenergy from particles, but also stressenergy from the electromagnetic field. This generates the nonlinearity.
Geometric formulation
The geometric view of the electromagnetic field is that it is the curvature 2form of a principal U(1)bundle, and acts on charged matter by holonomy. In this view, one of Maxwell's two equations, d F= 0, is a mathematical identity known as the Bianchi identity. This equation implies, by the Poincaré lemma, that there exists (at least locally) a 1form A satisfying F = d A. The other Maxwell equation is
where the curvature 2form F is known as the Faraday 2form in this context, J is the current 3form, the asterisk * denotes the Hodge star operator, and d is the exterior derivative operator. The dependence of Maxwell's equation (there is only one with any physical content in this language) on the metric of spacetime lies in the Hodge star operator. Written this way, Maxwell's equation is the same in any spacetime.
See also
 Electromagnetic wave equation
 Nonhomogeneous electromagnetic wave equation
 Formulation of Maxwell's equations in special relativity
 Theoretical motivation for general relativity
 Basic introduction to the mathematics of curved spacetime
 Electrovacuum solution
References
 Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0517029618.
 Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0716703440.
 Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0080181767.
 R. P. Feynman, F. B. Moringo, and W. G. Wagner (1995). Feynman Lectures on Gravitation. AddisonWesley. ISBN 0201627345.
External links
Notes
Categories: Fundamental physics concepts
 General relativity
 Partial differential equations

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