Maxwell's equations in curved spacetime

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In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (or more generally, spacetime with a non-Euclidean metric). They can be viewed as a generalisation of the vacuum Maxwell's equations as they are normally formulated in the local coordinates of flat spacetime, but general relativity dictates that the presence of electromagnetic fields themselves induce curvature in spacetime, so Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

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 in flat Euclidean or Minkowski space when using local coordinates that are not Euclidean. 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 generalisation.

However, this formulation of Maxwell's equations is only useful for the vacuum Maxwell's equations, also called the "microscopic" Maxwell's equations. For the macroscopic Maxwell's equations with inhomogeneous materials, the presence of the materials establishes a reference frame and the equations are no longer covariant.

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[edit] The equations

Maxwell's equations in curved spacetime can be obtained by replacing the partial derivatives in Maxwell's equations in flat spacetime with covariant derivatives. The source and source-free Maxwell equations in curved spacetime are respectively (in SI units):

F^{ab}{}_{;b} \, =\mu_0 J^a

and

F_{[ab;c]} \, = 0

where the semicolon indicates a covariant derivative (the square brackets indicate anti-symmetrization) and where Fab is the electromagnetic field tensor, Ja is the 4-current and μ0 is a constant called the permeability of free space. The equations can be written in cgs units by replacing the permeability with 4π / c.

Written out in full, the above two equations are

F^{ab}{}_{,b} + {\Gamma^a}_{\mu b} F^{\mu b} + {\Gamma^b}_{\mu b} F^{a \mu} = \mu_0 J^a ,

and

F_{ab,c} + F_{ca,b} + F_{bc,a} \, = 0

where the comma indicates a partial derivative (other notations include \frac{\partial}{\partial{x^c}}, \partial_c and sometimes Dc) and {\Gamma^a}_{\mu b} is a Christoffel symbol. This assumes the connection is torsion-free and metric compatible, as is done in general relativity (there are other approaches to how the manifold changes that have nothing to do with the metric).

If the covariant derivative acts on a covariant tensor rather than a contravariant tensor, the signs of the Christoffel symbols change

{F_{ab}}_{;a} \ \stackrel{\mathrm{def}}{=}\ D_a F_{ab} = \partial_a F_{ab} - {\Gamma^{\mu}}_{a a} F_{\mu b} - {\Gamma^{\mu}}_{b a} F_{a \mu} \!.

The second equality in the source-free Maxwell equation is the same in flat or in curved spacetime. This can be seen by noting that the Christoffel symbols are symmetric in their lower indices and the field tensor is antisymmetric. This also implies that the relationship to the 4-potential is unchanged from flat spacetime:

F^{ab} = \partial^b A^a - \partial^a A^b \,\!

[edit] Relationship between Christoffel symbols and the metric tensor

The relationship between the Christoffel symbols and the metric tensor for curved spacetime is

{{\Gamma}^a}_{bc} = g^{ad} {\Gamma}_{dbc}
{\Gamma}_{d b c} = {1 \over 2} \left ( { { \partial_{b} {g}}_{d c} } + { { \partial_{c} {g}}_{d b}} - { { \partial_d {g}}_{bc} } \right ) .

This expression permits the source equations to be rewritten as

J^ b = { 1 \over \sqrt{-g} }\partial_a \left ( \sqrt{-g} F^{ab} \right ) = D_a F^{ab} \!,

which, except for the terms in the square roots, is the same as in flat spacetime. Here, g \! is the determinant of the metric tensor. The square root of the negative of the determinant of the metric tensor is the Jacobian of the transformation between curved and flat spacetime. The Jacobian is proportional to the volume of an element of curved spacetime. Therefore, the sourced Maxwell equation is simply an expression for the divergence of the field tensor in curved spacetime.

[edit] Modification of other electromagnetic equations

As Maxwell's equations are modified in curved spacetime, any equations derived from them should be modified accordingly.

[edit] Charge conservation in curved spacetime

The continuity equation, the expression for charge conservation, in curved spacetime is:

0 = {J^ a}_{;a} \ \stackrel{\mathrm{def}}{=}\ D_a J^ a = { 1 \over \sqrt{-g} }\partial_a \left ( \sqrt{-g} J^{a} \right ) \!.

[edit] Lorentz force in curved spacetime

The Lorentz force law in curved spacetime becomes

m c u^a{}_{;b}u^b = m c \left (u^a{}_{,b} u^b + \Gamma^a_{bc} u^b u^c \right ) = q F^{ab} u_{b}

which is the same as in flat spacetime, except that the partial derivative is replaced with the covariant derivative. Here, m is the particle mass, q is the charge, and

u_{b} = g_{ba} u^a = g_{ba} { d x^{a} \over {d \tau} }

is the 4-velocity of the particle. Here, τ is c times the proper time of the particle.

The Lorentz force can also be written in terms of the Riemann tensor

m c u^a{}_{;b}u^b = m c \left (u^a{}_{,b} u^b + R^a_{bdc} u^bx^d u^c \right ) = q F^{ab} u_{b} .

[edit] Electromagnetic stress-energy tensor

The electromagnetic stress-energy tensor is the field strength tensor in flat spacetime with the derivative replaced by the covariant derivative and the Minkowski metric replaced by the general metric:

T^{ab} = \, F^{as} F_{s}{}^b + {1 \over 4} F_{st} F^{st} g^{ab}

from which it follows

T^{ab}{}_{;b} = \, F^{ab} J_b

The latter equation can be thought of as the rate per unit volume at which matter feeds 4-momentum into an electromagnetic field.

[edit] Electromagnetic wave equation

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

\Box F_{ab} \ \stackrel{\mathrm{def}}{=}\ F_{ab;}{}^d{}_d = \, -2 R_{acbd} F^{cd} + R_{ae}F^e{}_b - R_{be}F^e{}_a + J_{a;b} - J_{b;a}

where

Racbd

is the covariant form of the Riemann tensor and \Box is a generalization of the d'Alembertian operator for covariant derivatives.

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

\Box A^{a} = {{A^{a; }}^{b}}_{ b} = - \mu_0 J^{ a } + {R^{ a }}_{ b } A^{ b }

where

{R^{ a }}_{ b } \ \stackrel{\mathrm{def}}{=}\ {R^{ s }}_{ a s b }

is the Ricci curvature tensor. The generalization of the Lorenz gauge in curved spacetime

{A^{ a }}_{ ; a } = 0

has been assumed.

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.

[edit] 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 stress-energy tensor through the Einstein field equation

G_{ab} = \frac{8 \pi G} {c^4} T_{ab}

where

{G}_{a b} \ \stackrel{\mathrm{def}}{=}\ {R}_{a b} - {1 \over 2} {R} g_{a b}

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.

[edit] Geometric formulation

The geometric view of the electromagnetic field is that it is the curvature 2-form 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 1-form A satisfying F = d A. The other Maxwell equation is

d * \mathbf{F} =\mathbf{J}

where the curvature 2-form F is known as the Faraday 2-form in this context, J is the current 3-form, 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.

[edit] See also

[edit] External links

[edit] References

[1] Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-02961-8. 
[2] Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0. 
[3] Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7. 
[4] R. P. Feynman, F. B. Moringo, and W. G. Wagner (1995). Feynman Lectures on Gravitation. Addison-Wesley. ISBN 0-201-62734-5. 
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