|
|
| Line 1: |
Line 1: |
| {{for|background material used in this article|Covariant formulation of classical electromagnetism|Introduction to mathematics of general relativity}}
| | Greetings! I am Marvella and I really feel comfortable when individuals use the full title. One of the extremely best things in the world for me is to do aerobics and I've been doing it for quite a while. Hiring is my profession. Puerto Rico is exactly where he's been residing for many years and he will never transfer.<br><br>Check out my blog post; [https://hcbsales.com/node/3739 at home std test] |
| | |
| [[Image:Gravitation space source.png|thumb|200px|Induced spacetime curvature]]
| |
| | |
| In [[physics]], '''Maxwell's equations in curved spacetime''' govern the dynamics of the [[electromagnetic field]] in [[curvature|curved]] [[spacetime]] (where the [[Metric tensor (general relativity)|metric]] may not be the [[Minkowski metric]]) or where one uses an arbitrary (not necessarily [[Cartesian coordinate system|Cartesian]]) coordinate system. These equations can be viewed as a generalization of the [[Maxwell's equations#In vacuum|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,<ref>http://www.springerlink.com/content/h224886u7651v774/</ref> 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==
| |
| | |
| In [[general relativity]], the [[metric (general relativity)|metric]], ''g''<sub>αβ</sub>, is no longer a constant (like η<sub>αβ</sub> above) but can vary in space and time, and the equations of electromagnetism in a vacuum become:
| |
| :<math>F_{\alpha \beta} \, = \, \partial_{\alpha} A_{\beta} \, - \, \partial_{\beta} A_{\alpha} \,</math>
| |
| :<math>\mathcal{D}^{\mu\nu} \, = \, \frac{1}{\mu_{0}} \, g^{\mu\alpha} \, F_{\alpha\beta} \, g^{\beta\nu} \, \sqrt{-g} \,</math>
| |
| :<math>J^{\mu} \, = \, \partial_\nu \mathcal{D}^{\mu \nu} \,</math>
| |
| :<math>f_\mu \, = \, F_{\mu\nu} \, J^\nu \,</math>
| |
| | |
| where ''f''<sub>μ</sub> is the density of the [[Lorentz force]], ''g''<sup>αβ</sup> is the reciprocal of the [[metric tensor (general relativity)|metric tensor]] ''g''<sub>αβ</sub>, and ''g'' is the [[determinant]] of the metric tensor. Notice that ''A''<sub>α</sub> and ''F''<sub>αβ</sub> are (ordinary) tensors while <math> \mathcal{D}^{\mu\nu}</math>, ''J''<sup>ν</sup>, and ''f''<sub>μ</sub> are [[tensor density|tensor ''densities'']] of weight +1. Despite the use of [[partial derivative]]s, these equations are invariant under arbitrary curvilinear coordinate transformations. Thus if one replaced the partial derivatives with [[covariant derivative]]s, the extra terms thereby introduced would cancel out. (Cf. [[manifest covariance#Example]].)
| |
| | |
| ==The electromagnetic potential==
| |
| | |
| The [[Electromagnetic four-potential|electromagnetic potential]] is a covariant vector, ''A''<sub>α</sub> which is the undefined primitive of electromagnetism. As a covariant vector, its rule for transforming from one coordinate system to another is
| |
| :<math>\bar{A}_{\beta} = \frac{\partial x^{\gamma}}{\partial \bar{x}^{\beta}} A_{\gamma} \,.</math>
| |
| | |
| ==Electromagnetic field==
| |
| The [[Electromagnetic tensor|electromagnetic field]] is a covariant [[antisymmetric tensor]] of rank 2 which can be defined in terms of the electromagnetic potential by
| |
| | |
| :<math>F_{\alpha \beta} \, = \, \partial_{\alpha} A_{\beta} \, - \, \partial_{\beta} A_{\alpha} \,.</math>
| |
| | |
| To see that this equation is invariant, we transform the coordinates (as described in the [[classical treatment of tensors]])
| |
| :<math>\begin{align}
| |
| \bar{F}_{\alpha \beta} & = \frac{\partial \bar{A}_{\beta}}{\partial \bar{x}^{\alpha}} \, - \, \frac{\partial \bar{A}_{\alpha}}{\partial \bar{x}^{\beta}} \\
| |
| & = \, \frac{\partial}{\partial \bar{x}^{\alpha}} \left( \frac{\partial x^{\gamma}}{\partial \bar{x}^{\beta}} A_{\gamma} \right) \, - \, \frac{\partial}{\partial \bar{x}^{\beta}} \left( \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} A_{\delta} \right) \\
| |
| & = \, \frac{\partial^2 x^{\gamma}}{\partial \bar{x}^{\alpha} \, \partial \bar{x}^{\beta}} A_{\gamma} \, + \, \frac{\partial x^{\gamma}}{\partial \bar{x}^{\beta}} \frac{\partial A_{\gamma}}{\partial \bar{x}^{\alpha}} \, - \, \frac{\partial^2 x^{\delta}}{\partial \bar{x}^{\beta} \, \partial \bar{x}^{\alpha}} A_{\delta} \, - \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \frac{\partial A_{\delta}}{\partial \bar{x}^{\beta}} \\
| |
| & = \, \frac{\partial x^{\gamma}}{\partial \bar{x}^{\beta}} \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \frac{\partial A_{\gamma}}{\partial x^{\delta}} \, - \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \frac{\partial x^{\gamma}}{\partial \bar{x}^{\beta}} \frac{\partial A_{\delta}}{\partial x^{\gamma}} \\
| |
| & = \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \frac{\partial x^{\gamma}}{\partial \bar{x}^{\beta}} \, \left( \frac{\partial A_{\gamma}}{\partial x^{\delta}} \, - \, \frac{\partial A_{\delta}}{\partial x^{\gamma}} \right) \\
| |
| & = \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \frac{\partial x^{\gamma}}{\partial \bar{x}^{\beta}} \, F_{\delta \gamma} \ .
| |
| \end{align}</math>
| |
| | |
| This definition implies that the electromagnetic field satisfies
| |
| :<math>\partial_\lambda F_{\mu \nu} + \partial _\mu F_{\nu \lambda} + \partial_\nu F_{\lambda \mu} = 0 \,</math>
| |
| | |
| which incorporates [[Faraday's law of induction]] and [[Gauss's law for magnetism]]. This is seen by
| |
| | |
| :<math>\partial_\lambda F_{\mu \nu} + \partial _\mu F_{\nu \lambda} + \partial_\nu F_{\lambda \mu} \,</math>
| |
| :<math>= \, \partial_\lambda \partial_\mu A_\nu - \partial_\lambda \partial_\nu A_\mu +
| |
| \partial_\mu \partial_\nu A_\lambda - \partial_\mu \partial_\lambda A_\nu +
| |
| \partial_\nu \partial_\lambda A_\mu - \partial_\nu \partial_\mu A_\lambda \, = 0 \,.</math>
| |
| | |
| 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
| |
| :<math>F_{[\mu \nu ; \lambda]} \, = \, F_{[\mu \nu , \lambda]} \, = \, \frac{1}{6} \left( \partial_\lambda F_{\mu \nu} + \partial _\mu F_{\nu \lambda} + \partial_\nu F_{\lambda \mu} - \partial_\lambda F_{\nu \mu} - \partial _\mu F_{\lambda \nu} - \partial_\nu F_{\mu \lambda} \right) \,</math>
| |
| :<math>= \, \frac{1}{3} \left( \partial_\lambda F_{\mu \nu} + \partial _\mu F_{\nu \lambda} + \partial_\nu F_{\lambda \mu} \right) = 0 \,</math>
| |
| | |
| 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
| |
| :<math>F_{\alpha \beta ; \gamma} \, = \, F_{\alpha \beta , \gamma} - {\Gamma^{\mu}}_{\alpha \gamma} F_{\mu \beta} - {\Gamma^{\mu}}_{\beta \gamma} F_{\alpha \mu} \,</math>
| |
| | |
| where Γ<sup>α</sup><sub>β γ</sub> is the [[Christoffel symbol]] which is symmetric in its lower indices.
| |
| | |
| ==Electromagnetic displacement==
| |
| 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
| |
| :<math>\mathcal{D}^{\mu \nu} \, = \, \frac{1}{\mu_{0}} \, g^{\mu \alpha} \, F_{\alpha \beta} \, g^{\beta \nu} \, \sqrt{-g} \,.</math>
| |
| | |
| 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 [[magnetization]]-[[electric polarization|polarization]] tensor is non-zero, we have
| |
| :<math>\mathcal{D}^{\mu \nu} \, = \, \frac{1}{\mu_{0}} \, g^{\mu \alpha} \, F_{\alpha \beta} \, g^{\beta \nu} \, \sqrt{-g} \, - \, \mathcal{M}^{\mu \nu} \,.</math>
| |
| | |
| The transformation law for electromagnetic displacement is
| |
| :<math>\bar{\mathcal{D}}^{\mu \nu} \, = \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \frac{\partial \bar{x}^{\nu}}{\partial x^{\beta}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \,</math>
| |
| | |
| 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==
| |
| The electric current is the divergence of the electromagnetic displacement. In a vacuum,
| |
| :<math>J^{\mu} \, = \, \partial_{\nu} \mathcal{D}^{\mu \nu} \,.</math>
| |
| | |
| If magnetization-polarization is used, then this just gives the free portion of the current
| |
| :<math>J^{\mu}_{\text{free}} \, = \, \partial_{\nu} \mathcal{D}^{\mu \nu} \,.</math>
| |
| | |
| 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
| |
| :<math>\partial_{\mu} J^{\mu} \, = \, \partial_{\mu} \partial_{\nu} \mathcal{D}^{\mu \nu} = 0 \,</math>
| |
| | |
| because the partial derivatives [[Commutative property|commute]].
| |
| | |
| The Ampere-Gauss 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
| |
| :<math>\bar{J}^{\mu} \, = \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, J^{\alpha} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \,.</math>
| |
| | |
| Verification of this transformation law
| |
| | |
| :<math>\begin{align}
| |
| \bar{J}^{\mu} \, & = \, \frac{\partial}{\partial \bar{x}^{\nu}} \left( \bar{\mathcal{D}}^{\mu \nu} \right) \, = \, \frac{\partial}{\partial \bar{x}^{\nu}} \left( \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \frac{\partial \bar{x}^{\nu}}{\partial x^{\beta}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \right) \\
| |
| | |
| & = \, \frac{\partial^2 \bar{x}^{\mu}}{\partial \bar{x}^{\nu} \partial x^{\alpha}} \, \frac{\partial \bar{x}^{\nu}}{\partial x^{\beta}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \, + \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \frac{\partial^2 \bar{x}^{\nu}}{\partial \bar{x}^{\nu} \partial x^{\beta}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \\
| |
| | |
| & + \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \frac{\partial \bar{x}^{\nu}}{\partial x^{\beta}} \, \frac{\partial \mathcal{D}^{\alpha \beta}}{\partial \bar{x}^{\nu}} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \, + \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \frac{\partial \bar{x}^{\nu}}{\partial x^{\beta}} \, \mathcal{D}^{\alpha \beta} \, \frac{\partial}{\partial \bar{x}^{\nu}} \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \\
| |
| | |
| & = \, \frac{\partial^2 \bar{x}^{\mu}}{\partial x^{\beta} \partial x^{\alpha}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \, + \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \frac{\partial^2 \bar{x}^{\nu}}{\partial \bar{x}^{\nu} \partial x^{\beta}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \\
| |
| | |
| & + \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \frac{\partial \mathcal{D}^{\alpha \beta}}{\partial x^{\beta}} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \, + \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \frac{\partial \bar{x}^{\nu}}{\partial x^{\beta}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \frac{\partial \bar{x}^{\rho}}{\partial x^{\sigma}} \frac{\partial^2 x^{\sigma}}{\partial \bar{x}^{\nu} \partial \bar{x}^{\rho}}\\
| |
| | |
| & = \, 0 \, + \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \frac{\partial^2 \bar{x}^{\nu}}{\partial \bar{x}^{\nu} \partial x^{\beta}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \\
| |
| & + \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, J^{\alpha} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \, + \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \frac{\partial \bar{x}^{\rho}}{\partial x^{\sigma}} \frac{\partial^2 x^{\sigma}}{\partial x^{\beta} \partial \bar{x}^{\rho}} \\
| |
| | |
| & = \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, J^{\alpha} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \, + \, \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \, \mathcal{D}^{\alpha \beta} \, \det \left[ \frac{\partial x^{\sigma}}{\partial \bar{x}^{\rho}} \right] \left( \frac{\partial^2 \bar{x}^{\nu}} {\partial \bar{x}^{\nu} \partial x^{\beta}} \, + \, \frac{\partial \bar{x}^{\rho}}{\partial x^{\sigma}} \frac{\partial^2 x^{\sigma}}{\partial x^{\beta} \partial \bar{x}^{\rho}} \right)
| |
| | |
| \end{align}</math>
| |
| | |
| So all that remains is to show that
| |
| :<math>\frac{\partial^2 \bar{x}^{\nu}}{\partial \bar{x}^{\nu} \partial x^{\beta}} \, + \, \frac{\partial \bar{x}^{\rho}}{\partial x^{\sigma}} \frac{\partial^2 x^{\sigma}}{\partial x^{\beta} \partial \bar{x}^{\rho}} \, = \, 0 \,</math>
| |
| | |
| which is a version of a known theorem (see [[Inverse functions and differentiation#Higher derivatives]]).
| |
| | |
| :<math>\begin{align}
| |
| \frac{\partial^2 \bar{x}^{\nu}}{\partial \bar{x}^{\nu} \partial x^{\beta}} \, + \, \frac{\partial \bar{x}^{\rho}}{\partial x^{\sigma}} \frac{\partial^2 x^{\sigma}}{\partial x^{\beta} \partial \bar{x}^{\rho}} & = \, \frac{\partial x^{\sigma}}{\partial \bar{x}^{\nu}}
| |
| \frac{\partial^2 \bar{x}^{\nu}}{\partial x^{\sigma} \partial x^{\beta}} \, + \, \frac{\partial \bar{x}^{\nu}}{\partial x^{\sigma}} \frac{\partial^2 x^{\sigma}}{\partial x^{\beta} \partial \bar{x}^{\nu}} \\
| |
| | |
| & = \, \frac{\partial x^{\sigma}}{\partial \bar{x}^{\nu}}
| |
| \frac{\partial^2 \bar{x}^{\nu}}{\partial x^{\beta} \partial x^{\sigma}} \, + \, \frac{\partial^2 x^{\sigma}}{\partial x^{\beta} \partial \bar{x}^{\nu}} \frac{\partial \bar{x}^{\nu}}{\partial x^{\sigma}} \, = \, \frac{\partial}{\partial x^{\beta}} \left( \frac{\partial x^{\sigma}}{\partial \bar{x}^{\nu}} \, \frac{\partial \bar{x}^{\nu}}{\partial x^{\sigma}} \right) \\
| |
| | |
| & = \, \frac{\partial}{\partial x^{\beta}} \left( \, \frac{\partial \bar{x}^{\nu}}{\partial \bar{x}^{\nu}} \right) \, = \, \frac{\partial}{\partial x^{\beta}} \left( \mathbf{4} \right) \, = \, 0 \,.
| |
| \end{align}</math>
| |
| | |
| ==Lorentz force density==
| |
| The density of the [[Lorentz force]] is a covariant vector density given by
| |
| :<math>f_{\mu} \, = \, F_{\mu \nu} \, J^{\nu} \,.</math>
| |
| | |
| The force on a test particle subject only to gravity and electromagnetism is
| |
| :<math>\frac{d p_{\alpha}}{d t} \, = \, \Gamma^{\beta}_{\alpha \gamma} \, p_{\beta} \, \frac{d x^{\gamma}}{d t} \, + \, q \, F_{\alpha \gamma} \, \frac{d x^{\gamma}}{d t} \,</math>
| |
| | |
| where ''p''<sub>α</sub> is the linear 4-momentum of the particle, ''t'' is any time coordinate parameterizing the world line of the particle, Γ<sup>β</sup><sub>αγ</sub> 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 <math>d t/d \bar{t}</math> 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
| |
| :<math>\bar{\Gamma}^{\beta}_{\alpha \gamma} \, = \,
| |
| \frac{\partial \bar{x}^{\beta}}{\partial x^{\epsilon}} \,
| |
| \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \,
| |
| \frac{\partial x^{\zeta}}{\partial \bar{x}^{\gamma}} \,
| |
| \Gamma^{\epsilon}_{\delta \zeta} \,
| |
| +
| |
| \frac{\partial \bar{x}^{\beta}}{\partial x^{\eta}}\,
| |
| \frac{\partial^2 x^{\eta}}{\partial \bar{x}^{\alpha} \partial \bar{x}^{\gamma}} \,</math>
| |
| | |
| we get
| |
| | |
| :<math>\begin{align}
| |
| & \frac{d \bar{p}_{\alpha}}{d t} \, - \, \bar{\Gamma}^{\beta}_{\alpha \gamma} \, \bar{p}_{\beta} \, \frac{d \bar{x}^{\gamma}}{d t} \, - \, q \, \bar{F}_{\alpha \gamma} \, \frac{d \bar{x}^{\gamma}}{d t} \\
| |
| & = \, \frac{d}{d t} \left( \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, p_{\delta} \right) \, - \,
| |
| | |
| \left(
| |
| \frac{\partial \bar{x}^{\beta}}{\partial x^{\theta}} \,
| |
| \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \,
| |
| \frac{\partial x^{\iota}}{\partial \bar{x}^{\gamma}} \,
| |
| \Gamma^{\theta}_{\delta \iota} + \, \frac{\partial \bar{x}^{\beta}}{\partial x^{\eta}}\,
| |
| \frac{\partial^2 x^{\eta}}{\partial \bar{x}^{\alpha} \partial \bar{x}^{\gamma}}
| |
| \right) \, \frac{\partial x^{\epsilon}}{\partial \bar{x}^{\beta}} \, p_{\epsilon} \, \frac{\partial \bar{x}^{\gamma}}{\partial x^{\zeta}} \, \frac{d x^{\zeta}}{d t} \, - \, q \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, F_{\delta \zeta} \, \frac{d x^{\zeta}}{d t} \\
| |
| | |
| & = \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, \left(
| |
| \frac{d p_{\delta}}{d t} \, - \, \Gamma^{\epsilon}_{\delta \zeta} \, p_{\epsilon} \, \frac{d x^{\zeta}}{d t} \, - \, q \, F_{\delta \zeta} \, \frac{d x^{\zeta}}{d t} \right) + \frac{d}{d t} \left( \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \right) \, p_{\delta} \, - \,
| |
| \left( \frac{\partial \bar{x}^{\beta}}{\partial x^{\eta}}\,
| |
| \frac{\partial^2 x^{\eta}}{\partial \bar{x}^{\alpha} \partial \bar{x}^{\gamma}}
| |
| \right) \, \frac{\partial x^{\epsilon}}{\partial \bar{x}^{\beta}} \, p_{\epsilon} \, \frac{\partial \bar{x}^{\gamma}}{\partial x^{\zeta}} \, \frac{d x^{\zeta}}{d t} \\
| |
| | |
| & = \, 0 \, + \, \frac{d}{d t} \left( \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \right) \, p_{\delta} \, - \,
| |
| \frac{\partial^2 x^{\epsilon}}{\partial \bar{x}^{\alpha} \partial \bar{x}^{\gamma}} p_{\epsilon} \, \frac{d \bar{x}^{\gamma}}{d t} \, = \, 0 \ .
| |
| | |
| \end{align}</math>
| |
| | |
| ==Lagrangian==
| |
| In a vacuum, the [[Lagrangian]] for classical electrodynamics (in joules/meter<sup>3</sup>) is a scalar [[tensor density|density]]
| |
| | |
| :<math> \mathcal{L} \, = \, - \frac{1}{4 \mu_0} \, F_{\alpha \beta} \, F^{\alpha \beta} \, \sqrt{- g} \, + \, A_{\alpha} \, J^{\alpha} \,</math>
| |
| | |
| where
| |
| | |
| :<math>F^{\alpha \beta} = g^{\alpha \gamma} F_{\gamma \delta} g^{\delta \beta} \,.</math>
| |
| | |
| 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
| |
| :<math> \mathcal{L} \, = \, - \frac{1}{4 \mu_0} \, F_{\alpha \beta} \, F^{\alpha \beta} \, \sqrt{- g} \, + \,A_{\alpha} \, J^{\alpha}_{\text{free}} \, + \, \frac12 \, F_{\alpha \beta} \, \mathcal{M}^{\alpha \beta} \,.</math>
| |
| | |
| ==Electromagnetic stress-energy tensor==
| |
| {{main|Electromagnetic stress-energy tensor}}
| |
| | |
| As part of the source term in the [[Einstein field equations]], the electromagnetic [[stress-energy tensor]] is a covariant symmmetric tensor
| |
| :<math>T_{\mu \nu} \, = \, - \frac{1}{\mu_0} ( F_{\mu \alpha} g^{\alpha \beta} F_{\beta \nu} \, - \, \frac{1}{4} g_{\mu \nu} \, F_{\sigma \alpha} g^{\alpha \beta} F_{\beta \rho} g^{\rho \sigma} ) \,</math>
| |
| | |
| using a metric of signature (-,+,+,+). If using the metric with signature (+,-,-,-), the expression for <math>T_{\mu \nu}</math> will have opposite sign. The stress-energy tensor is trace-free
| |
| | |
| :<math>T_{\mu \nu} g^{\mu \nu} \, = \, 0 \,</math>
| |
| | |
| 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
| |
| :<math>\mathfrak{T}_{\mu}^{\nu} = T_{\mu \gamma} \, g^{\gamma \nu} \, \sqrt{-g}.</math>
| |
| | |
| From the equations above, one can show that
| |
| :<math>{\mathfrak{T}_{\mu}^{\nu}}_{; \nu} \, + \, f_{\mu} \, = \, 0 \,</math>
| |
| | |
| where the semicolon indicates a [[covariant derivative]].
| |
| | |
| This can be rewritten as
| |
| :<math>- {\mathfrak{T}_{\mu}^{\nu}}_{, \nu} \, = \, - \Gamma^{\sigma}_{\mu \nu} \mathfrak{T}_{\sigma}^{\nu} \, + \, f_{\mu} \,</math> | |
| | |
| 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
| |
| :<math>\begin{align}
| |
| {\mathfrak{T}_{\mu}^{\nu}}_{; \nu} \, + \, f_{\mu} \, & = \, - \frac{1}{\mu_0} ( F_{\mu \alpha ; \nu} g^{\alpha \beta} F_{\beta \gamma} g^{\gamma \nu} \, + \, F_{\mu \alpha} g^{\alpha \beta} F_{\beta \gamma ; \nu} g^{\gamma \nu} \, - \, \frac12 \delta_{\mu}^{\nu} \, F_{\sigma \alpha ; \nu} g^{\alpha \beta} F_{\beta \rho} g^{\rho \sigma} ) \sqrt{- g} \\
| |
| & + \frac{1}{\mu_{0}} \, F_{\mu \alpha} \, g^{\alpha \beta} \, F_{\beta \gamma ; \nu} \, g^{\gamma \nu} \, \sqrt{-g} \\
| |
| & = \, - \frac{1}{\mu_0} ( F_{\mu \alpha ; \nu} F^{\alpha \nu} \, - \, \frac12 F_{\sigma \alpha ; \mu} F^{\alpha \sigma} ) \sqrt{- g}\\
| |
| & = \, - \frac{1}{\mu_0} ( (- F_{\nu \mu ; \alpha} - F_{\alpha \nu ; \mu}) F^{\alpha \nu} \, - \, \frac12 F_{\sigma \alpha ; \mu} F^{\alpha \sigma} ) \sqrt{- g} \\
| |
| & = \, - \frac{1}{\mu_0} ( F_{\mu \nu ; \alpha} F^{\alpha \nu} - F_{\alpha \nu ; \mu} F^{\alpha \nu} \, + \, \frac12 F_{\sigma \alpha ; \mu} F^{\sigma \alpha} ) \sqrt{- g} \\
| |
| & = \, - \frac{1}{\mu_0} ( F_{\mu \alpha ; \nu} F^{\nu \alpha} - \frac12 F_{\alpha \nu ; \mu} F^{\alpha \nu} ) \sqrt{- g} \\
| |
| & = \, - \frac{1}{\mu_0} (- F_{\mu \alpha ; \nu} F^{\alpha \nu} \, + \, \frac12 F_{\sigma \alpha ; \mu} F^{\alpha \sigma} ) \sqrt{- g} \ ,
| |
| \end{align}</math>
| |
| | |
| 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 [[Formulation of Maxwell's equations in special relativity|special relativity form]] to
| |
| | |
| :<math> \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} </math>
| |
| | |
| where ''R<sub>acbd</sub>'' is the covariant form of the [[Riemann tensor]] and <math> \Box </math> is a generalization of the [[d'Alembertian]] operator for covariant derivatives. Using
| |
| | |
| :<math> \Box A^{a} = {{A^{a; }}^{b}}_{ b} </math>
| |
| | |
| Maxwell's source equations can be written in terms of the [[Four-potential|4-potential]] [ref 2, p. 569] as,
| |
| | |
| :<math> \Box A^{a} - {A^{b;a}}_{ b} = - \mu_0 J^{ a } </math>
| |
| | |
| or, assuming the generalization of the [[Lorenz gauge]] in curved spacetime
| |
| | |
| :<math> {A^{ a }}_{ ; a } = 0 \ ,</math>
| |
| :<math> \Box A^{a} = - \mu_0 J^{ a } + {R^{ a }}_{ b } A^{ b } </math>
| |
| | |
| where <math> R_{ a b } \ \stackrel{\mathrm{def}}{=}\ {R^{ s }}_{ a s b } </math> 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<sup>a</sup>'' plays the role of the 4-position.
| |
| | |
| == 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]]
| |
| | |
| :<math> G_{ab} = \frac{8 \pi G} {c^4} T_{ab} </math>
| |
| | |
| where
| |
| | |
| :<math> {G}_{a b} \ \stackrel{\mathrm{def}}{=}\ {R}_{a b} - {1 \over 2} {R} g_{a b} </math>
| |
| is the [[Einstein tensor]], ''G'' is the [[gravitational constant]], ''g<sub>ab</sub>'' 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 ==
| |
| | |
| In the differential geometric formulation of the electromagnetic field, the antisymmetric Faraday tensor can be considered as the [[differential form#Applications in physics|Faraday 2-form]] '''F'''. In this view, one of Maxwell's two equations is d'''F'''= '''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 [[differential form#Applications in physics|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]] <math>F(\nabla)</math> of a ''U''(1)-[[Connection (vector bundle)|connection]] <math>\nabla</math> on a [[principal bundle|principal ''U''(1)-bundle]] whose sections represent charged fields. The connection is much like the vector potential since every connection can be written as <math>\nabla = \nabla_0 + iA</math> for a "base" connection <math>\nabla_0</math> and '''F''' = '''F'''<sub>0</sub> + 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 like the Riemann tensor is the [[holonomy]] of the Levi Civita connection along an infinitessimal closed curve, the curvature of the connection [[holonomy]] of the U(1)-connection.
| |
| | |
| ==See also==
| |
| {{Wikipedia books|Maxwell's equations}}
| |
| * [[Electromagnetic wave equation]]
| |
| * [[Nonhomogeneous electromagnetic wave equation]]
| |
| * [[Mathematical descriptions of the electromagnetic field]]
| |
| * [[Formulation of Maxwell's equations in special relativity]]
| |
| * [[Theoretical motivation for general relativity]]
| |
| * [[Basic introduction to the mathematics of curved spacetime]]
| |
| * [[Electrovacuum solution]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{cite book | last=Einstein |first=A. |authorlink=Albert Einstein | title=Relativity: The Special and General Theory | location= New York | publisher=Crown| year=1961 | isbn=0-517-02961-8}}
| |
| *{{cite book | last1=Misner |first1=Charles W. |authorlink1=Charles W. Misner |last2=Thorne |first2=Kip S. |authorlink2=Kip Thorne |last3=Wheeler |first3=John Archibald |authorlink3=John Archibald Wheeler |title=[[Gravitation (book)|Gravitation]] |location=San Francisco |publisher=W. H. Freeman |year=1973 |isbn=0-7167-0344-0}}
| |
| *{{cite book | last1=Landau |first1=L. D. |authorlink1=Lev Landau |last2=Lifshitz |first2=E. M. |authorlink2=Evgeny Lifshitz | title=Classical Theory of Fields |edition=Fourth Revised English | location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7}}
| |
| *{{cite book | last1=Feynman |first1=R. P. |authorlink1=Richard Feynman |last2=Moringo |first2=F. B. |last3=Wagner |first3=W. G. | title=Feynman Lectures on Gravitation | publisher=Addison-Wesley | year=1995 | isbn=0-201-62734-5}}
| |
| | |
| ==External links==
| |
| * [http://www.iop.org/EJ/abstract/0264-9381/22/2/011/ Electromagnetic fields in curved spacetimes]
| |
| | |
| {{Physics-footer}}
| |
| {{tensors}}
| |
| | |
| [[Category:Concepts in physics]]
| |
| [[Category:General relativity|Maxwell's equations in curved spacetime]]
| |
| [[Category:Partial differential equations|Maxwell's equations in curved spacetime]]
| |
| [[Category:Maxwell's equations|Curved spacetime]]
| |