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{{About|the contribution of special relativity to the modern theory of classical electromagnetism|the contribution of classical electromagnetism to the development of special relativity|History of special relativity|a fully covariant discussion|Covariant formulation of classical electromagnetism}}
{{electromagnetism|cTopic=[[Covariant formulation of classical electromagnetism|Covariant formulation]]}}

The theory of [[special relativity]] plays an important role in the modern theory of [[classical electromagnetism]]. First of all, it gives formulas for how electromagnetic objects, in particular the [[electric field|electric]] and [[magnetic field]]s, are altered under a [[Lorentz transformation]] from one [[inertial frame]] of reference to another. Secondly, it sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. Third, it motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatible with special relativity.<ref>Questions remain about the treatment of accelerating charges: Haskell, "[http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm Special relativity and Maxwell's equations.]"</ref> Moreover, the apparent coincidences in which the same effect was observed due to different physical phenomena by two different observers would be shown to be not coincidental in the least by special relativity. In fact, half of Einstein's 1905 first paper on special relativity, "[[Annus Mirabilis Papers#Special relativity|On the Electrodynamics of Moving Bodies]]," explains how to transform Maxwell's equations.

==Transformation of the fields between inertial frames==

===The E and B fields===

This equation, also called the '''Joules-Bernoulli equation''', considers two [[inertial frames]]. As notation, the field variables in one frame are ''unprimed'', and in a frame moving relative to the unprimed frame at velocity '''v''', the fields are denoted with ''primes''. In addition, the fields ''parallel'' to the velocity '''v''' are denoted by <math>\stackrel{\mathbf {E}_{\parallel}}{}</math> while the fields perpendicular to '''v''' are denoted as <math>\stackrel{\mathbf {E}_{\bot}}{}</math>. In these two frames moving at relative velocity '''v''', the '''E'''-fields and '''B'''-fields are related by:<ref name=Chow3>
{{cite book 
|author=Tai L. Chow
|title=Electromagnetic theory
|year= 2006
|page =Chapter 10.21; p. 402–403 ff
|publisher=Jones and Bartlett 
|location=Sudbury MA
|isbn=0-7637-3827-1
|url=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn:0763738271#PPA368,M1}}
</ref>

:<math>\begin{align} 
& \mathbf {{E}_{\parallel}}' = \mathbf {{E}_{\parallel}}\\
& \mathbf {{B}_{\parallel}}' = \mathbf {{B}_{\parallel}}\\
& \mathbf {{E}_{\bot}}'= \gamma \left( \mathbf {E}_{\bot} + \mathbf{ v} \times \mathbf {B} \right) \\
& \mathbf {{B}_{\bot}}'= \gamma \left( \mathbf {B}_{\bot} -\frac{1}{c^2} \mathbf{ v} \times \mathbf {E} \right) 
\end{align} </math>

where 

:<math>\gamma \ \overset{\underset{\mathrm{def}}{}}{=} \ \frac{1}{\sqrt{1 - v^2/{c}^2}}</math>

is called the [[Lorentz factor]] and ''c'' is the [[speed of light]] in [[free space]]. The inverse transformations are the same except '''v''' → −'''v'''.

An equivalent, alternative expression is:<ref>{{Citation
|title=Physik: Elektrodynamik, relativistische Physik
|first1=Herbert
|last1=Daniel
|publisher=Walter de Gruyter
|year=1997
|isbn=3-11-015777-2
|chapter=4.5.1
|pages=360–361
|url=http://books.google.com/books?id=8vAC8YG41goC}}, [http://books.google.com/books?id=8vAC8YG41goC&pg=PA360 Extract of pages 360-361]
</ref>

:<math>\begin{align}
& \mathbf{E}' = \gamma \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right ) - \left ({\gamma-1} \right ) ( \mathbf{E} \cdot \mathbf{\hat{v}} ) \mathbf{\hat{v}} \\
& \mathbf{B}' = \gamma \left( \mathbf{B} - \frac {\mathbf{v} \times \mathbf{E}}{c^2} \right ) - \left ({\gamma-1} \right ) ( \mathbf{B} \cdot \mathbf{\hat{v}} ) \mathbf{\hat{v}}\\
\end{align} </math>

where '''v̂''' is the velocity [[unit vector]].

If one of the fields is zero in one frame of reference, that doesn't necessarily mean it is zero in all other frames of reference.  This can be seen by, for instance, making the unprimed electric field zero in the transformation to the primed electric field.  In this case, depending on the orientation of the magnetic field, the primed system could see an electric field, even though there is none in the unprimed system.

This does not mean two completely different sets of events are seen in the two frames, but that the same sequence of events is described in two different ways (see [[#Moving magnet and conductor problem|Moving magnet and conductor problem]] below). 

If a particle of charge ''q'' moves with velocity '''u''' with respect to frame S, then the Lorentz force in frame S is:

:<math>\mathbf{F}=q\mathbf{E}+q \mathbf{u} \times \mathbf{B}</math>

In frame S', the Lorentz force is:

:<math>\mathbf{F'}=q\mathbf{E'}+q \mathbf{u'} \times \mathbf{B'}</math>

If S and S' have aligned axes then:<ref>R.C.Tolman "Relativity Thermodynamics and Cosmology" pp25</ref>

:<math>\begin{align}
& u_x'=\frac{u_x+v}{1 + (v \ u_x)/c^2}\\
& u_y'=\frac{u_y/\gamma}{1 + (v \ u_x)/c^2}\\
& u_z'=\frac{u_z/\gamma}{1 + (v \ u_x)/c^2}
\end{align}</math>

A derivation for the transformation of the Lorentz force for the particular case '''u''' = '''0''' is given here.<ref>Force Laws and Maxwell's Equations http://www.mathpages.com/rr/s2-02/2-02.htm at MathPages</ref> A more general one can be seen here.<ref>http://www.hep.princeton.edu/~mcdonald/examples/EM/ganley_ajp_31_510_62.pdf</ref>

Component by component, for relative motion along the x-axis, this works out to be the following: 

:<math>\begin{align}
& E'_x = E_x & \qquad & B'_x = B_x \\
& E'_y = \gamma \left( E_y - v B_z \right)  & & B'_y = \gamma \left( B_y + \frac{v}{c^2} E_z \right) \\
& E'_z = \gamma \left( E_z + v B_y \right) & & B'_z = \gamma \left( B_z - \frac{v}{c^2} E_y \right). \\
\end{align}</math>

The transformations in this form can be made more compact by introducing the [[electromagnetic tensor]] (defined below), which is a [[covariance and contravariance of vectors|covariant tensor]].

===The D and H fields===

For the [[electric displacement]] '''D''' and [[magnetic field|magnetic intensity]] '''H''', using the [[constitutive relations#Electromagnetism|constitutive relations]] and the result for ''c''<sup>2</sup>:

:<math>\mathbf{D}=\epsilon_0\mathbf{E}\,, \quad \mathbf{B}=\mu_0\mathbf{H}\,,\quad c^2=\frac{1}{\epsilon_0\mu_0}\,, </math>

gives

:<math>\begin{align}
\mathbf{D}' & =\gamma \left( \mathbf{D}+\frac{1}{c^2}\mathbf{v}\times \mathbf{H} \right)+(1-\gamma )(\mathbf{D}\cdot \mathbf{\hat{v}})\mathbf{\hat{v}} \\
\mathbf{H}' & =\gamma \left( \mathbf{H}-\mathbf{v}\times \mathbf{D} \right)+(1-\gamma )(\mathbf{H}\cdot \mathbf{\hat{v}})\mathbf{\hat{v}} \\
\end{align}</math>

Analogously for '''E''' and '''B''', the '''D''' and '''H''' form the [[Covariant formulation of classical electromagnetism#Electromagnetic displacement tensor|electromagnetic displacement tensor]].

===The φ and A fields===

An alternative simpler transformation of the EM field uses the [[electromagnetic potential]]s - the [[electric potential]] φ and [[magnetic potential]] '''A''':<ref name="Physics Formulas 2010">The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.</ref>

:<math>\begin{align}
& \varphi' = \gamma (\varphi - v A_\parallel)\\
& A_\parallel' = \gamma (A_\parallel - v \varphi /c^2)\\
& A_\bot' = A_\bot
\end{align}</math>

where <math>\scriptstyle A_\parallel</math> is the parallel component of '''A''' to the direction of relative velocity between frames ''v'', and <math>\scriptstyle A_\bot</math> is the perpendicular component. These transparently resemble the characteristic form of other Lorentz transformations (like time-position and energy-momentum), while the transformations of '''E''' and '''B''' above are slightly more complicated. The components can be collected together as:

:<math>\begin{align}
\mathbf{A}' & = \mathbf{A} - \dfrac{\gamma \varphi}{c^2}\mathbf{v} + (\gamma-1) (\mathbf{A}\cdot\mathbf{\hat{v}})\mathbf{\hat{v}} \\ 
{\varphi}' & =\gamma \left( \varphi - \mathbf{A}\cdot \mathbf{v} \right) 
\end{align}</math>

===The ρ and J fields===

Analogously for the [[charge density]] ρ and [[current density]] '''J''',<ref name="Physics Formulas 2010"/>

:<math>\begin{align}
& J_\parallel' = \gamma ( J_\parallel - v\rho)\\
& \rho' = \gamma (\rho - v J_\parallel /c^2)\\
& J_\bot' = J_\bot
\end{align}</math>

Collecting components together:

:<math>\begin{align}
\mathbf{J}' & =\mathbf{J}-\gamma \rho \mathbf{v} +\left( \gamma -1 \right)(\mathbf{J}\cdot \mathbf{\hat{v}})\mathbf{\hat{v}} \\
{\rho }' & =\gamma ( \rho - \mathbf{J}\cdot \mathbf{v}/c^2) 
\end{align}</math>

===Non-relativistic approximations===

For speeds ''v'' ≪ ''c'', the relativistic factor γ ≈ 1, which yields:

:<math>\begin{align}
\mathbf{E}' & \approx \mathbf{E}+\mathbf{v}\times \mathbf{B} \\
\mathbf{B}' & \approx \mathbf{B}-\frac{1}{c^2}\mathbf{v}\times \mathbf{E} \\
\mathbf{j}' & \approx \mathbf{j}-\rho \mathbf{v}\\
\rho' & \approx \left( \rho -\frac{1}{c^2}\mathbf{j}\cdot \mathbf{v} \right) 
\end{align}</math>

so that there is no need to distinguish between the spatial and temporal coordinates in [[Maxwell's equations]].

==Relationship between electricity and magnetism==

{{cquote|One part of the force between moving charges we call the magnetic force. It is really one aspect of an electrical effect.|20px|20px|Richard Feynman<ref>[[The Feynman Lectures on Physics|Feynman Lectures]] vol. 2, ch. 1-1</ref>}}

===Deriving magnetism from electrostatics===
{{main|relativistic electromagnetism}}
The chosen reference frame determines if an electromagnetic phenomenon is viewed as an effect of electrostatics or magnetism.  Authors usually derive magnetism from electrostatics when special relativity and [[charge invariance]] are taken into account.  [[The Feynman Lectures on Physics]] (vol. 2, ch. 13-6) uses this method to derive the "magnetic" force on a moving charge next to a current-carrying wire.  See also Haskell,<ref>http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm</ref> Landau,<ref name=Landau>{{cite book 
|author=E M Lifshitz, L D Landau
|title=The classical theory of fields
|volume=Vol. 2 
|series=[[Course of Theoretical Physics]]
|edition=Fourth Edition
|publisher= Butterworth-Heinemann
|location=Oxford UK
|year=1980
|isbn=0-7506-2768-9
|url=http://worldcat.org/isbn/0750627689}}</ref> and Field.<ref>J H Field (2006) "Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its relationship to quantum electrodynamics". ''Phys. Scr.'' '''74''' 702-717</ref>

===Fields intermix in different frames===
The above transformation rules show that the electric field in one frame contributes to the magnetic field in another frame, and vice versa.<ref name=Chow>
{{cite book 
|author=Tai L. Chow
|title=Electromagnetic theory
|year= 2006
|page =395
|publisher=Jones and Bartlett 
|location=Sudbury MA
|isbn=0-7637-3827-1
|url=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn:0763738271#PPR6,M1}}
</ref> This is often described by saying that the electric field and magnetic field are two interrelated aspects of a single object, called the [[electromagnetic field]]. Indeed, the entire electromagnetic field can be encoded in a single rank-2 tensor called the [[electromagnetic tensor]]; see below.

===Moving magnet and conductor problem===
{{main|Moving magnet and conductor problem}}

A famous example of the intermixing of electric and magnetic phenomena in different frames of reference is called the "moving magnet and conductor problem", cited by Einstein in his 1905 paper on Special Relativity.

If a conductor moves with a constant velocity through the field of a stationary magnet, [[eddy current]]s will be produced due to a ''magnetic'' force on the electrons in the conductor. In the rest frame of the conductor, on the other hand, the magnet will be moving and the conductor stationary. Classical electromagnetic theory predicts that precisely the same microscopic eddy currents will be produced, but they will be due to an ''electric'' force.<ref>{{cite book 
|author=David J Griffiths
|title=Introduction to electrodynamics
|year= 1999
|edition=Third Edition
|pages=478–9
|publisher=Prentice Hall
|isbn=0-13-805326-X
|url=http://worldcat.org/isbn/013805326X}}
</ref>

==Covariant formulation in vacuum==

The laws and mathematical objects in classical electromagnetism can be written in a form which is [[manifestly covariant]].  Here, this is only done so for vacuum (or for the microscopic Maxwell equations, not using macroscopic descriptions of materials such as [[electric permittivity]]), and uses [[SI units]].

This section uses [[Einstein notation]], including [[Einstein summation convention]].  See also [[Ricci calculus]] for a summary of [[tensor]] index notations, and [[raising and lowering indices]] for definition of superscript and subscript indices, and how to switch between them. The [[Minkowski metric tensor]] η here has [[metric signature]] (+ − − −).

===Field tensor and 4-current===

{{main|Electromagnetic field tensor}}

The above relativistic transformations suggest the electric and magnetic fields are coupled together, in a mathematical object with 6 components: an [[Antisymmetric tensor|antisymmetric]] second-rank [[tensor]], or a [[bivector]].  This is called the [[electromagnetic field tensor]], usually written as ''F''<sup>μν</sup>.  In matrix form:<ref name="Griffiths, David J. 1998 557">{{cite book|page=557|author=Griffiths, David J.|title=Introduction to Electrodynamics|edition=3rd |publisher=Prentice Hall|year=1998|isbn=0-13-805326-X}}</ref> 

:<math>F^{\mu \nu} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{pmatrix}</math>

where ''c'' the [[speed of light]] - in [[natural units]] ''c'' = 1.

There is another way of merging the electric and magnetic fields into an antisymmetric tensor, by replacing '''E'''/''c'' → '''B''' and '''B''' → − '''E'''/''c'', to get the [[dual tensor]] ''G''<sup>μν</sup>.

:<math>G^{\mu \nu} = \begin{pmatrix} 0 & -B_x & -B_y & -B_z \\ B_x & 0 & E_z/c & - E_y/c \\ B_y & -E_z/c & 0 & E_x/c \\ B_z & E_y/c & -E_x/c & 0 \end{pmatrix}</math>

In the context of [[special relativity]], both of these transform according to the [[Lorentz transformation]] according to
:<math>F'^{\alpha \beta} = \Lambda^\alpha{}_\mu \Lambda^\beta{}_\nu F^{\mu \nu}</math>,
where Λ<sup>α</sup><sub>ν</sub> is the [[Lorentz transformation]] tensor for a change from one reference frame to another. The same tensor is used twice in the summation.

The charge and current density, the sources of the fields, also combine into the [[four-vector]] 

:<math>J^\alpha = \begin{pmatrix} c \rho & J_x & J_y & J_z \end{pmatrix}</math>

called the [[four-current]].

===Maxwell's equations in tensor form===

{{main|Covariant formulation of classical electromagnetism}}

Using these tensors, [[Maxwell's equations]] reduce to:<ref name="Griffiths, David J. 1998 557"/>

{{Equation box 1
|indent=:
|title='''Maxwell's equations ''' ''(Covariant formulation)''
|equation=<math> \begin{align} & \frac{\partial F^{\alpha \beta}}{\partial x^\alpha} = \mu_0 J^\beta\\
& \frac{\partial G^{\alpha \beta}}{\partial x^\alpha} = 0 
\end{align}</math> 
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

where the partial derivatives may be written in various ways, see [[Four-gradient|4-gradient]]. The first equation listed above corresponds to both [[Gauss's Law]] (for β = 0) and the [[Ampère's Law|Ampère-Maxwell Law]] (for β = 1, 2, 3).  The second equation corresponds to the two remaining equations, [[Gauss's law for magnetism]] (for β = 0) and [[Faraday's Law]] ( for β = 1, 2, 3). 

These tensor equations are [[manifestly-covariant]], meaning the equations can be seen to be covariant by the index positions. This short form of writing Maxwell's equations illustrates an idea shared amongst some physicists, namely that the laws of physics take on a simpler form when written using [[tensors]].

By lowering the indices on ''F''<sup>αβ</sup> to obtain ''F''<sub>αβ</sub> (see [[raising and lowering indices]]):

:<math>F_{\alpha\beta} = \eta_{\alpha\lambda} \eta_{\beta\mu} F^{\lambda\mu} </math>

the second equation can be written in terms of ''F''<sub>αβ</sub> as:

:<math> \epsilon^{\delta\alpha\beta\gamma} \dfrac{\partial F_{\beta\gamma}}{\partial x^\alpha} = \dfrac{\partial F_{\alpha\beta}}{\partial x^\gamma} + \dfrac{\partial F_{\gamma\alpha}}{\partial x^\beta} + \dfrac{\partial F_{\beta\gamma}}{\partial x^\alpha} = 0 </math>

where <math> \epsilon^{\alpha\beta\gamma\delta}</math> is the contravariant [[Levi-Civita symbol]]. Notice the [[cyclic permutation]] of indices in this equation: <math>\begin{array}{rc}
&  \scriptstyle{\alpha\,\, \longrightarrow \,\, \beta}  \\
&  \nwarrow_\gamma \swarrow  
\end{array}
</math>.

Another covariant electromagnetic object is the [[electromagnetic stress-energy tensor]], a covariant rank-2 tensor which includes the [[Poynting vector]], [[Maxwell stress tensor]], and electromagnetic energy density.

===4-potential===

{{main|4-potential}}

The EM field tensor can also be written<ref name =DJ_Griffiths>{{cite book |author=DJ Griffiths |title=Introduction to electrodynamics |publisher=Pearson/Addison-Wesley |year=1999 |location=Saddle River NJ |page=541 |isbn =0-13-805326-X}}</ref>

:<math> F^{\alpha \beta} = \frac {\partial A^{\beta}}{\partial x_{\alpha}} -  \frac {\partial A^{\alpha}}{\partial x_{\beta}} \, ,</math>

where

:<math> A^\alpha = (\varphi/c, A_x,A_y,A_z)\,, </math>

is the [[four-potential]] and 

:<math>x_\alpha = (ct,-x,-y,-z ) </math>

is the [[Four-vector#Four-position|four-position]]. 

Using the 4-potential in the Lorenz gauge, an alternative manifestly-covariant formulation can be found in a single equation (a generalization of an equation due to [[Bernhard Riemann]] by [[Arnold Sommerfeld]], known as the Riemann–Sommerfeld equation,<ref>
{{cite book
 | title = Collective Electrodynamics: Quantum Foundations of Electromagnetism
 | author = Carver A. Mead
 | publisher = MIT Press
 | isbn = 978-0-262-63260-7
 | pages = 37–38
 | url = http://books.google.com/?id=GkDR4e2lo2MC&pg=PA37&dq=Riemann+Summerfeld
 | date = 2002-08-07
}}</ref> or the covariant form of the Maxwell equations<ref>
{{cite book
 | title = High-field electrodynamics
 | author = Frederic V. Hartemann
 | publisher = CRC Press
 | year = 2002
 | isbn = 978-0-8493-2378-2
 | page = 102
 | url = http://books.google.com/?id=tIkflVrfkG0C&pg=PA102&dq=d%27Alembertian+covariant-form+maxwell-lorentz
}}</ref>):
{{Equation box 1
|indent=:
|title='''Maxwell's equations ''' ''(Covariant [[Lorenz gauge]] formulation)''
|equation=<math>\Box A^\mu  = \mu_0 J^\mu</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>\Box</math> is the [[d'Alembertian]] operator, or four-Laplacian. For a more comprehensive presentation of these topics, see [[Covariant formulation of classical electromagnetism]].

==See also==

* [[Maths of EM field]]

==Footnotes==

{{reflist}}

{{DEFAULTSORT:Classical Electromagnetism And Special Relativity}}
[[Category:Electromagnetism]]
[[Category:Special relativity]]