Classical electromagnetism and special relativity

Lorentz boost of an electric charge.

Top: The charge is at rest in frame F, so this observer sees a static electric field. An observer in another frame F′ moves with velocity v relative to F, and sees the charge move with velocity −v with an altered electric field E due to length contraction and a magnetic field B due to the motion of the charge.

Bottom: Similar setup, with the charge at rest in frame F′.

This equation, also called the Joules-Bernoulli equation, considers two inertial frames. The primed frame is moving relative to the unprimed frame at velocity v. Fields defined in the primed frame are indicated by primes, and fields defined in the unprimed frame lack primes. The field components parallel to the velocity v are denoted by ${\displaystyle \mathbf {E} _{\parallel }}$ and ${\displaystyle \mathbf {B} _{\parallel }}$ while the field components perpendicular to v are denoted as ${\displaystyle \mathbf {E} _{\perp }}$and ${\displaystyle \mathbf {B} _{\perp }}$. In these two frames moving at relative velocity v, the E-fields and B-fields are related by:[2]

${\displaystyle {\begin{aligned}\mathbf {E_{\parallel }} '&=\mathbf {E_{\parallel }} \\\mathbf {B_{\parallel }} '&=\mathbf {B_{\parallel }} \\\mathbf {E_{\bot }} '&=\gamma \left(\mathbf {E} _{\bot }+\mathbf {v} \times \mathbf {B} \right)\\\mathbf {B_{\bot }} '&=\gamma \left(\mathbf {B} _{\bot }-{\frac {1}{c^{2}}}\mathbf {v} \times \mathbf {E} \right)\end{aligned}}}$

where

${\displaystyle \gamma \ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$

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:[3]

${\displaystyle {\begin{aligned}\mathbf {E} '&=\gamma \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)-\left({\gamma -1}\right)(\mathbf {E} \cdot \mathbf {\hat {v}} )\mathbf {\hat {v}} \\\mathbf {B} '&=\gamma \left(\mathbf {B} -{\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}\right)-\left({\gamma -1}\right)(\mathbf {B} \cdot \mathbf {\hat {v}} )\mathbf {\hat {v}} \end{aligned}}}$

where ${\displaystyle \mathbf {\hat {v}} ={\frac {\mathbf {v} }{\Vert \mathbf {v} \Vert }}}$ is the velocity unit vector. With previous notations, one actually has ${\displaystyle (\mathbf {E} \cdot \mathbf {\hat {v}} )\mathbf {\hat {v}} =\mathbf {E} _{\parallel }}$ and ${\displaystyle (\mathbf {B} \cdot \mathbf {\hat {v}} )\mathbf {\hat {v}} =\mathbf {B} _{\parallel }}$.

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 below).

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

${\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {u} \times \mathbf {B} }$

In frame S', the Lorentz force is:

${\displaystyle \mathbf {F'} =q\mathbf {E'} +q\mathbf {u'} \times \mathbf {B'} }$

If S and S' have aligned axes then:[4]

${\displaystyle {\begin{aligned}u_{x}'&={\frac {u_{x}+v}{1+(v\ u_{x})/c^{2}}}\\u_{y}'&={\frac {u_{y}/\gamma }{1+(v\ u_{x})/c^{2}}}\\u_{z}'&={\frac {u_{z}/\gamma }{1+(v\ u_{x})/c^{2}}}\end{aligned}}}$

A derivation for the transformation of the Lorentz force for the particular case u = 0 is given here.[5] A more general one can be seen here.[6]

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

${\displaystyle {\begin{aligned}E'_{x}&=E_{x}&\qquad B'_{x}&=B_{x}\\E'_{y}&=\gamma \left(E_{y}-vB_{z}\right)&B'_{y}&=\gamma \left(B_{y}+{\frac {v}{c^{2}}}E_{z}\right)\\E'_{z}&=\gamma \left(E_{z}+vB_{y}\right)&B'_{z}&=\gamma \left(B_{z}-{\frac {v}{c^{2}}}E_{y}\right).\\\end{aligned}}}$

The transformations in this form can be made more compact by introducing the electromagnetic tensor (defined below), which is a covariant tensor.

For the electric displacement D and magnetic intensity H, using the constitutive relations and the result for c²:

${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} \,,\quad \mathbf {B} =\mu _{0}\mathbf {H} \,,\quad c^{2}={\frac {1}{\epsilon _{0}\mu _{0}}}\,,}$

gives

${\displaystyle {\begin{aligned}\mathbf {D} '&=\gamma \left(\mathbf {D} +{\frac {1}{c^{2}}}\mathbf {v} \times \mathbf {H} \right)+(1-\gamma )(\mathbf {D} \cdot \mathbf {\hat {v}} )\mathbf {\hat {v}} \\\mathbf {H} '&=\gamma \left(\mathbf {H} -\mathbf {v} \times \mathbf {D} \right)+(1-\gamma )(\mathbf {H} \cdot \mathbf {\hat {v}} )\mathbf {\hat {v}} \end{aligned}}}$

Analogously for E and B, the D and H form the electromagnetic displacement tensor.

An alternative simpler transformation of the EM field uses the electromagnetic potentials - the electric potential φ and magnetic potential A:[7]

${\displaystyle {\begin{aligned}\varphi '&=\gamma \left(\varphi -vA_{\parallel }\right)\\A_{\parallel }'&=\gamma \left(A_{\parallel }-{\frac {v\varphi }{c^{2}}}\right)\\A_{\bot }'&=A_{\bot }\end{aligned}}}$

where ${\displaystyle \scriptstyle A_{\parallel }}$ is the parallel component of A to the direction of relative velocity between frames v, and ${\displaystyle \scriptstyle A_{\bot }}$ 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:

${\displaystyle {\begin{aligned}\mathbf {A} '&=\mathbf {A} -{\frac {\gamma \varphi }{c^{2}}}\mathbf {v} +\left(\gamma -1\right)\left(\mathbf {A} \cdot \mathbf {\hat {v}} \right)\mathbf {\hat {v}} \\\varphi '&=\gamma \left(\varphi -\mathbf {A} \cdot \mathbf {v} \right)\end{aligned}}}$

Analogously for the charge density ρ and current density J,[7]

${\displaystyle {\begin{aligned}J_{\parallel }'&=\gamma \left(J_{\parallel }-v\rho \right)\\\rho '&=\gamma \left(\rho -{\frac {v}{c^{2}}}J_{\parallel }\right)\\J_{\bot }'&=J_{\bot }\end{aligned}}}$

Collecting components together:

${\displaystyle {\begin{aligned}\mathbf {J} '&=\mathbf {J} -\gamma \rho \mathbf {v} +\left(\gamma -1\right)\left(\mathbf {J} \cdot \mathbf {\hat {v}} \right)\mathbf {\hat {v}} \\\rho '&=\gamma \left(\rho -{\frac {\mathbf {J} \cdot \mathbf {v} }{c^{2}}}\right)\end{aligned}}}$

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

${\displaystyle {\begin{aligned}\mathbf {E} '&\approx \mathbf {E} +\mathbf {v} \times \mathbf {B} \\\mathbf {B} '&\approx \mathbf {B} -{\frac {1}{c^{2}}}\mathbf {v} \times \mathbf {E} \\\mathbf {J} '&\approx \mathbf {J} -\rho \mathbf {v} \\\rho '&\approx \left(\rho -{\frac {1}{c^{2}}}\mathbf {J} \cdot \mathbf {v} \right)\end{aligned}}}$

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

The chosen reference frame determines whether an electromagnetic phenomenon is viewed as an effect of electrostatics or magnetism or a combination of the two. 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[9] and Landau.[10]

The above transformation rules show that the electric field in one frame contributes to the magnetic field in another frame, and vice versa.[11] 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.

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 currents 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.[12]

The above relativistic transformations suggest the electric and magnetic fields are coupled together, in a mathematical object with 6 components: an antisymmetric second-rank tensor, or a bivector. This is called the electromagnetic field tensor, usually written as F^μν. In matrix form:[13]

${\displaystyle F^{\mu \nu }={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}}$

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^μν.

${\displaystyle G^{\mu \nu }={\begin{pmatrix}0&-B_{x}&-B_{y}&-B_{z}\\B_{x}&0&E_{z}/c&-E_{y}/c\\B_{y}&-E_{z}/c&0&E_{x}/c\\B_{z}&E_{y}/c&-E_{x}/c&0\end{pmatrix}}}$

In the context of special relativity, both of these transform according to the Lorentz transformation according to

${\displaystyle F'^{\alpha \beta }=\Lambda _{\mu }^{\alpha }\Lambda _{\nu }^{\beta }F^{\mu \nu }}$,

where Λ^α_ν 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

${\displaystyle J^{\alpha }=\left(c\rho ,J_{x},J_{y},J_{z}\right)}$

called the four-current.

Using these tensors, Maxwell's equations reduce to:[13]

where the partial derivatives may be written in various ways, see 4-gradient. The first equation listed above corresponds to both Gauss's Law (for β = 0) and the 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^αβ to obtain F_αβ (see raising and lowering indices):

${\displaystyle F_{\alpha \beta }=\eta _{\alpha \lambda }\eta _{\beta \mu }F^{\lambda \mu }}$

the second equation can be written in terms of F_αβ as:

${\displaystyle \epsilon ^{\delta \alpha \beta \gamma }{\dfrac {\partial F_{\beta \gamma }}{\partial x^{\alpha }}}={\dfrac {\partial F_{\alpha \beta }}{\partial x^{\gamma }}}+{\dfrac {\partial F_{\gamma \alpha }}{\partial x^{\beta }}}+{\dfrac {\partial F_{\beta \gamma }}{\partial x^{\alpha }}}=0}$

where ${\displaystyle \epsilon ^{\alpha \beta \gamma \delta }}$ is the contravariant Levi-Civita symbol. Notice the cyclic permutation of indices in this equation: ${\displaystyle {\begin{array}{rc}&\scriptstyle {\alpha \,\,\longrightarrow \,\,\beta }\\&\nwarrow _{\gamma }\swarrow \end{array}}}$.

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.

The EM field tensor can also be written[14]

${\displaystyle F^{\alpha \beta }={\frac {\partial A^{\beta }}{\partial x_{\alpha }}}-{\frac {\partial A^{\alpha }}{\partial x_{\beta }}}\,,}$

where

${\displaystyle A^{\alpha }=\left({\frac {\varphi }{c}},A_{x},A_{y},A_{z}\right)\,,}$

is the four-potential and

${\displaystyle x_{\alpha }=(ct,-x,-y,-z)}$

is the 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,[15] or the covariant form of the Maxwell equations[16]):

where ${\displaystyle \Box }$ is the d'Alembertian operator, or four-Laplacian. For a more comprehensive presentation of these topics, see Covariant formulation of classical electromagnetism.