Annus MirabilisEinstein in 1905
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On the Electrodynamics of Moving Bodies

Zur Elektrodynamik bewegter Körper

by A. Einstein

I. Kinematical Part.

II. Electrodynamical Part.

§ 6. Transformation of the Maxwell-Hertz Equations for Empty Space. On the Nature of the Electromotive Forces Occurring with Motion in a Magnetic Field.

Let the Maxwell-Hertz equations for empty space hold for the stationary system KK, so that we have: 1V∂X∂t=∂N∂y−∂M∂z,1V∂L∂t=∂Y∂z−∂Z∂y,1V∂Y∂t=∂L∂z−∂N∂x,1V∂M∂t=∂Z∂x−∂X∂z,1V∂Z∂t=∂M∂x−∂L∂y,1V∂N∂t=∂X∂y−∂Y∂x,\begin{aligned} \frac{1}{V} \frac{\partial X}{\partial t} &= \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} , &\qquad \frac{1}{V} \frac{\partial L}{\partial t} &= \frac{\partial Y}{\partial z} - \frac{\partial Z}{\partial y} , \\ \frac{1}{V} \frac{\partial Y}{\partial t} &= \frac{\partial L}{\partial z} - \frac{\partial N}{\partial x} , &\qquad \frac{1}{V} \frac{\partial M}{\partial t} &= \frac{\partial Z}{\partial x} - \frac{\partial X}{\partial z} , \\ \frac{1}{V} \frac{\partial Z}{\partial t} &= \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} , &\qquad \frac{1}{V} \frac{\partial N}{\partial t} &= \frac{\partial X}{\partial y} - \frac{\partial Y}{\partial x} , \end{aligned}Speed of lightElectric field, stationary systemTime in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemElectric field, stationary systemElectric field, stationary systemPosition in the stationary systemExplain this equation where (X,Y,Z)(X, Y, Z) denotes the vector of the electric force, and (L,M,N)(L, M, N) that of the magnetic force.

If we apply to these equations the transformation developed in § 3, by referring the electromagnetic processes to the coordinate system introduced there, moving with the velocity vv, we obtain the equations: 1V∂X∂τ=∂β(N−vVY)∂η−∂β(M+vVZ)∂ζ,1V∂β(Y−vVN)∂τ=∂L∂ζ−∂β(N−vVY)∂ξ,1V∂β(Z+vVM)∂τ=∂β(M+vVZ)∂ξ−∂L∂η,1V∂L∂τ=∂β(Y−vVN)∂ζ−∂β(Z+vVM)∂η,1V∂β(M+vVZ)∂τ=∂β(Z+vVM)∂ξ−∂X∂ζ,1V∂β(N−vVY)∂τ=∂X∂η−∂β(Y−vVN)∂ξ,\begin{aligned} \frac{1}{V} \frac{\partial X}{\partial \tau} &= \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \eta} - \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \tau} &= \frac{\partial L}{\partial \zeta} - \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \xi} , \\ \frac{1}{V} \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \tau} &= \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \xi} - \frac{\partial L}{\partial \eta} , \\ \frac{1}{V} \frac{\partial L}{\partial \tau} &= \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \zeta} - \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \eta} , \\ \frac{1}{V} \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \tau} &= \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \xi} - \frac{\partial X}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \tau} &= \frac{\partial X}{\partial \eta} - \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \xi} , \end{aligned}Speed of lightElectric field, stationary systemTime in the moving systemLorentz factorMagnetic field, stationary systemSpeed of the moving systemElectric field, stationary systemPosition in the moving systemMagnetic field, stationary systemElectric field, stationary systemPosition in the moving systemMagnetic field, stationary systemPosition in the moving systemExplain this equation where β=11−(vV)2.\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .Lorentz factorSpeed of the moving systemSpeed of lightExplain this equation

The principle of relativity now requires that the Maxwell-Hertz equations for empty space hold also in the system kk if they hold in the system KK, that is, that for the vectors of the electric and magnetic force ((X′,Y′Z′)(X', Y' Z') and (L′,M′,N′)(L', M', N')) of the moving system kk, which are defined in the moving system kk by their ponderomotive actions on electric or magnetic masses respectively, the equations hold: 1V∂X′∂τ=∂N′∂η−∂M′∂ζ,1V∂L′∂τ=∂Y′∂ζ−∂Z′∂η,1V∂Y′∂τ=∂L′∂ζ−∂N′∂ξ,1V∂M′∂τ=∂Z′∂ξ−∂X′∂ζ,1V∂Z′∂τ=∂M′∂ξ−∂L′∂η,1V∂N′∂τ=∂X′∂η−∂Y′∂ξ.\begin{aligned} \frac{1}{V} \frac{\partial X'}{\partial \tau} &= \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta} , &\qquad \frac{1}{V} \frac{\partial L'}{\partial \tau} &= \frac{\partial Y'}{\partial \zeta} - \frac{\partial Z'}{\partial \eta} , \\ \frac{1}{V} \frac{\partial Y'}{\partial \tau} &= \frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} , &\qquad \frac{1}{V} \frac{\partial M'}{\partial \tau} &= \frac{\partial Z'}{\partial \xi} - \frac{\partial X'}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial Z'}{\partial \tau} &= \frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} , &\qquad \frac{1}{V} \frac{\partial N'}{\partial \tau} &= \frac{\partial X'}{\partial \eta} - \frac{\partial Y'}{\partial \xi} . \end{aligned}Speed of lightElectric field, moving systemTime in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemElectric field, moving systemElectric field, moving systemPosition in the moving systemExplain this equation

Evidently the two systems of equations found for the system kk must now express exactly the same thing, since both systems of equations are equivalent to the Maxwell-Hertz equations for the system KK. Since, further, the equations of the two systems agree except for the symbols representing the vectors, it follows that the functions occurring at corresponding places in the systems of equations must agree up to a factor ψ(v)\psi(v) that is common to all the functions of the one system of equations, independent of ξ\xi, η\eta, ζ\zeta and τ\tau, and possibly dependent on vv. Thus the relations hold: X′=ψ(v)X,L′=ψ(v)L,Y′=ψ(v)β(Y−vVN),M′=ψ(v)β(M+vVZ),Z′=ψ(v)β(Z+vVM),N′=ψ(v)β(N−vVY).\begin{aligned} X' &= \psi(v) X , &\qquad L' &= \psi(v) L , \\ Y' &= \psi(v) \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \psi(v) \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \psi(v) \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \psi(v) \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}Electric field, moving systemUnknown scale factor: psi(v)Speed of the moving systemElectric field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, moving systemLorentz factorElectric field, stationary systemSpeed of lightMagnetic field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, stationary systemElectric field, moving systemMagnetic field, moving systemExplain this equation

If we now form the inverse of this system of equations, first by solving the equations just obtained, and second by applying the equations to the inverse transformation (from kk to KK), which is characterized by the velocity −v-v, it follows, taking into account that the two systems of equations so obtained must be identical: φ(v)⋅φ(−v)=1.\varphi(v) \cdot \varphi(-v) = 1 .Unknown scale factor: psi(v)Speed of the moving systemExplain this equation Further, it follows from reasons of symmetry[1)] that φ(v)=φ(−v);\varphi(v) = \varphi(-v) ;Unknown scale factor: psi(v)Speed of the moving systemExplain this equation hence φ(v)=1,\varphi(v) = 1 ,Unknown scale factor: psi(v)Speed of the moving systemExplain this equation and our equations take the form: X′=X,L′=L,Y′=β(Y−vVN),M′=β(M+vVZ),Z′=β(Z+vVM),N′=β(N−vVY).\begin{aligned} X' &= X , &\qquad L' &= L , \\ Y' &= \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}Electric field, moving systemElectric field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, moving systemLorentz factorElectric field, stationary systemSpeed of the moving systemSpeed of lightMagnetic field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, stationary systemElectric field, moving systemMagnetic field, moving systemExplain this equation For the interpretation of these equations we remark the following. Let there be a point-like quantity of electricity which, measured in the stationary system KK, is of magnitude “one”, that is, which, at rest in the stationary system, exerts the force of 1 dyne on an equal quantity of electricity at a distance of 1 cm. By the principle of relativity this electric mass is also of magnitude “one” when measured in the moving system. If this quantity of electricity is at rest relative to the stationary system, then by definition the vector (X,Y,Z)(X, Y, Z) is equal to the force acting on it. If the quantity of electricity is at rest relative to the moving system (at least at the instant in question), then the force acting on it, measured in the moving system, is equal to the vector (X′,Y′,Z′)(X', Y', Z'). Consequently the first three of the above equations can be put into words in the following two ways:

1. If a point-like electric unit pole moves in an electromagnetic field, there acts on it, besides the electric force, an “electromotive force”, which, neglecting terms multiplied by the second and higher powers of v/Vv/V, is equal to the vector product of the velocity of motion of the unit pole and the magnetic force, divided by the velocity of light. (Old mode of expression.)

2. If a point-like electric unit pole moves in an electromagnetic field, the force acting on it is equal to the electric force present at the location of the unit pole, which is obtained by transforming the field to a coordinate system at rest relative to the electric unit pole. (New mode of expression.)

Something analogous holds for the “magnetomotive forces”. One sees that in the theory developed here the electromotive force plays only the part of an auxiliary concept, which owes its introduction to the circumstance that the electric and magnetic forces have no existence independent of the state of motion of the coordinate system.

It is further clear that the asymmetry mentioned in the introduction, which arises when we consider the currents produced by the relative motion of a magnet and a conductor, disappears. The questions about the “seat” of electrodynamic electromotive forces (unipolar machines) also become pointless.

In conclusion I remark that in working on the problem treated here my friend and colleague M. Besso stood loyally by my side, and that I owe him several valuable suggestions.

Bern, June 1905.

(Received 30 June 1905.)