The Principle of Relativity · Albert Einstein
PART I § 2. The Limiting Case. The Fundamental Equations for Äther.
If we make use of (30) for non-magnetic bodies, and put accordingly H = B + [_u_, (D - E)], then in consequence of (C) in §8, (ε - 1) (E + [_u_, B]) = D - E + [_u_. [_u_, D - E]], _i.e._ for the direction of _u_, (ε - 1) (E + [_u_B])_{_u_} = (D - E)_{_u_} and for a perpendicular direction ū, (ε - 1) [E + (_u_B)]_{_u_} = (1 - _u²_) (D - E)_{_u_} _i.e._ it coincides with Lorentz’s assumption, if we neglect _u²_ in comparison to 1. Also to the same order of approximation, Lorentz’s form for J corresponds to the conditions imposed by the relativity principle [comp. (E) § 8]—that the components of J_{_u_}, J_{_ū_} are equal to the components of σ (E + [_u_ B]) multiplied by √(1 - _u²_) or 1 / √(1 - _u²_) respectively. §10. Fundamental Equations of E. Cohn. E. Cohn assumes the following fundamental equations. (31) Curl (M + [_u_ E]) = _d_E/_dt_ + u div. E + J - Curl [E - (_u_. M)] = _d_M/_dt_ + u div. M. (32) J = σ E, = ε E - [_u_ M], M = μ (_m_ + [_u_ E.]) where E M are the electric and magnetic field intensities (forces), E, M are the electric and magnetic polarisation (induction). The equations also permit the existence of true magnetism; if we do not take into account this consideration, div. M. is to be put = 0.
An objection to this system of equations, is that according to these, for ε = 1, μ = 1, the vectors force and induction do not coincide. If in the equations, we conceive E and M and not E - (U. M), and M + [U E] as electric and magnetic forces, and with a glance to this we substitute for E, M, E, M, div. E, the symbols _e_, M, E + [U M], _m_ - [_u_ _e_], ρ, then the differential equations transform to our equations, and the conditions (32) transform into J = σ(E + [_u_ M]) _e_ + [_u_, (_m_ - [_u_ _e_])] = ε(E + [_u_ M]) M - [_u_, (E + _u_ M)] = μ(_m_ - [_u_ _e_]) then in fact the equations of Cohn become the same as those required by the relativity principle, if errors of the order _u²_ are neglected in comparison to 1. It may be mentioned here that the equations of Hertz become the same as those of Cohn, if the auxiliary conditions are (33) E = εE, M = μM, J = σE. §11. Typical Representations of the Fundamental Equations.
In the statement of the fundamental equations, our leading idea had been that they should retain a covariance of form, when subjected to a group of Lorentz-transformations. Now we have to deal with ponderomotive reactions and energy in the electro-magnetic field. Here from the very first there can be no doubt that the settlement of this question is in some way connected with the simplest forms which can be given to the fundamental equations, satisfying the conditions of covariance. In order to arrive at such forms, I shall first of all put the fundamental equations in a typical form which brings out clearly their covariance in case of a Lorentz-transformation. Here I am using a method of calculation, which enables us to deal in a simple manner with the space-time vectors of the 1st, and 2nd kind, and of which the rules, as far as required are given below. A system of magnitudes _a__{_h_ _k_} formed into the matrix | _a₁₁_...................._a__{1 _q_} | | | | | | | | _a__{_p_ 1}..........._a__{_p_ _q_} | arranged in _p_ horizontal rows, and _q_ vertical columns is called a _p_ × _q_ series-matrix, and will be denoted by the letter A. If all the quantities _a__{_h_ _k_} are multiplied by C, the resulting matrix will be denoted by CA. If the roles of the horizontal rows and vertical columns be intercharged, we obtain a _q_ × _p_ series matrix, which will be known as the transposed matrix of A, and will be denoted by Ā.
Text from Project Gutenberg, public domain in the United States.