Hermann Minkowski

“ We look for those relations which make it possible for us — when proper fundamental data are given — to obtain the following quantities at every place and time, and therefore at every spacetime point as functions of x, y, z, t: — the vector of the electric force , the magnetic induction , the electrical induction , the magnetic force , the electrical space-density , the electric current (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process) , and lastly the vector , the velocity of matter. ”

“ I may here point out that by reforming mechanics, where instead of the Newtonian Relativity-Postulate with we assume a relativity-postulate with a finite c, then the axiomatic construction of Mechanics appears to gain considerably in perfection. ”

“ Since is always greater than we may call the quotient as the "Gain" (vorgehen) of the time over the proper-time of the material point and the law can then be thus expressed; — The kinetic energy of a material point is the product of its mass into the gain of the time over its proper-time.
The set of four equations (22) again shows the symmetry in x, y, z, it, which is demanded by the relativity postulate; to the fourth equation however, a higher physical significance is to be attached, as we have already seen in the analogous case in electrodynamics. ”

“ At every space-time point (as in § 18) , let a space time matrix of the 2nd kind be known
where are real magnitudes.
For a virtual displacement in a space-time sickle (with the previously applied designation) the value of the integral
extended over the whole range of the sickle, may be called the tensional work of the virtual displacement.
The sum which comes forth here, written in real magnitudes, is
we can now postulate the following minimum principle in mechanics. ”

“ If any space-time sickle be bounded, then for each virtual displacement in the sickle, the sum of the mass-works, and tension works shall always he an extremum for that process of the space-time line in the sickle which actually occurs. ”

“ A space time null point () will be kept fixed in a Lorentz transformation. The figure
which represents a hyperboloidal shell, contains the space-time points , and all points A', which after a Lorentz-transformation enter into the newly introduced system of reference as . The direction of a radius vector OA' drawn from to the point A' of (2) , and the directions of the tangents to (2) at A' are to be called normal to each other. ”

“ In the limitting case , the vector , and we obtain the ordinary equations in the theory of electrons.
APPENDIX. Mechanics and the Relativity-Postulate. It would be very unsatisfactory if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics.
Now many authors say that classical mechanics stand in opposition to the relativity postulate, which is taken to be the basic of the new Electro-dynamics. ”

“ The advantage of this method will be, as I expressly emphasize here, that we shall have to handle symbols which have a purely real appearance; we can however at any moment pass to real equations if it is understood that of the symbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those which have not at all the suffix 4, or have it twice denote real quantities. ”

“ Let us take a linear homogeneous transformation,
the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while , are purely imaginary, but is real and , and transforms into . The operation shall be called a general Lorentz transformation. ”

“ Further relations, which characterise the influence of existing matter for the most important case to which we limit ourselves, i.e. for isotopic bodies; — they are comprised in the equations where = dielectric constant, = magnetic permeability, = the conductivity of matter, all given as function of x, y, z, t. is here the conduction current. ”

“ In his theory [3] , Lorentz proceeds from certain equations, which must hold at every point of "Æther"; then by forming the average values over "physically infinitely small" regions, which however contain large numbers of electrons, the equations for electro-magnetic processes in moving bodies can be successfully built up. ”

“ The differential equations expressing the fundamental equations of electrodynamics for moving bodies, when written in and the vectors are exactly of the same form as the equations for moving bodies. The velocity of matter does not enter in these equations. In the vectorial way of writing, we have
The velocity of matter occurs only in the auxilliary equations which characterise the influence of matter on the basis of their characteristic constants . Let us now transform these auxilliary equations x, y, z into the original co-ordinates x, y, z, and t. ”