If we compare with this expression the previous formulæ[35] giving the
elementary law about the ponderomotive action of moving electric charges
upon each other, then we cannot but admit, that the relations which
occur here reveal the inner essence of full simplicity first in four
dimensions; but in three dimensions, they have very complicated
projections.
In the mechanics reformed according to the world-postulate, the
disharmonies which have disturbed the relations between Newtonian
mechanics and modern electrodynamics automatically disappear. I shall
now consider the position of the Newtonian law of attraction to this
postulate. I will assume that two point-masses _m_ and _m₁_ describe
their world-lines; a moving force-vector is exercised by _m_ upon _m₁_,
and the expression is just the same as in the case of the electron, only
we have to write +_mm₁_ instead -_ee₁_. We shall consider only the
special case in which the acceleration-vector of _m_ is always zero:
then _t_ may be introduced in such a manner that _m_ may be regarded as
fixed, the motion of _m_ is now subjected to the moving-force vector of
_m_ alone. If we now modify this given vector by writing
-([^.]1/√(1-(_v_²/_c²_)) instead of [._t_] ([._t_] = 1 up to magnitudes
of the order (1[^.]/_c²_)), then it appears that Kepler’s laws hold good
for the position (_x₁_, _y₁_, _z₁_), of _m₁_ at any time, only in place
of the time _t₁_, we have to write the proper time τ₁ of _m₁_. On the
basis of this simple remark, it can be seen that the proposed law of
attraction in combination with new mechanics is not less suited for the
explanation of astronomical phenomena than the Newtonian law of
attraction in combination with Newtonian mechanics.
Also the fundamental equations for electro-magnetic processes in moving
bodies are in accordance with the world-postulate. I shall also show on
a later occasion that the deduction of these equations, as taught by
Lorentz, are by no means to be given up.
The fact that the world-postulate holds without exception is, I believe,
the true essence of an electromagnetic picture of the world; the idea
first occurred to Lorentz, its essence was first picked out by Einstein,
and is now gradually fully manifest. In course of time, the mathematical
consequences will be gradually deduced, and enough suggestions will be
forthcoming for the experimental verification of the postulate; in this
way even those, who find it uncongenial, or even painful to give up the
old, time-honoured concepts, will be reconciled to the new ideas of time
and space,—in the prospect that they will lead to pre-established
harmony between pure mathematics and physics.
Footnote 30:
Planck, Zur Dynamik bewegter systeme, Ann. d. physik, Bd. 26, 1908, p.
1.
Footnote 31:
H. Minkowski; the passage refers to paper (2) of the present edition.
Footnote 32:
Minkowski—Mechanics, appendix, page 65 of paper (2). Planck—Verh. d.
D. P. G. Vol. 4, 1906, p. 136.
Footnote 33:
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