Relativistic physics has developed rapidly since Einstein's breakthrough publication. Minkowski has already presented many of his ideas in an internal talk at Göttingen; the papers cited here (the last of them will be prepared by Max Born after Minkowski's unexpected death) contain some of the technical details omitted from Raum und Zeit in deference to its broad audience.
"In the last-named field the first question that arises is : If a force
with components X, Y, Z, along the space-axes is applied at a world-
point P (x, y, z, t) where the velocity- vector is
,
as what
force is this to be conceived under any possible change of the
system of reference?"
We now move from kinematics, the geometrical study of motion, to dynamics, the physical study of force. Ironically, we will also see an improvement in the geometry.
"Now there exist tested lemmas about ponderomotive force in the electromagnetic field in the cases where the group G(c) is certainly to be allowed. These lemmas lead to the simple rule: On changing the system of reference, the said force is to be so applied in the new space coordinates that the vector pertaining thereto with the components
(where
is the work that the force divided by c² performs at the world-point), remains all unchanged. This vector is always normal to the velocity-vector at P. Such a vector belonging to a force at P shall be called a moving force-vector at P."
The most important point being introduced here is that all three-dimensional physical vector-quantities should be extended to four dimensions. Thus force, for example, must have a fourth component, call it T, a component along the t-axis. To maintain agreement with Newtonian mechanics, the four-dimensional force is normal (perpendicular) to the four-dimensional velocity.
"Now let the world-line running through P be described by a substance-point with a constant mechanical mass m."
We introduce point masses -- atoms or Lorentz electrons -- into the theory.
"Let m times the velocity-vector at P be called the momentum-vector at P, and m times the acceleration-vector at P be called the force-vector of the motion at P."
Thus generalising to four dimensions the elementary definitions of momentum and force from Newtonian mechanics.
"According to these definitions, the law describing the motion of a mass point with a given moving force-vector reads : The force-vector of the motion is equal to the moving force-vector. [H. Minkowski, Gesammelte Abhandlungen, Vol. II, p. 400. Compare also M. Planck, Verhandlungen der Physikalischen Gesellschaft, Vol. IV, 1906, p. 136.]
That is, f = ma in four dimensions too, and the four-dimensional force is the τ-derivative of the four-dimensional momentum.
"This statement summarizes four equations for the components along the four axes, of which the fourth (because both of the described vectors were a priori normal to the velocity-vector) can be regarded as a consequence of the first three. According to the above meaning of T the fourth equation undoubtedly expresses the law of energy. The kinetic energy of a point-mass is therefore to be defined as c² times the component of the momentum-vector along the t-axis. The expression for this is
which, after subtracting the additive constant mc² and neglecting quantities of the order 1/c² is the expression of kinetic energy in Newtonian mechanics ½mv²."
Energy is the fourth component of momentum. It is surprising (as some historians have noted) that Minkowski, having observed this, does not go on to show that mass is the non-Euclidean length of the momentum vector.
To show that the Newtonian kinetic-energy comes out of equation for the fourth component, expand the square-root in powers of (v/c²) and keep only low-order terms.
"In this the dependence of energy on the system of reference appears obvious. But since the t-axis can now be taken in the direction of any time-vector, the law of energy, on the other hand, formulated for every possible system of reference, contains the entire system of equations of motion. This fact retains its significance in the above-mentioned limiting case for c = ∞, also for the deductive development of the Newtonian mechanics, and in this sense it has already been noted by J. R. Schütz [ "Das Prinzip der absoluten Erhaltung der Energie" in Nachrichten der k. Gesellschaft der Wissenschaften zu Göttingen (mathematisch-physikalische Klasse), 1897, p. 110].
Another of Einstein's Göttingen precursors. The significance of this paragraph is its recognition that energy and momentum are still conserved in relativity.
"We can from the start so determine the relation of unit length to unit time, that the natural limit of velocity becomes c = 1. If we then introduce in place of t the new quantity s, defined as the square root of −1 times t, the quadratic differential expression becomes
thus completely symmetrical in x, y, z, s, and this symmetry now enters into every law which does not contradict the world-postulate."
At last! This is the key element of Minkowski's space-time, which distinguishes it from previous attempts to introduce a fourth dimension into physics, or even to identify time with such a dimension (as did H. G. Wells in 1895).
Time, Minkowski says, is not exactly the fourth dimension as usually understood, a fourth spatial-coordinate. Perhaps we should say this the other way around: the fourth dimension is not exactly time. Rather, the fourth dimension s is time multiplied by i, the square root of -1. With this change, there is no longer a need to use special units, as in the earlier parts of the talk, and everything becomes extremely simple. The geometry of space-time is the geometry of four-dimensional space with one dimension imaginary.
The law of Pythagoras then acquires a minus sign on one of the terms, and the length of, for example, the force vector with components X, Y, Z, iT is the square root of X² + Y² + Z² - T². "Circles" in the x-s plane become hyperbolas. The proper time τ itself can be replaced by a proper distance, σ = iτ.
One can only wonder why Minkowski does not make this substitution at the very beginning, as will most early Twentieth-Century relativity textbooks. Perhaps it is because of the lingering hostility to the imaginary which can be found in many Nineteenth-Century mathematical works. This attitude will resurface in the Twentieth, and i will disappear again from relativity, disguised as a unit vector in the time direction with negative square.
Minkowski could have taken that approach himself. In 1908 non-Euclidean geometry is well-established; any plane containing time and one space axis obeys the well-studied hyperbolic geometry. Minkowski is aware of this and has mentioned it in his earlier talk to his colleagues, but he does not refer it in his famous public address.
"Accordingly we can express the essence of this postulate very tersely in the mystical formula:
Space and time should be measured in identical units, only time is imaginary. This is the "conversion factor" approach to c mentioned earlier. One cannot fail to be impressed by the enormous extent, along the (imaginary) fourth dimension, of even short time intervals. Not only "substance-points" but also extended bodies like planets and galaxies have a thread-like appearance in Minkowski's world!
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Photo by Antony McCallum