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SPACE AND TIME: Part 4

PART 4: Minkowski's words are in **boldface**.

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,

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

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 = *i*τ.

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!

Along High Street, Windor, Berks.

Photo by Antony McCallum

TIME AND SPACE, by Hermann Minkowski