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SPACE AND TIME: Part 5

PART 5: Minkowski's words are in **boldface**.

The electrodynamics of moving bodies ought to have something to do with relativity!

**"Let us consider the world-line of such a point-electron with
charge e and introduce the proper time τ from any initial
point. To obtain the field determined by the electron at any
world-point P_{1} we construct the past-cone
P_{1} (see figure, right). This meets the
infinite world-line of the electron at a single point P because its
**[i.e. the world-line's]

To reiterate, the electron is instantaneously
at P. The electric field from the
electron travels at the speed of light along a light-cone,
and hits the other particle's world-line at P_{1}.

**"We construct
**[PQ]**the
tangent at P to the world-line and through
P _{1} the normal P _{1}Q to
this tangent. Let the scalar
[magnitude] of P_{1}Q be
r. Then, according to
the definition of a past cone we must take the scalar value of PQ
as r/c.
**

**
" Now the vector in the direction PQ of
length e/r represents in
its components along the x-, y-, z-axes the
vector potential
multiplied by c,
and in the component along the t-axis the
scalar potential
of the field produced by e for the world-point P_{1} .
This is the basis
of the fundamental laws established by A.
Liénard and E. Wiechert, **[A.
Liénard, "Champ électrique et magnétique produit par
une charge concentrée en un point
et animée d'un mouvement quelconque," in

A clear (but vector-calculus based)
discussion of the electric potential from a moving charge
can be found here,
but the details are not necessary.
The point which the reader should take away from this
paragraph is that the four fairly-complicated equations
describing the four components of the electromagnetic
potential can be viewed as a *single* equation in
four dimensional space.

Once the fundamental Minkowski insight of four-dimensionality
is gained, many things in physics become obvious. Electric
potential is one of numerous physically-important quantities
which come in groups of 3 + 1. For example, there is voltage, which
is just a number (so many volts), and there is also a closely-related
vector quantity, the vector potential, which has *x, y,* and
*z* components. Similarly, there is energy -- just a number,
so many joules -- and there is
momentum, a vector with three components. And so on, through the
whole of physics. All of these seem reminiscent of time (one dimension)
and space (three dimensions). If time and space are directions in
a single four-dimensional continuum,
each of these sets of 3 + 1 physical quantities might be a
four-dimensional vector, with the *t*-component being the
anomalous "just a number" member of the quartet.

**"In the description of the field itself produced by the electron
it is clearly seen that the separation of the field into electric and
magnetic forces is a relative one depending on the time axis of
reference. Both forces can be described together most luminously
after the analogy, however imperfect, of a force screw in mechanics."**

To describe how the electromagnetic field fits into the four-dimensional
world-view would take too long, but the general idea is that
the three components of the electric field and the three
components of the magnetic field are the six components of a
special object (called an antisymmetric tensor, a 2-form, or
a bivector depending on one's tribal affiliations) which arises
naturally in four dimensions. One might think of this as somehow
related to the six *planes* which can be constructed in
four-dimensional space: *xy, yz, zx, xt, yt, zt.*

In the next paragraph, which the non-physicist may wish to skip, Minkowski gives the four-dimensional formula for the force exerted by one charge on the other; the point, as before, is that this expression, complicated as it may look, is a vast simplification when compared to the four separate expressions in the standard literature.

The reader may wonder what the fourth component of force may be.
If we recall that force is the rate of change of momentum per
unit (proper) time, and that energy is the time-component of momentum,
we can see that the time-component of force must be (with
appropriate factors of *c* to get the units right) the
rate of change of energy per unit
(proper) time, that is, the (proper) *power*.

**"I shall now describe the ponderomotive effect of one
point-charge moving at will on another point-charge moving at will. Let
us take the world-line of the second point-electron of charge
e_{1},
passing through the world-point P . Let us determine P, Q, r as
before, then (see figure, right) construct the center M of the hyperbola of
curvature at P, and finally the normal MN from M to a straight
line through P parallel to QP_{1}. Let us next determine with P as
origin a system of reference with the t-axis in the direction of PQ,
the x-axis in the direction of QP_{1},
the y-axis in the direction of MN,
so that finally the direction of the z-axis is determined as normal
to the t-, x-, y-axes. Let the acceleration vector at P be
and the velocity-vector at P_{1} be
.
Now the action of
the moving force-vector of the first electron e moving at will on the
second electron e_{1} moving at will at P_{1}
is formulated thus:
**

**
in which the three relations between the components
of the vector
are:
and lastly, this vector
is normal to the velocity-vector at P _{1}
and through this circumstance alone is dependent on the latter
velocity-vector.
**

**
"If we compare this statement with the previous formulation
of the same fundamental law of the ponderomotive effect of moving
point-charges on each other,
**[found in K. Schwarzschild, *Nachrichten der k. Gesellschaft der
Wissenschaften zu Göttingen (mathematisch-physikalische Klasse), *
1903. p. 132, or H. A. Lorentz,
*Enzyklopädie der mathematischen Wissenschaften,*
Vol. V, Art. 14, p. 199]** we cannot but grant that the relations
here coming under observation do not manifest their intrinsic
character of utter simplicity except in four dimensions, but throw a very
complicated projection upon a tri-dimensional space preimposed
upon them."**

The "character of utter simplicity" only "manifest ... in four dimensions" is what makes relativity among the most beautiful of all scientific theories. Minkowski is the first relativist to perceive the wonderful æsthetic quality of the new physics.

**"In mechanics reformed according to the world-postulate the
disagreements which have caused friction between the Newtonian
mechanics and modern electrodynamics disappear of their own
accord. I shall touch upon the relation of the Newtonian law of
attraction to this postulate."**

This section of Minkowski's talk necessarily points to the future: the relativistic theory of gravity will only be developed by Einstein over the coming decade, and a unified theory of gravity and electromagnetism (at least one acceptable to the majority of physicists) will famously resist discovery even in the Twenty-first Century.

**"I shall assume that when two point-masses
m and m_{1} describe their world-lines,
a moving force-vector
acts from m on m_{1}
just as in the above expression in the case of
electrons, except that now mm_{1} is to be
substituted for -ee_{1}." **

Newton's law of gravity and Coulomb's law of electrostatic attraction are closely analogous. It is natural for Minkowski to begin with this analogy.

**"We shall now consider especially the particular case where the
acceleration-vector of m is constantly zero, in which case we can so
introduce t that m is conceived of as at rest,
and the motion of m_{1}
depends only on the moving force-vector proceeding from m. If we
modify this vector first by the factor
**

**
which, up to quantities of the order 1/ c² is equal to 1,
then it follows that for positions x_{1}, y_{1},
z_{1} of
m_{1}and their corresponding
time-positions, Kepler's laws would again obtain, except that in place
of the times t_{1}
the proper time τ_{1} of m would be substituted."
**

The full argument is found in H. Minkowski,
*Ges. Abhandlungen,* II, p. 403.**
**

**
"On the basis of this simple observation we can see that the
proposed law of attraction in conjunction with the new mechanics
would be no less suitable for explaining astronomical observations
than Newton's law of attraction in conjunction with the Newtonian mechanics."
**

**
**In fact, gravity will turn out to be a bit more complex than this.

**"The fundamental equations for electromagnetic processes in
ponderable bodies are likewise in complete harmony with the
world-postulate. Even the derivation of these equations, as taught by
Lorentz on the basis of conceptions of the electron theory, need not
for this end by any means be abandoned, as I shall show elsewhere.
**[H. Minkowski,
*Ges. Abhandlungen,* II, p. 405.]**"**

We come at last to the end of this historic talk:

**"The universal validity of the world-postulate is, I should believe,
the true core of an electromagnetic world-picture; first discovered
by Lorentz, then further developed by Einstein, it is now clearly
discernible. In the future development of its mathematical
consequences enough indications will be found for experimental
verification of the postulate to reconcile by the idea of a pre-established
harmony between pure mathematics and physics even those to whom
a surrender of old accustomed view-points is uncongenial or painful."
**

**
--- HERMANN MINKOWSKI.
**

The audience in Cologne applauds.

In the epilogue to this
edition of *Raum und Zeit*, we
will present
a brief synopsis of special relativity
from a more modern but still Minkowskian perspective.

TIME AND SPACE, by Hermann Minkowski