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the inertial system. If P' lies outside of the "light-cone" then PP' is space-like; in this case, by properly
choosing the inertial system, "l can be made to vanish.
By the introduction of the imaginary time variable, x4 = il , Minkowski has made the theory of
invariants for the four-dimensional continuum of physical phenomena fully analogous to the theory of
invariants for the three-dimensional continuum of Euclidean space. The theory of four-dimensional
tensors of special relativity differs from the theory of tensors in three-dimensional space, therefore,
only in the number of dimensions and the relations of reality.
A physical entity which is specified by four quantities, A½ , in an arbitrary inertial system of the
x1, x2, x3, x4, is called a 4-vector, with the components A½ , if the A½ correspond in their relations of
reality and the properties of transformation to the "x½ ; it may be space-like or timelike. The sixteen
quantities Aµ½ then form the components of a tensor of the second rank, if they transform according
to the scheme
A'µ½ = bµ±b½² A±²
It follows from this that the Aµ½ behave, with respect to their properties of transformation and their
properties of reality, as the products of the components, Uµ , V½ of two 4-vectors, (U) and (V). All
the components are real except those which contain the index 4 once, those being purely imaginary.
Tensors of the third and higher ranks may be defined in an analogous way. The operations of addition,
* 2
That material velocities exceeding that of light are not possible, follows from the appearance of the radical 1- v in
the special Lorentz transformation (29).
21
subtraction, multiplication, contraction and differentiation for these tensors are wholly analogous
to the corresponding operations for tensors in three-dimensional space.
Before we apply the tensor theory to the four-dimensional space-time continuum, we shall
examine more particularly the skew-symmetrical tensors. The tensor of the second rank has, in
general, 16 = 4.4 components. In the case of skew-symmetry the components with two equal
indices vanish, and the components with unequal indices are equal and opposite in pairs. There
exist, therefore, only six independent components, as is the case in the electromagnetic field. In
fact, it will be shown when we consider Maxwell's equations that these may be looked upon as
tensor equations, provided we regard the electromagnetic field as a skew-symmetrical tensor.
Further, it is clear that a skew-symmetrical tensor of the third rank (skew-symmetrical in all
pairs of indices) has only four independent components, since there are only four combinations
of three different indices.
We now turn to Maxwell's equations (19a), (19b), (20a), (20b), and introduce the notation:*
Æ31 Æ12 Æ14 Æ24 Æ34
ñø
ôøÆ23
(30a)
òøh h31 h12 - iex - iey - iez
ôø 23
óø
J1 J2 J3 J4
ñø
ôø
(31)
òø1 1 1
ôøc ix c iy c iz iÁ
óø
with the convention that Ƶ½ shall be equal to -Ƶ½ . Then Maxwell's equations may be
combined into the forms
"Ƶ½
= Jµ (32)
"x½
"Ƶ½ "ƽà "Æõ
+ + = 0 (33)
"xà "xµ "x½
as one can easily verify by substituting from (30a) and (31). Equations (32) and (33) have a
tensor character, and are therefore co-variant with respect to Lorentz transformations, if the Ƶ½
and the Jµ have a tensor character, which we assume. Consequently, the laws for transforming
these quantities from one to another allowable (inertial) system of co-ordinates are uniquely
determined. The progress in method which electro-dynamics owes to the theory of special
relativity lies principally in this, that the number of independent hypotheses is diminished. If
we consider, for example, equations (19a) only from the standpoint of relativity of direction, as
we have done above, we see that they have three logically independent terms. The way in
which the electric intensity enters these equations appears to be wholly independent of the way
"eµ
in which the magnetic intensity enters them; it would not be surprising if instead of , we
"l
"2eµ
had, say, , or if this term were absent. On the other hand, only two independent terms
"l2
appear in equation (32). The electromagnetic field appears as a formal unit; the way in which
*
In order to avoid confusion from now on we shall use the three-dimensional space indices, x, y, z instead of 1, 2, 3, and
we shall reserve the numeral indices 1, 2, 3, 4 for the four-dimensional space-time continuum.
22
the electric field enters this equation is determined by the way in which the magnetic field
enters it. Besides the electromagnetic field, only the electric current density appears as an inde-
pendent entity. This advance in method arises from the fact that the electric and magnetic fields
lose their separate existences through the relativity of motion. A field which appears to be
purely an electric field, judged from one system, has also magnetic field components when
judged from another inertial system. When applied to an electromagnetic field, the general law
of transformation furnishes, for the special case of the special Lorentz transformation, the
equations
ñø
ôøe' = ex h 'x = hx
x
ôø
ôø
ôøe' = ey - vhz h 'y = hy + vez
(34)
òø
y
1- v2 1- v2
ôø
ôø
ez + vhy hz - vey
ôø
e'z = h 'z =
ôø
1- v2 1- v2
óø
If there exists with respect to K only a magnetic field, h, but no electric field, e, then with respect
to K' there exists an electric field e' as well, which would act upon an electric particle at rest relatively
to K'. An observer at rest relatively to K would designate this force as the Biot-Savart force, or the
Lorentz electromotive force. It therefore appears as if this electromotive force had become fused with
the electric field intensity into a single entity.
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