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volume of electricity,
k = Á e + i x h (35)
in which i is the vector velocity of electricity, with the velocity of light as the unit. If we introduce
Jµ and Ƶ according to (30a) and (31), we obtain for the first component the expression
Æ12J2 +Æ13J3 +Æ14J4
Observing that Æ11 , vanishes on account of the skewsymmetry of the tensor (Æ ), the components of k
are given by the first three components of the four-dimensional vector
Kµ = Ƶ½ J½ (36)
and the fourth component is given by
(37)
K4 = Æ41J1 +Æ42J2 +Æ43J3 = i exix + eyiy + eziz = i»
()
There is, therefore, a four-dimensional vector of force per unit volume; whose first three components,
k1, k2, k3 are the ponderomotive force components per unit volume, and whose fourth component is
the rate of working of the field per unit volume, multiplied by -1 .
23
l
l2
l
l1
O x1
Fig. 2
A comparison of (36) and (35) shows that the theory of relativity formally unites the
ponderomotive force of the electric field, Á e , and the Biot-Savart or Lorentz force i x h.
Mass and Energy. An important conclusion can be drawn from the existence and significance of
the 4-vector Kµ . Let us imagine a body upon which the electromagnetic field acts for a time. In the
symbolic figure (Fig. 2) Ox1 designates the x1-axis, and is at the same time a substitute for the three
space axes Ox1, Ox2, Ox3; Ol designates the real time axis. In this diagram a body of finite extent is
represented, at a definite time l, by the interval AB; the whole space-time existence of the body is
represented by a strip whose boundary is everywhere inclined less than 45° to the l-axis. Between the
time sections, l = l1, and l = l2, but not extending to them, a portion of the strip is shaded. This
represents the portion of the space-time manifold in which the electromagnetic field acts upon the
body, or upon the electric charges contained in it, the action upon them being transmitted to the body.
We shall now consider the changes which take place in the momentum and energy of the body as a
result of this action.
We shall assume that the principles of momentum and energy are valid for the body. The change
in momentum, "Ix, "Iy, "Iz, and the change in energy, "E, are then given by the expressions
l1
1
"Ix = dl dxdydz = K1dx1dx2dx3dx4
+" +"k +"
l0 x
i
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
l1
1 # 1
"E = dl
+" +"»dxdydz = õø i K4dx1dx2dx3dx4
l0
i !#
Since the four-dimensional element of volume is an invariant, and (K1, K2, K3, K4) forms a 4-vector,
the four-dimensional integral extended over the shaded portion transforms as a 4-vector, as does also
the integral between the limits l1, and l2, because the portion of the region which is not shaded
contributes nothing to the integral. It follows, therefore, that "Ix, "Iy, "Iz, i"E form a 4-vector. Since
the quantities themselves may be presumed to transform in the same way as their increments, we infer
that the aggregate of the four quantities
Ix, Iy, Iz, iE
has itself vector character; these quantities are referred to an instantaneous condition of the body (e.g.
at the time l = l1).
This 4-vector may also be expressed in terms of the mass m, and the velocity of the body,
considered as a material particle. To form this expression, we note first, that
24
2 2 2 2 2
-ds2 = dÄ = - dx1 + dx2 + dx3 - dx4 = dl2 1- q2 (38)
() ( )
is an invariant which refers to an infinitely short portion of the four-dimensional line which represents
the motion of the material particle. The physical significance of the invariant dÄ may easily be given.
If the time axis is chosen in such a way that it has the direction of the line differential which we are
considering, or, in other terms, if we transform the material particle to rest, we shall have dÄ = dl; this
will therefore be measured by the light-seconds clock which is at the same place, and at rest relatively
to the material particle. We therefore call r the proper time of the material particle. As opposed to dl,
dÄ is therefore an invariant, and is practically equivalent to dl for motions whose velocity is small
compared to that of light. Hence we see that
dxÃ
uà = (39)
dÄ
has, just as the dx½,, the character of a vector; we shall designate (uÃ) as the four-dimensional vector
(in brief, 4-vector) of velocity. Its components satisfy, by (38), the condition
2
(40)
"u =-1
Ã
We see that this 4-vector, whose components in the ordinary notation are
qx qy qz i
,,, (41)
1- q2 1- q2 1- q2 1- q2
is the only 4-vector which can be formed from the velocity components of the material particle
which are defined in three dimensions by
dx dy dz
qx = , qy = , qz =
dl dl dl
We therefore see that
dxµ
ëø öø
m (42)
ìø ÷ø
dÄ
íø øø
must be that 4-vector which is to be equated to the 4-vector of momentum and energy whose
existence we have proved above. By equating the components, we obtain, in threedimensional
notation,
mqx
ñøI =
x
ôø
1- q2
ôø
ôø. . . . .
ôø
(43)
òø. . . . .
ôø
ôø
m
ôøE =
1- q2
ôø
óø
25
 We recognize, in fact, that these components of momentum agree with those of
classical mechanics for velocities which are small compared to that of light. For large velocities
the momentum increases more rapidly than linearly with the velocity, so as to become infinite
on approaching the velocity of light.
If we apply the last of equations (43) to a material particle at rest (q = 0), we see that the
energy, Eo, of a body at rest is equal to its mass. Had we chosen the second as our unit of time,
we would have obtained
E0 = mc2 (44)
Mass and energy are therefore essentially alike; they are only different expressions for the same
thing. The mass of a body is not a constant; it varies with changes in its energy.* We see from
the last of equations (43) that E becomes infinite when q approaches l, the velocity of light. If [ Pobierz całość w formacie PDF ]

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