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we could simply refer to a general component as ø;, and say that ¿ could either
be z, , or z, and that these are the three components; that is, imagine that ¿ is
any one of three directions, ø, , or z. The notation that we use for four-vecftors
is analogous to this: we write ø„ for the four-vector, and / stands for the ƒour
possible directions Ý, ø, , Or Z.
W© could, of course, use any notation we want; do not laugh at notations;
Invent them, they are powerful. In fact, mathematics is, to a large extent,
Invention of better notations. “The whole idea of a four-vector, In fact, is an
improvement in notation so that the transformations can be remembered easily.
A„, then, is a general four-vector, but for the special case oŸ momentum, the ?¿
is identified as the energy, ø„ is the momentum in the z-direction, øy is that in
the -direction, and ø; is that in the z-direction. To add four-vectors, we add
the corresponding componenfs.
TỶ there is an equation among four-vectors, then the equation is true for cœch
cơmponen‡. Eor instance, 1f the law of conservation of three-vector momentum is
to be true in particle collisions, i.e., if the sum of the momenta for a large number
of interacting or colliding particles is to be a constant, that must mean that the
sums of all momenta in the z-direction, in the -direction, and in the z-direction,
for all the particles, must each be constant. 'This law alone would be impossible
in relativity because it 1s Zncomjplete; it is like talking about only bwo of the
components of a three-vector. lt is incomplete because iŸ we rotate the axes, we
mix the various componentfs, so we must include all three components in our law.
Thus, in relativity, we must complete the law of conservation of momentum by
extending it to include the #ữne component. This is øbsolutel necessar to go
with the other three, or there cannot be relativistic invariance. 'Phe conseruation
öƒ energụ 1s the fourth equation which goes with the conservation of momentum
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to make a valid four-vector relationship in the geometry of space and time. Thus
the law of conservation of energy and momentum in four-dimensional notation is
» Pụ — » Đụ (17.13)
particles particles
in out
or, in a slightly diferent notation
À pụ, = À ` Địu, (17.14)
where ? = l, 2,... refers to the particles goïng into the collision, 7 = 1, 2,...
refers to the particles coming out of the collision, and / = ø, , z, or ứ. You say,
“In which axes?” It makes no diference. The law is true for each component,
USINE đn axes.
In vector analysis we discussed one other thing, the dot produect oŸ bwo vecbors.
Let us now consider the corresponding thing in space-time. In ordinary rotation
we discovered there was an unchanged quantity #2 + #2 + z2. In four dimensions,
we find that the corresponding quantity is £ — #7 — 2 — z2 (Eq. 17.3). How can
we write that? One way would be to write some kind of four-dimensional thing
with a square dot between, like A„ L-] „; one of the notations which is actually
used is
3) A,A,=A?— A?— A?— A2. (17.15)
The prime on 3` means that the first term, the “time” term, is positive, but the
other three terms have minus signs. This quantity, then, will be the same in any
coordinate system, and we may call it the square of the length of the four-vector.
For instance, what is the square of the length of the four-vector momentum of
a single particle? This will be equal to gøý — Ø2 — Ø2 — Ø2 or, in other words,
E2 — p2, because we know that p¿ is . What is #2 — p?? It must be something
which is the same in every coordinate system. In particular, it must be the same
for a coordinate system which is moving right along with the particle, in which
the particle is standing still. If the particle is standing still, it would have no
mmomentum. So in that coordinate system, iÈ is purely i%s energy, which is the
same as its rest mass. Thus #3 — p° = m. So we see that the square of the
length of this vector, the four-vector momentum, is equal to mã.
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From the square of a vector, we can go on to invent the “dot product,” or the
product which is a scalar: iŸ ø„ is one four-vector and b„ is another four-vector,
then the scalar product is
» dubu = đi — ayÙy — quby — g„by. (17.16)
Tt is the same in all coordinate systems.
Finally, we shall mention certain things whose rest mass mọ is zero. A photon
of light, for example. A photon is like a particle, in that ib carries an energy
and a momentum. The energy of a photon is a certain constant, called Planeck”s
constant, times the frequency of the photon: # = hz. Such a photon also carries
a momentum, and the momentum of a photon (or of any other particle, in fact)
is 5 divided by the wavelength: p = h/^A. But, for a photon, there is a delnite
relationship bebween the frequency and the wavelength: = c/A. (The number
of waves per second, times the wavelength of each, is the distance that the light
goes in one second, which, oŸ course, is c.) Thus we see immmediately that the
energy of a photon must be the momentum times c, or 1Í c=— 1, the energụ œnd
momentưm are cqual. That 1s to say, the rest mass 1s zero. Let us look at that
again; that is quite curious. lf it is a particle of zero rest mass, what happens
when it stops? ϣ neuer stopsf Tt always goes at the speed c. The usual formula
for energy is mo/V 1 — 02. Ñow can we say that rmọ = 0Ö and 0 = 1, so the energy
is 0? We cannof say that it is zero; the photon really can (and does) have energy
even though it has no rest mass, but this it possesses by perpetually going at the
speed of lightl
W© also know that the momentum of any particle is equal to its total energy
tỉmes is velocity: iŸ e = 1, p = 0# or, in ordinary units, p = 0/c?. Eor any
particle moving at the speed of light, p = ifc = 1. The formulas for the energy
of a photon as seen from a moving system are, of course, given by Eq. (17.12),
but for the momentum we must substitute the energy times é (or times 1 in this
case). The different energies after transformation means that there are diferent
frequencies. 'Phis is called the Doppler eÑfect, and one can calculate it easily from
Eq. (17.12), using also = p and = hứ.
As Minkowski said, “Space of itself, and time of itself will sink into mere
shadows, and only a kind of union between them shall survive.”
--- Trang 333 ---
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18-1 The center of mass
In the previous chapters we have been studying the mechanics oŸ points, or
small particles whose internal structure does not concern us. For the next few
chapters we shall study the application of NÑewton's laws to more complicated
things. When the world becomes more complicated, it also becomes more
interesting, and we shall ñnd that the phenomena associated with the mechanics
of a more complex object than just a point are really quite striking. Of course