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'This circumstance ïs called “failure of simultaneity at a distance,” and to make
the idea a little clearer let us consider the following experiment.
Suppose that a man moving ïn a space ship (system ,5”) has placed a clock at
each end of the ship and is interested in making sure that the two clocks are in
synchronism. How can the clocks be synchronized? There are many ways. One
way, involving very little calculation, would be frst to locate exactly the midpoint
between the clocks. hen from this station we send out a light signal which will
go both ways at the same speed and will arrive at both clocks, clearly, at the
same time. 'Phis simultaneous arrival of the signals can be used to synchronize
the clocks. Let us then suppose that the man in Š” synchronizes his clocks by
this particular method. Let us see whether an observer in system Š would agree
that the two clocks are synchronous. The man in Š” has a right to believe they
are, because he does not know that he is moving. But the man in Š reasons that
since the ship is moving forward, the clock in the front end was running away
trom the light signal, hence the light had to go more than halfway in order to
catch up; the rear clock, however, was advancing to meet the light signal, so this
distance was shorter. Therefore the signal reached the rear clock frst, although
the man in S5” thought that the signals arrived simultaneously. We thus see that
when a man in a space ship thinks the times at two locations are simultaneous,
cqual values of # in his coordinate system must correspond to đjferent values
of # in the other coordinate systeml
1ã-7 EFour-vectors
Let us see what else we can discover in the Lorentz transformation. Ït is
interesting to note that the transformation between the #'s and £”s is analogous
in form to the transformation of the #ø's and s that we studied in Chapter 11
for a rotation oŸ coordinates. We then had
, :
; #cos ổ -Ƒ sìn ổ, (15.8)
ự —=cos8 — zsin0,
in which the new #“ mixes the old z and , and the new ˆ also mixes the old
+ and ; similarly, in the Lorentz transformation we fnd a new zø“ which is a
mixture of z and ý, and a new £ which is a mixture of ¿ and z. So the Lorentz
transformation is analogous to a rotation, only it is a “rotation” in spœce and
time, which appears to be a strange concept. A check of the analogy to rotation
--- Trang 298 ---
can be made by calculating the quantity
a2 + 2 + z2 — c?U2 = x? + 02+ z? — c?£. (15.9)
In this equation the first three terms on each side represent, in three-dimensional
geometry, the square of the distance between a point and the origin (surface
of a sphere) which remains unchanged (invariant) regardless of rotation of the
coordinate axes. Similarly, Øq. (15.9) shows that there is a certain combination
which includes time, that is invariant to a Lorentz transformation. Thus, the
analogy to a rotation is complete, and is of such a kind that vectors, 1.e., quantities
involving “components” which transform the same way as the coordinates and
time, are also useful in connection with relativity.
'Thus we contemplate an extension of the idea of vectors, which we have so far
considered to have only space components, to ineclude a time component. That
1s, we expect that there will be vectors with four components, three of which are
like the components of an ordinary vector, and with these will be associated a
fourth component, which is the analog of the time part.
This concept will be analyzed further in the next chapters, where we shall
fínd that ïf the ideas of the preceding paragraph are applied to momentum,
the transformation gives three space parts that are like ordinary momentum
components, and a fourth component, the time part, which is the energ.
15-8 Relativistic dynamics
W©S are now ready to investigate, more generally, what form the laws of
mechanics take under the Lorentz transformation. [We have thus far explained
how length and time change, but not how we get the modifed formula for ?n
(Eq. 15.I). We shall do this in the next chapter.]} To see the consequences
of Hinstein's modification of m for Newtonian mechanics, we start with the
Newtonian law that force is the rate of change of momentum, or
Ƒ' = d(mo)/dt.
Momentum is still given by rm, but when we use the new ?n this becomes
ÐĐ= 1ẽU= ——————. 15.10
v1— 032/c2 ( )
--- Trang 299 ---
This is Einstein's modification of Newton”s laws. Under this modification, 1Í
action and reaction are still equal (which they may not be in detail, but are in
the long run), there will be conservation oŸ momentum in the same way as before,
but the quantity that is being conserved is not the old z2 with its constant mass,
but instead is the quantity shown in (15.10), which has the modifed mass. When
this change is made in the formula for momentum, conservation of momentum
still works.
Now let us see how momentum varies with speed. In NÑewtonian mechanics it
is proportional to the speed and, according (15.10), over a considerable range of
speed, but small compared with e, it is nearly the same in relativistic mechanics,
because the square-root expression differs only slightly from 1. But when 0 is
almost equal to e, the square-root expression approaches zero, and the momentum
therefore goes toward infnity.
'What happens i a constant force acts on a body for a long time? In Newtonian
mechanies the body keeps picking up speed until it goes faster than light. But
this is impossible in relativistic mechanics. In relativity, the body keeps picking
up, not speed, but momentum, which can continually increase because the mass
1s increasing. After a while there is practically no acceleration in the sense of a
change of velocity, but the momentum continues to increase. Of course, whenever
a force produces very little change in the velocity of a body, we say that the body
has a great deal oŸ inertia, and that is exactly what our formula for relativistic
mass says (see lq. 15.10)—it says that the inertia is very great when ø is nearly
as great as c. Ás an example of this efect, to defect the high-speed electrons
in the synchrotron that is used here at Caltech, we need a magnetic ñeld that
1s 2000 times stronger than would be expected on the basis of Newton”s laws.
In other words, the mass of the electrons in the synchrotron is 2000 times as
great as their normal mass, and is as great as that of a protonl That m should
be 2000 times mọ means that 1 — ø2/c2 must be 1/4,000,000, and that means
that 0 difers from c by one part in 8,000,000, so the electrons are getting pretty
close to the speed of light. If the electrons and light were both to start from
the synchrotron (estimated as 700 feet away) and rush out to Bridge Lab, which
would arrive first? The light, of course, because light always travels faster.* How
mụuch earlier? 'Phat is too hard to tell—instead, we tell by what distance the
light is ahead: ¡it is about 1/1000 of an inch, or hì the thickness of a piece of