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V1—u2/c2:
Next we discuss the interesting problem of the addition of velocities in relativity.
We© recall that one of the original puzzles was that light travels at 186,000 mi/sec
in all systems, even when they are in relative motion. 'Phis is a special case of
--- Trang 309 ---
the more general problem exermplified by the following. Suppose that an object
inside a space ship is going at 100,000 mi/sec and the space ship itself is goïing at
100,000 mi/sec; how fast is the object inside the space ship moving from the point
of view of an observer outside? We might want to say 200,000 mi/sec, which is
faster than the speed of light. This is very unnerving, because it is not supposed
to be going faster than the speed of lightl "The general problem is as follows.
Let us suppose that the object inside the ship, from the point of view of
the man inside, is moving with velocity 0, and that the space ship Itself has a
velocity œ with respect to the ground. We want to know with what velocity 0;
this object is moving from the point of view of the man on the ground. 'Thịs is,
of course, still but a special case in which the motion is in the z-direction. “There
will also be a transformation for velocities in the ¿-direction, or for any angle;
these can be worked out as needed. Inside the space ship the velocity 1s ơ„,
which means that the displacement z” is equal to the velocity times the tỉme:
+ = Uy. (16.3)
Now we have only to calculate what the position and time are from the point of
view of the outside observer for an object which has the relation (16.2) between
+“ and #. So we simply substitute (16.3) into (16.2), and obtain
... (16.4)
v1—u?/c2
But here we fnd z expressed in terms of f“. In order to get the velocity as seen
by the man on the outside, we must divide hús distance by hás từne, not by the
other rmmans timef So we must also calculate the £#me as seen from the outside,
which 1s
# ;# 2
"x. “dã (16.5)
v1— 1u2/c2
Now we must fnd the ratio of z to , which is
H5 + -Ƒ Đại
=—==——a, 16.6
"`" /c 066)
the square roots having cancelled. 'This is the law that we seek: the resultant
velocity, the “summing” of two velocities, is not just the algebraic sum oŸ wo
--- Trang 310 ---
velocities (we know that it cannot be or we get in trouble), but is “corrected”
by 1+ uo/c.
Now let us see what happens. Suppose that you are moving inside the space
ship at half the speed of light, and that the space ship itself is goiïng at half the
specd of light. 'Thus % is 2C and 0 is Ọ€, but ïn the denominator œo is one-fourth,
so that
›€C + ›C 4e
“` 1+1. 5
So, in relativity, “half” and “half” does not make “one,” it makes only “4/5” Of
course low velocities can be added quite easily in the familiar way, because so
long as the velocities are small compared with the speed of light we can forget
about the (1 + Ͽ/e2) factor; but things are quite diferent and quite interesting
at high velocity.
Let us take a limiting case. Jjust for fun, suppose that inside the space ship
the man was observing lgh# ?tsejf In other words, ø = c, and yet the space ship
is moving. How will it look to the man on the ground7? 'Phe answer will be
tu -+EC u+C
" 1+ ue/c2 _..aẽ.
Therefore, if something is moving at the speed of light inside the ship, it will
appear to be moving at the speed of light from the point of view of the man
on the ground tool "This is good, for it is, in fact, what the Einstein theory of
relativity was desiegned to do in the frst place—so it had beffer workl
Of course, there are cases in which the motion is not in the direction of the
uniform translation. For example, there may be an object inside the ship which
is just moving “upward” with the velocity 0y; with respect to the ship, and the
ship is moving “horizontally.” Now, we simply go through the same thing, only
using #s instead of z's, with the result
Ụ=1/ = 0t,
so that 1 0„: = 0,
Uụ = : =0wW1— u2/c2. (16.7)
Thus a sidewise velocity is no longer ⁄, but 0yv⁄/1— u2/c?. We found this
result by substituting and combining the transformation equations, but we can
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ưN 'Ộ IN
ZN ¡y4 NÊGHT
£ N lư h N
# X f 1 \
: "u N # h Lư N
Fig. 16-1. Trajectorles described by a light ray and particle inside a
moving clock.
also see the result directly from the principle of relativity for the following reason
(it is always good to look again to see whether we can see the reason). We have
already (Pig. 15-3) seen how a possible clock might work when it is moving; the
light appears to travel a% an angle at the speed c in the fxed system, while 1t
simply goes vertically with the same speed in the moving system. We found that
the 0erfical componen¿ of the velocity in the ñxed system is less than that of
light by the facbor 4⁄1 — w2/c2 (see Bq. 15.3). But now suppose that we let a
material particle go back and forth in this same “clock,” but at some integral
fraction 1/ø of the speed of light (Eig. 16-1). Then when the particle has gone
back and forth once, the light will have gone exactly ? times. That is, each “click”
of the “particle” elock will coincide with each øœth “click” of the light clock. 7s
ƒact tmwust si be true tuhen the tuhole sụstem ¡s mmoving, because the physical
phenomenon of coinecidence will be a coincidenee in any frame. Therefore, since
the speed ey is less than the speed of light, the speed „ of the particle must be
slower than the corresponding speed by the same square-root ratiol That is why
the square root appears in any vertical velocity.
16-4 Relativistic mass
We learned in the last chapter that the mass of an object increases with
velocity, but no demonstration of this was given, in the sense that we made no
arguments analogous to those about the way clocks have to behave. However,
we cøn show that, as a consequence of relativity plus a few other reasonable