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the oxygen, we could ñnd out how much energy would be liberated when carbon |
and oxygen form carbon dioxide. The only trouble here is that the diferences In |
mmasses are so small that it is technically very difcult to do. |
Now let us turn to the question of whether we should add mạc? to the kinetic |
energy and say from now on that the total energy of an objeect is me2. First, iŸ we |
can still see the component pieces of rest mass ?mọ inside M, then we could say |
that some of the mass Ä⁄ of the compound object is the mechanical rest mass of |
the parts, part of it is kinetic energy of the parts, and part of it is potential energy |
of the parts. But we have discovered, in nature, particles of various kinds which |
--- Trang 318 --- |
undergo reactions just like the one we have treated above, in which with all the |
study ín the world, we cannot sec the parts ứnside. For instance, when a K-meson |
disintegrates into two pions it does so according to the law (16.11), but the idea |
that a K is made out of 2 s is a useless idea, because it also disintegrates into |
'Therefore we have a neu idea: we do not have to know what things are made |
of inside; we cannot and need not identify, inside a particle, which of the energy 1s |
rest energy of the parts into which it is goïng to disintegrate. It is not convenient |
and often not possible to separate the total me? energy of an object into rest |
energy of the inside pieces, kinetic energy of the pieces, and potential energy of |
the pieces; instead, we simply speak of the £oføœÏ energu of the particle. We “shift |
the origin” of energy by adding a constant mọc? to everything, and say that the |
total energy of a particle is the mass in motion times c2, and when the object is |
standing still, the energy is the mass at rest times cẺ. |
Finally, we fnd that the velocity 0, momentum ?, and total energy # are |
related in a rather simple way. That the mass in motion at speed 0 is the mass mo |
at rese divided by 4⁄1 — 02/c?, surprisingly enough, is rarely used. Instead, the |
following relations are easily proved, and turn out to be very useful: |
E2 — P?c? = mặc" (16.13) |
Pc= EuÍc. (16.14) |
--- Trang 319 --- |
Spereco- Time© |
17-1 The geometry of space-time |
The theory of relativity shows us that the relationships of positions and |
times as measured in one coordinate system and another are not what we would |
have expected on the basis of our intuitive ideas. It is very important that we |
thoroughly understand the relations oŸ space and time implied by the Lorentz |
transformation, and therefore we shall consider this matter more deeply in this |
chapter. |
The Lorentz transformation between the positions and tỉimes (#,, z,È) as |
measured by an observer “standing still,” and the corresponding coordinates and |
tìme (4, /, z, ) measured inside a “moving” space ship, moving with velocity u |
; % — tu |
# =———p, |
v1—*2/c2 |
Ụ =U, |
17.1 |
Xa (11) |
TH... a |
v1—*2/c2 |
Let us compare these equations with Eq. (11.5), which also relates measurements |
in two systems, one of which in this instance is ro/a£#ed relative to the other: |
+ = #øcos 0 + 1 sin 0, |
= cosØ — #sin 0, (17.2) |
z' =z. |
In this particular case, Moe and Jjoe are measuring with axes having an angle Ø |
between the z- and z-axes. In each case, we note that the “primed” quantities |
--- Trang 320 --- |
are “mixtures” of the “unprimed” ones: the new # is a mixture oŸ # and , and |
the new # is also a mixture oŸ z and . |
An analogy is useful: When we look at an object, there is an obvious thing we |
might call the “apparent width,” and another we might call the “depth” But the |
two ideas, width and depth, are not ƒundœmenfal properties of the object, because |
1ƒ we step aside and look at the same thing from a different angle, we get a different |
width and a diferent depth, and we may develop some formulas for computing the |
new ones rom the old ones and the angles involved. Equations (17.2) are these |
formulas. One might say that a given depth is a kind of “mixture” of all depth |
and all width. If it were impossible ever to move, and we always saw a given |
object from the same position, then this whole business would be irrelevant——we |
would always see the “true” width and the “true” depth, and they would appear |
to have quite diferent qualities, because one appears as a subtended optical angle |
and the other involves some focusing of the eyes or even intuition; they would |
seem to be very different things and would never get mixed up. lt is because we |
can walk around that we realize that depth and width are, somehow or other, |
Jjust two different aspects of the same thiỉng. |
Can tue no‡ look at the Loren‡z transƒformations ín the sơme tua? Here aÌso |
we have a mixture—of positions and the time. A diference between a space |
mmneasurement and a time measurement produces a new space measurement. Ïn |
other words, in the space measurements of one man there is mixed in a little bit |
of the time, as seen by the other. Our analogy permits us to generate this idea: |
The “reality” of an object that we are looking at is somehow greater (speaking |
crudely and intuitively) than its “width” and its “depth” because £#e depend |
upon ho we look at it; when we move to a new position, our brain immediately |
recalculates the width and the depth. But our brain does not immediately |
recaleulate coordinates and time when we move at high speed, because we have |
had no efective experience of going nearly as fast as light to appreciate the |
fact that time and space are also of the same nature. It is as though we were |
always stuck in the position oŸ having to look at just the width of something, |
not beïng able to move our heads appreciably one way or the other; if we could, |
we understand now, we would see some of the other man ”s tine—we would see |
“behind,” so to speak, a little bít. |
Thus we shall try to think of objects in a new kind of world, of space and time |
mixed together, in the same sense that the objects in our ordinary space-world are |
real, and can be looked at from different directions. We shall then consider that |
obJects occupying space and lasting for a certain length of time occupy a kind of |
--- Trang 321 --- |
(a)| /Œ) |
X0 x |
Fig. 17-1. Three particle paths in space-time: (a) a particle at rest |
at x = xo; (b) a particle which starts at x = xo and moves with constant |
speed; (c) a particle which starts at high speed but slows down; (d) a |
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