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Unformatted text preview: 17 Space Time 17—1 The geometry of spacetime 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 of 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 times (x, y, z, t) as
measured by an observer “standing still,” and the corresponding coordinates and
time (x’, y’, z’, 1’) measured inside a “moving” space ship, moving with velocity
u are , _ x — ut
x/l — u2/c2 ’
y, 2, k:
H (17.1) t —— ux/c2 — V1 — u2/c2. Let us compare these equations with Eq. (11.5), which also relates measurements
in two systems, one of which in this instance is rotated relative to the other: “a
 I x = xcosa + ysin6, I y = ycos 6 — xsin 6, (17.2)
2’ = z. In this particular case, Moe and Joe are measuring with axes having an angle 9 between the x’ and xaxes. In each case, we note that the “primed” quantities are “mixtures” of the “unprimed” ones: the new x’ is a mixture of x and y, and the new y’ is also a mixture of x and y. 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 fundamental properties of the object, because
if we step aside and look at the same thing from a different angle, we get a different
width and a different depth, and we may develop some formulas for computing the
new ones from 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 different 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. It is because we can walk
around that we realize that depth and width are, somehow or other, just two differ
ent aspects of the same thing. Can we not look at the Lorentz transformations in the same way? Here also we
have a mixture—of positions and the time. A difference between a space measure
ment and a time measurement produces a new space measurement. In 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 17—1 171 The geometry of spacetime
17—2 Spacetime intervals 17—3 Past, present, and future
17—4 More about fourvectors 17—5 Fourvector algebra Ci SLOW x, x Fig. 17—1. Three particle paths in
spacetime: (a) a particle at rest at
x = x0; (b) a particle which starts at
x = x0 and moves with constant speed;
(c) a particle which starts at high speed
but slows down. (0) NOT CORRECT (b) CORRECT Fig. 17—2. Two views of a disinfe
grating particle. an object that we are looking at is somehow greater (speaking crudely and intui
tively) than its “width” and its “depth” because they depend upon how 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 recalculate coordinates
and time when we move at high speed, because we have had no effective 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 of having
to look at just the width of something, not being able to move our heads appreci
ably one way or the other; if we could, we understand now, we would see some of
the other man’s time—we would see “behind,” so to speak, a little bit. 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 spaceworld
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 a “blob” in a new kind of world, and that we look at this “blob” from different
points of View when we are moving at different velocities. This new world, this
geometrical entity in which the “blobs” exist by occupying position and taking up a
certain amount of time, is called spacetime. A given point (x, y, z, t) in spacetime
is called an event. Imagine, for example, that we plot the xpositions horizontally,
y and z in two other directions, both mutually at “right angles” and at “right
angles” to the paper (1), and time, vertically. Now, how does a moving particle,
say, look on such a diagram? If the particle is standing still, then it has a certain
x, and as time goes on, it has the same x, the same x, the same x; so its “path” is
a line that runs parallel to the taxis (Fig. 17—1 a). On the other hand, if it drifts
outward, then as the time goes on x increases (Fig. 17—1 b). So a particle, for ex
ample, which starts to drift out and then slows up should have a motion something
like that shown in Fig. 17—1(c). A patricle, in other words, which is permanent
and does not disintegrate is represented by a line in spacetime. A particle which
disintegrates would be represented by a forked line, because it would turn into
two other things which would start from that point. What about light? Light travels at the speed c, and that would be represented
by a line having a certain ﬁxed slope (Fig. 17—1 d). Now according to our new idea, if a given event occurs to a particle, say if it
suddenly disintegrates at a certain spacetime point into two new ones which follow
some new tracks, and this interesting event occurred at a certain value of x and a
certain value of t, then we would expect that, if this makes any sense, we just have
to take a new pair of axes and turn them, and that will give us the new I and the
new x in our new system, as shown in Fig. l7—2(a). But this is wrong, because
Eq. (17.1) is not exactly the same mathematical transformation as Eq. (17.2).
Note, for example, the difference in sign between the two, and the fact that one is
written in terms of cos 0 and sin 6, while the other is written with algebraic quanti
ties. (Of course, it is not impossible that the algebraic quantities could be written as
cosine and sine, but actually they cannot.) But still, the two expressions are very
similar. As we shall see, it is not really possible to think of spacetime as a real,
ordinary geometry because of that difference in sign. In fact, although we shall
not emphasize this point, it turns out that a man who is moving has to use a set of
axes which are inclined equally to the light ray, using a special kind of projection
parallel to the x’ and t’axes, for his x’ and t’, as shown in Fig. 17—2(b). We shall
not deal with the geometry, since it does not help much; it is easier to work with
the equations. 172 Spacetime intervals Although the geometry of spacetime is not Euclidean in the ordinary sense,
there is a geometry which is very similar, but peculiar in certain respects. If this
idea of geometry is right, there ought to be some functions of coordinates and time
which are independent of the coordinate system. For example, under ordinary
rotations, if we take two points, one at the origin, for simplicity, and the other one
somewhere else, both systems would have the same origin, and the distance from 1 7—2 here to the other point is the same in both. That is one property that is inde
pendent of the particular way of measuring it. The square of the distance is
x2 + y2 + 22. Now what about spacetime? It is not hard to demonstrate that
we have here, also, something which stays the same, namely, the combination czt2 — x2 — y2 — z2 is the same before and after the transformation: c2tr2 _ 12 12 x — y — z’2 = cztz — x2 — y2 — 22. (17.3) This quantity is therefore something which, like the distance, is “real” in some
sense; it is called the interval between the two spacetime points, one of which is,
in this case, at the origin. (Actually, of course, it is the interval squared, just as
x2 + y2 + 22 is the distance squared.) We give it a different name because it is
in a different geometry, but the interesting thing is only that some signs are reversed
and there is a c in it. Let us get rid of the c; that is an absurdity if we are going to have a wonderful
space with x’s and y’s that can be interchanged. One of the confusions that could
be caused by someone with no experience would be to measure widths, .say, by the
angle subtended at the eye, and measure depth in a different way, like the strain on
the muscles needed to focus them, so that the depths would be measured in feet
and the widths in meters. Then one would get an enormously complicated mess of
equations in making transformations such as (17.2), and would not be able to see
the clarity and simplicity of the thing for a very simple technical reason, that the
same thing is being measured in two dilTerent units. Now in Eqs. (17.1) and (17.3)
nature is telling us that time and space are equivalent; time becomes space; they
should be measured in the same units. What distance is a “second”? It is easy to
ﬁgure out from (17.3) what it is. It is 3 X 108 meters, the distance that light would
go in one second. In other words, if we were to measure all distances and times
in the same units, seconds, then our unit of distance would be 3 X 108 meters,
and the equations would be simpler. Or another way that we could make the units
equal is to measure time in meters. What is a meter of time? A meter of time
is the time it takes for light to go one meter, and is therefore 1/3 X 10‘8 sec, or
3.3 billionths of a second! We would like, in other words, to put all our equations
in a system of units in which c = 1. If time and space are measured in the same
units, as suggested, then the equations are obviously much simpliﬁed. They are x’ = x — ut ,
m
y’ = y,
z, = 2’ (17.4)
,I = £19..
V1 — u2
t'2 _ x’2 __ y’2 _ 2,2 ___ t2 _ x2 _ yz _ 22' (175) If we are ever unsure or “frightened” that after we have this system with c = l
we shall never be able to get our equations right again, the answer is quite the
opposite. It is much easier to remember them without the c’s in them, and it is
always easy to put the c’s back, by looking after the dimensions. For instance, in
V1 — u2, we know that we cannot subtract a velocity squared, which has units,
from the pure number 1, so we know that we must divide u2 by c2 in order to make
that unitless, and that is the way it goes. The diﬂ‘erence between spacetime and ordinary space, and the character of
an interval as related to the distance, is very interesting. According to formula
(17.5), if we consider a point which in a given coordinate system had zero time, and
only space, then the interval squared would be negative and we would have an
imaginary interval, the square root of a negative number. Intervals can be either
real or imaginary in the theory. The square of an interval may be either positive
or negative, unlike distance, which has a positive square. When an interval is
imaginary, we say that the two points have a spacelike interval between them 17—3 LIGHT  CONE LIGHTCONE Fig. 17—3. The spacetime
surrounding a point at the origin. reg ion (instead of imaginary), because the interval is more like space than like time.
On the other hand, if two objects are at the same place in a given coordinate system,
but differ only in time, then the square of the time is positive and the distances are
zero and the interval squared is positive; this is called a timelike interval. In our
diagram of spacetime, therefore, we would have a representation something like
this: at 45° there are two lines (actually, in four dimensions these will be “cones,”
called light cones) and points on these lines are all at zero interval from the origin.
Where light goes from a given point is always separated from it by a zero interval,
as we see from Eq. (17.5). Incidentally, we have just proved that if light travels
with speed c in one system, it travels with speed c in another, for if the interval is
the same in both systems, i.e., zero in one and zero in the other, then to state that
the propagation speed of light is invariant is the same as saying that the interval
18 zero. 17—3 Past, present, and future The spacetime region surrounding a given spacetime point can be separated
into three regions, as shown in Fig. 17—3. In one region we have spacelike inter
vals, and in two regions, timelike intervals. Physically, these three regions into
which spacetime around a given point is divided have an interesting physical
relationship to that point: a physical pbject or a signal can get from a point in
region 2 to the event 0 by moving along at a speed less than the speed of light.
Therefore events in this region can affect the point 0, can have an inﬂuence on it
from the past. In fact, of course, an object at P on the negative taxis is precisely
in the “past” with respect to 0; it is the same spacepoint as 0, only earlier. What
happened there then, affects 0 now. (Unfortunately, that is the way life is.) An
other object at Q can get to 0 by moving with a certain speed less than c, so if this
object were in a space ship and moving, it would be, again, the past of the same
spacepoint. That is, in another coordinate system, the axis of time might go
through both 0 and Q. So all points of region 2 are in the “past” of 0, and any
thing that happens in this region can affect 0. Therefore region 2 is sometimes
called the aﬂective past, or affecting past; it is the locus of all events which can
affect point 0 in any way. Region 3, on the other hand, is a region which we can affect from 0, we can
“hit” things by shooting “bullets” out at speeds less than c. So this is the world
whose future can be affected by us, and we may call that the aﬂective future. Now
the interesting thing about all the rest of spacetime, i.e., region 1, is that we can
neither affect it now from 0, nor can it affect us now at 0, because nothing can go
faster than the speed of light. Of course, what happens at R can affect us later;
that is, if the sun is exploding “right now,” it takes eight minutes before we know
about it, and it cannot possibly affect us before then. What we mean by “right now” is a mysterious thing which we cannot deﬁne
and we cannot affect, but it can affect us later, or we could have affected it if we
had done something far enough in the past. When we look at the star Alpha
Centauri, we see it as it was four years ago; we might wonder what it is like “now.”
“Now” means at the same time from our special coordinate system. We can only
see Alpha Centauri by the light that has come from our past, up to four years ago,
but we do not know what it is doing “now”; it will take four years before what it
is doing “now” can affect us. Alpha Centauri “now” is an idea or concept of our
mind; it is not something that is really deﬁnable physically at the moment, because
we have to wait to observe it; we cannot even deﬁne it right “now.” Furthermore,
the “now” depends on the coordinate system. If, for example, Alpha Centauri
were moving, an observer there would not agree with us because he would put
his axes at an angle, and his “now” would be a diﬂerent time. We have already
talked about the fact that simultaneity is not a unique thing. There are fortune tellers, or people who tell us they can know the future, and
there are many wonderful stories about the man who suddenly discovers that he
has knowledge about the affective future. Well, there are lots of paradoxes pro
duced by that because if we know something is going to happen, then we can make 17—4 sure we will avoid it \by doing the right thing at the right time, and so on. But
actually there is no fortune teller who can even tell us the present! There is no one
who can tell us what is really happening right now, at any reasonable distance,
because that is unobseryable. We might ask ourselves this question, which we
leave to the student to try to answer: Would any paradox be produced if it were suddenly to become possible to know things that are in the spacelike intervals of
region 1? 17—4 More about fourvectors Let us now return to our consideration of the analogy of the Lorentz trans
formation and rotations of the space axes. We have learned the utility of collecting
together other quantities which have the same transformation properties as the
coordinates, to form what we call vectors, directed lines. In the case of ordinary
rotations, there are many quantities that transform the same way as x, y, andz
under rotation: for example, the velocity has three components, an x, y, and
z—component; when seen in a "different coordinate system, none of the components
is the same, instead they are all transformed to new values. But, somehow or other,
the velocity “itself” has a greater reality than do any of its particular components,
and we represent it by a directed line. We therefore ask: Is it or is it not true that there are quantities which transform,
or which are related, in a moving system and in a nonmoving system, in the same
way as x, y, z, and t? From our experience with vectors, we know that three of
the quantities, like x, y, 2, would constitute the three components of an ordinary
spacevector, but the fourth quantity would look like an ordinary scalar under
space rotation, because it does not change so long as we do not go into a moving
coordinate system. Is it possible, then, to associate with some of our known
“threevectors” a fourth object, that we could call the “time component,” in such a
manner that the four objects together would “rotate” the same way as position
and time in spacetime? We shall now show that there is, indeed, at least one such
thing (there are many of them, in fact): the three components of momentum, and the
energy as the time component, transform together to make what we call a “four—
vector.” In demonstrating this, since it is quite inconvenient to have to write c’s
everywhere, we shall use the same trick concerning units of the energy, the mass,
and the momentum, that we used in Eq. (17.4). Energy and mass, for example:
differ only by a factor c2 which is merely a question of units, so we can say energy
is the mass. Instead of having to write the c2, we put E = m, and then, of course,
if there were any trouble we would put in the right amounts of c so that the units
would straighten out in the last equation, but not in the intermediate ones. Thus our equations for energy and momentum are E=m=m/\/l—1)2,
W (17.6) mv = mov/x/l  122. P Also in these units, we have
E2 — p2 = mg. (17.7) For example, if we measure energy in electron volts, what does a mass of l electron
volt mean? It means the mass whose rest energy is 1 electron volt, that is, moc2
is one electron volt. For example, the rest mass of an electron is 0.511 X 106 ev. Now what would the momentum and energy look like in a new coordinate
system? To ﬁnd out, we shall have to transform Eq. (17.6), which we can do
because we know how the velocity transforms. Suppose that, as we measure it, an
object has a velocity 2), but we look upon the same object from the point of view
of a space ship which itself is moving with a velocity u, and in that system we use a
prime to designate the corresponding thing. In order to simplify things at ﬁrst,
we shall take the case that the velocity v is in the direction of u. (Later, we can do the
more general case.) What is v’, the velocity as seen from the space ship? It is the 1 7—5...
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 Spring '09
 LeeKinohara
 Physics

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