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 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 $(x',y',z',t')$ measured inside a “moving” space ship, moving                         with velocity $u$ are                         \begin{equation}                         \begin{aligned}                         x'&=\frac{x-ut}{\sqrt{1-u^2/c^2}},\\                         y'&=y,\\[2ex]                         z'&=z,\\                         t'&=\frac{t-ux/c^2}{\sqrt{1-u^2/c^2}}.                         \end{aligned}                         \label{Eq:I:17:1}                         \end{equation}                         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:                         \begin{equation}                         \begin{alignedat}{4}                         &x'&&=x&&\cos\theta+y&&\sin\theta,\\                         &y'&&=y&&\cos\theta-x&&\sin\theta,\\                         &z'&&=z&&.                         \end{alignedat}                         \label{Eq:I:17:2}                         \end{equation}                         In this particular case, Moe and Joe are measuring with axes having an                         angle $\theta$ between the $x'$- and $x$-axes. 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 different                         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 measurement 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 an object that we                         are looking at is somehow greater (speaking crudely and intuitively)                         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                         appreciably 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 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 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                         space-time. A given point $(x,y,z,t)$ in space-time is called an                         event. Imagine, for example, that we plot the $x$-positions                         horizontally, $y$ and $z$ in two other directions, both mutually at                         “right angles” and at “right angles” to the paper (!), 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 $t$-axis                         (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 example, which starts to drift out and then slows up                         should have a motion something like that shown in                         Fig. 17–1(c). A particle, in other words, which is permanent                         and does not disintegrate is represented by a line in space-time. 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 space-time 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 $t$ and the                         new $x$ in our new system, as shown in Fig. 17–2(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 $\cos\theta$ and $\sin\theta$, while the other is                         written with algebraic quantities. (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                         space-time 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.