Space-time intervals
Although the geometry of space-time 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 here to the other point is the same in both. That                         is one property that is independent of the particular way of measuring                         it. The square of the distance is $x^2 + y^2 + z^2$. Now what about                         space-time? It is not hard to demonstrate that we have here, also,                         something which stays the same, namely, the combination $c^2t^2 - x^2 -                         y^2 - z^2$ is the same before and after the transformation:                         \begin{equation}                         \label{Eq:I:17:3}                         c^2t'^2\!-x'^2\!-y'^2\!-z'^2\!=c^2t^2\!-x^2\!-y^2\!-z^2\!.                         \end{equation}                         This quantity is therefore something which, like the distance, is                         “real” in some sense; it is called the interval between the two                         space-time points, one of which is, in this case, at the origin.                         (Actually, of course, it is the interval squared, just as $x^2 + y^2 +                         z^2$ 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                         different 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 $3\times10^8$ 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 $3\times10^8$ 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                         $1/3\times10^{-8}$ 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                                                  \begin{gather}                         \begin{aligned}                         x'&=\frac{x-ut}{\sqrt{1-u^2}},\\                         y'&=y,\\[1.5ex]                         z'&=z,\\                         t'&=\frac{t-ux}{\sqrt{1-u^2}}.                         \end{aligned}                         \label{Eq:I:17:4}\\[2.25ex]                         t'^2\!-x'^2\!-y'^2\!-z'^2\!=t^2\!-x^2\!-y^2\!-z^2\!.                         \label{Eq:I:17:5}                         \end{gather}                                                  If we are ever unsure or “frightened” that after we have this system                         with $c=1$ 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 $\sqrt{1 - u^2}$, we know                         that we cannot subtract a velocity squared, which has units, from the                         pure number $1$, so we know that we must divide $u^2$ by $c^2$ in order                         to make that unitless, and that is the way it goes. The difference between space-time 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                         space-like interval between them (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 time-like interval. In our diagram of space-time, therefore, we                         would have a representation something like this: at $45^\circ$ 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 is zero.                         