S19. Properties of spacetime intervals

In the Euclidean geometry, the separation between two points is the measure of the physical distance. The quantity measured is purely spatial and its always positive. However, in the spacetime geometry the points measured are the events of the system whose coordinates includes both the spatial and the temporal measurement. The measured invariant interval between the two events is represented by $\Delta s^2$ and is defined as,

$\Delta s^2$=$-c \Delta t^2$+ $\Delta x^2$$\Delta y^2$$\Delta z^2$

Unlike the ordinary distance between two events $E_1$ and $E_2$, the interval $\Delta s^2$ can be positive, zero and negative. These three different possibilities have their own names:

1. $\Delta s^2<0:$ $E_1$ and $E_2$ are separated by a time-like interval.
2. $\Delta s^2=0:$ $E_1$ and $E_2$ are separated by a light-like interval.
3. $\Delta s^2>0:$ $E_1$ and $E_2$ are separated by a space-like interval.

What these different possibilities represent is best illustrated on a spacetime diagram. Suppose event $O$ occcurs at the spaectimepoint $(0,0)$ in some frame of reference $S$. We can divide the spacetime diagram into two regions as illustrated in the figure below: the shaded region lying between the world lines of photons passing through $(0,0)$, and the unshaded region lying outside these world lines. Note that if we added a further space axis, in the $Y$ direction say, the world lines of the photons passing through will lie on a cone with its vertex at $O$. This cone is known as 'light cone'. Then events such as $Q$ will lie 'inside the light cone', events such as P 'outside the light cone', and events such as $R$ 'on the light cone'. 

The point $Q$ within the light cone (the shaded region) is separated from $O$ by a time-like interval. A signal travelling at a speed less than $c$ can reach $Q$ from $O$. The point $R$ on the edge of the light cone is separated from $O$ by light-like interval, and a signal moving at the speed $c$ can reach $R$ from $O$. The point $P$ is outside the light cone. No signal can reach $P$ from $O$.

The physical meaning of these three possibilities can be seen if we consider whether or not the event $O$ can in some way affect the events $P,Q, or R$. In order for one event to physically affect another some sort of signal must make its way from one event to the other. The signal can be any kind: a flash of light created at $O$, a massive particle emitted at $O$, a piece of paper with a message on it and placed in a bottle. Whatever it is, in order to be present at the other event and hence to either affect it (or even to cause it) this signal must travel the distance $\Delta x$ in time $\Delta t$, i.e. with speed $\Delta x/\Delta t$.

We can now look at what this will mean for each of the events $P, Q, R$. Firstly, for event $P$ we find that speed $\Delta x/\Delta t>c$. Thus, the signal must travel faster than the speed of light, which is not possible. Consequently event $O$ cannot affect, or cause event $P$. Secondly, for event $Q$ we find that speed $\Delta x/\Delta t<c$ so the signal will travel at a speed less than the speed of light, so event $O$ can affect (or cause) event $Q$. Finally, for $R$ by means of a signal travelling at the speed o light. In short,

1. Two events separated by a space-like interval cannot affect one another.
2. Two events separated by a time-like interval can affect one another.

Thus, returning to our spacetime diagram, we have:

All the events that can be influenced by $O$ constitute the future of event $O$ while all events that can influence $O$ constitute the past event $O$.

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