### Invariant times and lengths

Newton et al. thought that time is universal across all frames of reference, whether they are stationary, moving at uniform speed, or accelerating.

They thought that if you synchronise two clocks, or indeed whole collections of clocks, then they will remain synchronised regardless of their individual motions. As we have seen, this is not true. Moving clocks tick off time at a different rate than stationary clocks.

We shall show that the **proper time** between two space-time events, defined to be the time as measured by a clock on a world-line connecting those two events, is an invariant quantity on which all observers agree.

In contrast, **coordinate time** is the time between two events as measured by any other clock. Coordinate time depends on the motion of the clock relative to the clock measuring proper time, hence observers in other frames will *not* agree on its value.

Similarly, Newton would have thought that the length of an object, a rod say, would be a universal property of the object.

We shall show that only the **proper length** of an object, which can be defined to be the length as measured by an observer at rest relative to the object, is invariant, and the **coordinate length**, as measured by observers moving relative to the rod, is not the same for all.

### Separation

Given an event, $E \rightarrow (t, x)$, as measured in some frame $O$, then the square of the **separation**, $S$, between $E$ and the origin of $O$ is given by $$S^2 = t^2 - x^2$$

As we have seen in the last section, this quantity (which we previously referred to as proper time) is an invariant quantity. That is, if $E \rightarrow (t', x')$ in some other frame, $O'$, then the separation as measured in $O'$, called $S'$ say, satisfies $S'^2 = S^2$.

Nb. If we set $c \neq 1$, then $E \rightarrow (ct, x)$ and so the separation (squared) is $S^2 = c^2 t^2 - x^2$.

We can use separation to distinguish *three* regions of space-time:

#### Time-like events

These satisfy $S^2 \gt 0$. A time-like event can always be intersected by a world-line (of an object with a non-*zero* mass) through the origin.

Another way to say it is that time-like events (events with time-like separation) can never be measured to be simultaneous, in any frame.

The separation of time-like events is related to the **proper time** between those events by $$\tau = \sqrt{S^2} = \sqrt{t^2 - x^2}$$

#### Space-like events

These are events satisfying $S^2 \lt 0$. Space-like events are characterised by the fact that there *always* exists a frame in which they are measured to be simultaneous.

Space-like events can never be intersected by world-lines of any objects, massive or massless.

The separation of space-like events is related to the **proper distance** between those events by $$\lambda = \sqrt{-S^2} = \sqrt{x^2 - t^2}$$

#### Light-like events

These satisfy $S^2 = 0$, meaning that light-like can always be intersected by light-paths - world-lines of photons, and only by light-paths.

In normal space, if the distance between two positions is *zero* then the positions are the same, but in space-time, we can have two non-equal light-like events with separation *zero*.

Figure 4.4 shows the three regions of space-time.

With respect to $O$, $E_1$ is time-like, $E_2$ is light-like and $E_3$ is space-like.

The light-like region is called the **light cone** of $O$.

The time-like regions of $O$ can be split into the **absolute future** of $O$ ($S^2 \gt 0, t \gt 0$) and the **absolute past** of $O$ ($S^2 \gt 0, t \lt 0$), whilst the space-like region is often referred to as the **absolute elsewhere** of $O$.

Nb. In Galilean relativity, the future of space and the past of space would constitute all of space-time.