# Light-like Interval

In SR, the interval $I$ between two spacetime events is called light-like if $I=0$. Griffiths in his Introduction to Electrodynamics book says that [page 503],

If $I=0$ we call the interval light-like, for this is the relation that holds when the two events are connected by a signal traveling at the speed of light.

What does this statement mean?

• ... exactly what it says? – Emilio Pisanty Sep 13 '13 at 0:44

In special relativity there is an important quantity called the proper time, $\tau$, which is the time measured by a freely moving observer. If you have two spacetime points ($t_1$, $x_1$, $y_1$, $z_1$) and ($t_2$, $x_2$, $y_2$, $z_2$), and we define $\Delta t = t_2 - t_1$, $\Delta x = x_2 - x_1$, and so on, then the proper time is given by:

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

The quantity $I$ referred to in your book is the proper time.

It should be fairly obvious that for anything travelling at the speed of light $\Delta \tau^2$ is zero, hence the statement that the interval is light-like if it's zero. $\Delta \tau^2$ is positive if you can move between the two events without travelling faster than light, and it's negative (i.e. $\Delta \tau$ is imaginary) if moving between the two events requires faster than light travel.

The proper time is an invarient, that is every observer in every inertial frame will measure the same proper time between the two events. From this simple principle you can derive all the weird behaviour seen in special relativity like time dilation and length contraction.

Let's say events $A$ and $B$ are separated by a light-like interval. If a photon is emitted from the location of $A$ at the same time that $A$ happens, and the photon is pointed directly at the location of $B$, it will arrive at $B$ at the same time that event $B$ happens.

If separated by a time-like interval, you could replace the photon by some slower-moving particle that travels from $A$ to $B$. If by a space-like interval, no particle will travel from $A$ to $B$ in the manner described above because it would have to move faster than the speed of light.

Hence $A$ could have caused $B$ is they are time-like separated but never when they are space-like separated. Light-like separation is the boundary between those cases; $A$ could have caused $B$, but just barely.

It should be instructive to consider an answer which does not involve any particular assignment of coordinates to events, but only measurable physical quantities.

First of all, let's look at the question being quoted selectively as follows:

[...] we call the interval light-like [...] when the two events are connected by a signal [...] of light

This, itself, should be self-evident; and it is indeed valid in GR just as well.

So it remains to be explained why those light-like intervals are conveniently assigned the magnitude value $0$;
and, along the way, which other types of intervals there are, how they are called, and how their magnitudes are valued.

Well -- there are also pairs of events which are not connected (and cannot be connected) by any signal at all; intervals between such pairs of events are called "space-like", and such a pair of events itself are called "space-like separated".

On the other hand, there are pairs of events which are (or at least, could be) even connected by "stuff", i.e. connected by someone travelling on some particular route (and therefore generally connected also by others travelling on some other particular routes as well); intervals between such pairs of events are called "time-like", and such a pair of events itself are called "time-like separated".

Considering any particular pair of such time-like separated events, with various travellers having jointly taken off from one event and having all met again at the other event, the durations (a.k.a. "proper times") of these travellers from departure from one event until arrival at the other event are (generally) different from each other. The longest duration taken by any of those travellers is (taken as) the magnitude of the time-like interval separating these two events. (This also applies in GR.)

Since there are no travellers at all connecting events which are light-like separated (but, as the name indicates, only light signals), there is no corresponding way to assign some duration value to the magnitude of any light-like interval.

The discussion of pairs of space-like separated events is easier being restricted to SR; so we may consider pairs of participants who are at rest to each other. Considering any particular pair of space-like separated events, there are (or at least, there could be) many different pairs of participants such that one of them had been at one event, and the other participant had been at the other event under consideration. The distances (a.k.a. "proper lengths") between those distinct pairs of participants are (generally) different from each other. The shortest distance determined by any of those pairs is (taken as) the magnitude of the space-like interval separating these two events. This happens to be the distance between the pair of participants who find that one of them had been at one event simultaneous to the other having been at the other event.

But for light-like separated events there are no pairs of participants such that one had been at one of these events simultaneous to the other participant having been at the other event. So there is no corresponding way to assign some distance value to the magnitude of any light-like interval either.

What's left therefore, as a convenient possibility, is to assign the magnitude $0$ equally to all light-like intervals.

$I=0$ is equivalent to say $c^2(\Delta t)^2=(\Delta x)^2+(\Delta y)^2+(\Delta z)^2$. What this mean? The meaning is that something travels with the speed of light. Consider that $\Delta$ represents the interval of two events, that "something" must be a signal traveling with $c$.