Causality Preserved

One important consequence of Einstein's theories of relativity is the breaking of absolute notions of time. This means that people can disagree about whether two events occur simultaneously, or whether the first occurred before the second, or whether the second occurred before the first.

This seems to potentially entail some disastrous consequences (frequently explored in time-travel fiction). For instance, suppose I head out in a spaceship relativistically in such a way that, when I return home, I return before I left. Then, I rush inside my house and destroy my spaceship, preventing me from leaving in the first place!

It seems as though I have broken causality, as I just performed an action and then prevented myself from performing the action; but this prevention was only possible because I performed the action in the first place! (This is a variant of something called the Grandfather paradox; there are other examples of breaking causality as well, such as the Bootstrap paradox.)

Are such things actually possible? It turns out that, under the theory of relativity, this is not possible: causality is still preserved. The reason that causality-breaking has become so popular in science fiction is that most people derive "time travel" as a consequence of relativity (since simultaneity is broken). However, there are still restrictions on this time travel, which most people are not familiar with, because to understand these, you need to examine the Lorentz transformation mathematically (which, of course, most people don't).

We will demonstrate this mathematically using the Lorentz transformation. Let $(x, y, z, t)$ be the spacetime displacement between two events. We define the spacetime interval $s^2$ between these two events as follows: $$s^2 = (ct)^2 - (x^2+y^2+z^2)$$ Sometimes, you will see the right side with the opposite sign convention. Either way is fine; it does not really matter. Also, $x, y, z, t$ should technically all have the Delta symbol in front of them (as in $\Delta x, \Delta y, \Delta z, \Delta t$), but we have discarded these for convenience. Note that, assuming that spacetime is Euclidean (a notion which agrees with our observations), the second term on the right side is the square of the spatial distance between the two events.

We claim that this spacetime interval is an invariant of the Lorentz transformation. To prove this, let $S$ and $S'$ be two inertial frames, with $S'$ traveling at a velocity $v$ relative to $S$ in the $x$ direction (we can always choose our coordinates such that this is true). We first write out the spacetime intervals in both of our inertial frames, which we have labelled $s^2$ and $s'^2$: $$s^2 = (ct)^2 - (x^2+y^2+z^2)$$ $$s'^2 = (ct')^2 - (x'^2 + y'^2 + z'^2)$$ The following are the equations that define the Lorentz transformation: $$t' = \gamma\left(t-\frac{v}{c^2}x\right)$$ $$x' = \gamma\left(x-vt\right)$$ $$y' = y$$ $$z' = z$$ Now, we calculate the spacetime interval in the $S'$ frame: $$s'^2 = c^2\gamma^2 \left(t-\frac{v}{c^2}x\right)^2 - \gamma^2\left(x-vt\right)^2 - y^2 - z^2$$ $$s'^2 = \frac{c^4}{c^2-v^2}\left(t^2-\frac{2vtx}{c^2}+\frac{v^2}{c^4}x^2\right)-\frac{c^2}{c^2-v^2}\left(x^2-2vtx+v^2t^2\right) - y^2 - z^2$$ $$s'^2 = t^2\left(\frac{c^4-c^2v^2}{c^2-v^2}\right) +2vtx\left(\frac{-c^2+c^2}{c^2-v^2}\right) + x^2\left(\frac{v^2-c^2}{c^2-v^2}\right) - y^2 - z^2$$ $$s'^2 = c^2t^2 - x^2 - y^2 - z^2$$ $$s'^2 = s^2$$ We see that the two spacetime intervals are equal, and for arbitrarily chosen inertial frames! Thus, the spacetime interval is the same for all inertial frames of reference.

So what does this mean for causality? Well, since everything travels slower than light, if one event affects the other, then: $$(ct)^2 \gt x^2 + y^2 + z^2$$ $$s^2 \gt 0$$ Such a separation between two events ($s^2\gt 0$) is called a time-like separation. (It is called this because there is some inertial reference frame in which the two events occur at the same location and are separated only by time.) Since all observers can agree on the spacetime interval, all of them can agree on whether the separation is time-like. As shown above, one event can affect the other if and only if their separations are time-like. Thus, all inertial reference frames will agree on whether one event can cause the other.

If $s^2\lt 0,$ then the separation is called space-like. (It is called this because there is some inertial reference frame in which the two events occur at the same time and are separated only by location.) If two events have a space-like separation, then it is impossible for one event to affect the other without something traveling faster than the speed of light, which is not possible.

Finally, if $s^2 = 0,$ then the separation is called light-like. (It is called this because a beam of light can start from the first event when it occurs and reach the other event right as it occurs.)

Given an event $A,$ the set of all points in spacetime that have a light-like separation from $A$ ($s^2 = 0$) is called the light cone of $A.$ This light cone determines the relation of event $A$ to the rest of the universe. The set of points in spacetime "outside" the light cone of $A$ ($s^2\lt 0$) is called the elsewhere of $A.$ Events occurring at these points in spacetime cannot affect $A,$ nor can they be affected by $A.$ Finally, the set of points in spacetime "inside" the light cone of $A$ ($s^2\gt 0$) is called the past and future of $A$ (depending on whether the event is before or after $A$--if the separation is time-like, then all observers will agree on which event occurs first). Events in the past of $A$ can affect $A,$ and events in the future of $A$ can be affected by $A.$

Causality must be preserved. But what is causality?

For instance, consider 2 events, hammering an egg and the egg breaking. The egg breaking must always occur after the egg is hammered, and the 2 events cant be reversed.