Relativity of simultaneity

The relativity of simultaneity is the statement that two events that are simultaneous in inertial frame are not necessarily simultaneous in another.

Thought experiment#

Imagine a train cart traveling at some constant speed along a smooth, straight track. In the center of the cart, a light bulb is attached to the ceiling, initially off. When someone on the train turns the light on, the light spreads out in all directions at the speed of light $c$, according Einstein's second postulate. Since the light bulb is place in the exact center of the cart, the light will reach the ends of the cart simultaneously.

When someone on the ground looks at the light being turned on, something different happens. As the train rolls forward, the back of the cart moves towards the light, while the front moves away. The light needs to travel less space to reach the back than it does to reach the front. In Galilean relativity this would not in practice make a difference, as the velocity of the train would be added to that of light, so the speed towards the back would be lower, canceling out the benefit of having to travel less space. But because the second postulate states that $c$ is universal, no such speed change occurs and there is a difference. The light reaches the back first and the front after. In other words, they are no longer simultaneous events from the perspective of the person on the ground. This is called relativity of simultaneity:

From Lorentz transformations#

The relativity of simultaneity can be seen quite easily by applying a Lorentz transformation. Say we have two events $A$ and $B$. The former occurs in $x_{A}=0$ and at $t_{A}=0$, while the latter in $x_{B}=b$ and at $t_{B}=0$¹. These are obviously simultaneous in the initial frame of reference since both occur at $t_{A}=t_{B}=0$, but doing a Lorentz transformation into another inertial frame shows that the new time coordinates are $t_{A}'=0$ and $t_{B}'=-\gamma \beta b/c$. Since these are no longer the same, they are not simultaneous in the new frame.

Ordering#

Due to this phenomenon, it is possible for the order in which two events occur to differ depending on the frame of reference. This is not always possible. For this to be allowed, the events must be spacelike. Timelike events have their ordering set in stone, as by definition they must occur at different times. Reordering them would require them to be simultaneous in some frame (the one in which they "cross over"): no Lorentz transformation will allow that. This obligation essentially holds the entirety of causality on its back, as if all events could be reordered, no one could ever say that "this happened before that", as there would always be someone in a different frame that could prove you wrong. In other words, only timelike events are causally linked to each other. Spacelike events are not.