# Relativity of simultaneity - under what conditions would two events happen simultaneously for frames $S$ and $S'$ when $S'$ is moving relative to $S$?

I feel like I have a fundamental confusion with the following exercise:

I'm confident with my answer in (a), which is that $B$ must be within the light cone of $A$, or that $B$ is causally connected to $A$.

However, with (b), I'm confused, perhaps by the wording, or perhaps by my understanding.

• If $A$ and $B$ both occur on the $x$-axis in $S$, this implies they occur simultaneously in $S$, otherwise they wouldn't both be on the $x$-axis.
• In order for the two events to be simultaneous in frame $S'$, they must be both be on its $x'$-axis, so, the only way it can be simultaneous for both observers is if $S$ and $S'$ are not moving relative to eachother so that their worldlines are parallel.

However, $S'$ is clearly worded as moving to the right of $S$, so where am I going wrong here?

• $S'$ is moving with velocity v with respect to $S$. What must v be for them not to be moving with respect to each other? But the problem says $y = z = 0$. Not $y = z = t = 0$. Dec 9, 2017 at 16:06
• But if they both occur on the same $x$-axis for $S$, how can they not be at the same time? They should be on the same $ct$ cut ($ct = 0$). Dec 9, 2017 at 16:16
• A occurs before B in S. A and B are simultaneous is S'. A and B are on the X axis at different times in S. S' is moving along the x axis with respect to S. So y and z coordinates are the same in both. A and B are on the X axis at the same time in S'. Dec 9, 2017 at 16:26
• Duplicate: physics.stackexchange.com/questions/366727/…. Dec 9, 2017 at 17:10
• Pick an event $E$. The space of events that are simultaneous with $E$ in a given frame has codimension 1 in spacetime, so the space of events that are simultaneous with $E$ in two different frames has (in general) codimension 2. Dec 9, 2017 at 17:50