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

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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?

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  • $\begingroup$ $S'$ 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 $y = z = t = 0$. $\endgroup$
    – mmesser314
    Dec 9, 2017 at 16:06
  • $\begingroup$ 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 ($ct = 0$). $\endgroup$
    – sangstar
    Dec 9, 2017 at 16:16
  • $\begingroup$ 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'. $\endgroup$
    – mmesser314
    Dec 9, 2017 at 16:26
  • $\begingroup$ Duplicate: physics.stackexchange.com/questions/366727/…. $\endgroup$ Dec 9, 2017 at 17:10
  • $\begingroup$ 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. $\endgroup$
    – WillO
    Dec 9, 2017 at 17:50

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