Does the Relativity of Simultaneity Depend on the Axis Along Which Events Occur?

[darkchild]

I was reviewing this concept last night, and it occurred to me that the statement

Two spatially separated events simultaneous in one reference frame are not simultaneous in any other inertial frame moving relative to the first

is not specific enough. If the simultaneous (as seen in S) events occur along an axis that is perpendicular to the direction of motion of the second reference frame S', won't they also appear simultaneous in S'?

It seems like the paragraphs and paragraphs used to explain this phenomenon are a complicated way of saying that events that are simultaneous in one frame S are not simultaneous in another frame moving with respect to the first S' because the moving frame S' is moving toward one of the events and away from the other, thus inducing an observed time delay between the closer event and the further event, in which case the axis along which the events occur is a crucial factor.

 

[quZz]

Yes, you're quite right
$$\Delta t' = \gamma (\Delta t - (\textbf{V} \cdot \Delta \textbf{r})/c^2)$$

 

[GRDixon]

A crucial point is that the clocks at rest in inertial frame K' are not all synchronized, according to the clocks at rest in K. In SRT only the clocks in a plane normal to the velocity vector of K' relative to K are synchronized in the opinion of both frames. If K' moves in the positive x-direction of K, and if t1 and t2 are equal, K contends that the events at the respective clocks occurred simultaneously. But K' contends that the clocks are not synchronized (although he agrees that they read the same thing when the space-time coordinates of the two events were measured using the grid/clocks of K). Consequently, K' contends that the two events did not happen simultaneously.