In order To synchronize clock should the observers has same plane of simultaneity?

Suppose we have to observer $$A$$ and $$B$$ with their time given by $$t_A$$ and $$t_B$$.To synchronize their clocks observer $$A$$ send lights at $$t_A$$ towards $$B$$. $$B$$ receives the light at $$t_B$$ and send it back to $$A$$,and $$A$$ receives the light at $$t_A'$$.Finaly $$A$$ sends a message to $$B$$ with instructions that his clock should have $$\frac{t_A+t_B}{2}$$ in $$t_B$$.

Now suppose we have two events, $$e_A$$ in the worldline of $$A$$ and $$e_B$$ on the worldline of $$B$$. Suppose also that these two observers agree that these events are simultaneous, than they can synchronize their clocks by setting their clocks to zero for these events.

My question is, given another event $$e_c$$, and suppose that $$e_c$$ is simultaneous to $$e_A$$ from the point of view of $$A$$, are $$e_A$$ and $$e_c$$ simultaneous from the point of view of $$B$$?

• With two observers only, remember that they will ways be on "the same plane" .This is basic geometry. – Sidarth May 29 at 5:19
• @Sidarth Well, they aren't actually planes, they are spaces. But 4D diagrams aren't easy to draw or understand, so it's customary in Special Relativity diagrams to suppress 1 or 2 space dimensions, so we get lines or planes of simultaneity. Please see en.wikipedia.org/wiki/Relativity_of_simultaneity – PM 2Ring May 29 at 5:56
• Are A & B at rest relative to each other? – PM 2Ring May 29 at 5:57
• They are at rest in a uniform rotating frame – amilton moreira May 29 at 5:59
• @amiltonmoreira If they are in a uniform rotating frame, then they are not inertial observers. So, you cannot blindly use special relativity – silverrahul May 29 at 6:21