Two different events?

Quick question about a relativity calculation. Scenario: conveyor belt going at 0.5c. At t= 0 Observer A is on the floor at x = 0 next to, but not on the conveyor belt. Observer A sees Observer B on the conveyor belt at t = 0 x = 0 smashing through a opaque sugar glass pane wearing body clock suit which is displaying the time as t'=0 and stamping observer B's forehead with that time. I'll refer to that as Event X. Observer B agrees with Observer A that its clock was saying t'=0, and that t'=0 has been stamped on its forehead at Event X. Observer C according to Observer A was at x = c on the conveyor belt at t = 0. With my calculations, Observer C would be stating that Event X happened at t' = -0.57735. Presumably this would mean that the B's clock would have been displaying t'=-0.57735 as it smashed through the opaque sugar glass, and presumably that would suggest -0.57735 had been stamped on the forehead. Which would seem to describe a different event. Have I understood this correctly?

 

I may have made a mistake in the calculation.

gamma = 1/sqrt(1 - ((v * v) / (c * c)))
= 1/sqrt(1 - ((0.5c * 0.5c) / (c * c))
= 1/sqrt(1 - 0.25)
= 1/sqrt(0.75)
= 1.1547 (approx)

t' = gamma * (t - ((v * x)/(c * c)))

So x and t are from Observer A's perspective, and used to calculate the perspectives of Observer B and Observer C

So for Observer B:
t'= 1.1547 (0 - ((0.5c * 0) / (c * c)))
t'= 0

for Observer C:
t'0 = 1.1547 (0 - ((0.5c * c) / (c * c)))
t'0 = 1.1547 (-0.5)
t'0 = -0.57735

Have I made a mistake?

 

Sorry I have I can see it, the Event X is at x = 0 and so the calculation for Observer C is inappropriate.

 

Yes thanks you're right :)

 

What if the body clock suit also stamped on the Observer B's forehead the time it saw on Observer A's body clock? I'm assuming what I was using as the Observer C calculation
t' = 1.1547 (0 - ((0.5c * c) / (c * c)))
t' = 1.1547 (-0.5)
t' = -0.57735

was when Observer C is saying t = 0 at x = 0, yet wouldn't there be evidence on Observer B's head that from Observer C's rest frame the event of Observer A's clock equalling t = 0 was at t' = 0, or is it that they are both equally right concerning what time it was when t = 0?

 

Or is it when t = 0 at x = c?

 

Sorry I think I've been using the equations with one at rest at x = 0, t = 0 and the other at velocity v at x, and then using the equations for an account of how one sees the other. Which as you can see has lead to some confusion on my part with me flip flopping over what the x represents. But I can now see that t' and t are both considered the same for all events in their respective rest frames, and the equation I used for observer C just shows that if there had of been an observer D at rest with observer A at x = c then observer C would say its clock wasn't in synch with observer A's.