I ran into a serious problem with the Lorentz transformation and time dilation. In the standard configuration you have one observer S and another one S' with their x-axis aligned. I assume S to be at rest and S' to be moving in direction of the x-axis at a speed v = 0.5*c (the relativistic gamma is then g = 1.15). Each observer has a clock and they meet when both clocks show 0 s.
Now suppose a firecracker goes off at the origin of S. This happens at a time when the clock of S reads 5 s. So he assigns the event the coordinates x = 0 m and t = 5 s. I want to know what coordinates observer S' assigns the same event. According to the Lorentz transformation, these are x' = -862,500,000 m and t' = 5.75 s.
(Just to check, I inserted x' = -862,500,000 m and t' = 5.75 s into the inverse Lorentz transformation and got x = 0 m and t = 5 s as expected)
So far, so good. But I'm having trouble interpreting this. So observer S says 5 s passed between their meeting and the explosion of the firecracker, observer S' says 5.75 s passed between their meeting and the explosion of the firecracker. That's also fine. But this means that S says the clock of S' is ticking faster while S' says the clock of S is running slower. Shouldn't both say that the clock of the other is ticking at a slower rate? Where is the problem in my logic, what do I misunderstand?
I initially expected that when insert x = 0 m and t = 5 s into the Lorentz transformation, I get t' < t (clock runs slower), but this doesn't happen!
Would be fantastic if somebody could help me here. It's a very exciting topic, but I feel like I have hit a dead end. No matter how I try to resolve it, I always get the same problem. One observer sees time dilation, the other time "acceleration", but I know that it should always be time dilation (moving clocks run slower, the mantra of SR).