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).

## Best Answer

"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?"

You are forgetting to take into account the relativity of simultaneity, which is relevant because each observer is supposed to assign time to events using readings on a network of synchronized clocks at rest at different locations along their own rulers (the clocks synchronized using the Einstein synchronization convention), so that each event's time can be judged by a clock that was right next to the event when it happened. That way you don't have to worry about light delays--for example, if I see light from an explosion 5 light-years away in 2005, and light from an explosion 10 light-years away in 2010, I can look at the clock in my system that was next to each event when it happened, and see that each explosion happened next to a clock that read t=2000 at the moment it happened, so I judge that these events happened simultaneously in my frame despite the fact that I

sawthem at different times. This page has an illustration of such a lattice of rulers and clocks, which give physical meaning to the position and time coordinates assigned by a given inertial frame:In your example, there are two events that the observer in S' is assigning coordinates to--the first is the event of the clock belonging to frame S reading t=0, the second is the event of the firecracker going off. S' has a clock #1 at position x'=0 that was next to the first event when it happened, and that clock read t'=0 at that moment; and S' has a separate clock #2 at position x'=-865,426,282 m that was next to the second event when it happened, and that clock read t'=5.77 s (my numbers are slightly different than yours, it looks like you had some roundoff error). The key to understand why this does

notcontradict the fact that S should see all of the clocks in S' running slow is that from the perspective of frame S, these two clocks in S' arenotsynchronized, due to the relativity of simultaneity. If the two clocks are a distance L apart in S' and synchronized in that frame, then from the perspective of frame S which judges S' to have velocity v, the back clock's time will be ahead of the front clock's time by an amount Lv/c^2 at any given moment. So in this case, with L=865,426,282 m and v = 0.5c = 149,896,229 m/s, frame S will say that at any given moment, clock #2's time is ahead of clock #1's time by the amount (865,426,282)*(149,896,229)/(299,792,458)^2 = 1.4385 seconds. So at t=0 in frame S, when clock #1 reads t'= 0 s, clock #2 reads t'= 1.4434 s. Then 5 seconds later in frame S, each of the clocks in S' has only advanced forward by 4.3301 s, because time dilation says that the clocks in S' must be running slow in frame S. But since clock #2 already had a head start of 1.4434 s, at t = 5 in frame S, clock #2 will read 1.4434 + 4.3301 = 5.7735 s, and that's the moment at which the firecracker goes off next to it.