Many people have difficulties to understand how time dilation can be consistent with he special relativity principle: "According to the observer on the Earth, the observer at the space ship is aging slower, but according to the observer at the space ship, the observer on the Earth is aging slower. Who is really aging slower?" Those asking this type of questions have not really grasped the relatvitivity of simultaneity. I think it is better to explain this using a thought experiment with a train than with a space ship, because a space ship has a quite small extension and it is difficult to intuitively associate it with an entire coordinate system. A train, on the other hand, is perceived as "long", with many cars, and easier to associate with a coordinate system. The disadvantage is that such a thought experiment is even more unrealistic than a space ship thought experiment; we need an absurdely fast and long train and an absurdely long platform. Still, I think the benefits outweigh the disadvantages. So, here comes the thought experiment (where we use the Lorentz transformation to go from one sytsem to the other one): Consider a train running east, passing, without stopping, a platform extending in the east-west direction. The velocity of the train, relative to the platform, is 0.87c (= sqrt(3)/2 c, making the factor sqrt(1-v^2/c^2) = 1/2). There are two clocks at the west and east ends of the platform, respectively. They are synchronized, all observers at the platform agree that these two clocks show the same time. Also, there are two clocks in the train, one in the front, in the locomotive, and one in the back, in the last car. They are synchronized too, all observers on the train agree that these two clocks show the same time. The train driver is, of course, located in the locomotive all the time. The station master is located at the west end of the platform and remains there during all the time when train passes. Assume that at the event when the locomotive arrives at the west end of the platform, the clock at the west end of the platform and the clock in the locomotive both show t=0 (I prefer not specifying time units, it just becomes ridicolously unrealistic otherwise). Also, assume that at the event when the locomotive arrives to the east end of the platform, the clock at the east end shows t=2. Then, the clock in the locomotive shows t=1. Finally, assume that at the event when the back of the train arrives to the west end of the platform, the clock at the west end of the platform also shows t=2. According to an observer at the platform, these two events are simultaneous (t=2 for both). The station master would therefore say that the train driver has aged 1 unit while himself has aged 2 units, so the train driver has aged slower than himself. But, by the special principle of relativity, the train driver can, with same right, say that the station master ages slower than himself. So, how can that be? Didn't we just show that the train driver is really aging slower than the station master? Well, no. What we showed was that the train driver is aging slower than the station master according to the station master (and other observers on the platform). The solution lies the fact that simultaneity is not absolute. The two events when the locomotive arrives to the east end and the back of the train arrives to the west end of the platform are simultaneous (t=2), but only according to an abserver on the platform. Recall that at the first of these events, the clock in the locomotive shows t=1. But what does the clock in the last car of the train show at the second of these events? Answer: t=4! An observer on the train would therefore say that the two events are far from simultaneous. The latter event would, according to an abserver of the train, be simultaneous to an event when t=4 in the locomotive, and then the locomotive is far east of the entire platform. The train driver then would say that the station master is aging slower than himself (2 units for the station master and 4 units for the train driver). Tom summarize: Event 1: Locomotive arrives to east end. Clock at east end shows 2, clock in locomotive shows 1 Event 2: Last car arrives to west end. Clock at west end shows 2, clock in last car shows 4. According to an observer at the platform, the two evenst are simultaneous, and the train driver has been aging slower than the station master. According to an observer on the train, the two events are not simultaneos, and also, the station master is aging slower than the train driver. They both say that the other one is aging slower, and they are equally right, and that is no contradiction.