One frequently hears the affirmation that Einstein established that time is relative. A more precise statement would be that observers in relative motion will attribute different values to the interval of time between two events. For example, if an observer compares to his own clock the clock carried by an other observer in motion with respect to him, he will conclude that the moving clock is running slow, for the interval between ticks of the moving clock is larger than the interval between ticks of his own clock. The ratio between these intervals is a simple function of the velocity of one observer with respect to the other.
It should be stressed that the effect is reciprocal. For the second observer, it is the clock carried by the first observer which is moving and is, therefore, running slow.
Time dilation is verified experimentally, for example in the decay times of unstable particles produced by cosmic rays when they impinge on the Earth's atmosphere. One can imagine that such particles possess an "internal clock'' which determines their proper lifetime. In the referential of the Earth, the particles propagate with a velocity close (although inferior) to the speed of light. The time they take to traverse the atmosphere, measured in the referential of the Earth, is larger than their proper lifetime so that, if time dilation did not occur, the particles would vanish before they could be detected in laboratories installed on the Earth's surface. But thanks to time dilation, the lifetime of these particles in the Earth's referential is much larger than their proper lifetime and, if such a particle moves with velocity sufficiently close to the speed of light, it will survive long enough to cross the whole atmosphere and be detected at ground level.
Here is the link to the text “Concepts of Special Relativity” in PDF format. Time dilation is analyzed in Chapter 5. As mentioned above, the approach known as k-calculus is adopted, so that time dilation is first derived in terms of Bondi's k factor. The more familiar expression, in terms of the so-called Lorentz factor is then deduced, by using results obtained in Chapter 4 to express the k factor in terms of the relative velocity.
A numerical example of time dilation can be found in Appendix A.3.
In order to understand clearly the concepts and the nomenclature used, it is recommended to begin by reading the first three chapters of the text.
An animation software written in the Java language is available. Time dilation is illustrated by the fourth animation proposed.
If you are interested in a particular topic of Special Relativity, here is a list of the topics covered in the text and in the software. Clicking on an item in this list, will open a page which briefly introduces that topic and indicates the parts of the text and of the software in which it is treated.