Spiral Modern Physics (D’Alessandris)

Time Dilation

Einstein developed much of his understanding of relativity through the use of gedanken, or thought, experiments. In a gedanken experiment, Einstein would imagine an experiment that could not be performed due to technological limitations, and “perform” the experiment in his head. By analyzing the results of these experiments, he was lead to a deeper understanding of his theory. In developing his understanding of the relativity of time, Einstein imagined what he considered the simplest possible clock, consisting of a mirror on the floor and a mirror on the ceiling of a train car, and a light pulse bouncing back and forth between the mirrors. Each reflection of the light pulse serves as a “tick” of the clock.

Now imagine two observers and the train car traveling to the right at speed . A passenger on the train will see the light pulse travel a vertical path from floor to ceiling. An observer at rest on the earth, however, will see the train displaced to the right during the time it takes the pulse to travel from floor to ceiling, and hence see the light pulse follow an angled path.

Using Length Contraction

The star Vega is approximately 25 light-years from Earth (as measured by observers on Earth). a. How fast must a spaceship travel in order to reach Vega in 30 years, as measured on Earth? b. How fast must a spaceship travel in order to reach Vega in 30 years, as measured on the spaceship? The distance between two events, for example leaving Earth and arriving at Vega, depends on who makes the measurements. Within any particular reference system, the familiar results of classical physics are valid. However, in comparing results between observers in different reference systems, a method of relating one observer’s measurements to another is needed.

Lorentz Transformation

The time dilation and length contraction relationships actually have very limited applicability. To use the time dilation relationship, one of the two observers must measure a proper time, where the two events occur at exactly the same spatial point. To use the length contraction relationship, one of the two observers must measure a proper length, where the two events must be at rest with respect to the observer. What if you want to compare measurements concerning more general events? To do this requires the Lorentz Transformation, which allows you to transform the spacetime coordinates of an event in one inertial reference system to any other inertial reference system.