Imagine that you are standing on the Earth while our friend is traveling across our solar system at a high speed, as shown in the accompanying sketch. You set off a flashbulb that emits sudden bright flash of light. The radiation moves away from you at the same speed at all directions, and thus you see an expanding spherical shell of light. What does your high-speed friend see?

Einstein argued that this person must also see light moving away from her at the same speed in all directions, and thus she also sees an expanding spherical shell of light.By requiring that both people observe a spherical shell, Einstein derived a series of equations to relate specific measurements of time and distance between two people. These equations are named the Lorentz transformations, after the famous Dutch physicist Hendrik Antoon Lorentz (a contemporary of Einstein who developed these equations independently but did not grasp their true meaning.) These equations tell us exactly how a moving person's clock slow down and how rulers shrink.

To appreciate the Lorentz transformations, again imagine that you are on Earth while a friend is moving at a speed v with respect to you. Suppose that you both observe the same phenomenon on Earth - say, the beating of your heart or the ticking of your watch, which appears to occur over an interval of time. According to your clock (which is not moving relative to the phenomenon), the phenomenon lasts for T0 seconds. This is called the proper time of the phenomenon. But according to your friend's clock (which is moving relative to the phenomenon), the same phenomenon lasts for a different length of time, T seconds. The Lorentz transformation for time tells us that these two time intervals are related by:

Lorentz transformation for time

T = time interval measured by an observer moving relative to the phenomenon

T0 = time interval measured by an observer not moving relative to the phonomenon

v = speed of the moving observer

c = speed of light

EXAMPLE: Suppose that your friend is moving at 98% of the speed of light. Then, v/c = 0.98 so that

=> T = 5T0

The Lorentz transformation for time is plotted in the accompanying graph, which shows how 1 second measured on a stationary clock is stretched out when measured using a clock carried by a moving observer. Note that significant differencesEXAMPLE: Fast-moving protons from interstellar space frequently collide with atoms in the Earth's upper atmosphere. When they do, they can create unstable particles called muons (pronounced "mewons") that decay in an average time of 2.2 x 10 to power -6 seconds. Such muons typically move at 99.9% of the speed of light and are formed at an altitude of 10kn. As measured by an observer on the Earth, the time that a muon would take to reach the Earth's surface is

This is 15 times longer than the life expectancy of a muon, so it would seem that muon would never reach the Earth's surface before decaying. In fact, these muons are detected by experiments on the surface! The reason is that as seen by an Earth observer, the muon is a "moving clock", and hence its decay is slowed down by time dilation. To an Earth observer, the actual lifetime of a muon is

Thus, as measured by an Earth observer, muons live more than long enough for them to reach the surface. The detection at the Earth's surface of muons from the upper atmosphere is compelling evidence for the reality of the time dilation.