Simple Inference of Time Dilation Due To Relative Velocity
Time dilation can be inferred from the observed fact of the constancy of the speed of light in all reference frames.
This constancy of the speed of light means, counter to intuition, that speeds of material objects and light are not additive. It is not possible to make the speed of light appear greater by approaching at speed towards the material source that is emitting light. It is not possible to make the speed of light appear less by receding from the source at speed. From one point of view, it is the implications of this unexpected constancy that take away from constancies expected elsewhere.
Consider a simple clock consisting of two mirrors A and B, between which a light pulse is bouncing. The separation of the mirrors is L and the clock ticks once each time the light pulse hits a given mirror.
In the frame where the clock is at rest (diagram at right), the light pulse traces out a path of length 2L and the period of the clock is 2L divided by the speed of light
From the frame of reference of a moving observer traveling at the speed v (diagram at lower right), the light pulse traces out a longer, angled path. The second postulate of special relativity states that the speed of light is constant in all frames, which implies a lengthening of the period of this clock from the moving observer's perspective. That is to say, in a frame moving relative to the clock, the clock appears to be running more slowly. Straightforward application of the Pythagorean theorem leads to the well-known prediction of special relativity:
The total time for the light pulse to trace its path is given by
The length of the half path can be calculated as a function of known quantities as
Substituting D from this equation into the previous and solving for Δt' gives:
and thus, with the definition of Δt:
which expresses the fact that for the moving observer the period of the clock is longer than in the frame of the clock itself.
Read more about this topic: Time Dilation
Famous quotes containing the words simple, inference, time, due and/or relative:
“Whenever a person strives, by the help of dialectic, to start in pursuit of every reality by a simple process of reason, independent of all sensuous information—never flinching, until by an act of the pure intelligence he has grasped the real nature of good—he arrives at the very end of the intellectual world.”
—Plato (c. 427–347 B.C.)
“I shouldn’t want you to be surprised, or to draw any particular inference from my making speeches, or not making speeches, out there. I don’t recall any candidate for President that ever injured himself very much by not talking.”
—Calvin Coolidge (1872–1933)
“From time immemorial the men of the town have been famous seamen, and have divided their energies between fishing and hating the English.”
—Willa Cather (1876–1947)
“I wish to speak with all respect of persons, but sometimes I must pinch myself to keep awake, and preserve the due decorum. They melt so fast into each other, that they are like grass and trees, and it needs an effort to treat them as individuals.”
—Ralph Waldo Emerson (1803–1882)
“To revolt is a natural tendency of life. Even a worm turns against the foot that crushes it. In general, the vitality and relative dignity of an animal can be measured by the intensity of its instinct to revolt.”
—Mikhail Bakunin (1814–1876)