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:
“The Cairo conference ... is about a complicated web of education and employment, consumption and poverty, development and health care. It is also about whether governments will follow where women have so clearly led them, toward safe, simple and reliable choices in family planning. While Cairo crackles with conflict, in the homes of the world the orthodoxies have been duly heard, and roundly ignored.”
—Anna Quindlen (b. 1952)
“Rules and particular inferences alike are justified by being brought into agreement with each other. A rule is amended if it yields an inference we are unwilling to accept; an inference is rejected if it violates a rule we are unwilling to amend. The process of justification is the delicate one of making mutual adjustments between rules and accepted inferences; and in the agreement achieved lies the only justification needed for either.”
—Nelson Goodman (b. 1906)
“This form, this face, this life
Living to live in a world of time beyond me; let me
Resign my life for this life, my speech for that unspoken,
The awakened, lips parted, the hope, the new ships.”
—T.S. (Thomas Stearns)
“But let my due feet never fail
To walk the studious cloister’s pale,
And love the high embowed roof,
With antic pillars massy proof,
And storied windows richly dight,
Casting a dim, religious light.”
—John Milton (1608–1674)
“Three elements go to make up an idea. The first is its intrinsic quality as a feeling. The second is the energy with which it affects other ideas, an energy which is infinite in the here-and-nowness of immediate sensation, finite and relative in the recency of the past. The third element is the tendency of an idea to bring along other ideas with it.”
—Charles Sanders Peirce (1839–1914)