Relativity Calculator
§ Assume: S' system is moving "inside" stationary S system with velocity v and is carrying the following clock consisting of a light-flash source and a receiving photocell. One "clock tick" consists of a roundtrip light-flash and photocell reception:
suggested source of diagram: Richard Feynman's " Lectures on Physics - Vol. I "
Employing simple algebra for the above Stationary Clock (a) in S' system versus Moving Clock (b) in S system, we get the following:
So as
.
What this all means is that as viewed from within Stationary system S, time in Moving system S'
will appear to move more slowly - i.e., relative units of time become comparatively expanded - as relative velocity
increases as between Stationary system S and Moving system S'.
Inside the Moving system S', rest time
however moves at a "normal rate".
In conclusion, any "moving clock" moving at a uniform velocity in an inertial ( non - accelerating ) frame of reference relative to a stationary observer's clock will therefore appear to run slower!
§ Time dilation:
In special relativity, a uniformly moving clock relative to a stationary clock in an inertial frame of reference present at two events will show time measured that is shorter than that of the stationary clock at the same two events. Since all motion is relative in special relativity, time dilation is reciprocal; that is, an observer at either of the two clocks will be stationary relative to the other clock and therefore it will reciprocally be the other clock which will be time dilated to the ( relatively stationary ) observer.
In general relativity, on the other hand, where the Principle of Equivalence as between gravitational attraction and acceleration is invoked, the phenomenon of time dilation of clocks running slower at lower potentials of a gravitational field ( where by definition there is greater spacetime metric curvature ) will be agreed upon by any other stationary observer of these same clocks throughout the gravity field, notwithstanding differences in altitude. In other words, clocks at ground level to a planet's gravity field will run slower than those clocks at higher altitude and that all observers, at whatever altitude, will agree with these same time dilation results. Hence, in general relativity time dilation is not reciprocal.
This concept of relativistic time dilation in special relativity is also shown in this American Museum of Natural History - "A Matter of Time" movie as well as PBS's NOVA Science program of the 1971 time dilation experiment aboard a transatlantic British Airways flight:
source: American Museum of Natural History - "A Matter of Time"
[ note: for those who cannot view this page whole, see quicktime movie ]
source: PBS's NOVA Science - "Time Dilation Experiment, 1971"
[ note: for those who cannot view this page whole, see quicktime or flash movie ]
Observed Facts: Flying eastward and then westward from the stationary U.S. Naval Observatory, General Relativity predicted a time loss of 40 ± 23 nonoseconds on the eastward bound and a time gain of 275 ± 21 nanoseconds for the westward trip; the actual eastward loss was 59 ± 10 nanoseconds going eastward but gained 273 ± 7 nanoseconds for the westward trip!!
§ Another derivation partially using both (c) above and referring back to "Albert A. Michelson & the Aether-Part II":
Please also refer back to "Albert A. Michelson & the Aether-Part II" as the mathematics is identical although the problem herein is subtly different.
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