Light Clock
Physics
World
Back to Special
Relativity
The
phenomena of time dilation can be derived from the two postulates of special
relativity, namely
(1) Principle of
Relativity - The laws of physics are the same in all inertial frames of
reference
(2) Light Postulate - The
speed of light in an inertial frame of reference is independent of the source
Consider the following setup. There is a clock attached to a mirror. The clock/mirror remains at a constant distant L from another mirror which is parallel to the first. See diagram below.
Such
an apparatus is called a Light Clock. The distance L is adjusted so that
the time interval between clock ticks will equal the time the light takes to
travel from the clock to the mirror and back to the clock again. Let S represent
the frame of reference in which the clock is at rest. Let "t"
equal the time between clock
ticks as measured in S. t
is therefore the time between
the two events defined as
Event
1: Light is emitted
Event
2: Light is detected.
as
measured in S. t is
called the proper time, i.e. it is the time between two events
as measured on ideal clock which passes through both events. Let S' be a frame
of reference in which the clock is moving, with speed v, to the right as
shown in the figure above. Let "t" represent the time between these
two events as measured in S'. Symmetry
and the Principle of relativity imply that the distance L remains unchanged due
to the motion. From the diagram above and the Pythagorean theorem it follows
that
However by definition of t we also know that
Therefore
Solving for t gives
Therefore
the time, t, between the same two events as measured in the frame S' will
be larger. E.g. if the clock is moving at 99% of the speed of light and if t = 1 second
then t = 7.09 seconds. Hence moving
clocks run slower than the same clock at rest. Since there is nothing inherent
in the clock that we relied on in the derivation above this implies that it is
time itself that is running slower!!