“Detectors are set up high on a mountain and at the base of the mountain. The distance between detectors is known, and by detecting the number of muons at both positions, relevant data are obtained. We just need to measure the number of muons per unit time at both positions. The muons act like precision clocks, and they have a known half-life.”

– Kognity, Inc

You can use the applet below to try the experiment yourself. Note that you’ll need Adobe Flash.

Instead of being intrinsic physical properties of a moving object, time dilation and length contraction are symmetric properties, being applicable to any objects from two different IRFs. These relational properties are solely the result of coordinate transformations, and thus deal with kinematics, not dynamics.

Time Dilation

Definition: Proper Time is the time interval {\Delta}t_0 measured between two events that take place in the same IRF, at rest to the observer. This definition holds true in any IRF.

When comparing time measurements between IRFs, proper time is always the shortest possible time interval. A moving clock’s time is seen to run slower than that of a stationary clock’s, though this is all relative to an observer. This slowing down of time is time dilation, meaning time extension.

Definition: Dilated time is the time interval {\Delta}t measured for a timepiece in relativistic motion in relation to the observer. Near the speed of light, time slows down.

The equation to calculate for dilated time is: {\Delta}t=\gamma{\Delta}t_0, with the time interval of the stationary clock as {\Delta}t_0, and that of the moving clock as {\Delta}t.

Length Contraction

Definition: Proper Length is the invariant quantity of length measured of an object in the same IRF as the observer. This definition holds true in any IRF.

Proper length is always the longest possible measurement made of an object when comparing reference frame measurements, and is always longer than or equal to a contracted length, meaning shrunken length.

Definition: An entirely relative concept, Contracted Length is the measured length of an object in relativistic motion to the observer. Though this may seem like an optical trick, it is not. The object does not shrink, yet the measured size is real for the observer.

The strangeness of relativity.

The contracted length of an object in relation to its proper length is given by the equation L=\frac{L_0}{\gamma}, where L is the contracted length, and L_0 is the proper length. Note that L_0=v{\Delta}t, where t is the dilated time.

Reality Check

In real life, we do not see time slowing or meter-sticks shrinking to centimetre-sticks simply because we do not go fast enough. Recall that at low speeds \gamma\approx 1, and only as v approaches c does \gamma\to1. This follows that L\to1. Note that length is a 1D concept, so length contraction only occurs along the x-axis.

Time and length in physics are all measured values, and measurements cannot be made of moving objects without reference frames. Time dilation and length contraction are real, just objective and quantitative, not subjective.

Spacetime intervals, defined as {\Delta}s^2=(c{\Delta}t)^2-{\Delta}x^2 for 1D motion in a reference frame S, is also an invariant quantity. This means that {\Delta}s^2={\Delta}(s^{\prime})^2, which follows that (ct^{\prime})^2-(x^{\prime})^2=(ct)^2-(x)^2.

The following chart describes the values of {\Delta}s^2:

Note that velocity values are vectors, and that the speed u^{\prime}, is always with respect to S^{\prime} where S^{\prime} is in relative motion to S.

The Galilean equations are modified as follows to be applicable at light speed:
u^{\prime} = u-v becomes u^{\prime}= \frac {u-v}{1-\frac{uv}{c^2}}.
u=u^{\prime}+v becomes u=\frac{u^{\prime}+v}{1+\frac{u’v}{c^2}}.

These modified equations allow us to use Galileo’s velocity additions with the speed of light.

The gamma factor is defined as \gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}. Note that v is defined as the relative speed between two inertial frames, and c is the speed of light. Using this gamma factors, the following Galilean equations can be used:

x^{\prime}=x-vt becomes x^{\prime}=\gamma(x-vt).
The inverse, x=x^{\prime}+vt is rewritten as x=x^{\prime}+vt^{\prime}, and it becomes x=\gamma(x^{\prime}+vt^{\prime}).
A reminder that the speed of light is the same in each reference frame: c=xt and c^{\prime}=x^{\prime}t^{\prime}.

We can see from the graph below that as v approaches 0, the \gamma factor approaches 1, meaning that these equations work in Newtonian mechanics, as \gamma is very close to 1 in regular situations. It is also evident that, as v approaches c, \gamma approaches \infty. From the graph below of \gamma against speed, the speed of light is the asymptote for v, so an accelerated body can never reach c.

Note that the gamma factor has no units, and {\alpha}{\beta}% of c is the same as 0.{\alpha}{\beta}c.

So, how do we synchronize clocks? We can use a flash of light at the midpoint between 2 clocks, with photocells to detect light and in the same IRF. This would synchronize the 2 clocks.

However, let’s assume that 2 observers are in different IRFs. One is with the 2 clocks to be synchronized, the other is far away, and their relative velocity is nonzero. The observer from one IRF would see that the clocks in the second IRF are not synchronized, but the other observer would see that they are.

Who’s right? Both are objective and correct because of asymmetric relationship of space and time between 2 frames of reference.

Postulate 1: Laws of physics are same in all IRFs.

This means that there is no preferred frame of reference, no absolute or unique position in the universe, and no experiment that can show that an observer is at rest or moving at a constant velocity.
Thus, all uniform motion is relative.

Postulate 2: Speed of light in a vacuum is constant/invariant, denoted c, for all IRFs

This is an extension of Maxwell’s Theory of Electromagnetism, where Einstein states that the generic velocity v = \frac{d}{t} doesn’t really work for light because of this postulate.

Though today we see Einstein’s work as a total revolution in relativity and nuclear physics, Einstein himself thinks his postulates and theories are just a completion of Newtonian physics, not a revolution.

Definition: An invariant quantity or property of a system remains unchanged as the system undergoes transformations.

One of the important conclusions of Maxwell’s theory of electromagnetism is that the speed of light is invariant in vacuum. This also means that the value for c is independent of any relative motion between source and observer.

The table below summarizes the understandings of 4 renowned physicists on addition close to the speed of light: