Before reading this have a look at the file
called 'Time dilation'. Effects similar to those described there have been observed in the real
world, but because of the very large velocity of light they are much more difficult to see.

The slowing down of time **(time dilation)** has been
noticed in atomic clocks that have been carried in satellites but a natural phenomenon that
supports time dilation is the decay of fast moving muons as they travel downwards through the
atmosphere.

High-energy cosmic ray protons entering the upper atmosphere interact
with the nuclei of oxygen and nitrogen atoms to give a group of pions and these then decay into
muons that then move off at a speed of up to 0.994 c. These muons are formed at a height of
between ten and fifteen thousand metres above the ground.

The half-life of a muon is
2.2 microseconds and so even moving at 0.994 c they would only expect to travel about 660 m
before half of them decayed. Muons formed at, say 12000 m would take 40 <μs or about 20 half lives to reach the ground. This would mean that only
1/2^{20} of the original number would be detected. The fact that the proportion is much
higher than this means that the muons are living longer.

At 0.994c the formula for time
dilation gives the half-life for the muons to be 20 μs. This means
that at 0.994c the proportion reaching the ground should be 0.25. This means that the number of
muons per second detected at the ground is much greater than expected.

(Note: in Figure 1
only pion decays that result in muons are shown. A μ_{o} decays resulting in gamma rays, electrons and positrons are
omitted)

Mass increase with velocity and the 'cosmic speed limit'

The increase of
mass becomes a problem in high-energy accelerators where as the particles approach the
speed of light they become more and more difficult to accelerate further. Even the electrons in
our colour television tubes are moving so fast that their actual masses are some 21 per cent
heavier than those of electrons at rest. Beta particle emitted in radioactive decay are moving at
90% of the speed of light and are 2.4 times heavier than an electron at rest.

Using the
formula for the relativistic mass of a particle you can see that there is kind of cosmic speed limit
for matter.

Since: m = m_{o}/(1 –
v^{2}/c^{2})^{1/2} close to that of light the mass of a particle becomes
very high and at v = c the limit of further increase in speed is reached. The mass of the particle
is converted into energy.

Where v

This can be proved using the formula for relativistic velocity addition:

Relativistic velocity addition for two bodies moving at speeds v_{A} and v_{B} relative to an external reference frame:

Relative speed = [v_{A} + v_{B}]/(1 + [v_{A} x v_{B}]/c^{2})

For the two photons v

Relative speed = [c + c]/(1 + [c x c]/c

However this is different if you are in a frame of reference outside both photons and observing them approaching each other. You now observe the approach velocity to be 2c. This does not contradict the special theory of relativity because neither photon is moving faster than the speed of light (c).

You have probably used
Einstein's famous equation relating matter to energy, E = mc^{2}. This equation can be
deduced starting with the first equation of special relativity:

(See:
Mass energy equation)

Imagine an experiment where you shine a laser beam onto the surface
of the Moon so that the beam hits one edge of the visible disc. Now swing the laser through an angle of
0.5^{o}, the angle subtended by the Moon at the Earth, in 0.01 s. This would mean that the beam
would have travelled across the lunar surface – a distance of 3 500 000 m. This means that the beam is
moving across the Moon's surface at 3500000/0.01 = 3.5x10^{8} ms^{-1}.

There is no
problem here – the result does not contradict the theory of relativity. No individual photons make that
traverse since as the beam moves different photons arrive at different places.