Lorentz Transformations

Time Dilation

Moving clocks record their own proper time. (The proper time is the time recorded by a clock, which moves along with the considered object.) The proper time interval recorded by a clock moving with constant speed v relative to an inertial frame A is given by

,

where is the coordinate time interval recorded by clocks at rest in A, i.e.,

.

Hence for all and moving clocks "run slow." This is the phenomenon of time dilation.

Muons are elementary particles that can be produced when primary cosmic rays hit the atmosphere of the Earth. The muons are created at an altitude of around 15 km and the lifetime of the muons (i.e., the time which the muons live in their own rest frame) is approximately ₀ = 2.210^-6 s. In classical Newtonian mechanics, this would mean that the muons could in average move approximately c₀ = 660 m before they decay and would not be observed on Earth. However, a large fraction of the muons do reach the surface of the Earth.

How can this be explained?

Well, this can be explained in principle in two ways - by either using length contraction or time dilation. Assume that the muons move with a speed v close to that of light, e.g., v = 0.999 c.

Time dilation: In the frame of the Earth, the lifetime of the muons will be = ₀ (v), which is approximately 22 ₀. This means that the muons move the distance v = 0.99922660 m (approximately 15 km) in the frame of the Earth, which is in principle the thickness of the atmosphere.

Length contraction: In the rest frame of a muon, the thickness of the atmosphere is about 10 km/22 = 450 m. But during the lifetime of the muon, the Earth will move the distance v₀ = 0.999660 m (approximately 660 m) in the frame of the muon, which is longer than 450 m.