Time dilation of moving particles

Time dilation as predicted by special relativity is often verified by means of particle lifetime experiments. According to special relativity, the rate of a clock C traveling between two synchronized laboratory clocks A and B, as seen by a laboratory observer, is slowed down relative to the laboratory clock rates. Since any periodic process can be considered a clock, the lifetimes of unstable particles such as muons must also be affected, so that moving muons should have a longer lifetime than resting ones. A variety of experiments confirming this effect have been performed both in the atmosphere and in particle accelerators. Another type of time dilation experiments is the group of Ives–Stilwell experiments measuring the relativistic Doppler effect.Eckhause et al. (1965)Nordin (1961)Meyer et al. (1963) Time dilation as predicted by special relativity is often verified by means of particle lifetime experiments. According to special relativity, the rate of a clock C traveling between two synchronized laboratory clocks A and B, as seen by a laboratory observer, is slowed down relative to the laboratory clock rates. Since any periodic process can be considered a clock, the lifetimes of unstable particles such as muons must also be affected, so that moving muons should have a longer lifetime than resting ones. A variety of experiments confirming this effect have been performed both in the atmosphere and in particle accelerators. Another type of time dilation experiments is the group of Ives–Stilwell experiments measuring the relativistic Doppler effect. The emergence of the muons is caused by the collision of cosmic rays with the upper atmosphere, after which the muons reach Earth. The probability that muons can reach the Earth depends on their half-life, which itself is modified by the relativistic corrections of two quantities: a) the mean lifetime of muons and b) the length between the upper and lower atmosphere (at Earth's surface). This allows for a direct application of length contraction upon the atmosphere at rest in inertial frame S, and time dilation upon the muons at rest in S′. Length of the atmosphere: The contraction formula is given by L = L 0 / γ {displaystyle L=L_{0}/gamma } , where L0 is the proper length of the atmosphere and L its contracted length. As the atmosphere is at rest in S, we have γ=1 and its proper Length L0 is measured. As it is in motion in S′, we have γ>1 and its contracted length L′ is measured. Decay time of muons: The time dilation formula is T = γ   T 0 {displaystyle T=gamma T_{0}} , where T0 is the proper time of a clock comoving with the muon, corresponding with the mean decay time of the muon in its proper frame. As the muon is at rest in S′, we have γ=1 and its proper time T′0 is measured. As it is moving in S, we have γ>1, therefore its proper time is shorter with respect to time T. (For comparison's sake, another muon at rest on Earth can be considered, called muon-S. Therefore, its decay time in S is shorter than that of muon-S′, while it is longer in S′.)