Time dilation of moving particles

Relationship between the Lorentz factor γ and the relative speed.

The time dilation of moving particles can be determined by measuring the lifetime or decay time of particles. The time dilation according to the special theory of relativity says that if a clock C is moved between two synchronized clocks A and B, which are idle in a laboratory, it will lag behind the two clocks. Since in principle every periodic process can be called a clock, this applies e.g. B. also for the lifetimes or decay times of unstable particles such as muons . This means that moving muons should have a longer lifespan than those at rest. To demonstrate this effect, measurements are carried out in the atmosphere as well as in particle accelerators with different types of particles, with all results confirming the time dilation. Other experiments on time dilation belong to the group of Ives-Stilwell experiments (see also tests of the special theory of relativity ).

Atmospheric tests

Minkowski diagram

a) View in p

b) view in S ′

c) View in an inertial system in which S and S ′ fly in opposite directions at the same speed. (In order not to let the differences become too extreme, only 0.7c was used instead of 0.995c in all diagrams.)

theory

It is measured that when the cosmic radiation hits the molecules of the upper air layers , muons are formed at a height of 9 to 12 kilometers . They are one of the main components of secondary cosmic rays and move towards the earth's surface at almost the speed of light. The experiment depends on the half-life of the muons, which in turn is influenced by the relativistic corrections of the following quantities: a) the mean lifespan of the muons moving relative to the earth and b) the extent from the upper to the lower atmospheric layer (at the earth's surface). This measurement situation enables the application of the formulas for time dilation and length contraction that are valid in all inertial systems according to the principle of relativity to the objects relevant for this experiment, namely the atmosphere at rest in S and the muons at rest in S ′:

Time dilation and length contraction

Expansion of the atmosphere : The contraction formula is where L _{0 is} the rest length of the atmosphere and its moved-contracted length. Since the atmosphere is at rest in S, its rest length is measured. Since it is moved in S ′, it holds and its contracted length is measured. ${\displaystyle L=L_{0}/\gamma }$ ${\displaystyle L}$ ${\displaystyle \gamma =1}$ ${\displaystyle L=L_{0}}$ ${\displaystyle \gamma >1}$ ${\displaystyle L'=L_{0}/\gamma }$

Lifetime of the muon : The time dilation formula is or . Here is the mean resting life (proper time ) of the muon and the corresponding time in systems moving relative to the muon. Since the muon rests in S ′ , the proper time is measured and the proper time is measured. Since it is moved in S , from which it follows that the measured time is greater than , the latter is therefore dilated with respect to . [For comparison purposes, another muon resting on earth with the same decay time can be considered. In S this muon S decays faster than muon S ′, while this is exactly the opposite in S ′.] ${\displaystyle T=T_{0}\cdot \gamma }$ ${\displaystyle T_{0}=T/\gamma }$ ${\displaystyle T_{0}}$ ${\displaystyle T}$ ${\displaystyle \gamma =1}$ ${\displaystyle T'=T_{0}}$ ${\displaystyle \gamma >1}$ ${\displaystyle T}$ ${\displaystyle T_{0}}$ ${\displaystyle T}$

• In S, muon-S ′ has a longer lifetime than muon-S. Thus, muon-S ′ has enough time to penetrate the rest length of the atmospheric layers to the surface of the earth.
• In S ′, muon-S has a longer lifetime than muon-S ′. However, this is not a problem, because the atmosphere here is moving and thus contracted in relation to its rest length. This means that even the shorter lifespan of muon S ′ is sufficient until the moving atmosphere has passed and the earth's surface has arrived.

Minkowski diagram

A muon is created at the zero point (event A) by collision of radiation with the upper end of the atmosphere. It rests in S ′, its world line is the ct ′ axis. The upper end of the atmosphere rests in S, its world line is the ct axis. All events that occurred in S and S ′ at the same time as the muon formation are located on the x-axis or x 'axis. At event D earth and muon meet. Since the earth is at rest in S, its world line (identical to the lower end of the atmosphere) is drawn through D parallel to the ct-axis until it intersects both the x′-axis and the x-axis.

Times : The time between two events that lie on the world line of a single clock is called proper time . It is one of the fundamental invariants of the theory of relativity. Since the formation of the muon at A and the meeting with the earth at D lie on the world line of the muon, only a clock that moves with the muon and is therefore at rest in S ′ can display the proper time T ′ ₀ = AD . Due to the invariance of AD, the muon clock in the earth system S must show exactly the same time between the events, and because the muon clock is moving here, T ′ ₀ = AD is dilated with respect to the clocks resting in S. This can be recognized from the significantly longer time spans T = BD = AE displayed by the S clocks parallel to the ct axis .

Lengths : Event B, where the world line of the earth intersects the x-axis, corresponds in S to the position of the earth at the time the muons were formed. Event C, where its world line intersects the x ′ axis, corresponds in S ′ to the position of the earth at the time of muon formation. Length L ₀ = AB in S is clearly longer than L ′ = AC in S ′.

Experiments

Result of the Frisch-Smith experiment . Decay curves according to the formulas for and . ${\displaystyle M_{\mathrm {Newton} }}$ ${\displaystyle M_{\mathrm {SRT} }}$

A special filter arrangement enables the measurement to be restricted to those muons that move at a certain speed. The comparison of the measured numbers makes it possible to determine the mean lifetime as well as the half-life of the fast moving muons. For atmospheric measurements, the number of measured muons at higher altitudes, at sea level, is the transit time in the rest system of the earth to cross this distance, and the mean resting life of the muons is: ${\displaystyle N}$ ${\displaystyle M}$ ${\displaystyle Z}$ ${\displaystyle T_{0}}$

${\displaystyle {\begin{aligned}M_{\mathrm {Newton} }&=N\exp \left[-Z/T_{0}\right]\\M_{\mathrm {SRT} }&=N\exp \left[-Z/\left(\gamma T_{0}\right)\right]\end{aligned}}}$

Rossi Hall

Bruno Rossi and David Hall (1941) were the first to carry out such experiments. The detectors were located in Echo Lake (3240 m) and Denver (1616 m) in Colorado , with an altitude difference of 1624 m. The muons moved at speeds of over 0.99 c, and their momentum and number were measured. It was found that the range of the muons depends on their relativistic momentum, which in turn is in accordance with the longer lifetime due to the time dilation of the faster moving muons. Conversely, the mean resting life of the muons to about 2.4 µs could be determined from the pulse and time dilation formula, in qualitative agreement with the laboratory results known at the time (modern experiments specified this value to ≈ 2.2 µs).

Fresh Smith

A similar experiment was carried out with increased precision by David H. Frisch and Smith (1963). Approximately 563 muons per hour were observed at Mount Washington . By determining their kinetic energy, a speed between 0.995 c and 0.9954 c was found. The destination was at sea level in Cambridge with an altitude difference of 1907 m, which at this speed would result in a flight time of 6.4  microseconds . If there were no time dilation, with an average resting life of 2.2 µs, only 27 muons per hour would reach the goal. In fact, however, about 412 muons per hour arrived at the target, from which Frisch and Smith concluded a time dilation factor of 8.8 ± 0.8.

Frisch and Smith showed that this is in agreement with the time dilation of the special theory of relativity: The time dilation factor of the muons on the mountain is approximately 10.2 at the indicated speeds of 0.995 c to 0.9954 c. In contrast, the velocity of the muons at sea level, determined from the kinetic energy, gave values ​​between 0.9881 c and 0.9897 c (since they had been slowed down somewhat by the air), and the corresponding time dilation factor was consequently reduced to approximately 6.8. From the mountain (≈ 10.2) to sea level (≈ 6.8) there is an average time dilation factor of 8.4 ± 2, which corresponds to the empirical value within the scope of the measurement accuracy (see above formulas and figure for calculating the Decay curves).

More experiments

Since then, such experiments, which determine the lifetime and time dilation of muons in the atmosphere, have been routinely carried out in experiments for undergraduate studies , see for example Easwar et al. (1991), or Coan et al. (2006).

Accelerator tests

Time dilation and CPT theorem

However, tests of particle decay times and verifications of time dilation are mainly carried out in particle accelerators , with significantly greater accuracy than in atmospheric tests. In addition, the CPT theorem was confirmed by comparing the lifetimes of positive and negative particles . It says that the decay times of particles and their antiparticles must be the same. A deviation from this would result in a violation of the Lorentz invariance and thus the special theory of relativity.

Pion	Kaon	Muon
Durbin et al. (1952) Eckhaus et al. (1965) Nordberg et al. (1967) Greenburg et al. (1969) Ayres et al. (1971)	Burrowes et al. (1959) Nordin (1961) Boyarski et al. (1962) Lobkowicz, et al. (1969) Ott et al. (1971) Skjeggestad, et al. (1971) Geweniger et al. (1974) Carithers et al. (1975)	Lundy (1962) Meyer et al. (1963) Eckhaus et al. (1963) Balandin et al. (1974)

In modern particle accelerators, the time dilation together with the relativistic energy-momentum relationship is already routinely confirmed and is necessary when analyzing collision experiments.

Twin paradox

Bailey et al . (1977) examined the lifespan of positive and negative muons in the storage ring of CERN . They sent the particles on a circular path so that they came back to the starting point several times. The confirmation of the time dilation achieved in this experiment is also a confirmation of the twin paradox , namely the statement that a clock returning to the starting point slows down compared to the clock that was left behind. Further tests also measure the time dilation caused by gravity , see Hafele-Keating experiment and repetitions.

Clock hypothesis

In the experiments above, the time dilation was measured during particle decay when the particles were not subjected to acceleration. In contrast, Bailey et al . (1977) also confirmed the so-called “clock hypothesis”, according to which the acceleration does not influence the time dilation. In their experiment, the muons were exposed to a continuous transverse acceleration of up to ∼10 ¹⁸ g, but they received the same value for the time dilation as in the other experiments. This has also been demonstrated by Roos et al. (1980) confirmed which decaying Σ-baryons subjected to a longitudinal acceleration between 0.5 and 5.0 × 10 ¹⁵ g, and also received the usual value for the time dilation.

Web links

• T. Roberts; S. Schleif; JM. Dlugosz: What is the experimental basis of Special Relativity? . In: Usenet Physics FAQ . University of California, Riverside . 2007. Retrieved May 21, 2011.
• Time Dilation - An Experiment With Mu-Mesons

Individual evidence

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