1000 muons produced at an altitude of 3.0 km travel toward the ground at 98% of the speed of light.
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高さ3.0 kmで生まれたミュー粒子の何％が地上に届くか？
同じ速さで鉛直から60°傾いて飛ぶと、何％が地上に届くか？
The equations explain the phenomenon
1The principle of relativity is the starting point.The laws of physicsin every inertial framehave the same form.……
2Maxwell's equations in vacuum determine the speed of electromagnetic waves.c=1√(ε₀μ₀)≒2.998×10⁸ m/s……
3The principle of relativity also applies to electromagnetism.The speed of light in every inertial frame=c……
4The round-trip travel time of light is calculated in the inertial frame in which the clock is at rest.Δt₀=2Lc……
The mirror separation perpendicular to the direction of motion is L in the ground frame as well. This property is used here; only lengths parallel to the direction of motion contract, as shown in step 13.
5The round-trip travel time in the moving light clock is calculated in the ground inertial frame.Δt=γ Δt₀，γ=1√(1 − v²/c²)……
6The value of γ is calculated for different speeds.γ=1√(1 − v²/c²)≒5.03（v=0.98c）……
The mean lifetime τ is the average lifetime of many particles. Individual particles decay at different times; they do not all decay at τ.Over each equal time interval, the same fraction of the particles present at the start of that interval survives. The number remaining therefore decreases exponentially, with N/N₀ = 1/e at t₀ = τ, as indicated by the dark blue point in the graph.
7The muon decay equation is expressed in terms of time measured on the ground.N=N₀ e−t/(γτ)……
8The fraction reaching the ground is calculated.NN₀=e−h/(γ v τ)≒0.397……
9The atmospheric thickness required to give the same survival fraction is calculated in the inertial frame moving with the particles.L=hγ≒597 m……
10The fraction reaching the ground is calculated when oblique motion doubles the distance traveled.NN₀=(e−h/(γ v τ))²≒0.157……
The following advanced topics cover simultaneity, the Lorentz transformation, energy, and GPS.
11Light is emitted at the center of a train, and its arrival times at the front and rear walls are compared.tfront − trear=2ℓ vc² − v²……
12The transformation for x′ is taken to be linear, and its coefficient is determined from the behavior of light.x′=γ(x − v t)，γ=1√(1 − v²/c²)……
13The transformation for t′ is derived.x′=γ(x − v t)，t′=γ(t − v xc²)……
14The relation between velocities measured in different inertial frames is derived.u=u′ + v1 + u′ v/c²……
15Kinetic energy is derived from the momentum p = γ m v, giving an expression for the total energy.K=(γ − 1) m c²，E=K+m c²=γ m c²……
16The time dilation of a GPS satellite clock is calculated below.Δt≒−v²2c² × 1 day≒-7.2 μs……
What this chapter has shown
Applying the principle of relativity to Maxwell's equations in vacuum implies that the speed of light is c in every inertial frame. As a result, while a time t elapses on the ground, a time t/γ elapses for a moving particle. Its mean lifetime in the ground frame is γτ, and the fraction reaching the ground can be calculated from the flight time and the decay equation.
39.7 %The fraction reaching the ground is shown
At the same speed, a trajectory inclined at 60° to the vertical doubles the distance traveled. The fraction reaching the ground is then the square of the original fraction, or about 16%.
In everyday lifeThe high speed of GPS satellites causes their clocks to lose about 7.2 μs in 1 day. This time dilation is included in the corrections applied to GPS clocks.
Detailed summary
Applying the principle of relativity to electromagnetism implies that the speed of light is c in every inertial frame. A light clock shows that while a moving clock measures an interval Δt₀, an interval γΔt₀ elapses in the ground frame. Muons produced at an altitude of 3.0 km have a mean lifetime of 11.0 μs as measured in the ground frame, allowing about 40% to reach the ground. In the inertial frame moving with the muons, the atmosphere contracts to a thickness of 597 m, giving the same survival fraction. These results are unified by the Lorentz transformation, which also implies that adding velocities never yields a speed greater than c and that even an object at rest has energy E=mc².