Controls
Scenario
Muon Detection
Earth → Neptune
Reference frame
Earth frame
Muon frame
Speed \(\beta = v/c\)
—
Lorentz factor \(\gamma\)
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Time (this frame)
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Length (this frame)
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Earth frame: The muon travels the full 10 km atmosphere
(\(L_0\) — proper length, both ends at rest in this frame).
The muon's internal clock runs slow, so we observe it surviving longer than its lab
half-life (dilated time \(t = \gamma\,\tau_0\)).
Muon frame: The muon is at rest — it ticks its own proper
lifetime \(\tau_0 = 2.197\,\mu\text{s}\). The atmosphere rushes past, but its thickness
is contracted to \(L = L_0/\gamma\). Either way, the muon just barely reaches
sea level.
Earth frame: Earth and Neptune are at rest, so the
Earth–Neptune distance is the proper length \(L_0 = 29.1\,\text{AU}\).
The ship's clocks run slow; we observe dilated travel time
\(t = L_0 / v = \gamma\,\tau_0\).
Ship frame: The ship is at rest — it measures its own
proper travel time \(\tau_0 = t/\gamma\). Neptune rushes toward it, but the
Earth–Neptune gap is contracted to \(L = L_0/\gamma\), consistent with
\(\tau_0 = L/v\).