Dilemma in race between muon and light
I have a dilemma concerning my understanding of Special Relativity. Maybe I am understanding or calculating something wrong and would hear so.
The problem is based on muons created in the upper atmosphere by cosmic rays. Basically, what happens when we time the difference between such a muon and light created the same place at the same time.
Since the muon travels at 0.994c, it arrives 301 nanoseconds later than the light. However this is from observation on earth. From the muon's observation, the point of creation and the destination on the earth's surface are moving at 0.994c and the distance is contracted from 15 Km to 1638 m, so the light arrives 33 nanoseconds earlier.
Further, from the muon's observation, a clock on earth will experience time dilation and should only record 3.6 nanoseconds of time. So how much time does the earth clock record?
Detailed Description: In practice, identifying one muon and whether it would not decay in the journey would be problematic. Since we are using observations from the muon, replace it with a spaceship traveling at the (now arbitrary) velocity of 0.994c. If you do not like crashing the spaceship into the surface of the earth, or worry about general theory effects from the earth or sun, move the thought experiment to space, say the midpoint between Sol and Alpha Centauri.
A point m, is 15.0 Km above a point on the surface of the earth, e. At point me is a space station, Muse.
A space ship traveling at 298 m/μs constant velocity passes point m and then point e.
When the spaceship passes point m, the Muse Space Station turns on a light. A photographic plate at point e collects light only from the Muse Space Station. This will be our clock, and we will measure how long an exposure to light is indicated by the photographic plate at the time where the space ship reaches point e. 275
(Perhaps when the space ship passes we close a shutter on the plate.)
The light takes 50.03461 μs to reach point e. The spaceship takes 50.03461 μs to reach point e, 301 nanoseconds later. So the photographic plate is exposed to light for 301 ns.
However, from the space ship's reference, the space ship is not moving, but points m and e are moving at 298 m/μs constant velocity. When point m reaches the space ship, it turns on a light.
Because of length contraction predicted by special relativity, the distance between points m and e is 1637 m. The light from point m takes 5.49540 μs to reach point e. Point e reaches the ("stationary") space ship after 5.46254 μs. So the photographic plate is exposed to light for 32.9 ns.
So how much has the exposure to light darkened the photographic plate? As much as expected for 301 ns or 33 ns?
Further, since from the space ship's observation, the photographic plate at point e is moving at a speed close to c. So it should experience time dilation according to special relativity.
If instead of leaving the light on when point m passes the space station, the light at point m was only turned on for 1 nanosecond, the clock timing that nanosecond would be slower observed from the space station, and would be on for 9 ns. Then the 9 ns of light would only expose the plate at e as much as for 1 ns of light, because of the time dilation at point e. So, observed from the space ship for every 9 units of time that the plate at point e is exposed to light, it only reacts as much as for one unit of time. So the 33 ns of light when point e reaches the space ship, the plate should only show 3.6 ns of exposure.
Since we are comparing times in microseconds, and subtracting to get a time difference in nanoseconds, to get four digit precision in our result, we need to start with and carry seven digit precision in our thought experiment, even when the input was arbitrary.
Earth Observation:
Distance of segment em: 15 000.000 000 m
Speed of muon space ship: 298.000 000 m/μs
Speed of Light: 299.792 458 m/μs
Time for space ship to reach e 50.335 57 μs
Time before light reaches e 50.034 61 μs
Time of exposure of p-plate 301.0 ns
Muon Space Ship Observation:
Distance of segment em: <PHONE_NUMBER> m
Rel Speed of e & m to muon space ship:
298.000 000 m/μs
Speed of Light: 299.792 458 m/μs
Time for space ship to reach e 5. 562