1 atmospheric tests
1.1 theory
1.2 experiments
1.2.1 rossi–hall experiment
1.2.2 frisch-smith experiment
atmospheric tests
theory
the emergence of muons caused collision of cosmic rays upper atmosphere, after muons reach earth. probability muons can reach earth depends on half-life, modified relativistic corrections of 2 quantities: a) mean lifetime of muons , b) length between upper , lower atmosphere (at earth s surface). allows direct application of length contraction upon atmosphere @ rest in inertial frame s, , time dilation upon muons @ rest in s′.
time dilation , length contraction
length of atmosphere: contraction formula given
l
=
l
0
/
γ
{\displaystyle l=l_{0}/\gamma }
, l0 proper length of atmosphere , l contracted length. atmosphere @ rest in s, have γ=1 , proper length l0 measured. in motion in s′, have γ>1 , contracted length l′ measured.
decay time of muons: time dilation formula
t
=
t
0
⋅
γ
{\displaystyle t=t_{0}\cdot \gamma }
, t0 proper time of clock comoving muon, corresponding mean decay time of muon in proper frame. muon @ rest in s′, have γ=1 , proper time t′0 measured. moving in s, have γ>1, therefore proper time shorter respect time t. (for comparison s sake, muon @ rest on earth can considered, called muon-s. therefore, decay time in s shorter of muon-s′, while longer in s′.)
in s, muon-s′ has longer decay time muon-s. therefore, muon-s has sufficient time pass proper length of atmosphere in order reach earth.
in s′, muon-s has longer decay time muon-s′. no problem, since atmosphere contracted respect proper length. therefore, faster decay time of muon-s′ suffices in order passed moving atmosphere , reached earth.
minkowski diagram
the muon emerges @ origin (a) collision of radiation upper atmosphere. muon @ rest in s′, worldline ct′-axis. upper atmosphere @ rest in s, worldline ct-axis. upon axes of x , x′, events present simultaneous in s , s′, respectively. muon , earth meeting @ d. earth @ rest in s, worldline (identical lower atmosphere) drawn parallel ct-axis, until intersects axes of x′ , x.
time: interval between 2 events present on worldline of single clock called proper time, important invariant of special relativity. origin of muon @ , encounter earth @ d on muon s worldline, clock comoving muon , resting in s′ can indicate proper time t′0=ad. due invariance, in s agreed clock indicating time between events, , because in motion here, t′0=ad shorter time t indicated clocks resting in s. can seen @ longer intervals t=bd=ae parallel ct-axis.
length: event b, worldline of earth intersects x-axis, corresponds in s position of earth simultaneous emergence of muon. c, earth s worldline intersects x′-axis, corresponds in s′ position of earth simultaneous emergence of muon. length l0=ab in s longer length l′=ac in s′.
experiments
results of frisch-smith experiment. curves computed
m
n
e
w
t
o
n
{\displaystyle m_{\mathrm {newton} }}
,
m
s
r
{\displaystyle m_{\mathrm {sr} }}
.
if no time dilation exists, muons should decay in upper regions of atmosphere, however, consequence of time dilation present in considerable amount @ lower heights. comparison of amounts allows determination of mean lifetime half-life of muons.
n
{\displaystyle n}
number of muons measured in upper atmosphere,
m
{\displaystyle m}
@ sea level,
z
{\displaystyle z}
travel time in rest frame of earth muons traverse distance between regions, ,
t
0
{\displaystyle t_{0}}
mean proper lifetime of muons:
m
n
e
w
t
o
n
=
n
exp
[
−
z
/
t
0
]
m
s
r
=
n
exp
[
−
z
/
(
γ
t
0
)
]
{\displaystyle {\begin{aligned}m_{\mathrm {newton} }&=n\exp \left[-z/t_{0}\right]\\m_{\mathrm {sr} }&=n\exp \left[-z/\left(\gamma t_{0}\right)\right]\end{aligned}}}
rossi–hall experiment
in 1940 @ echo lake (3240 m) , denver in colorado (1616 m), bruno rossi , d. b. hall measured relativistic decay of muons (which thought mesons). measured muons in atmosphere traveling above 0.99 c (c being speed of light). rossi , hall confirmed formulas relativistic momentum , time dilation in qualitative manner. knowing momentum , lifetime of moving muons enabled them compute mean proper lifetime – obtained ≈ 2.4 µs (modern experiments improved result ≈ 2.2 µs).
frisch-smith experiment
a more precise experiment of kind conducted david h. frisch , smith (1963), measured approximately 563 muons per hour in 6 runs on mount washington. measuring kinetic energy, mean muon velocities between 0.995 c , 0.9954 c determined. target located in cambridge, massachusetts difference in height of 1907 m, should traversed muons in 6994640000000000000♠6.4 µs. assuming mean lifetime of 2.2 µs, 27 muons reach location if there no time dilation. however, approximately 412 muons per hour arrived in cambridge, resulting in time dilation factor of 7000880000000000000♠8.8±0.8.
frisch , smith showed in agreement predictions of special relativity: time dilation factor muons on mount washington traveling @ 0.995 c 0.9954 c approximately 10.2. kinetic energy , velocity diminished until reached cambridge 0.9881 c , 0.9897 c due interaction atmosphere, reducing dilation factor 6.8. between start (≈ 10.2) , target (≈ 6.8) average time dilation factor of 7000840000000000000♠8.4±2 determined them, in agreement measured result within margin of errors (see above formulas , image computing decay curves).
other experiments
since then, many measurements of mean lifetime of muons in atmosphere , time dilation have been conducted in undergraduate experiments.
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