Implementation of a non-numeric spacetime model with the Minkowski metric
View the Project on GitHub vgurianov/srt
Module mms.experiment1 is simulated kinematic relativistic effects.
Estimated calculation for π+ meson (pion):
Time we measure in metre of light time, i.e. 3.335640×10−9 seconds.
Lifetime is τ0=2.6×10−8 seconds or 7.8 metres of light time.
β=v/c=0.5.
Moving pion has a longer decay time τ=τ0√1−β2=3.0×10−8 seconds or 9.0 metres of light time.
Distance d=c×β×τ=4.5 metres,
i.e. we must consider grid 10 x 10.
Parameters of experiment:
count_tick= 8, size_tick= 10
Particle_velosety= 5 ,i.e v/c = 0.5
Time count = 80
nu_t = 10.0 , nu_x = 10.0 , nu_m = 1.0
mass = 1 , lightVel = 1.0
In table is depicted result of simulation.
Trajectory of particle and time of the particle
+----+-----+-----+------+------+-----+
| Tw | x | t | ta | err% | tp |
+----+-----+-----+------+------+-----+
| 0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 1 | 0.5 | 1.2 | 1.12 | 7.33 | 1.0 |
| 2 | 1.0 | 2.3 | 2.24 | 2.86 | 2.0 |
| 3 | 1.5 | 3.4 | 3.35 | 1.37 | 3.0 |
| 4 | 2.0 | 4.5 | 4.47 | 0.62 | 4.0 |
| 5 | 2.5 | 5.6 | 5.59 | 0.18 | 5.0 |
| 6 | 3.0 | 6.8 | 6.71 | 1.37 | 6.0 |
| 7 | 3.5 | 7.9 | 7.83 | 0.94 | 7.0 |
+----+-----+-----+------+------+-----+
Column Tw is the number of tick of model time. Column x - is the coordinate of the particle in moment Tw. Column tp is the time of the particle. We can observe dilation of time. In the particle, tp units of time elapse, but, in the rest frame of reference t units of time are registered . Column ta is analytic calculation to formula ta=√s2+x2 . Value ta compare with t
ϵ=|ta−tta|×100It is column err%.
This result is depicted in Fig.1. Black “o” is the result of the measurement. The red line is analytical value.
Figure 1. Minkowski spacetime diagram for β=0.5
We processing of data and calculate of incline k (green line)
Experimental error of measurement t is 0.05
Point couple method (d=4)
0 20 45 2.25
1 20 44 2.2
2 20 45 2.25
3 20 45 2.25
Couple count = 4
Measurement incline k_ar= 2.24
k_ar = 2.24 +/- 0.012
Analytical incline k_an= 2.24, k_err%= 0.12
In case of small velocity, plot depicted in Fig.2.
Figure 2. Minkowski spacetime diagram for β=0.3
Description: the class is a simulation model
Bases: mms.Composite
def __init__(self, size_tick, count_tick, particle_velosety, observer)
Name | Type | Description |
---|---|---|
size_tick | int | size of time tact |
count_tick | int | count of tacts |
particle_velosety | int | inicial speed particle |
observer | Table instance | Detector and recorder |
Operations:
def interaction(self, car) Description: none interaction Parameters: “car” is “Currer” instance
Description: new procedures join to processor of data
Bases: ResacherInstruments.DataProcessing
def __init__(self, observer,particle_velosety, sizeTick, countTick)
Name | Type | Description |
---|---|---|
observer | Table instance | Detector and recorder |
particle_velosety | int | initial speed particle |
size_tick | int | size of time tact |
count_tick | int | count of tacts |
def incline(self)
Description: incline k calculate and error
Parameters: None
Algorithm:
The analytical value of incline kan can deduce from the formula of the invariant interval
s2=c2t2−x2
Let t′ be fix moment of time then distance be x=vt′.
We have s=ct′ if x=0.
We obtain
ct=√s2+x2=√(cx/v)2+x2=x√1+c2/v2 .
The result is
aan=√1+c2/v2
The experiment data processing is
Δti=ti−ti−4
Δxi=xi−xi−4
kar=1N∑ni=4Δti/Δxi,
wher N is count of couples.
The standard deviation is
skar=√Var(kar)/(N−1), where Var(kar) is variance and N is count of pair.
The confidence interval is
dkar=skar/√N
Then
kar=kar±dkar
Description: rewrite procedure xtPrintPrettyTable Bases: print_results.TablePrint