Python/physics/lorenz_transformation_four_vector.py

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Lorenz transformation - physics (#6097) * Add files via upload * Changed print to f-string Also printed out results in a math notation * Add files via upload * Fixes: #4710 provided return type * File exists in another pull request * imported radians from math * Updated file according to pre-commit test * Updated file * Updated gamma * Deleted duplicate file * removed pi * reversed tests * Fixed angle condition * Modified prints to f-string * Update horizontal_projectile_motion.py * Update horizontal_projectile_motion.py * Fixes #4710 added exceptions and tests * Added float tests * Fixed type annotations * Fixed last annotation * Fixed annotations * fixed format * Revert "fixed format" This reverts commit 5b0249ac0a0f9c36c3cfbab8423eb72925a73ffb. Undo changes @wq * Revert "Fixed annotations" This reverts commit c37bb9540834cb77e37822eb376a5896cda34778. * Revert "Fixed last annotation" This reverts commit e3678fdeadd23f1bfca27015ab524efa184f6c79. * Revert "Fixed type annotations" This reverts commit 3f2b238c34cd926b335d1f6f750e009f08e8f270. * Revert to 4e2fcaf6fb * Fixing errors found during pre-commit * Added gauss law * Implemented Lorenz tranformation with four vector * pre-commit fixes * flake8 fixes * More flake8 fixes * Added blank space for flake8 * Added reference * Trailing whitespace fix * Replaced argument u with velocity (descriptive name fix) * Added tests for functions + moved velocity check to beta function * Modified condition to 'not symbolic' in the transform function * trainling whitespace fix * Added type hint for 'smybolic' argument in transform function * Changed reference to avoid pre-commit fails because of spelling issue related to the URL * Added tests for gamma and transformation_matrix functions * Fixed transformation_matrix tests * Fixed tests on beta and gamma functions
2022-05-16 14:26:19 +00:00
"""
Lorenz transformation describes the transition from a reference frame P
to another reference frame P', each of which is moving in a direction with
respect to the other. The Lorenz transformation implemented in this code
is the relativistic version using a four vector described by Minkowsky Space:
x0 = ct, x1 = x, x2 = y, and x3 = z
NOTE: Please note that x0 is c (speed of light) times t (time).
So, the Lorenz transformation using a four vector is defined as:
|ct'| | γ -γβ 0 0| |ct|
|x' | = |-γβ γ 0 0| *|x |
|y' | | 0 0 1 0| |y |
|z' | | 0 0 0 1| |z |
Where:
1
γ = ---------------
-----------
/ v^2 |
/(1 - ---
-/ c^2
v
β = -----
c
Reference: https://en.wikipedia.org/wiki/Lorentz_transformation
"""
from __future__ import annotations
from math import sqrt
import numpy as np # type: ignore
from sympy import symbols # type: ignore
# Coefficient
# Speed of light (m/s)
c = 299792458
# Symbols
ct, x, y, z = symbols("ct x y z")
ct_p, x_p, y_p, z_p = symbols("ct' x' y' z'")
# Vehicle's speed divided by speed of light (no units)
def beta(velocity: float) -> float:
"""
>>> beta(c)
1.0
>>> beta(199792458)
0.666435904801848
>>> beta(1e5)
0.00033356409519815205
>>> beta(0.2)
Traceback (most recent call last):
...
ValueError: Speed must be greater than 1!
"""
if velocity > c:
raise ValueError("Speed must not exceed Light Speed 299,792,458 [m/s]!")
# Usually the speed u should be much higher than 1 (c order of magnitude)
elif velocity < 1:
raise ValueError("Speed must be greater than 1!")
return velocity / c
def gamma(velocity: float) -> float:
"""
>>> gamma(4)
1.0000000000000002
>>> gamma(1e5)
1.0000000556325075
>>> gamma(3e7)
1.005044845777813
>>> gamma(2.8e8)
2.7985595722318277
>>> gamma(299792451)
4627.49902669495
>>> gamma(0.3)
Traceback (most recent call last):
...
ValueError: Speed must be greater than 1!
>>> gamma(2*c)
Traceback (most recent call last):
...
ValueError: Speed must not exceed Light Speed 299,792,458 [m/s]!
"""
return 1 / (sqrt(1 - beta(velocity) ** 2))
def transformation_matrix(velocity: float) -> np.array:
"""
>>> transformation_matrix(29979245)
array([[ 1.00503781, -0.10050378, 0. , 0. ],
[-0.10050378, 1.00503781, 0. , 0. ],
[ 0. , 0. , 1. , 0. ],
[ 0. , 0. , 0. , 1. ]])
>>> transformation_matrix(19979245.2)
array([[ 1.00222811, -0.06679208, 0. , 0. ],
[-0.06679208, 1.00222811, 0. , 0. ],
[ 0. , 0. , 1. , 0. ],
[ 0. , 0. , 0. , 1. ]])
>>> transformation_matrix(1)
array([[ 1.00000000e+00, -3.33564095e-09, 0.00000000e+00,
0.00000000e+00],
[-3.33564095e-09, 1.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
1.00000000e+00]])
>>> transformation_matrix(0)
Traceback (most recent call last):
...
ValueError: Speed must be greater than 1!
>>> transformation_matrix(c * 1.5)
Traceback (most recent call last):
...
ValueError: Speed must not exceed Light Speed 299,792,458 [m/s]!
"""
return np.array(
[
[gamma(velocity), -gamma(velocity) * beta(velocity), 0, 0],
[-gamma(velocity) * beta(velocity), gamma(velocity), 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1],
]
)
def transform(
velocity: float, event: np.array = np.zeros(4), symbolic: bool = True
) -> np.array:
"""
>>> transform(29979245,np.array([1,2,3,4]), False)
array([ 3.01302757e+08, -3.01302729e+07, 3.00000000e+00, 4.00000000e+00])
>>> transform(29979245)
array([1.00503781498831*ct - 0.100503778816875*x,
-0.100503778816875*ct + 1.00503781498831*x, 1.0*y, 1.0*z],
dtype=object)
>>> transform(19879210.2)
array([1.0022057787097*ct - 0.066456172618675*x,
-0.066456172618675*ct + 1.0022057787097*x, 1.0*y, 1.0*z],
dtype=object)
>>> transform(299792459, np.array([1,1,1,1]))
Traceback (most recent call last):
...
ValueError: Speed must not exceed Light Speed 299,792,458 [m/s]!
>>> transform(-1, np.array([1,1,1,1]))
Traceback (most recent call last):
...
ValueError: Speed must be greater than 1!
"""
# Ensure event is not a vector of zeros
if not symbolic:
# x0 is ct (speed of ligt * time)
event[0] = event[0] * c
else:
# Symbolic four vector
event = np.array([ct, x, y, z])
return transformation_matrix(velocity).dot(event)
if __name__ == "__main__":
import doctest
doctest.testmod()
# Example of symbolic vector:
four_vector = transform(29979245)
print("Example of four vector: ")
print(f"ct' = {four_vector[0]}")
print(f"x' = {four_vector[1]}")
print(f"y' = {four_vector[2]}")
print(f"z' = {four_vector[3]}")
# Substitute symbols with numerical values:
values = np.array([1, 1, 1, 1])
sub_dict = {ct: c * values[0], x: values[1], y: values[2], z: values[3]}
numerical_vector = [four_vector[i].subs(sub_dict) for i in range(0, 4)]
print(f"\n{numerical_vector}")