Python/physics/lorentz_transformation_four_vector.py
Tianyi Zheng 57c12fab28
Fix mypy errors in lorentz_transformation_four_vector.py (#8075)
* updating DIRECTORY.md

* Fix mypy errors in lorentz_transformation_four_vector.py

* Remove unused symbol vars

* Add function documentation and rewrite algorithm explanation

Previous explanation was misleading, as the code only calculates Lorentz
transformations for movement in the x direction (0 velocity in the y and
z directions) and not movement in any direction

* updating DIRECTORY.md

* Update error message for speed

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
2023-01-26 08:13:03 +01:00

190 lines
6.3 KiB
Python
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

"""
Lorentz transformations describe the transition between two inertial reference
frames F and F', each of which is moving in some direction with respect to the
other. This code only calculates Lorentz transformations for movement in the x
direction with no spacial rotation (i.e., a Lorentz boost in the x direction).
The Lorentz transformations are calculated here as linear transformations of
four-vectors [ct, x, y, z] described by Minkowski space. Note that t (time) is
multiplied by c (the speed of light) in the first entry of each four-vector.
Thus, if X = [ct; x; y; z] and X' = [ct'; x'; y'; z'] are the four-vectors for
two inertial reference frames and X' moves in the x direction with velocity v
with respect to X, then the Lorentz transformation from X to X' is X' = BX,
where
| γ -γβ 0 0|
B = |-γβ γ 0 0|
| 0 0 1 0|
| 0 0 0 1|
is the matrix describing the Lorentz boost between X and X',
γ = 1 / √(1 - v²/c²) is the Lorentz factor, and β = v/c is the velocity as
a fraction of c.
Reference: https://en.wikipedia.org/wiki/Lorentz_transformation
"""
from math import sqrt
import numpy as np
from sympy import symbols
# Coefficient
# Speed of light (m/s)
c = 299792458
# Symbols
ct, x, y, z = symbols("ct x y z")
# Vehicle's speed divided by speed of light (no units)
def beta(velocity: float) -> float:
"""
Calculates β = v/c, the given velocity as a fraction of c
>>> 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 or equal to 1!
"""
if velocity > c:
raise ValueError("Speed must not exceed light speed 299,792,458 [m/s]!")
elif velocity < 1:
# Usually the speed should be much higher than 1 (c order of magnitude)
raise ValueError("Speed must be greater than or equal to 1!")
return velocity / c
def gamma(velocity: float) -> float:
"""
Calculate the Lorentz factor γ = 1 / √(1 - v²/c²) for a given velocity
>>> 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 or equal to 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.ndarray:
"""
Calculate the Lorentz transformation matrix for movement in the x direction:
| γ -γβ 0 0|
|-γβ γ 0 0|
| 0 0 1 0|
| 0 0 0 1|
where γ is the Lorentz factor and β is the velocity as a fraction of c
>>> 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 or equal to 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.ndarray | None = None) -> np.ndarray:
"""
Calculate a Lorentz transformation for movement in the x direction given a
velocity and a four-vector for an inertial reference frame
If no four-vector is given, then calculate the transformation symbolically
with variables
>>> transform(29979245, np.array([1, 2, 3, 4]))
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 or equal to 1!
"""
# Ensure event is not empty
if event is None:
event = np.array([ct, x, y, z]) # Symbolic four vector
else:
event[0] *= c # x0 is ct (speed of light * time)
return transformation_matrix(velocity) @ 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
sub_dict = {ct: c, x: 1, y: 1, z: 1}
numerical_vector = [four_vector[i].subs(sub_dict) for i in range(4)]
print(f"\n{numerical_vector}")