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lorentz_transformation_four_vector.py
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lorentz_transformation_four_vector.py
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"""
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 spatial 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
| y -γβ 0 0|
B = |-γβ y 0 0|
| 0 0 1 0|
| 0 0 0 1|
is the matrix describing the Lorentz boost between X and X',
y = 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 y = 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:
| y -γβ 0 0|
|-γβ y 0 0|
| 0 0 1 0|
| 0 0 0 1|
where y 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}")