mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-27 15:01:08 +00:00
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>
This commit is contained in:
Tianyi Zheng
GitHub
parent
c00af459fe
commit
57c12fab28
|
|
@ -557,6 +557,7 @@
|
|||
* [Gamma Recursive](maths/gamma_recursive.py)
|
||||
* [Gaussian](maths/gaussian.py)
|
||||
* [Gaussian Error Linear Unit](maths/gaussian_error_linear_unit.py)
|
||||
* [Gcd Of N Numbers](maths/gcd_of_n_numbers.py)
|
||||
* [Greatest Common Divisor](maths/greatest_common_divisor.py)
|
||||
* [Greedy Coin Change](maths/greedy_coin_change.py)
|
||||
* [Hamming Numbers](maths/hamming_numbers.py)
|
||||
|
|
|
|||
|
|
@ -1,39 +1,33 @@
|
|||
"""
|
||||
Lorentz 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 Lorentz 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
|
||||
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.
|
||||
|
||||
NOTE: Please note that x0 is c (speed of light) times t (time).
|
||||
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
|
||||
|
||||
So, the Lorentz transformation using a four vector is defined as:
|
||||
| γ -γβ 0 0|
|
||||
B = |-γβ γ 0 0|
|
||||
| 0 0 1 0|
|
||||
| 0 0 0 1|
|
||||
|
||||
|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
|
||||
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 __future__ import annotations
|
||||
|
||||
from math import sqrt
|
||||
|
||||
import numpy as np # type: ignore
|
||||
from sympy import symbols # type: ignore
|
||||
import numpy as np
|
||||
from sympy import symbols
|
||||
|
||||
# Coefficient
|
||||
# Speed of light (m/s)
|
||||
|
|
@ -41,79 +35,77 @@ 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:
|
||||
"""
|
||||
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 1!
|
||||
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]!")
|
||||
|
||||
# Usually the speed u should be much higher than 1 (c order of magnitude)
|
||||
raise ValueError("Speed must not exceed light speed 299,792,458 [m/s]!")
|
||||
elif velocity < 1:
|
||||
raise ValueError("Speed must be greater than 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 1!
|
||||
|
||||
>>> gamma(2*c)
|
||||
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]!
|
||||
ValueError: Speed must not exceed light speed 299,792,458 [m/s]!
|
||||
"""
|
||||
return 1 / (sqrt(1 - beta(velocity) ** 2))
|
||||
return 1 / sqrt(1 - beta(velocity) ** 2)
|
||||
|
||||
|
||||
def transformation_matrix(velocity: float) -> np.array:
|
||||
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],
|
||||
|
|
@ -123,16 +115,14 @@ def transformation_matrix(velocity: float) -> np.array:
|
|||
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!
|
||||
|
||||
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]!
|
||||
ValueError: Speed must not exceed light speed 299,792,458 [m/s]!
|
||||
"""
|
||||
return np.array(
|
||||
[
|
||||
|
|
@ -144,44 +134,39 @@ def transformation_matrix(velocity: float) -> np.array:
|
|||
)
|
||||
|
||||
|
||||
def transform(
|
||||
velocity: float, event: np.array = np.zeros(4), symbolic: bool = True # noqa: B008
|
||||
) -> np.array:
|
||||
def transform(velocity: float, event: np.ndarray | None = None) -> np.ndarray:
|
||||
"""
|
||||
>>> transform(29979245,np.array([1,2,3,4]), False)
|
||||
array([ 3.01302757e+08, -3.01302729e+07, 3.00000000e+00, 4.00000000e+00])
|
||||
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]))
|
||||
>>> 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]))
|
||||
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!
|
||||
ValueError: Speed must be greater than or equal to 1!
|
||||
"""
|
||||
# Ensure event is not a vector of zeros
|
||||
if not symbolic:
|
||||
|
||||
# x0 is ct (speed of ligt * time)
|
||||
event[0] = event[0] * c
|
||||
# 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)
|
||||
|
||||
# Symbolic four vector
|
||||
event = np.array([ct, x, y, z])
|
||||
|
||||
return transformation_matrix(velocity).dot(event)
|
||||
return transformation_matrix(velocity) @ event
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
|
|
@ -197,9 +182,8 @@ if __name__ == "__main__":
|
|||
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)]
|
||||
# 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}")
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user