diff --git a/physics/__init__.py b/physics/__init__.py new file mode 100644 index 000000000..e69de29bb diff --git a/physics/lorenz_transformation_four_vector.py b/physics/lorenz_transformation_four_vector.py new file mode 100644 index 000000000..6c0d5f9d1 --- /dev/null +++ b/physics/lorenz_transformation_four_vector.py @@ -0,0 +1,205 @@ +""" +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}")