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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 commit5b0249ac0a. Undo changes @wq * Revert "Fixed annotations" This reverts commitc37bb95408. * Revert "Fixed last annotation" This reverts commite3678fdead. * Revert "Fixed type annotations" This reverts commit3f2b238c34. * Revert to4e2fcaf6fb* 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
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Aviv Faraj
GitHub
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physics/__init__.py
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physics/__init__.py
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physics/lorenz_transformation_four_vector.py
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physics/lorenz_transformation_four_vector.py
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"""
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Lorenz transformation describes the transition from a reference frame P
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to another reference frame P', each of which is moving in a direction with
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respect to the other. The Lorenz transformation implemented in this code
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is the relativistic version using a four vector described by Minkowsky Space:
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x0 = ct, x1 = x, x2 = y, and x3 = z
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NOTE: Please note that x0 is c (speed of light) times t (time).
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So, the Lorenz transformation using a four vector is defined as:
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|ct'| | γ -γβ 0 0| |ct|
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|x' | = |-γβ γ 0 0| *|x |
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|y' | | 0 0 1 0| |y |
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|z' | | 0 0 0 1| |z |
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Where:
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1
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γ = ---------------
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-----------
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/ v^2 |
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/(1 - ---
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-/ c^2
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v
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β = -----
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c
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Reference: https://en.wikipedia.org/wiki/Lorentz_transformation
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"""
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from __future__ import annotations
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from math import sqrt
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import numpy as np # type: ignore
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from sympy import symbols # type: ignore
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# Coefficient
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# Speed of light (m/s)
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c = 299792458
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# Symbols
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ct, x, y, z = symbols("ct x y z")
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ct_p, x_p, y_p, z_p = symbols("ct' x' y' z'")
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# Vehicle's speed divided by speed of light (no units)
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def beta(velocity: float) -> float:
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"""
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>>> beta(c)
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1.0
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>>> beta(199792458)
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0.666435904801848
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>>> beta(1e5)
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0.00033356409519815205
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>>> beta(0.2)
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Traceback (most recent call last):
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...
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ValueError: Speed must be greater than 1!
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"""
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if velocity > c:
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raise ValueError("Speed must not exceed Light Speed 299,792,458 [m/s]!")
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# Usually the speed u should be much higher than 1 (c order of magnitude)
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elif velocity < 1:
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raise ValueError("Speed must be greater than 1!")
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return velocity / c
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def gamma(velocity: float) -> float:
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"""
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>>> gamma(4)
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1.0000000000000002
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>>> gamma(1e5)
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1.0000000556325075
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>>> gamma(3e7)
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1.005044845777813
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>>> gamma(2.8e8)
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2.7985595722318277
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>>> gamma(299792451)
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4627.49902669495
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>>> gamma(0.3)
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Traceback (most recent call last):
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...
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ValueError: Speed must be greater than 1!
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>>> gamma(2*c)
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Traceback (most recent call last):
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...
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ValueError: Speed must not exceed Light Speed 299,792,458 [m/s]!
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"""
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return 1 / (sqrt(1 - beta(velocity) ** 2))
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def transformation_matrix(velocity: float) -> np.array:
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"""
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>>> transformation_matrix(29979245)
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array([[ 1.00503781, -0.10050378, 0. , 0. ],
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[-0.10050378, 1.00503781, 0. , 0. ],
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[ 0. , 0. , 1. , 0. ],
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[ 0. , 0. , 0. , 1. ]])
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>>> transformation_matrix(19979245.2)
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array([[ 1.00222811, -0.06679208, 0. , 0. ],
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[-0.06679208, 1.00222811, 0. , 0. ],
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[ 0. , 0. , 1. , 0. ],
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[ 0. , 0. , 0. , 1. ]])
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>>> transformation_matrix(1)
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array([[ 1.00000000e+00, -3.33564095e-09, 0.00000000e+00,
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0.00000000e+00],
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[-3.33564095e-09, 1.00000000e+00, 0.00000000e+00,
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0.00000000e+00],
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[ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00,
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0.00000000e+00],
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[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
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1.00000000e+00]])
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>>> transformation_matrix(0)
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Traceback (most recent call last):
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...
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ValueError: Speed must be greater than 1!
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>>> transformation_matrix(c * 1.5)
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Traceback (most recent call last):
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...
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ValueError: Speed must not exceed Light Speed 299,792,458 [m/s]!
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"""
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return np.array(
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[
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[gamma(velocity), -gamma(velocity) * beta(velocity), 0, 0],
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[-gamma(velocity) * beta(velocity), gamma(velocity), 0, 0],
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[0, 0, 1, 0],
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[0, 0, 0, 1],
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]
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)
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def transform(
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velocity: float, event: np.array = np.zeros(4), symbolic: bool = True
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) -> np.array:
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"""
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>>> transform(29979245,np.array([1,2,3,4]), False)
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array([ 3.01302757e+08, -3.01302729e+07, 3.00000000e+00, 4.00000000e+00])
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>>> transform(29979245)
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array([1.00503781498831*ct - 0.100503778816875*x,
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-0.100503778816875*ct + 1.00503781498831*x, 1.0*y, 1.0*z],
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dtype=object)
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>>> transform(19879210.2)
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array([1.0022057787097*ct - 0.066456172618675*x,
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-0.066456172618675*ct + 1.0022057787097*x, 1.0*y, 1.0*z],
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dtype=object)
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>>> transform(299792459, np.array([1,1,1,1]))
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Traceback (most recent call last):
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...
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ValueError: Speed must not exceed Light Speed 299,792,458 [m/s]!
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>>> transform(-1, np.array([1,1,1,1]))
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Traceback (most recent call last):
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...
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ValueError: Speed must be greater than 1!
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"""
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# Ensure event is not a vector of zeros
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if not symbolic:
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# x0 is ct (speed of ligt * time)
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event[0] = event[0] * c
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else:
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# Symbolic four vector
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event = np.array([ct, x, y, z])
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return transformation_matrix(velocity).dot(event)
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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# Example of symbolic vector:
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four_vector = transform(29979245)
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print("Example of four vector: ")
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print(f"ct' = {four_vector[0]}")
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print(f"x' = {four_vector[1]}")
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print(f"y' = {four_vector[2]}")
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print(f"z' = {four_vector[3]}")
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# Substitute symbols with numerical values:
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values = np.array([1, 1, 1, 1])
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sub_dict = {ct: c * values[0], x: values[1], y: values[2], z: values[3]}
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numerical_vector = [four_vector[i].subs(sub_dict) for i in range(0, 4)]
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print(f"\n{numerical_vector}")
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