Lorenz transformation - physics

physics/lorenz_transformation_four_vector.py

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+"""
+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}")