physics.lorentz_transformation_four_vector

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

Attributes

c

four_vector

Functions

beta(→ float)

Calculates β = v/c, the given velocity as a fraction of c

gamma(→ float)

Calculate the Lorentz factor y = 1 / √(1 - v²/c²) for a given velocity

transform(→ numpy.ndarray)

Calculate a Lorentz transformation for movement in the x direction given a

transformation_matrix(→ numpy.ndarray)

Calculate the Lorentz transformation matrix for movement in the x direction:

Module Contents

physics.lorentz_transformation_four_vector.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!

physics.lorentz_transformation_four_vector.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]!

physics.lorentz_transformation_four_vector.transform(velocity: float, event: numpy.ndarray | None = None) → numpy.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!
physics.lorentz_transformation_four_vector.transformation_matrix(velocity: float) → numpy.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]!
physics.lorentz_transformation_four_vector.c = 299792458
physics.lorentz_transformation_four_vector.four_vector