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cclauss merged 9 commits into from
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Fix mypy errors in lorentz_transformation_four_vector.py#8075

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updating DIRECTORY.md
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8b44f10
Merge branch 'TheAlgorithms:master' into master
tianyizheng02 Jan 2, 2023
7d8e71e
Fix mypy errors in lorentz_transformation_four_vector.py
tianyizheng02 Jan 4, 2023
2b800d6
Remove unused symbol vars
tianyizheng02 Jan 4, 2023
9999dd9
Add function documentation and rewrite algorithm explanation
tianyizheng02 Jan 4, 2023
916559a
Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2
tianyizheng02 Jan 7, 2023
87e4807
Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2
tianyizheng02 Jan 12, 2023
1b111c7
updating DIRECTORY.md
Jan 12, 2023
0471337
Update error message for speed
tianyizheng02 Jan 26, 2023
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1 change: 1 addition & 0 deletions DIRECTORY.md
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Original file line numberDiff line numberDiff line change
Expand Up@@ -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)
Expand Down
144 changes: 64 additions & 80 deletions physics/lorentz_transformation_four_vector.py
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Original file line numberDiff line numberDiff line change
@@ -1,119 +1,111 @@
"""
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

NOTE: Please note that x0 is c (speed of light) times t (time).

So, the Lorentz 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
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.

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

| γ -γβ 0 0|
B = |-γβ γ 0 0|
| 0 0 1 0|
| 0 0 0 1|

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)
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],
Expand All@@ -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(
[
Expand All@@ -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__":
Expand All@@ -197,9 +182,8 @@ def transform(
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}")
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Fix `mypy` errors in `lorentz_transformation_four_vector.py` by tianyizheng02 · Pull Request #8075 · TheAlgorithms/Python · GitHub
Skip to content
Fix mypy errors in lorentz_transformation_four_vector.py#8075

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 Merged

cclauss
merged 9 commits into

from

Jan 26, 2023
 Merged


Fix mypy errors in lorentz_transformation_four_vector.py#8075
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f7ba9da

updating DIRECTORY.md

Jan 2, 2023
8b44f10

Merge branch 'TheAlgorithms:master' into master


tianyizheng02 Jan 2, 2023
7d8e71e

Fix mypy errors in lorentz_transformation_four_vector.py


tianyizheng02 Jan 4, 2023
2b800d6

Remove unused symbol vars


tianyizheng02 Jan 4, 2023
9999dd9

Add function documentation and rewrite algorithm explanation


tianyizheng02 Jan 4, 2023
916559a

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 7, 2023
87e4807

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 12, 2023
1b111c7

updating DIRECTORY.md

Jan 12, 2023
0471337

Update error message for speed


tianyizheng02 Jan 26, 2023

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DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -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)
Expand Down

144 changes: 64 additions & 80 deletions
physics/lorentz_transformation_four_vector.py

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -1,119 +1,111 @@
"""
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

NOTE: Please note that x0 is c (speed of light) times t (time).

So, the Lorentz 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
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.

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

 | γ -γβ 0 0|
B = |-γβ γ 0 0|
 | 0 0 1 0|
 | 0 0 0 1|

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)
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],
Expand All@@ -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(
 [
Expand All@@ -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__":
Expand All@@ -197,9 +182,8 @@ def transform(
 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}")

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Fix `mypy` errors in `lorentz_transformation_four_vector.py` by tianyizheng02 · Pull Request #8075 · TheAlgorithms/Python · GitHub
Skip to content
Fix mypy errors in lorentz_transformation_four_vector.py#8075

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 Merged

cclauss
merged 9 commits into

from

Jan 26, 2023
 Merged


Fix mypy errors in lorentz_transformation_four_vector.py#8075
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f7ba9da

updating DIRECTORY.md

Jan 2, 2023
8b44f10

Merge branch 'TheAlgorithms:master' into master


tianyizheng02 Jan 2, 2023
7d8e71e

Fix mypy errors in lorentz_transformation_four_vector.py


tianyizheng02 Jan 4, 2023
2b800d6

Remove unused symbol vars


tianyizheng02 Jan 4, 2023
9999dd9

Add function documentation and rewrite algorithm explanation


tianyizheng02 Jan 4, 2023
916559a

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 7, 2023
87e4807

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 12, 2023
1b111c7

updating DIRECTORY.md

Jan 12, 2023
0471337

Update error message for speed


tianyizheng02 Jan 26, 2023

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DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -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)
Expand Down

144 changes: 64 additions & 80 deletions
physics/lorentz_transformation_four_vector.py

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@@ -1,119 +1,111 @@
"""
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

NOTE: Please note that x0 is c (speed of light) times t (time).

So, the Lorentz 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
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.

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

 | γ -γβ 0 0|
B = |-γβ γ 0 0|
 | 0 0 1 0|
 | 0 0 0 1|

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)
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],
Expand All@@ -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(
 [
Expand All@@ -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__":
Expand All@@ -197,9 +182,8 @@ def transform(
 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}")

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Highlight search terms from Google/DuckDuckGo/Bing referrer
(function() {
var ref = document.referrer;
var terms = [];
if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {
var url = new URL(ref);
var q = url.searchParams.get('q') || url.searchParams.get('p');
if (q) {
terms = q.split(/\s+/).filter(function(t) { return t.length > 2; });
}
}
if (terms.length === 0) return;
var style = document.createElement('style');
style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';
document.head.appendChild(style);
function highlight(node) {
if (node.nodeType === 3) { // text node
var text = node.textContent;
var found = false;
terms.forEach(function(term) {
var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\]\\]/g, '\\') + ')', 'gi');
if (regex.test(text)) {
found = true;
var frag = document.createDocumentFragment();
var parts = text.split(regex);
parts.forEach(function(part, i) {
if (i % 2 === 0) {
frag.appendChild(document.createTextNode(part));
} else {
var span = document.createElement('span');
span.className = 'userscript-highlight';
span.textContent = part;
frag.appendChild(span);
}
});
node.parentNode.replaceChild(frag, node);
}
});
} else if (node.nodeType === 1 && node.childNodes) { // element
var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];
if (!skipTags.includes(node.tagName)) {
Array.from(node.childNodes).forEach(highlight);
}
}
}
highlight(document.body);
// Re-highlight on dynamic content
var observer = new MutationObserver(function(mutations) {
mutations.forEach(function(m) {
m.addedNodes.forEach(function(node) {
if (node.nodeType === 1 || node.nodeType === 3) highlight(node);
});
});
});
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
Fix `mypy` errors in `lorentz_transformation_four_vector.py` by tianyizheng02 · Pull Request #8075 · TheAlgorithms/Python · GitHub
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Fix mypy errors in lorentz_transformation_four_vector.py#8075

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 Merged

cclauss
merged 9 commits into

from

Jan 26, 2023
 Merged


Fix mypy errors in lorentz_transformation_four_vector.py#8075
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f7ba9da

updating DIRECTORY.md

Jan 2, 2023
8b44f10

Merge branch 'TheAlgorithms:master' into master


tianyizheng02 Jan 2, 2023
7d8e71e

Fix mypy errors in lorentz_transformation_four_vector.py


tianyizheng02 Jan 4, 2023
2b800d6

Remove unused symbol vars


tianyizheng02 Jan 4, 2023
9999dd9

Add function documentation and rewrite algorithm explanation


tianyizheng02 Jan 4, 2023
916559a

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 7, 2023
87e4807

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 12, 2023
1b111c7

updating DIRECTORY.md

Jan 12, 2023
0471337

Update error message for speed


tianyizheng02 Jan 26, 2023

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1 change: 1 addition & 0 deletions
DIRECTORY.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
Expand Up@@ -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)
Expand Down

144 changes: 64 additions & 80 deletions
physics/lorentz_transformation_four_vector.py

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -1,119 +1,111 @@
"""
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

NOTE: Please note that x0 is c (speed of light) times t (time).

So, the Lorentz 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
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.

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

 | γ -γβ 0 0|
B = |-γβ γ 0 0|
 | 0 0 1 0|
 | 0 0 0 1|

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)
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],
Expand All@@ -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(
 [
Expand All@@ -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__":
Expand All@@ -197,9 +182,8 @@ def transform(
 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}")

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Strip utm_, fbclid, gclid, etc. from all links on page
(function() {
var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',
'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',
'ref', 'ref_src', 'source', 'medium', 'campaign'];
function cleanUrl(url) {
try {
var u = new URL(url, window.location.origin);
var changed = false;
trackingParams.forEach(function(p) {
if (u.searchParams.has(p)) {
u.searchParams.delete(p);
changed = true;
}
});
return changed ? u.toString() : url;
} catch (e) {
return url;
}
}
function cleanLinks() {
document.querySelectorAll('a[href]').forEach(function(a) {
var clean = cleanUrl(a.href);
if (clean !== a.href) a.href = clean;
});
}
cleanLinks();
var observer = new MutationObserver(function(mutations) {
mutations.forEach(function(m) {
m.addedNodes.forEach(function(node) {
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Fix `mypy` errors in `lorentz_transformation_four_vector.py` by tianyizheng02 · Pull Request #8075 · TheAlgorithms/Python · GitHub
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Fix mypy errors in lorentz_transformation_four_vector.py#8075

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Fix mypy errors in lorentz_transformation_four_vector.py#8075
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f7ba9da

updating DIRECTORY.md

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8b44f10

Merge branch 'TheAlgorithms:master' into master


tianyizheng02 Jan 2, 2023
7d8e71e

Fix mypy errors in lorentz_transformation_four_vector.py


tianyizheng02 Jan 4, 2023
2b800d6

Remove unused symbol vars


tianyizheng02 Jan 4, 2023
9999dd9

Add function documentation and rewrite algorithm explanation


tianyizheng02 Jan 4, 2023
916559a

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 7, 2023
87e4807

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 12, 2023
1b111c7

updating DIRECTORY.md

Jan 12, 2023
0471337

Update error message for speed


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Expand Up@@ -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)
Expand Down

144 changes: 64 additions & 80 deletions
physics/lorentz_transformation_four_vector.py

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Original file line numberDiff line numberDiff line change
@@ -1,119 +1,111 @@
"""
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

NOTE: Please note that x0 is c (speed of light) times t (time).

So, the Lorentz 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
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.

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

 | γ -γβ 0 0|
B = |-γβ γ 0 0|
 | 0 0 1 0|
 | 0 0 0 1|

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)
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],
Expand All@@ -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(
 [
Expand All@@ -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__":
Expand All@@ -197,9 +182,8 @@ def transform(
 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}")

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Auto-enable theater mode on YouTube
(function() {
function tryTheater() {
var btn = document.querySelector('button[aria-label="Theater mode"], ytd-player #player button[title="Theater mode"]');
if (btn && !btn.classList.contains('activated')) {
btn.click();
}
}
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// Try after navigation (SPA)
var lastUrl = location.href;
setInterval(function() {
if (location.href !== lastUrl) {
lastUrl = location.href;
setTimeout(tryTheater, 500);
}
}, 1000);
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observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
Fix `mypy` errors in `lorentz_transformation_four_vector.py` by tianyizheng02 · Pull Request #8075 · TheAlgorithms/Python · GitHub
Skip to content
Fix mypy errors in lorentz_transformation_four_vector.py#8075

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 Merged

cclauss
merged 9 commits into

from

Jan 26, 2023
 Merged


Fix mypy errors in lorentz_transformation_four_vector.py#8075
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f7ba9da

updating DIRECTORY.md

Jan 2, 2023
8b44f10

Merge branch 'TheAlgorithms:master' into master


tianyizheng02 Jan 2, 2023
7d8e71e

Fix mypy errors in lorentz_transformation_four_vector.py


tianyizheng02 Jan 4, 2023
2b800d6

Remove unused symbol vars


tianyizheng02 Jan 4, 2023
9999dd9

Add function documentation and rewrite algorithm explanation


tianyizheng02 Jan 4, 2023
916559a

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 7, 2023
87e4807

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 12, 2023
1b111c7

updating DIRECTORY.md

Jan 12, 2023
0471337

Update error message for speed


tianyizheng02 Jan 26, 2023

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DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -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)
Expand Down

144 changes: 64 additions & 80 deletions
physics/lorentz_transformation_four_vector.py

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -1,119 +1,111 @@
"""
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

NOTE: Please note that x0 is c (speed of light) times t (time).

So, the Lorentz 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
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.

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

 | γ -γβ 0 0|
B = |-γβ γ 0 0|
 | 0 0 1 0|
 | 0 0 0 1|

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)
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],
Expand All@@ -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(
 [
Expand All@@ -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__":
Expand All@@ -197,9 +182,8 @@ def transform(
 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}")

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Remove or un-stick sticky/fixed headers that block content
(function() {
function unstick() {
document.querySelectorAll('header, nav, [role="banner"], .header, .navbar, .sticky, .fixed-top, [style*="position: fixed"], [style*="position:sticky"]').forEach(function(el) {
if (el.style.position === 'fixed' || el.style.position === 'sticky' || getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {
el.style.position = 'static';
el.style.top = 'auto';
el.style.zIndex = 'auto';
}
});
}
unstick();
var observer = new MutationObserver(unstick);
observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });
})();
}
} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
})();
Fix `mypy` errors in `lorentz_transformation_four_vector.py` by tianyizheng02 · Pull Request #8075 · TheAlgorithms/Python · GitHub
Skip to content
Fix mypy errors in lorentz_transformation_four_vector.py#8075

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cclauss
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Jan 26, 2023
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Fix mypy errors in lorentz_transformation_four_vector.py#8075
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f7ba9da

updating DIRECTORY.md

Jan 2, 2023
8b44f10

Merge branch 'TheAlgorithms:master' into master


tianyizheng02 Jan 2, 2023
7d8e71e

Fix mypy errors in lorentz_transformation_four_vector.py


tianyizheng02 Jan 4, 2023
2b800d6

Remove unused symbol vars


tianyizheng02 Jan 4, 2023
9999dd9

Add function documentation and rewrite algorithm explanation


tianyizheng02 Jan 4, 2023
916559a

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 7, 2023
87e4807

Merge branch 'TheAlgorithms:master' into fix-mypy-errs-2


tianyizheng02 Jan 12, 2023
1b111c7

updating DIRECTORY.md

Jan 12, 2023
0471337

Update error message for speed


tianyizheng02 Jan 26, 2023

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1 change: 1 addition & 0 deletions
DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -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)
Expand Down

144 changes: 64 additions & 80 deletions
physics/lorentz_transformation_four_vector.py

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Original file line numberDiff line numberDiff line change
@@ -1,119 +1,111 @@
"""
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

NOTE: Please note that x0 is c (speed of light) times t (time).

So, the Lorentz 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
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.

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

 | γ -γβ 0 0|
B = |-γβ γ 0 0|
 | 0 0 1 0|
 | 0 0 0 1|

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)
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],
Expand All@@ -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(
 [
Expand All@@ -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__":
Expand All@@ -197,9 +182,8 @@ def transform(
 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}")