kerr_newman_2.py

"""
Kerr metric in Boyer Lindquist coordinates
Newman, J. Math. Phys., v6, p918, (1965)
Stephani (19.19) p213
"""
from sympy import cos, sin, symbols, zeros

coords = symbols("t r theta phi", real=True)
variables = symbols("a M Q", constant=True)
functions = ()
t, r, th, ph = coords
a, M, Q = variables
expr = a ** 2 * cos(th) ** 2 + r ** 2
metric = zeros(4)
metric[0, 0] = -(a ** 2 - sin(th) ** 2 * a ** 2 + r ** 2 - (2 * M * r - Q ** 2)) / expr
metric[1, 1] = expr / (r ** 2 - 2 * M * r + Q ** 2 + a ** 2)
metric[2, 2] = expr
metric[3, 3] = (
    (
        -a ** 4 * sin(th) ** 2
        + a ** 4
        + 2 * r ** 2 * a ** 2
        - a ** 2 * sin(th) ** 2 * r ** 2
        + 2 * a ** 2 * sin(th) ** 2 * M * r
        - a ** 2 * sin(th) ** 2 * Q ** 2
        + r ** 4
    )
    * sin(th) ** 2
    / expr
)
metric[0, 3] = metric[3, 0] = (
    -(2 * a * sin(th) ** 2 * M * r - a * sin(th) ** 2 * Q ** 2) / expr
)