Kerr−Newman metric

This OCaml code computes the Kerr−Newman metric as given here.

code id	Wiki id	meaning
bSI		3 physical constants in SI units
gu		Same units geometrized
c	c	speed of light
bG	G	gravitational constant
ke	1/4πε₀c⁴	Coulomb’s constant
bM	M	Mass of BH
bQ	Q	charge on BH
bJ	J	angular momentum
dtau2	dτ	τ is proper time, measured locally. dtau2 is a function of coordinates and differential vector.
rs	r_s	Schwarzschild radius
rQ2	r_Q²	a squared measure of the charge.
a	α	common subexpression
r	r	Schwarzschild radius, the r coordinate.
th	θ	Latitude = (90−180*θ/π)°, a coordinate
dt	dt	delta t where t is the coordinate time.
dr	dr	delta r
dth	dθ	delta θ
dp	dφ	delta longitude. Longitude = 180*φ/π.
rh2	ρ²	common subexpression
del	Δ	common subexpression

t, r, φ and θ are the coördinates.
θ is actually 0 and π at the poles and π/2 at the equator. It determines the latitude.

Three physical constants play a role here, c, bG and ke. These constants can be expressed in either SI units of geometrized units.

Three constants describe the particular black hole: M, Q and J.

The function dtau2 is of the relevant coordinates, r and th, and the differential vector, (dt, dr, dth, dp), separating the two events whose distance the metric defines.

Cartesian Coordinates

This to compute r² from x, y, z and a.

The plan:

Choose a random point by its polar coordinates,
choose a random small 4D displacement,
use dtau2 to compute length of displacement,
Transform polar coordinates via p2C to Cartesian,
Compute covariant tensor with gC,
Compute length of displacement via ds² = g_ijdxⁱdx^j,
Compare computed lengths.

Elephant

The Elephant in the room is whether these solutions satisfy Einstein’s equations.
start? Start!