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use super::coordinates::{CoordinateSystem, Point};
use super::tensors::{
ContravariantIndex, CovariantIndex, InnerProduct, InvTwoForm, Tensor, TwoForm,
};
use crate::inner;
use crate::typenum::consts::{U0, U1, U2, U3};
use crate::typenum::{Exp, Pow, Unsigned};
use generic_array::ArrayLength;
pub trait MetricSystem: CoordinateSystem
where
<Self as CoordinateSystem>::Dimension: Pow<U2> + Pow<U3>,
Exp<<Self as CoordinateSystem>::Dimension, U2>: ArrayLength<f64>,
Exp<<Self as CoordinateSystem>::Dimension, U3>: ArrayLength<f64>,
{
fn g(point: &Point<Self>) -> TwoForm<Self>;
fn inv_g(point: &Point<Self>) -> InvTwoForm<Self> {
Self::g(point).inverse().unwrap()
}
fn dg(point: &Point<Self>) -> Tensor<Self, (CovariantIndex, (CovariantIndex, CovariantIndex))> {
let d = Self::dimension();
let mut result = Tensor::zero(point.clone());
let h = Self::small(point);
for j in 0..d {
let mut x = point.clone();
x[j] = x[j] - h;
let g1 = Self::g(&x);
x[j] = x[j] + h * 2.0;
let g2 = Self::g(&x);
for coord in g1.iter_coords() {
let index = [coord[0], coord[1], j];
result[&index[..]] = (g2[&*coord] - g1[&*coord]) / (2.0 * h);
}
}
result
}
fn covariant_christoffel(
point: &Point<Self>,
) -> Tensor<Self, (CovariantIndex, (CovariantIndex, CovariantIndex))> {
let dg = Self::dg(point);
let mut result =
Tensor::<Self, (CovariantIndex, (CovariantIndex, CovariantIndex))>::zero(point.clone());
for i in result.iter_coords() {
result[&*i] =
0.5 * (dg[&*i] + dg[&[i[0], i[2], i[1]][..]] - dg[&[i[1], i[2], i[0]][..]]);
}
result
}
fn christoffel(
point: &Point<Self>,
) -> Tensor<Self, (ContravariantIndex, (CovariantIndex, CovariantIndex))> {
let ig = Self::inv_g(point);
let gamma = Self::covariant_christoffel(point);
<InvTwoForm<Self> as InnerProduct<
Tensor<Self, (CovariantIndex, (CovariantIndex, CovariantIndex))>,
U1,
U2,
>>::inner_product(ig, gamma)
}
}
impl<T> Tensor<T, ContravariantIndex>
where
T: MetricSystem,
T::Dimension: Pow<U1> + Pow<U2> + Pow<U3> + Unsigned,
Exp<T::Dimension, U1>: ArrayLength<f64>,
Exp<T::Dimension, U2>: ArrayLength<f64>,
Exp<T::Dimension, U3>: ArrayLength<f64>,
{
pub fn square(&self) -> f64 {
let g = T::g(self.get_point());
let temp = inner!(_, _; U1, U2; g, self.clone());
*inner!(_, _; U0, U1; temp, self.clone())
}
pub fn normalize(&mut self) {
let len = self.square().abs().sqrt();
for i in 0..T::Dimension::to_usize() {
self[i] /= len;
}
}
}
impl<T> Tensor<T, CovariantIndex>
where
T: MetricSystem,
T::Dimension: Pow<U1> + Pow<U2> + Pow<U3> + Unsigned,
Exp<T::Dimension, U1>: ArrayLength<f64>,
Exp<T::Dimension, U2>: ArrayLength<f64>,
Exp<T::Dimension, U3>: ArrayLength<f64>,
{
pub fn square(&self) -> f64 {
let g = T::inv_g(self.get_point());
let temp = inner!(_, _; U1, U2; g, self.clone());
*inner!(_, _; U0, U1; temp, self.clone())
}
pub fn normalize(&mut self) {
let len = self.square().abs().sqrt();
for i in 0..T::Dimension::to_usize() {
self[i] /= len;
}
}
}