arRay: A repository from Sbozzolo

arRay will be a General-Relativistic Ray-Tracing code in Dyalog APL.

Why?

APL (A Programming Language) is an array programming language. This is a completely different (and for the most part obscure) paradigm compared to most other languages out there. APL shines in producing terse and efficient code when used for manipulations of matrices. I think that problem of General-Relativistic Ray-Tracing can be cast in these terms, so I wanted to see what it could come out.

Note: I started learning APL last in December 2022, so this code will likely be terrible.

General-Relativistic Ray-Tracing

According to General Relativity, light does not travel on straight lines and its motion is influenced by massive bodies (such as our Sun). Light rays travel on null geodesics, which are special curves in spacetime. General-relativistic ray-tracing consists in finding such curves given the initial position of the camera.

In equations

Let $x^\mu(\lambda)$ be the four-position of a photon with $\lambda$ affine parameter. We define its momentum as

$$ k^\mu = \frac{d x^\mu}{d \lambda}.$$

$k^\mu$ is also function of $\lambda$. Given that we are describing a null geodesic, $k^\mu$ has to be null, so $k^\mu k_\mu = 0$. Performing ray-tracing means solving the geodesic equation:

$$ \frac{d k^\mu}{d \lambda} = -\Gamma^\mu_{\alpha\beta} k^\alpha k^\beta . $$

$\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols and depend on the spacetime and the four-position in the spacetime. The geodesic equation is a set of ordinary differential equations (ODEs) in $\lambda$, which have to be coupled with the definition of the four-momentum to find the position of the photon. So, we have to solve eight ODEs (four of which are trivial). We represent each photon with a 8-element vector, where the first four element is $x^\mu$, and the last 4 are $k^\mu$. Then, starting from some initial data on our camera, we solve the ODEs until we reach the black hole or some outer domain.

Initial data

First, we need to generate the initial data. Our initial data will consist of a grid of photons initially on our camera. Then, we integrate the photons backward in time towards the black hole. Each photon is described by the 8-element state that contains its four-position and four-momentum.

For the initial data, we need to specify five parameters: d, the distance from the black hole, i and j, the inclination and azimuthal angle of the camera, N and F are the number of pixels and the field-of-view (we assume a square camera). From these variables, we define a Cartesian coordinate system $(\alpha, \beta)$ onto the camera.

For a given $(\alpha, \beta)$, the initial four-position will be

Evolution

In this first implementation, we solve the ODEs with a forward Euler's method, which is the simplest way to solve an ODE. With this method, we move from the $i$-th step $x^\mu(\lambda_i), k^\mu(\lambda_i)$ at $\lambda = \lambda_i$ to the following at $\lambda_{i+1} = \lambda + \Delta \lambda$ with

$$ x^\mu(\lambda_{i + 1}) = x^\mu(\lambda_{i}) + \Delta \lambda k^\mu(\lambda_{i}), $$

$$ k^\mu(\lambda_{i + 1}) = k^\mu(\lambda_{i}) -\Delta \lambda \Gamma^\mu_{\alpha\beta}(\lambda_{i}) k^\alpha(\lambda_{i}) k^\beta(\lambda_{i}). $$

In this, we defined $ \Gamma^\mu_{\alpha\beta}(\lambda_{i}) = \Gamma^\mu_{\alpha\beta}(x^\mu(\lambda_{i})) $.

Stopping criterion

For each photon, we set two termination conditions: either the photon gets very close to the horizon, or it goes too far. So, we define two parameters that control whether to continue the integration or not depending on the spatial position of the photon with respect to the horizon.