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-## Runge-Kutta 4 (RK4) Integration — Explained
-
-When we talk about geodesics in curved spacetime, we’re dealing with a system of differential equations that describe how a particle—or in our case, a ray of light—moves. These equations are usually too complicated to solve exactly, so we turn to numerical methods. One of the most popular choices is the **fourth-order Runge-Kutta method (RK4)**.
-
-Think of RK4 like taking careful steps along a winding mountain trail. At each step, instead of just looking straight ahead, RK4 takes a few “sneak peeks” along the way to estimate the path more accurately.
-
-### The Idea in Simple Terms
-
-Suppose you know where you are at a particular moment and you know the slope of your path (the derivative). A naive method like **Euler’s method** would take a single step using that slope and call it a day. But if the slope changes a lot along your step, Euler can easily go off-track.
-
-RK4 improves on this by taking **four evaluations** of the slope at carefully chosen points:
-
-1. **Start of the step** — check the slope right where you are (`k1`).
-2. **Halfway in, using the first slope** — imagine taking a mid-step to see if the slope changes (`k2`).
-3. **Halfway in, using the second slope** — another mid-step with a slightly better estimate (`k3`).
-4. **End of the step** — take a peek at the slope at the far end of your step (`k4`).
-
-Then RK4 combines all these slopes in a weighted average:
-
-$$
-y_{n+1} = y_n + \frac{h}{6} (k_1 + 2 k_2 + 2 k_3 + k_4)
-$$
-
-This weighted combination gives a very accurate estimate of where you should be at the next step.
-
-### Why RK4 Works Well for Geodesics
-
-In the context of geodesics:
-
-* Each ray has six “pieces of information”: position `(r, θ, φ)` and velocity `(dr/dλ, dθ/dλ, dφ/dλ)`.
-* The RK4 method allows us to update all six components **simultaneously**, while keeping the accumulated error small.
-* Because spacetime curvature can change dramatically near a black hole, RK4 is especially helpful: it’s stable enough to handle strong curvature without requiring tiny steps everywhere.
-
-### A Visual Analogy
-
-Imagine you’re rowing a boat down a twisting river:
-
-* **Euler**: You look at the current direction, row a fixed distance, and hope for the best. You’ll likely drift off course if the river bends sharply.
-* **RK4**: You peek ahead four times along your intended path and adjust your stroke accordingly. You stay much closer to the true river path, even around tight bends.
-
-### Accuracy
-
-* RK4 is called **fourth-order** because the error per step scales with $h^5$, and the total accumulated error scales roughly with $h^4$ (where $h$ is your step size).
-* This means you can take reasonably large steps without losing accuracy, which is critical when simulating millions of rays efficiently.
-
-### Connecting to Code
-
-In your code, each `k` evaluation corresponds to calculating how the ray’s position and velocity would change at different “guesses” along the step. The final weighted combination moves the ray forward accurately in spacetime.
-
-By using RK4, we’re essentially giving each ray a **very careful and informed nudge**, instead of blindly pushing it along, which is why the results are both stable and accurate—even near the extreme curvature of a black hole.
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