Solving for the Equations of Motion in the Schwarzschild Metric

This article outlines a straightforward approach to solving the equations of motion needed to simulate the trajectories of light and massive particles around a non-rotating Schwarzschild black hole.

We begin with the Schwarzschild metric in spherical coordinates:

Here, ds represents the infinitesimal spacetime interval between two nearby points in Schwarzschild spacetime, and r is the distance from the particle or photon to the center of the black hole. Additionally, the Schwarzschild radius Rs is defined as:

Where G is the universal gravitational constant, M is the mass of the black hole, and c is the speed of light.

Since G, M, and c are constants — and our goal is to solve for the dynamic variables in the equations of motion — we can simplify the system by setting G=M=c=1. This normalization also sets the Schwarzschild radius Rs​ to 1. We can reintroduce the physical constants later if needed for dimensional accuracy:

Another simplification arises from the spherical symmetry of the Schwarzschild metric: any rotation around theta yields an equivalent result. Therefore, we can constrain the motion to a single plane without loss of generality. By choosing this plane to be the equatorial plane, we set theta to π / 2​, which makes sin⁡²(θ) = 1, simplifying the metric and eliminating any dependence on theta:

At this point, we must determine the type of particle whose motion we’re solving: massive particles or massless photons. For photons, which follow null geodesics, the spacetime interval satisfies ds² = 0. For particles with mass, which follow time-like geodesics, ds² < 0, and is often normalized to ds² = −1 . We’ll begin by solving the equations of motion for a photon:

To simplify our expressions, we adopt Newton’s notation for differentiation: a variable with an overdot denotes differentiation with respect to an affine parameter λ. This parameter serves as a generalization of proper time for massless particles such as photons:

And for clarity we will also substitute in a function, f(r):

Where

Since the metric does not depend explicitly on t or ϕ, they each correspond to a conserved quantity. We will find E, the energy conserved quantity, and L, the angular momentum conserved quantity.

To do so, we will treat the metric as a Lagrangian, setting ds² to 𝐿

Then, because energy is conserved with the energy per unit mass, we will take the partial derivative of 𝐿 with respect to the derivative of t:

Which gives us the energy conservation quantity. Then, we will take the partial derivative of 𝐿 with respect to the derivative of phi to solve for the angular momentum conserved quantity:

We now have the two conserved quantities E, and L of the metric.

Let’s now rearrange and solve for the respective derivatives

We now have our first two equations. Let’s solve for the derivative of r by plugging these back into the simplified metric equation:

And now let’s rearrange to solve for the derivative of r:

We now have the three equations of motion:

In order to model the motion of particles or photons, we need the second derivative function of r, which will act as the acceleration function. Taking this second derivative gives:

We’ll call this p and therefore our equations of motion become:

And just like that, we now have the equations necessary to model the motion of a particle or photon near a Schwarzschild black hole.