Parabolic projectile motion in the Paradigm of General Relativity

Question

Apologies if this may sound like a basic question or if somehow if has been asked in another form in this forum, but I could not find a similar question in SE.

In the formalism of general relativity, my understanding is that the curvature of spacetime is governed by the content of stress-energy tensor, and this is formalised by the Einstein Field Equations. Under some conditions (small fields), it is possible to linearize Einstein's equations into a form that very much looks like Newton's laws. My question is the following:

I struggle to "connect the dots" between, on one hand, the formalism of Einstein's Field Equations, and , on another hand, very simple trajectories such as, say, a parabolic projectile motion. In textbooks, major examples we often find, are the deviation of light trajectory by an astronomical-scale mass, such as a star, and such examples are "easy" to follow (conceptually, not mathematically): the light follows the geodesics, which are defined by the spacetime curvature, which itself depends on the stellar mass/energy (encoded in the stress-energy tensor). My difficulty in trying to understand the case of the parabolic projectile motion, is that the trajectory "crosses" the ground (let's say I throw a stone from $y=0$ and it arrives at $y=0$ for simplicity) but I can't mathematically picture what kind of geodesics is that, or how to see it through a "general relativity" perspective.

My background: I studied General Relativity several years ago, as undergraduate, the course was very mathematical and although I passed it very well, sadly I felt like I substantially missed the point from a physics point of view, so I definitely acknowledge gaps in my current understanding. I am re-reading my old material for intellectual curiosity and I am dismayed that I didn't ask myself such questions many years ago. I am happy with "technical" answer, or suggestions of specific reading, etc.

Answer 1

Preface: if you’re interested in learning about basically anything relating to GR and already have an decent understanding of differential geometry, Gravitation is an excellent textbook I would recommend (maybe you’ve already read through it), and it has great explanations of Newtonian approximations.

---

The geodesic equation, in a coordinate/holonomic basis (most coordinate bases you’ll deal with), is

$$\frac{\text{d}^2X^\alpha}{\text{d}\tau^2}=-\Gamma^\alpha_{\mu\nu}\frac{\text{d}X^\mu}{\text{d}\tau}\frac{\text{d}X^\nu}{\text{d}\tau},$$

for position 4-vector $X$, which relates the 4-acceleration (LHS) to the connection coefficients $\Gamma_{\mu\nu\lambda}$ and 4-velocities (RHS). Remember that Einstein’s convention means $\mu$ and $\nu$ are summed over. The connection coefficients are spacetime-dependent,

$$\Gamma_{\mu\nu\lambda}=\frac{1}{2}(g_{\mu\nu,\lambda}+g_{\mu\lambda,\nu}-g_{\nu\lambda,\mu}),\qquad\Gamma^\alpha_{\mu\nu}=g^{\alpha\lambda}\Gamma_{\lambda\mu\nu}$$

in a holonomic basis, where $g$ is the metric. In Schwarzschild spacetime, the Christoffel symbols take different forms; in the Newtonian approximation, they pretty much reduce to Newton’s gravitation in spherical coordinates.

You can approximate the geodesic under Newtonian-ish gravity by launching a particle with known initial position/velocity and integrating

$$\frac{\text{d}X}{\text{d}t}\approx-\nabla U$$

for the gravitational potential $U$ as normal. Trying to visualize this in Newtonian gravity is something I still haven’t really found out an excellent way of doing. In Schwarszchild spacetime, though, where gravity is very strong, one way to visualize it is as the derivative of the radial component of the metric:

Black dotted line: radial component of metric; red line: its derivative w.r.t. radius, which is a component of gravitational acceleration; black area: region within the event horizon.

From a GR perspective, you can see now that the connection coefficient $\Gamma^r_{rr}$ has the rough shape of $1/r^2$, like gravity; this means that there’s at least one component of gravity that behaves like that. Now, when you’re integrating the geodesic equation above, you’ll be integrating this force (among others) that will apply to test particles following the curve along which you are integrating.

In classical mechanics, you’re much farther away from the event horizon, which is where the forces deviate from Newtonian gravity; in that case, the $1/r^2$ approximation just gets more and more accurate. On Earth, spacetime curvature is there, but it’s so weak that gravity is only a mere 9.8 meters per second squared; out here, very far away from the Earth’s hypothetical event horizon, that curvature is well-approximated by Newtonian gravity.

Hint: use approximations when dealing with Newtonian-scale weak gravity. It’s a lot of extra work to deal with the other tiny pieces of relativity that are still there but negligible, and it doesn’t usually add much.