Does a Thrown Ball Reveal the Curvature of Spacetime?
The question sounds innocent: you throw a ball, it traces a neat parabola, and you wonder whether you have just watched Einstein at work. The difficulty is that “curvature” is doing two very different jobs in this story. One sense concerns the visible curvature of a spatial trajectory drawn by an object as time passes. The other concerns the invariant curvature of four-dimensional spacetime itself. These are related, but they are not the same thing. The confusion arises when they are either collapsed into one another, or artificially torn apart. The clean way forward is to contrast three observer stances and be explicit about what each can and cannot infer.
1. The ground-based observer: “The ball falls.”
You stand on the Earth and watch the ball arc downward. In your coordinates, the path through space is approximately parabolic. This is not in dispute. The question is what that parabola represents.
In general relativistic terms, you are not an inertial observer. The Earth’s surface is prevented from free fall by internal stresses in matter; your accelerometer reads roughly 1g. Your frame is therefore non-inertial. When you describe the motion of freely falling objects from such a frame, inertial effects appear that look exactly like a gravitational force.
From this standpoint, the ball’s parabolic trajectory is the spatial trace of a geodesic described in a non-inertial coordinate system. This is a perfectly legitimate description, but it does not by itself settle the question of spacetime curvature. Curved spatial paths can arise either because spacetime is curved or because the observer’s frame is accelerating. From the ground alone, those possibilities are not disentangled.
2. The local free-fall observer: “The ball is (almost) straight.”
Now imagine you are launched alongside the ball, sharing its free-fall motion. In your immediate neighbourhood you are very close to an inertial frame. Over short distances and times, the ball does not exhibit any strong downward acceleration relative to you. Its motion is close to uniform; its worldline appears nearly straight in your local coordinates.
This is the equivalence principle in its proper domain: local, approximate, and powerful. It guarantees that along any freely falling worldline one can choose coordinates in which gravitational effects largely disappear. This is technically represented by the ability to set the connection coefficients (the Christoffel symbols) to zero at a specific point.
But this does not mean spacetime is flat. What vanishes locally are the connection coefficients, not the curvature. Curvature reveals itself only when you consider a region rather than a point or a single worldline. If you watch another freely falling object nearby, you will eventually observe relative acceleration between the two. Those tidal effects cannot be transformed away. They are the signature of the Riemann curvature tensor.
3. The deep-space inertial observer: “The ball follows a geodesic of curved spacetime.”
Now consider an inertial observer far from the Earth, equipped with a telescope, floating freely and not accelerating. This observer is the “smoking gun” witness. Because they know they themselves are inertial, any deviation they see in the ball’s path cannot be attributed to their own coordinate acceleration.
This observer does not need to appeal to accelerating coordinate systems to explain what they see. They observe a ball moving in the gravitational field sourced by the Earth, and in general relativity that means the ball follows a geodesic of the Earth’s spacetime geometry. When this observer asks, “What curve does this geodesic trace in three-dimensional space as a function of my time coordinate?”—referring to the coordinate time t of a distant clock—the answer in the weak-field, low-velocity regime is: approximately a parabola.
That near-parabolic shape is not an illusion, nor is it a coordinate trick. It is precisely how timelike geodesics in the Earth’s weak gravitational field project into ordinary space when described using a reasonable global time coordinate. In this limited but perfectly legitimate sense, the parabolic trajectory is explained by spacetime curvature. The Earth’s mass curves spacetime; free particles follow geodesics; those geodesics, when viewed spatially, look parabolic to an excellent approximation.
Why the parabola is still not “the curvature”
Here is the crucial distinction that must be made sharply. Although the near-parabolic trajectory is a consequence of spacetime curvature, it is not itself a direct measure of that curvature. A single geodesic, by itself, does not encode the Riemann tensor. Different curved spacetimes can support geodesics whose spatial projections look very similar over short distances.
What curvature controls, invariantly and unavoidably, is the relative motion of nearby geodesics. If the deep-space observer watches two balls thrown with slightly different initial conditions, they will see their separation evolve in a way that cannot be mimicked in flat spacetime by any choice of coordinates. That relative acceleration—tidal deviation—is the coordinate-independent manifestation of spacetime curvature.
“Mostly timelike rather than spacelike curvature”
This phrase gestures at a genuine feature of the weak-field regime. Near Earth, the dominant deviation of the metric from flatness lies in the time-time component (g00). This encodes gravitational time dilation and reproduces the Newtonian potential in the appropriate limit. For slow-moving objects (where velocity v is much less than the speed of light c), this time-warping largely governs their motion.
Spatial curvature is also present, but its effects on slow projectiles are subleading. If the ball were thrown at relativistic speeds, or if we were observing the path of light, the spatial components of the curvature would become just as prominent as the temporal ones. It is one unified spacetime geometry, but the ball’s low velocity makes it primarily sensitive to the temporal "stretch" of the metric.
The answer, stated precisely
A thrown ball near Earth follows a timelike geodesic of the Earth’s weakly curved spacetime. A ground-based observer describes that geodesic as a parabola because they are in a non-inertial frame. A local free-fall observer sees the motion as nearly straight because curvature is negligible over small regions. A deep-space inertial observer sees neither illusion nor fiction: they see a genuine geodesic of a curved spacetime whose spatial projection is approximately parabolic.
Addendum: Why do different observers see the same shape?
It may seem counter-intuitive that the earthbound observer (in a non-inertial frame) and the deep-space observer (in an inertial frame) both conclude the ball follows an approximate parabola. One might expect such different perspectives to yield different geometries. However, they converge because the Earth’s gravitational potential and the ground’s physical acceleration are numerically and geometrically coupled.
For the ground-based observer, the parabola is an inertial effect. Because the ground is constantly pushing you upward, you are in an accelerating frame. In such a frame, a free particle appears to accelerate in the opposite direction. Your description follows the Newtonian kinematic: z(t) = z0 + vz0t - ½gt2. This is a "fictitious" force result, but the mathematical plot is a literal parabola.
For the deep-space observer, the interpretation is inverted. They see the ball following a worldline that is as "straight" as the curved geometry allows (a geodesic). Because they are observing a slow-moving object in a weak field, the metric component governing time dilation (g00 ≈ 1 + 2Φ/c2) dominates the math. When they project this 4D worldline onto their 3D spatial grid, the resulting equation for the trajectory yields the exact same ½gt2 relationship.
Ultimately, they must converge because of the Equivalence Principle. If the "fictitious" parabola seen on the ground did not match the "geometric" parabola seen from space, an observer could distinguish between gravity and acceleration simply by throwing a ball. The fact that they see the same shape is not a coincidence; it is a requirement of the symmetry between acceleration and gravity that lies at the heart of General Relativity.
