How spacetime describes a fall

So far, we have seen that freely falling objects can draw together while neither feels a supporting force. We described their motion as following the straightest available paths, called geodesics. To understand that description, we need to include time in the picture.

A path includes when, as well as where

Imagine taking a photograph of a stone once every second. Each photograph records its position at a particular time. Arrange those records on a diagram with height across the page and time running upwards.

A stone held at one height makes a vertical line: its position stays the same as the seconds pass. A released stone makes a line that bends towards lower heights as it falls. A stone thrown upwards first gains height, then returns.

Each line is a worldline, a record of an object’s position through time. Even the stone on the table has a worldline. Drawing one does not mean it is secretly moving through space.

What geometry adds

On an ordinary map, the scale tells us how to turn a line on the page into a distance. A spacetime description must also tell us how much time a clock records along a journey, and how signals and free objects can travel between events.

These relationships are part of its geometry. Near Earth, they differ from those in a region far from gravitating bodies. For example, clocks held at different heights do not run at exactly the same rate. We will measure that difference in the next lesson.

A grid drawn on a screen is only a way to label positions and times. Its appearance alone does not tell us which path is a geodesic. A line that looks straight on the page may require support to follow, while a curved-looking line may describe free fall.

Following the free path

Hold the stone at a fixed height, then release it. Before release, your hand supplies a force that keeps it there. After release, the stone follows the free path set by the surrounding geometry and the motion it had when you let go.

In a small freely falling room alongside it, the stone initially stays nearby. Relative to the supported ground, its downward speed increases. Both viewpoints describe the same sequence of events.

The straightest-path rule applies through spacetime. It does not require the stone to keep a constant height or trace a straight line on a ground-based drawing.

Why there is more than one free path

The geometry does not specify one trajectory that every object must join. Starting motion matters too. A stone released from rest falls downwards. One thrown upwards rises before falling. With suitable sideways speed, an object can orbit Earth.

Once released, each is in free fall, provided air resistance and other forces are negligible. The different paths come from different starting velocities in the same surrounding geometry.

What determines the geometry?

Einstein’s field equations relate matter and energy to spacetime geometry. For the examples here, Earth’s mass is the main source we include. Once the geometry is known, the free-path equations tell us how an object with a given starting position and velocity will move.

These are two parts of the explanation: determining the geometry, then determining a path through it. Near Earth, their predictions closely match Newton’s familiar description of gravitational attraction. They also predict effects that Newton’s theory does not, including differences between clocks held at different heights.

Calling a path a geodesic does not by itself calculate a fall. We need the geometry and the starting conditions. The visualisations use those ingredients to show why remaining at a fixed height requires support, while a released stone follows a different path.