Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved four-dimensional (4-D) spacetime geometry around the star onto three-dimensional (3-D) space.

In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress–energy tensor (representing matter, for instance). In other words, a freely moving or falling particle always moves along a geodesic. They are used to study the geometry of the metric and appear, for example, in the geodesic equation. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. General Differential Geometry MathWorld Contributors Salamin MathWorld Contributors Rowland, Todd Christoffel Symbol The Christoffel symbols are tensor -like objects derived from a Riemannian metric. How to Define a Metric How to Define a Tensor Computing the Christoffel Symbols The Riemann Tensor, The Ricci Tensor, The Ricci Scalar, and The Einstein Tensor The Stress-Energy Tensor Einstein’s Field Equations 2 GR Calculations in Specific Bases Using Mathematica.nb.

Figure 13: The open and flat FLRW metrics represent Universes that expand forever.

In general relativity, a geodesic generalizes the notion of a 'straight line' to curved spacetime. An interresting 'method' that allows you to know the acceleration vector with respect to any coordinate system is just a matter of recognize some key formulas. When we know the metric g, we can calculate the Christoffel symbols.

Christoffel symbols for flat frw metric