for the acceleration can be generalized, yielding the geodesic equation where is the 4-position of the particle and is the curvature tensor given by where is the Kronecker delta function, and we have the constraints and . It is easily verified that circular orbits satisfy the geodesic equation. The geodesic equation is actually more general. Circular orbits are a particular solution of the equation. Solutions other than circular orbits are permissible and valid. Ricci curvature tensor and trace The Ricci curvature tensor is a special curvature tensor given by the contraction . The trace of the Ricci tensor, called the scalar curvature, is . The geodesic equation in a local coordinate system Consider the situation in which there are now two particles in nearby circular polar orbits of the earth at radius and speed . The particles execute simple harmonic motion about the earth and with respect to each other. They are at their maximum distance from each other as they cross the equator. Their trajectories intersect at the poles. Imagine we have a spacecraft co-moving with one of the particles. The ceiling of the craft, the direction, coincides with the direction. The front of the craft is in the direction, and the direction is to the left of the craft. The spacecraft is small compared with the size of the orbit so that the local frame is a local Lorentz frame. The 4-separation of the two particles is given by . In the local frame of the spacecraft the geodesic equation is given by where and is the curvature tensor in the local frame. Geodesic equation as a covariant derivative The equation of motion for a particle in flat spacetime and in the absence of forces is . If we require a particle to travel along a geodesic in curved spacetime, then the analogous expression in curved spacetime is where the derivative on the left is the covariant derivative, which is the generalization of the normal derivative to a derivative in curved spacetime. Here is a Christoffel symbol. The curvature is related to the Christoffel symbol by . Metric tensor in the local frame The interval in the local frame is where is the angle with the axis (longitude) and is the angle with the axis (latitude). This gives a metric of in the local frame. The inverse of the metric tensor is defined such that where the term