the Hilbert stress-energy, and remembering that A+B=1 removes the ambiguity in A, B, and C. and . This gives . The field equation can be written where . This is the Einstein field equation that describes curvature of spacetime that results from stress-energy density. This equation, along with the geodesic equation have been motivated by the kinetics and dynamics of a particle orbiting the earth in a circular orbit. They are true in general. Solving the Einstein field equation Solving the Einstein field equation requires an iterative process. The solution is represented in the metric tensor . Typically there is an initial guess for the tensor. The guess is used to calculate Christoffel symbols, which are used to calculate the curvature. If the Einstein field equation is not satisfied, the process is repeated. Solutions occur in two forms, vacuum solutions and non-vacuum solutions. A vacuum solution is one in which the stress–energy tensor is zero. The relevant vacuum solution for circular orbits is the Schwarzschild metric. There are also a number of exact solutions that are non-vacuum solutions, solutions in which the stress tensor is non-zero. Solving the geodesic equation Solving the geodesic equations requires knowledge of the metric tensor obtained through the solution of the Einstein field equation. Either the Christoffel symbols or the curvature are calculated from the metric tensor. The geodesic equation is then integrated with the appropriate boundary conditions. Electrodynamics in curved spacetime Maxwell's equations, the equations of electrodynamics, in curved spacetime are a generalization of Maxwell's equations in flat spacetime (see Formulation of Maxwell's equations in special relativity). Curvature of spacetime affects electrodynamics. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. The sourced and source-free equations become (cgs units): , and where is the 4-current, is the field strength tensor, is the Levi-Civita symbol, and is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations. The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and the Ampère's law with Maxwell's correction. The second equation is an expression of the homogeneous equations, Faraday's law of induction and Gauss's law for magnetism. The electromagnetic wave equation is modified from the equation in flat spacetime in two ways, the derivative is replaced with the covariant derivative and