This article will use the Einstein summation convention. The theory of general relativity required the adaptation of existing theories of physical, electromagnetic, and quantum effects to account for non-Euclidean geometries. These physical theories modified by general relativity are described below. Classical mechanics and special relativity Classical mechanics and special relativity are lumped together here because special relativity is in many ways intermediate between general relativity and classical mechanics, and shares many attributes with classical mechanics. In the following discussion, the mathematics of general relativity is used heavily. Also, under the principle of minimal coupling, the physical equations of special relativity can be turned into their general relativity counterparts by replacing the Minkowski metric (ηab) with the relevant metric of spacetime (gab) and by replacing any partial derivatives with covariant derivatives. In the discussions that follow, the change of metrics is implied. Inertia Inertial motion is motion free of all forces. In Newtonian mechanics, the force F acting on a particle with mass m is given by Newton's second law, , where the acceleration is given by the second derivative of position r with respect to time t . Zero force means that inertial motion is just motion with zero acceleration: The idea is the same in special relativity. Using Cartesian coordinates, inertial motion is described mathematically as: where is the position coordinate and τ is proper time. (In Newtonian mechanics, τ ≡ t, the coordinate time). In both Newtonian mechanics and special relativity, space and then spacetime are assumed to be flat, and we can construct a global Cartesian coordinate system. In general relativity, these restrictions on the shape of spacetime and on the coordinate system to be used are lost. Therefore, a different definition of inertial motion is required. In relativity, inertial motion occurs along timelike or null geodesics as parameterized by proper time. This is expressed mathematically by the geodesic equation: where is a Christoffel symbol. Since general relativity describes four-dimensional spacetime, this represents four equations, with each one describing the second derivative of a coordinate with respect to proper time. In the case of flat space in Cartesian coordinates, we have , so this equation reduces to the special relativity form. Gravitation For gravitation, the relationship between Newton's theory of gravity and general relativity is governed by the correspondence principle: General relativity must produce the same results as gravity does for the cases where Newtonian physics has