on the right is the Kronecker delta. The transformation of the infinitesimal 4-volume is where g is the determinant of the metric tensor. The differential of the determinant of the metric tensor is . The relationship between the Christoffel symbols and the metric tensor is . Principle of least action in general relativity The principle of least action states that the world line between two events in spacetime is that world line that minimizes the action between the two events. In classical mechanics the principle of least action is used to derive Newton's laws of motion and is the basis for Lagrangian dynamics. In relativity it is expressed as between events 1 and 2 is a minimum. Here S is a scalar and is known as the Lagrangian density. The Lagrangian density is divided into two parts, the density for the orbiting particle and the density of the gravitational field generated by all other particles including those comprising the earth, . In curved spacetime, the "shortest" world line is that geodesic that minimizes the curvature along the geodesic. The action then is proportional to the curvature of the world line. Since S is a scalar, the scalar curvature is the appropriate measure of curvature. The action for the particle is therefore where is an unknown constant. This constant will be determined by requiring the theory to reduce to Newton's law of gravitation in the nonrelativistic limit. The Lagrangian density for the particle is therefore . The action for the particle and the earth is . We find the world line that lies on the surface of the sphere of radius r by varying the metric tensor. Minimization and neglect of terms that disappear on the boundaries, including terms second order in the derivative of g, yields where is the Hilbert stress–energy tensor of the field generated by the earth. The relationship, to within an unknown constant factor, between the stress-energy and the curvature is . Stress–energy tensor Newton's law of gravitation Newton's Law of Gravitation in non-relativistic mechanics states that the acceleration on an object of mass due to another object of mass is equal to where is the gravitational constant, is a vector from mass to mass and is the magnitude of that vector. The time t is scaled with the speed of light c . The acceleration is independent of . For definiteness. consider a particle of mass