the same as for a flat spacetime is that stretching or compressing an elastic material in a spherically symmetric manner (radially) will not change the angular distance between two points. Thus, the metric can be put in the form: with and as yet undetermined functions of . Note that if or is equal to zero at some point, the metric would be singular at that point. Calculating the Christoffel symbols Using the metric above, we find the Christoffel symbols, where the indices are . The sign denotes a total derivative of a function. Using the field equations to find A(r) and B(r) To determine and , the vacuum field equations are employed: Hence: where a comma is used to set off the index that is being used for the derivative. The Ricci curvature is diagonal in the given coordinates: where the prime means the r derivative of the functions. Only three of the field equations are nontrivial and upon simplification become: (the fourth equation is just times the second equation). Subtracting the first and third equations produces: where is a non-zero real constant. Substituting into the second equation and tidying up gives: which has general solution: for some non-zero real constant . Hence, the metric for a static, spherically symmetric vacuum solution is now of the form: Note that the spacetime represented by the above metric is asymptotically flat, i.e. as , the metric approaches that of the Minkowski metric and the spacetime manifold resembles that of Minkowski space. Using the weak-field approximation to find K and S The geodesics of the metric (obtained where is extremised) must, in some limit (e.g., toward infinite speed of light), agree with the solutions of Newtonian motion (e.g., obtained by Lagrange equations). (The metric must also limit to Minkowski space when the mass it represents vanishes.) (where is the kinetic energy and is the Potential Energy due to gravity) The constants and are fully determined by some variant of this approach; from the weak-field approximation one arrives at the result: where is the gravitational constant, is the mass of the gravitational source and is the speed of light. It is found that: and Hence: and So, the Schwarzschild metric may finally be written in the form: Note that: is the definition of the Schwarzschild radius for an object of mass , so the Schwarzschild metric may be rewritten in the alternative form: which