### The Schwarzschild Solution ↑

As might be expected, the calculation of the Schwarzschild metric is a little brutal. This included in accordance with mathematical and scientific rigour. If one wants to properly grasp the meaning of the equations of general relativity, one should follow through a calculation, even if one does not want to check every minus sign oneself!In spherical coordinates, (τ,

*r*, θ, φ), with an origin at the centre of an isolated spherically symmetric, non-rotating, gravitating body, such as a planet or a star, ignoring the gravity of other stellar objects, the metric is

*k*=

*e*

^{λ}, which will simplify calculation,

*r = x*

^{1}, the non-vanishing partial derivatives of the metric are

, | |

, |

The Ricci tensor is

*R*

_{22},

Einstein’s field equation says this must vanish. So,

*C*is a constant of integration. Thus,

*C*= −2

*GM*. Thus the metric outside of an isolated, spherically symmetric, non-rotating, gravitating body of mass

*M*is

*Schwarzschild metric*.

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