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
Put k = e^−λ, which will simplify calculation,
Using
Using prime to denote differentiation with respect to r = x¹, the non-vanishing partial derivatives of the metric are
The non-vanishing Christoffel symbols,
are
On raising the first index,

The Ricci tensor is
If one calculates all the terms, one finds a diagonal matrix. However, there is only one free parameter in the metric, and it is sufficient to calculate only one component of the Ricci tensor,
Calculate the terms individually
Substitute into R₂₂,


Einstein’s field equation says this must vanish. So,
where C is a constant of integration. Thus,
In the weak field limit, Newton’s law of gravity states,
So, the constant of integration is C = −2GM. Thus the metric outside of an isolated, spherically symmetric, non-rotating, gravitating body of mass M is
This is the Schwarzschild metric.

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