Deriving the Schwarzschild solution

The Schwarzschild solution describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object. Of the solutions to the Einstein field equations, it is considered by some to be one of the simplest and most useful. The Schwarzschild solution describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object. Of the solutions to the Einstein field equations, it is considered by some to be one of the simplest and most useful. Working in a coordinate chart with coordinates ( r , θ , ϕ , t ) {displaystyle left(r, heta ,phi ,t ight)} labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is a smooth function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum. For the purposes of this article, these assumptions may be stated as follows (see the relevant links for precise definitions): The first simplification to be made is to diagonalise the metric. Under the coordinate transformation, ( r , θ , ϕ , t ) → ( r , θ , ϕ , − t ) {displaystyle (r, heta ,phi ,t) ightarrow (r, heta ,phi ,-t)} , all metric components should remain the same. The metric components g μ 4 {displaystyle g_{mu 4}} ( μ ≠ 4 {displaystyle mu eq 4} ) change under this transformation as: But, as we expect g μ 4 ′ = g μ 4 {displaystyle g'_{mu 4}=g_{mu 4}} (metric components remain the same), this means that: Similarly, the coordinate transformations ( r , θ , ϕ , t ) → ( r , θ , − ϕ , t ) {displaystyle (r, heta ,phi ,t) ightarrow (r, heta ,-phi ,t)} and ( r , θ , ϕ , t ) → ( r , − θ , ϕ , t ) {displaystyle (r, heta ,phi ,t) ightarrow (r,- heta ,phi ,t)} respectively give: