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Yes. The reason is the following. For any diagonal, 4-D spacetime, such as what you are considering, the de Sitter-Schwarzschild metric, the Riemann tensor has 20 independent components. To do General Relativity, one must obtain the Ricci tensor which is obtained from contracting the Riemann tensor to obtain 10 independent components: $R_{ab} - ...


The reason is because you need a hypersurface similar to a Cauchy surface in order to be able to define thermodynamic quantities of the black hole like temperature, mass etc. For example - if you want to calculate the mass of a black hole theoretically, you would first have to construct a global hypersurface at the asymptotic limit of spacetime and then ...

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