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How would one derive the Schwarzschild metric using the full machinery of differential geometry, using the component approach as little as possible?

Something along these lines: Begin with a manifold $\mathscr{M}^4$ on which a metric $ds^2$ of Lorentz signature is defined. Assume $\mathscr{M}^4$ to be spherically symmetric in the sense that to any $3\times 3$ rotation matrix $A$ there corresponds a mapping (rotation) of $\mathscr{M}^4$, also called $A$ ($A: \mathscr{M}^4\to \mathscr{M}^4$: $\mathscr{P} \to A\mathscr{P}$, for all points $\mathscr{P}$), that preserves the lengths of all curve. Using the Lie derivative we find...

Natanael
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1 Answers1

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As has been pointed out in the comments, it's not entirely clear how you intend to specify a metric without the use of some set of coordinates. That said, a couple of common GR texts have non-standard approaches to the Schwarzchild metric that you might find interesting.

  • Misner, Thorne, and Wheeler's Gravitation has a fairly detailed sidebar (Box 23.3, "Rigorous Derivation of the Spherically Symmetric Line Element") which starts with the assumption of a manifold $M^4$ on which there exists a set of automorphisms that preserve length of curves, and which automorphisms (treated as a group) are isomorphic to $SO(3)$. They then show how these assumptions lead to a natural definition of the coordinates $t$ and $r$ in which the metric can be taken to be diagonal. Note that this is not specific to Schwarzschild, but could apply to any spherically symmetric situation (even dynamical ones rather than static ones.)

  • Wald's General Relativity uses a restricted version of the above argument (assuming staticity from the get-go) to show how the coordinates $t$ and $r$ are obtained from geometric considerations. He then uses the orthonormal tetrad formalism (instead of the more conventional coordinate-component method) to obtain the differential equations which the metric must satisfy.

Michael Seifert
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