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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...

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    $\begingroup$ To write down a metric is to give its components in some coordinates...so you have to tell us what the "component approach is." $\endgroup$ – Ryan Unger Jan 12 '16 at 17:17
  • $\begingroup$ Well one can state all the conditions and such in coordinate free language. And choose coordinates as late as possible maybe. $\endgroup$ – Natanael Jan 12 '16 at 17:29
  • $\begingroup$ Wikipedia, derivation of Schwarzschild solution. It gives an intuitive and easy to follow explanation. What you ask for is impossible as to solve the field equations, to pick out a coordinate at the end of the day. $\endgroup$ – Horus Jan 14 '16 at 12:43
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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.

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  • $\begingroup$ Yes! It was exactly something along those lines I was looking for. Thank you. $\endgroup$ – Natanael Jan 12 '16 at 18:21
  • $\begingroup$ Concerning spherical symmetry, see also Birkhoff's theorem, cf. e.g. this Phys.SE post. $\endgroup$ – Qmechanic Jan 12 '16 at 18:38

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