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All of the physics books that I've seen which discuss General Relativity do so in terms of coordinates - the tensor calculus - even though the naturally relevant entities are invariant under general coordinate change. Modern Differential geometry dispenses with coordinates.

Is there a book out there that discusses GR wholly in terms of differential geometry?

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  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/15002/2451 $\endgroup$ – Qmechanic Feb 13 '13 at 9:13
  • $\begingroup$ I'd say this is adequately covered in the duplicate link. Of the references in there, Sachs and Wu takes a very coordinate free approach. $\endgroup$ – twistor59 Feb 13 '13 at 9:43
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    $\begingroup$ I strongly suggest heeding at least some of Ron's answer there. Physicist's notation is not as coordinate centric as thought by mathematicians who've never used it. Used properly, coordinate invariance is manifest in all your Einstein-notated equations. Every GR physicist fully understands differential geometry, but we choose a notation that actually enables physical problems to be solved, and all physical problems must be cast into a particular basis in the end anyway. $\endgroup$ – user10851 Feb 13 '13 at 16:41