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How does one start from the Einstein Hilbert action and show that in a small neighborhood the metric must be Lorentzian (equivalence principle)?

Qmechanic
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More Anonymous
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    You mean you want to show Normal coordinates always exist? https://en.wikipedia.org/wiki/Normal_coordinates – Guliano Jan 31 '23 at 04:23
  • @Guliano yes that would suffice – More Anonymous Jan 31 '23 at 04:25
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    I believe this is a general mathematical result applicable to any (pseudo-)Riemannian manifold, so the action has nothing to do with it. – Guliano Jan 31 '23 at 04:29
  • Ah i wasn't aware. Feel free to share an answer (explicitly stating the broadness) – More Anonymous Jan 31 '23 at 04:31
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    I think this reference only uses Riemannian manfiolds: http://staff.ustc.edu.cn/~wangzuoq/Courses/16S-RiemGeom/Notes/Lec14.pdf, but according to the wiki there should also be a proof for pseudo-Riemannian ones. – Guliano Jan 31 '23 at 04:35
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    Possible duplicate: https://physics.stackexchange.com/q/647687/2451 – Qmechanic Jan 31 '23 at 05:45
  • Note that not all gravity theories need to respect equivalence principle, like the Brans-Dicke action violates strong equivalence principle – KP99 Jan 31 '23 at 15:42

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