As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be constructed in the neighbourhood of some point on a manifold if that manifold is locally flat, corresponding to the vanishing of the Riemann tensor at the point whose neighbourhood we are considering. Now, a sphere is not locally flat, but in general relativity, the equivalence principle states that within a sufficiently small neighbourhood of a given point special relativity should apply. In special relativity one considers inertial frames, defined as frames in which one can construct an orthonormal canonical basis, such that the metric tensor is given by ##\eta_{\mu\nu}=\text{diag}\left(-1,1,1,1\right)##, such that the line element is of the form ##ds^{2}=-(dx^{0})^{2}+ (dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2}##.(adsbygoogle = window.adsbygoogle || []).push({});

My question, given the above, is how it is possible to reconcile this when one considers applying these ideas to the surface of the Earth?

Maybe I have misunderstood things, regardless I'm feeling a bit confused. I realise that one can always approximate to Cartesian coordinates within a sufficiently small neighbourhood of a point, but shouldn't this be exact (rather than an approximation) in order for the equivalence principle to work?!

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# Non-Euclidean geometry and the equivalence principle

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