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Any physical quantity $K(t,x,x')$ on a maximally symmetric spacetime only depends on the geodesic distance between the points $x$ and $x'$.

Why is this so?

N.B.:

This statement is different from the statement that

The geodesic distance on any spacetime is invariant under an arbitrary coordinate transformation of that spacetime.

nightmarish
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    This statement is incorrect, geodesic distance is invariant under every coordinate transform in every space-time. Perhaps you should check and cite the source for more context. – Void Dec 11 '17 at 20:51
  • see my edit to question! – nightmarish Dec 11 '17 at 21:00
  • Crossposted on MSE and MO. – AccidentalFourierTransform Dec 11 '17 at 22:39
  • @void: I don't think that's quite right; geodesic distance isn't invariant under diffeomorphisms, but only under isometries; I assume that isometries is what the OP is talking about. – Mozibur Ullah Dec 11 '17 at 23:56
  • @MoziburUllah So you are really telling me that the notion of geodesic distance is dependent on the coordinate system in which you compute it? – Void Dec 12 '17 at 10:11
  • @void: not at all; I'm saying that diffeomorphisms are transformations of the manifold that don't take into account geodesic distance, in other words they ignore the euclidean structure on a manifold; whereas isometries do - thats why they preserve distance; its only being more precise with language. – Mozibur Ullah Dec 12 '17 at 12:00
  • @MoziburUllah In relativity, geodesic distance is the extremal length of curves connecting the two points measured by the pseudo-Riemannian metric, not any Euclidean notion of distance. This length is diffeomorphism invariant. See wiki. – Void Dec 12 '17 at 12:11
  • @void: can you point out where the wiki makes this claim - I don't see it? If they do make this claim then they're wrong. A smooth manifold need not have any riemannian structure, and so no distances can be defined; yet it still has diffeomorphisms. perhaps you're conflating diffeomorphisms with isometries? An isometry is after all a diffeomorphism (the reverse does not hold). – Mozibur Ullah Dec 12 '17 at 12:20

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