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I often hear that spacetime corresponds to a hyperbolic geometry. The distance metric is not Euclidean, it is hyperbolic, and Lorentz transformations correspond to hyperbolic rotations. Hyperbolic geometry has a negative Gaussian curvature. At the same time, I also hear that Minkowski is flat and has vanishing Riemann tensor. How do I reconcile these two apparently contradictory statements about Minkowski curvature ?

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    $\begingroup$ It has a vanishing Riemann tensor. $\endgroup$ – InertialObserver Sep 28 '18 at 23:48
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    $\begingroup$ Minkowski does not have negative gaussian curvature. Where did you get this idea? $\endgroup$ – AccidentalFourierTransform Sep 28 '18 at 23:55
  • $\begingroup$ thanks for the answer. i made the "negative curvature" statement because distances in special relavity are hyperbolic distances, and lorentz transformations are hyperbolic rotations. thus, special relavity seems to correspond to hyperbolic geometry, which has negative constant curvature. i'm obviously confusing things, since that hyperbolic geometry apparently has nothing to do with the geometry of space-time (or of minkowski spacetime) itself ? $\endgroup$ – marjimbel Sep 30 '18 at 4:06
  • $\begingroup$ Hi marjimbel welcome to StackExchange and Physics SE in particular. I upvoted your question since it is a fair question that you made in good faith. I also did it as a welcome gift to compensate the actions of 2 "superior" members who down voted your very first question, for no particular reason. They clearly violated our code of conduct. In Italian people use to say - citing Dante's Divine Commedy - "Non ti curar di lor ma guarda e passa " (Let us not speak of them, but look, and pass) $\endgroup$ – magma Sep 30 '18 at 8:27
  • $\begingroup$ thank you @magma :) . i'm honestly confused about this. people make statements like "the geometry of Minkowski 4-space is hyperbolic" all the time. perhaps those statements are wrong (if so i'd like to understand why). you see them all the time even in other SE questions ( ie. math.stackexchange.com/questions/497504/… ) . i'm just trying to understand the two things $\endgroup$ – marjimbel Sep 30 '18 at 13:33
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I'm not sure where you got that impression, as it's simply not right.

I'm guessing you read a 3D hyperbola embedded in 4D Minkowski space has constant negative curvature. That's true, but it doesn't mean Minkowski space itself has curvature. Similarly, you can embed a 3D sphere in a 4D Euclidean space, but that doesn't mean the Euclidean space has curvature.

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  • $\begingroup$ thanks for the answer. i made the "negative curvature" statement because distances in special relavity are hyperbolic distances, and lorentz transformations are hyperbolic rotations. thus, special relavity seems to correspond to hyperbolic geometry, which has negative constant curvature. i'm obviously confusing things, since that hyperbolic geometry apparently has nothing to do with the geometry of space-time (or of minkowski spacetime) itself ? $\endgroup$ – marjimbel Sep 30 '18 at 4:06
  • $\begingroup$ @marjimbel That reasoning’s simply too vague. Just because the signature has a minus sign in it doesn’t make Minkowski space a hyperbola. For example, distances on your standard hyperbola are positive but distances in Minkowski space can be positive or negative. $\endgroup$ – knzhou Sep 30 '18 at 9:15
  • $\begingroup$ you would concede that the geometry of spacetime is not euclidean though? so if it's not euclidean, then what is it ? also, i think for hyperbolic geometry distances can be negative @knzhou link . i've often read that spacetime corresponds to a hyperbolic geometry link. i'm surprised at the reaction here. i'm just trying to reconcile the correspondence with hyperbolic geometry and minkowski being flat $\endgroup$ – marjimbel Sep 30 '18 at 13:42
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    $\begingroup$ @marjimbel I think we're going in circles, and the confusion will disappear if you just write down some math. Edit your question to say exactly what you think in equations, not words. $\endgroup$ – knzhou Sep 30 '18 at 16:24
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    $\begingroup$ @marjimbel The mistake you are making is analogous to thinking that the Euclidean metric is invariant under non-hyperbolic (i.e. normal) rotations which leave spheres fixed, so therefore Euclidean space must have spherical geometry and positive curvature. $\endgroup$ – knzhou Sep 30 '18 at 16:25

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