Which is the metric of Euclidean 3-space in spherical coordinates. Now additionally to this being a hyperbolic equation, the hypersurfaces $d\Sigma^2$ can have curvature, especially constant positive (closed sphere), zero (Euclidean), or constant negative (open hyperboloid) curvature.Ī flat universe is, for example, given by Where $d\Sigma^2$ describes the spatial hypersurface. If you take these assumptions and allow time dependence of spatial coordinates on time, you end up with the FLRW metric: Galaxy clusters are such freely falling observers. Their world lines are perpendicular to the spatial hypersurfaces and their proper time matches the synchronous time $\tau = t$. have fixed spatial coordinates $\mathbf x$. Freely falling observers are comoving, i.e. There you have a synchronous time $t$ and spatial hypersurfaces for given $t$ ("time slices"). This kind of hyperbolic is not the point of the question.Ĭosmology: If we assume large scale homogeneity and isotropy (Copernican principle), we can choose isochronous coordinates. You can use a Lorentz transform to switch to different coordinates $dt$ and $dx$ while the line element $ds$ is invariant, analogous to the hyperbola The answer to this question explains this for the Minkowski space, where a Lorentz transform is a hyperbolic rotation. Changing the frame of reference of an observer is done by a Lorentz transformation which preserves the light cone and is closely related to the concept of causality. Lorentz transformations: Lorentz covariance is the mathematical expression of Einstein's principle of the constancy of the speed of light. TL DR: Spacetime is expanding exponentially and if you choose an unconventional hypersurface, that hypersurface is hyperbolic. There are three kinds of different hyperbolic in this question and the previous answer by AtmosphericPrisonEscape: the behavior of Lorentz transformations, the spatial hypersurfaces in cosmology (which seem to be flat), and the hyperbolic appearance of spacetime when tracing world lines in a universe with accelerated expansion.
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