GNU Astronomy Utilities

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9.1.2 Extending distance concepts to 3D

The concepts of Distance on a 2D curved space are here extended to a 3D space that might be curved in a 4D space. We can start with the generic infinitesimal distance in a static 3D universe, but this time not in spherical coordinates instead of polar coordinates. $$\theta$$ is shown in Figure 9.2, but here we are 3D beings, positioned on $$O$$ (the center of the sphere) and the point $$O$$ is tangent to a 4D-sphere. In our 3D space, a generic infinitesimal displacement will have the distance: $$ds_s^2=dx^2+dy^2+dz^2=dr^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2).$$Like our 2D friend before, we now have to assume an abstract dimension which we cannot visualize. Let’s call the fourth dimension $$w$$, then the general change in coordinates in the full four dimensional space will be: $$ds_s^2=dr^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2)+dw^2.$$But we can only work on a 3D curved space, so following exactly the same steps and conventions as our 2D friend, we arrive at: $$ds_s^2={dr^2\over 1-Kr^2}+r^2(d\theta^2+\sin^2{\theta}d\phi^2).$$In a non-static universe (with a scale factor a(t), the distance can be written as: $$ds^2=c^2dt^2-a^2(t)[d\chi^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2)].$$