## 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. We can start with the generic infinitesimal distance in a static 3D universe, but this time 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 correspond to the following distance in spherical coordinates:

$$ds_s^2=dx^2+dy^2+dz^2=dr^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2).$$

Like the 2D creature before, we now have to assume an abstract dimension which we cannot visualize easily. 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:

$$ds^2=c^2dt^2-a^2(t)[d\chi^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2)].$$