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My question refers to page 10 of this document. Specifically, when using spherical polar coordinates in cosmology, why does the author of this work choose the origin of the coordinate system to be at the north pole, as opposed to the centre of the sphere? Is this more useful to cosmologists, if so: why?

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  • $\begingroup$ Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$ – Qmechanic Apr 12 at 23:07
  • $\begingroup$ You can edit the question and do that if you so wish. $\endgroup$ – wrb98 Apr 13 at 21:59
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It is true, as the answer of benrg says, that the sphere in question does not necessarily have to be embedded in a space of higher dimension in which its centre would be a point in that higher-dimensional space. But one could in principle imagine that it was so embedded, so there does exist a coordinate system in which the sphere's centre in that higher-dimensional space can act as origin of coordinates. However, all the coordinate locations between that origin and the actual 3-sphere are not part of the 3-sphere one wants to discuss. So even after placing an origin there, we still have the issue of constructing suitable coordinates on the 3-sphere itself.

Let's drop one dimension to make the illustration. Now our job is to construct coordinates on the surface of a sphere. The angles $\theta,\phi$ of the spherical polar coordinate system could be used, but they also include an initial step in which an arbitrary place on the sphere's surface is taken to act as a pole for the definition of $\theta$, and an arbitrary 'zero' direction is required for the definition of $\phi$. So they do not avoid this issue of picking a place to call 'north pole'.

To learn this type of geometry it is helpful to get used to constructing coordinates directly on the surface, without reference to any other place---because, after all, those other places (inside or outside our sphere) do not have anything to do with the surface we want to discuss, and for all we know they may not even exist.

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  • $\begingroup$ I was looking back at this yesterday, and was wondering if you can come up with an analog for the surface of the hyperboloid $x^2 + y^2 - z^2 = R^2$? That is, a justified parameterisation with the reference point being on the surface as opposed to at the origin (which does not lie on the surface in question). If you can find one, I would very much appreciate it. $\endgroup$ – wrb98 Apr 23 at 1:22
  • $\begingroup$ Should be $-R^2$ there. $\endgroup$ – wrb98 Apr 23 at 1:29
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The sphere has no poles really. It's fully symmetric with no distinguished points. The author of these notes has chosen to call the arbitrary origin point (normally taken to be our location) the "north pole", possibly to clarify that it's a point on the surface of the sphere and not its center.

The origin can't be the center because there is no center: the 3-sphere is an intrinsically curved 3-dimensional space and isn't embedded in a higher-dimensional space where a center could be defined.

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