I want to compute the components of the Riemann curvature tensor (for a case similar to the Schwarzschild solution) in 4 + 1 dimensions, but I want to use a higher-dimensional analogue of spherical coordinates. I first want to investigate a metric for the Euclidean case, i.e. "flat" space-time with no matter or energy present. How would I write this Euclidean metric using a higher-dimensional analogue of spherical coordinates?
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$\begingroup$ This link might help: en.wikipedia.org/wiki/N-sphere#Spherical_coordinates $\endgroup$– G. PailyJul 29, 2014 at 19:01
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$\begingroup$ Thanks. I had already found that page, but I've been looking for a confirmation of it. en.wikipedia.org/wiki/… also has some information, but, again, I've been looking for a non-Wikipedia source. $\endgroup$– HDE 226868Jul 29, 2014 at 19:04
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$\begingroup$ So, this?: mathworld.wolfram.com/Hypersphere.html :-D . If you look around equation 12 he gives essentially the same coordinates as the wiki article, he just numbers them differently. Confirmation is given by equation 16. Then you just need to construct $dx^{1}$, $dx^{2}$, etc. $\endgroup$– G. PailyJul 29, 2014 at 19:10
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$\begingroup$ Thanks again. This is exactly what I need. The initial discussion is also helpful. $\endgroup$– HDE 226868Jul 29, 2014 at 19:19
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$\begingroup$ Hi HDE 226868. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. $\endgroup$– Qmechanic ♦Mar 19, 2015 at 9:42
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Aren't you basically asking to the formula of higher dimensional vectors to compute yourself being in the sphere at the center with appropriate lines to all vector points?
I think I get what your asking for I asked the same questions when I was younger
Anyways here's a hyperlink to 6th dimensional operation. The operator is used in quantum physics, but clearly they are talking about bosons and fermions. The reason I am referencing this is because this goes in the quantum level from one to six dimensions as far as the x y and the z but they use all this jargon now referencing your sphere of influence that you were talking about you could always plug in the sphere formula into the equation
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2$\begingroup$ Minor comment to the post (v2): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/1310.6260 $\endgroup$– Qmechanic ♦Sep 10, 2015 at 20:33