# How to integrate a function across a 3d-hypersphere (brane) embedded in 4d-space?

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1. Oct 1, 2014

### muinon

1. The problem statement, all variables and given/known data

This is not homework but relevant to an interaction model that I am developing.

I assume an even distribution of sources on a spherical 3d-hypersurface, the sources having a potential as a function of pairwise 4-distance between any two sources. I wish to obtain the sum effect in terms of potential's gradient, which at any point of 3d-sphere's surface supposedly is perpendicular to the surface, parallel to the local radius of the hypersphere.

While integrating across a 2d spherical surface in a 3d space is not a problem, I should probably consider what kind of a metric there is. Is it at all legitimate to create a space where 3d-spherical surface encloses a bulk, across which I can calculate the strengths of individual pairwise interactions? Or does such space become impossible only when introducing a metric where at least one term has a negative sign (typically of a timelike kind - t^2 * c^2)

If we choose a metric that forms a hyperboloid of two sheets, for instance, is there a way to integrate across it? The size of the surface could (must?) be infinite but the integral at any single point of it might be finite (depending on the function, too!) If surface IS infinite, I suppose I'll have to assume a global sum of source "charges" that is evenly distributed on the surface rather than an infinite charge, don't I? E.g. that "sum charge" being something similar to the total mass of the universe if the interaction were to be gravity.

Please first ask relevant questions to obtain more details if the problem-setting appears too vague.

Thank you!

2. Relevant equations

3. The attempt at a solution

2. Oct 6, 2014