Notes

Chapter 9: Fundamental Physics

Section 15: The Phenomenon of Gravity


Sphere volumes

In ordinary flat Euclidean space the area of a 2D circle is π r^2, and the volume of a 3D sphere 4/3 π r^3. In general, the volume of a sphere in d-dimensional Euclidean space is s[d] r^d where s[d] = π^(d/2)/(d/2)! (the surface area is d s[d] r^d-1). (The function s[d] has a maximum around d=5.26, then decreases rapidly with d.)

If instead of flat space one considers a space defined by the surface of a 3D sphere—say with radius a—one can ask about areas of circles in this space. Such circles are no longer flat, but instead are like caps on the sphere—with a circle of radius r containing all points that are geodesic (great circle) distance less than r from its center. Such a circle has area

2 Pi a^2 (1- Cos[r/a]) = π r^2 (1 - r^2/(12 a^2) + r^4/(360a^4) - …)

In the d-dimensional space corresponding to the surface of a (d+1)-dimensional sphere of radius a, the volume of a d-dimensional sphere of radius r is similarly given by

d s[d] a^d Integrate[Sin[θ]^(d-1), {θ,0,r/a}] = s[d] r^d (1 - d*((d - 1)/(6 (d+2)))*(r^2/a^2) + ((d*(5*d^2 - 12*d + 7))/(360*(d + 4)))*(r^4/a^4) + …)

where

Integrate[Sin[x]^(d-1),x] = -Cos[x] Hypergeometric2F1[1/2, (2 - d)/2, 3/2, Cos[x]^2]

In an arbitrary d-dimensional space the volume of a sphere can depend on position, but in general it is given by

s[d] r^d (1 - RicciScalar/(6(d+2)) r^2 + …)

where the Ricci scalar curvature is evaluated at the position of the sphere. (The space corresponding to a (d+1)-dimensional sphere has RicciScalar = d(d-1)/a^2.) The d=2 version of this formula was derived in 1848; the general case in 1917 and 1939. Various derivations can be given. One can start from the fact that the volume density in any space is given in terms of the metric by Sqrt[Det[g]]. But in normal coordinates the first non-trivial term in the expansion of the metric is proportional to the Riemann tensor, yet the symmetry of a spherical volume makes it inevitable that the Ricci scalar is the only combination of components that can appear at lowest order. To next order the result is

s[d] r^d (1 - RicciScalar/(6(d+2)) r^2 +(5 RicciScalar^2-3 RiemannNorm + 8 RicciNorm-18 Laplacian[RicciScalar])/(360 (d+2)(d+4))r^4 + …)

where the new quantities involved are

RicciNorm = Norm[RicciTensor,{g, g}] RiemannNorm = Norm[Riemann,{g,g,g,Inverse[g]}] Norm[t_, gl_] :=Tr[Flatten[t Dual[t, gl]]] Dual[t_,gl_]:= Fold[Transpose[#1.Inverse[#2], RotateLeft[Range[TensorRank[t]]]] &, t, Reverse[gl]] Laplacian[f_] := Inner[D, Sqrt[Det[g]]Inverse[g].Map[D[f, #] &, p], p]/Sqrt[Det[g]]

In general the series in r may not converge, but it is known that at least in most cases only flat space can give a result that shows no correction to the basic r^d form. It is also known that if the Ricci tensor is non-negative, then the volume never grows faster than r^d.


From Stephen Wolfram: A New Kind of Science [citation]