Read the last section of this page, on Cauchy Data and Predictions.
For the T4 universe, we found that if we had Cauchy data across the width of the universe at a single instant of time (where the time axis could be any of the four coordinates that wrapped around the 4-torus), we could determine the values of the finite number of coefficients of a free wave, and thus reconstruct its history for all time.
For S4, we can do the same thing with Cauchy data on any “great 3-sphere”, i.e. any 3-sphere of radius R. Assuming the geometry allows sourceless waves, all the spherical harmonics Yj, k, lm(ξ, ψ, θ, φ) that satisfy the sourceless wave equation will share the same value of l. We choose a coordinate system in which ξ=π/2 on the 3-sphere for which we have data. For those harmonics that reach a maximum or minimum at ξ=π/2, we can find their coefficients from the field’s value on the 3-sphere, while those harmonics that are zero there will have maxima or minima in their derivatives in the ξ direction, and we can find their coefficients from the field’s derivative. So from Cauchy data on the 3-sphere, we can reconstruct the entire history of the solution.
What if we have data on a smaller 3-sphere, which we could describe as a hypersurface ξ=ξ0 for some ξ0 < π/2? So long as we actually know the value of ξ0, the factors Ξkl(ξ) and ∂ξΞkl(ξ) will be known quantities on the 3-sphere (and they will never both be zero at once), so in principle we should always be able to compute all the coefficients of the solution.
This leads to the curious observation that in principle we could reconstruct the entire solution from Cauchy data on even the smallest 3-sphere. After all, such a 3-sphere is a boundary of two finite regions: its interior as normally construed, and also the rest of the S4 universe, just as the Arctic Circle is a boundary for the region around the north pole and also for the remainder of the Earth’s surface. But in practice, for ξ much less than π/2 the values of Ξkl(ξ) become extremely small compared to the values at π/2, and also the peaks of the other factors in the harmonics become increasingly close, to the point where extrapolating outwards from a small 3-sphere to the whole universe would demand a prohibitive degree of accuracy in the data.
