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.
