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On mass and mean curvature

Consider physics in a 4-sphere universe not with mathematical precision but from the perspective of a sentient rational being attempting to make sense of a four (space) dimensional universe.  An extremely simple model of gravity (for which much more rigorous approaches are possible) is that we have the de Sitter spacetime for S4 universe:  $S^4(1/h)\times\mathbf{R}$ and we consider the Einstein-Hilbert action coupled with matter terms: $S_{EH}(\Sigma) = \int_{\Sigma} \sqrt{-g}(R + L_{matter})$ where the matter Lagrangian is defined for spinor sections of $S^4(1/h)$ in a sensible manner.  The variation calculus is technically very similar to the usual situation in general relativity and the Euler-Lagrange equations are the gravitational field equations:

$R_{ij} - Rg_{ij} + \Lambda g_{ij} = -\kappa T_{ij}$

where $T_{ij} = 1/\sqrt{-g} \delta L_{matter}/\delta g^{ij}$, and we are getting the $\Lambda$ term from the sectional curvature $h^2 of the ambient$latex S^4(1/h)\$.  While mathematically similar to the general relativity situation, the interpretation is quite different for what gravity means in this situation.  Gravity is the choice of a specific hypersurface of a fixed 4-sphere and is inherently three dimensional, while the spinor fields that concern the gauge forces are inherently four dimensional.

Note that the above analysis is static but does not produce any problems with an addition of space-independent absolute time.  Since the universe is compact, absolute time is justified.  One can also deform variable time smoothly to produce an absolute time, so an absolute time model is reasonable.

Suppose $\Sigma$ is the three-dimensional physical universe hypersurface that is picked out by solving the above gravitational variational problem.  We denote by $S\Sigma$ and $SS^4(1/h)$ to be the usual spinor bundles and there is a well-known identification $SS^4(1/h)$ with $S\Sigma \oplus S\Sigma$.  As a sanity check one can easily see that without matter terms we would recover the equator $S^3(1/h)$.  Suppose that the matter consists of a single pair of massive fermions represented by spinor fields on $\Gamma(S\Sigma)$ satisfying $D\phi = +/- m\phi$.  We can assume that the positive mass is greater than $h$ and then we can use the Lawn-Roth theorem to isometrically immerse $\Sigma$ into a 4-sphere of radius $1/h$ such that $m-h$ appears as mean curvature of the immersion.  In fact, in this case, because the spinor $\phi_+$ in $\Gamma(S\Sigma)$ is a positive eigenfunction and the ambient space has no harmonic spinors, we have a unique extension to $S^4(1/h)$ by a theorem of Hijazi et. al. telling us that the boundary value problem for the Atiyah-Patodi-Singer boundary conditions have unique solution.  This is a very useful theorem because it tells us that positive mass spinors in $\Sigma$ have unique extensions solving a Dirac equation to a hemisphere.