The Einstein picture of gravity in its standard formulation seeks critical points of the Einstein-Hilbert action over the space of metrics over a fixed background physical space. We can also consider the same action over the hypersurfaces of a fixed background (the important background being a scaled 4-sphere) which leads to the same equations for the Euler-Lagrange equations but the latter approach gives us a natural extrinsic geometic picture and the energy-momentum tensor is trace-free but otherwise similar to a second fundamental form. Indeed, in 2003 B. Morel provided an explicit identification between the energy-momentum tensor of a Dirac eigenspinor and the second fundamental form of an isometric immersion (in a warped product which seems to apply to the case of hypersurfaces of a sphere): MorelEnergyMomentumTensorSecondFundamentalForm. Lawn and Roth have shown that two Dirac eigenspinors satisfying Dirac equations on a spin three-manifold with eigenvalues for a constant where is a function can be used to embed the manifold into a 4-sphere of radius a function of . These mathematical results point to the intuitive identification of mass and mean curvature: one would like to say that the Einstein-Hilbert action for an ambient space including matter Lagrangians will minimize on hypersurfaces that deviate from ’empty space’ the equator in such a way as to pick up mean curvature from the massive particles.

A relation between mass and mean curvature obviously is not sensible unless there is an ambient manifold in which the physical universe is immersed, but assuming one accepts the empirical evidence marshalled to show that the actual universe has four macro space dimensions, is compact and is in fact a scaled 4-sphere, the issue of this identification has solid grounds.

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