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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 $+/- 2/3(H+k)$ for a constant $k$ where $H$ is a function can be used to embed the manifold into a 4-sphere of radius a function of $k$.  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 $S^4(1/h)$ including matter Lagrangians will minimize on hypersurfaces that deviate from ’empty space’ the equator $S^3(1/h)$ in such a way as to pick up mean curvature from the massive particles.