## Mass is mean curvature

April 29, 2014 by zulfahmed

An intuitive identification of mass would be to consider it as the possible non-negative eigenvalues of the Dirac operator in the physical universe which we assume is a hypersurface of S4(1/h). The analysis of Dirac operators for hypersurfaces performed by Hijazi et. al. HijaziDiracHypersurfaces show that the first nonnegative eigenvalue of the Dirac operator has the lower bound where is the mean curvature of the hypersurface . Thus geometrically speaking, it is reasonable to develop the rule of thumb that mass is mean curvature. This gives us an intuitive picture of what happens geometrically in a physical hypersurface of S4(1/h): matter represented by eigenspinors of the Dirac operator gets heavier with mean curvature of the hypersurface geometry. The Einsteinian picture of gravity in terms of a metric of the three-manifold is sharpened here: the mean curvature of the metric increases with heavier objects such as Dirac eigenspinors with high eigenvalues. This intuitive picture is marred by my current lack of understanding regarding why Dirac eigenspinors should have strong localization properties to serve as models for particles but if one accepts the identification then the relationship between hypersurface geometry and mass of a fermion (Dirac eigenspinor) is clearer.

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