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The general setting is that we have a 1-form $A$ representing electromagnetic potential in a scaled S4 and we would like to understand how gravity is described in a hypersurface using a Lagrangian density.  The standard or established approach is based on the Einstein-Hilbert action on the entire spacetime manifold but we would like to consider the situation where we define a functional in the space of possible submanifolds that substitutes for the Einstein-Hilbert action:
$L = \int_X \sqrt{-g}( R + \kappa L_M ) where$latex X\$ is a fixed three-dimensional space (we focus on space aspects and consider time separately) and $L_M$ is a matter lagrangian that can be constructed out of terms from $F=dA$.  We would like the understand the right Lagrangian where in the case $L_M=0$ then the right extremum is simply a minimal submanifold of S4 and even the great sphere (the ’empty universe’).  We want the Euler-Lagrange equations of such an action to be equivalent to the gravitational field equations in the form:
$R_{ij} - Rg_{ij} = -\kappa T^{ij}$