Dynamics of Extended Objects: the Einstein-Hilbert Drop
  The Einstein-Hilbert worldspace action is used to investigate the dynamics of extended object. In the Robertson-Walker worldspace, this is seen to introduce a pressureless density which could contribute to dark matter. Such pressureless energy density, present from the very beginning, should have enormous consequences on large scale structure formation in the early Universe. Generalizing the idea to complexified internal co-ordinates, it becomes possible to gauge the action with U(1) symmetry. A trivial solution of this theory is Einstein's general relativity and source free Maxwell theory. Generally, the equations of motion of the gauge fields are Maxwell equations with source terms that include these fields themselves. The internal co-ordinates, under vacuum domination with negative pressure, obey an EOM that is a hyperbolic wave equation of a charged scalar field that interacts with the gauge fields and gravity in a disperso-conductive medium ; under matter domination with positive pressure however, it is an elliptic potential equation. Since the hyperbolic to elliptic transition can be made by introducing imaginary time, this result supports the view that time is actually complex, becoming Minkowskian in vacuum and Euclidean in matter. A supersymmetric version of the action can also be immediately written down. 
U. Khanal