Surface tension with normal curvature in curved space-time

With an aim to include the contribution of surface tension in the action of the boundary, we define the tangential pressure in terms of surface tension and Normal curvature in a more naturally geometric way. For a thin shell approximation of a static spherically symmetric surface and for weak and slowly varying fields, the negative tangential pressure $\tau _{\alpha \beta }$ is chosen to be analogous to $S_{\alpha \beta },$ where $S_{\alpha \beta }$ is the classical surface tension. First, by a suitable choice of the enveloping surfaces, we show that the negative tangential pressure is independent of the four-velocity of a very thin hyper-surface. Second, using suitable definition of the normal curvature for such a surface layer, we relate the 3-pressure of a surface layer to the normal curvature and the surface tension. Third, using the fact that the tangential pressure on the surface layer is independent of the four-velocity and a central force interaction, we relate the surface tension $S_{\alpha \beta }$ to the energy of the surface layer. Four, we show that the delta like energy flows across the hypersurface will be zero for such a representation of intrinsic 3-pressure. Five, for the weak field approximation and for static spherically symmetric configuration, we deduce the classical Kelvin’s relation between surface tension, pressure difference and mean curvature from this sort of representation of negative tangential pressure $\tau _{\alpha \beta }$ in terms of surface tension $S_{\alpha \beta }$ and the normal curvature. Six, using the representation of tangential pressure in terms of surface tension and normal curvature, we write a modified action for the boundary having contributions both from surface tension and normal curvature of the surface layer. Also we propose a method to find the physical action assuming a reference background, where the background is not flat. (The $g_{\mu \nu }^{+}$ or just $g_{\mu \nu }$ has been chosen to represent the metric coefficent of the hypersurface of $V_{+}$ space which is time-like surface layer here. The $g_{\mu \nu }^{-}$ represents the metric coefficient of the space like hypersurface of $V_{-}$ space.)