Possible fluid interpretation and tidal force equation on a generic null hypersurface in Einstein-Cartan theory

TL;DR

This paper derives the evolution and tidal force equations on a null hypersurface in Einstein-Cartan theory, proposing a fluid interpretation akin to a generalized Navier-Stokes fluid, extending gravitational dynamics understanding.

Contribution

It introduces a fluid-like interpretation of the Hajicek 1-form evolution in Einstein-Cartan theory and derives a tidal force equation on null hypersurfaces.

Findings

Evolution equation related to projected Einstein tensor on null surface

Fluid interpretation similar to Cosserat generalization of Navier-Stokes

Derived tidal force equation in Einstein-Cartan theory

Abstract

The dynamical evolution of the Hajicek $1$-form is derived in Einstein-Cartan (EC) theory. We find that like Einstein theory of gravity, the evolution equation is related to a projected part of the Einstein tensor $(\hat{G}_{ab})$ on a generic null surface $\mathcal{H}$, particularly $\hat{G}_{ab}l^a q^b_{~c}$, where $l^a$ and $q^a_{~c}$ are the outgoing null generators of $\mathcal{H}$ and the induced metric to a transverse spatial cross-section of $\mathcal{H}$ respectively. Under the {\it geodesic constraint} a possible fluid interpretation of this evolution equation is then proposed. We find that it has the structure which is reminiscent to the {\it Cosserat generalization} of the Navier-Stokes fluid provided we express the dynamical evolution equation of the Hajicek $1$-form in a set of coordinates adapted to $\mathcal{H}$ and in a local inertial frame. An analogous viewpoint can also be built under the motive that the usual material derivative for fluids should be replaced by the Lie derivative. Finally, the tidal force equation in EC theory on the null surface is also derived.