Spacelike immersions in certain Lorentzian manifolds with lightlike foliations

TL;DR

This paper investigates spacelike submanifolds within generalized Schwarzschild spacetimes, providing explicit formulas for their mean curvature and characterizations, especially focusing on those lying in lightlike foliations related to black hole models.

Contribution

It offers new explicit formulas and characterizations for spacelike submanifolds in generalized Schwarzschild spacetimes with lightlike foliations, including special cases like Schwarzschild and Reissner-Nordström.

Findings

Derived explicit mean curvature formulas for submanifolds

Characterized slices within lightlike foliations

Analyzed the case where the warping function is the radial coordinate

Abstract

The generalized Schwarzschild spacetimes are introduced as warped manifolds where the base is an open subset of $\mathbb{R}^2$ equipped with a Lorentzian metric and the fiber is a Riemannian manifold. This family includes physically relevant spacetimes closely related to models of black holes. The generalized Schwarzschild spacetimes are endowed with involutive distributions which provide foliations by lightlike hypersurfaces. In this paper, we study spacelike submanifolds immersed in the generalized Schwarzschild spacetimes, mainly, under the assumption that such submanifolds lie in a leaf of the above foliations. In this scenario, we provide an explicit formula for the mean curvature vector field and establish relationships between the extrinsic and intrinsic geometry of the submanifolds. We have derived several characterizations of the slices, and we delve into the specific case where the warping function is the radial coordinate in detail. This subfamily includes the Schwarzschild and Reissner-Nordstr\"om spacetimes.