# Spherically symmetric solutions of the $\lambda$-R model

## TL;DR

This paper derives spherically symmetric solutions in the mbda-R model, revealing non-static, non-asymptotically flat geometries with a rich structure differing from standard Schwarzschild solutions in general relativity.

## Contribution

It provides the first detailed phase space analysis and explicit solutions of the mbda-R model under spherical symmetry, highlighting key differences from general relativity.

## Key findings

- Solutions include Schwarzschild geometry at mbda=1
- Most solutions are non-static and not asymptotically flat
- The Ricci scalar is generally time-dependent and non-zero

## Abstract

We derive spherically symmetric solutions of the classical \lambda-R model, a minimal, anisotropic modification of general relativity with a preferred foliation and two local degrees of freedom. Starting from a 3 + 1 decomposition of the four-metric in a general spherically symmetric ansatz, we perform a phase space analysis of the reduced model. We show that its constraint algebra is consistent with that of the full \lambda-R model, and also yields a constant mean curvature or maximal slicing condition as a tertiary constraint. Although the solutions contain the standard Schwarzschild geometry for the general relativistic value \lambda = 1 or for vanishing mean extrinsic curvature K, they are in general non-static, incompatible with asymptotic flatness and parametrized not only by a conserved mass. We show by explicit computation that the four-dimensional Ricci scalar of the solutions is in general time-dependent and nonvanishing.