In this work, we try to obtain a stable Lemaître-Tolman-Bondi (LTB) static universe, which is spherically symmetric and radially inhomogeneous. However, this is not an easy task, and fails in general relativity (GR) and various modified gravity theories, because the corresponding LTB static universes must reduce to the Friedmann-Robertson-Walker (FRW) static universes. We find a way out in a new type of modified gravity theory, in which the conservation of energy and momentum is broken. In this work, we have proposed a novel modification to the original Rastall gravity. In some sense, our Rastall-like gravity is essentially different from GR and the original Rastall gravity. In this Rastall-like gravity, LTB static solutions have been found. The stability of LTB static universe against both the homogeneous and the inhomogeneous scalar perturbations is also discussed in details. We show that a LTB static universe can be stable in this Rastall-like gravity.

在这项工作中，我们尝试获得一个稳定的Lemaître-Tolman-Bondi（LTB）静态宇宙，该球面对称且径向不均匀。但是，这并不是一件容易的事，并且在广义相对论（GR）和各种经过修改的重力理论中都失败了，因为相应的LTB静态宇宙必须简化为Friedmann-Robertson-Walker（FRW）静态宇宙。我们发现了一种新型的修正重力理论，其中打破了能量和动量守恒的方法。在这项工作中，我们提出了对原始Rastall重力的新颖修改。从某种意义上说，我们类似Rastall的重力与GR和原始Rastall重力有本质区别。在这种类似Rastall的重力中，已经发现了LTB静态解。还详细讨论了LTB静态宇宙对均质和非均质标量摄动的稳定性。我们证明了LTB静态宇宙在这种类似Rastall的引力中可以稳定。