Lorentz invariance and diffeomorphism invariance in modified gravity

研究领域：数学物理在过去的十年里，大量的引力修正理论被提出和研究。其中包括修改的远程平行重力模型。在一个适当的限制，这些理论成为等价于爱因斯坦的广义相对论（GR），在这种情况下，人们谈到广义相对论（TEGR）的远程平行等价物。虽然GR是，通过建设，在局部洛伦兹变换下不变，TEGR只是准不变的意义上说，它的行动是局部洛伦兹不变的边界项。这并不影响该理论的场方程，它与GR的场方程是等价的。本项目旨在深入理解局部洛伦兹变换和局部坐标变换在修正引力模型中的作用。在前6-12个月，学生将学习各种修正的引力理论。这些理论将包含在坐标变换下不是局部洛仑兹不变或不变的模型。在第二年，学生将推导出其中一些理论和以前没有研究过的模型的运动方程。一个重要的里程碑将是显示这些方程减少到以前研究的模型在一定的限制。下一步，这些方程将在球对称的背景下，在宇宙学中进行研究。学生将致力于构造这些方程的显式解，并研究它们的性质。通过研究，学生可以研究以下有趣的问题：第一，去除初始奇点的正则宇宙学模型的存在性。第二，正则黑洞解的存在性。

Research Area: Mathematical PhysicsA large number of modified theories of gravity has been proposed and studied in the last decade. Amongst those are modified teleparallel gravity models. In a suitable limit these theories become equivalent to Einstein's theory of General Relativity (GR) in which case one speaks of the teleparallel equivalent of general relativity (TEGR). While GR is, by construction, invariant under local Lorentz transformations, TEGR is only quasi-invariant in the sense that its action is locally Lorentz invariant only up to a boundary term. This does not affect the field equations of the theory which are equivalent to those of GR.This project aims to develop an in depth understanding of the role of local Lorentz transformations and local coordinate transformations in the context of modified gravity models.In the first 6-12 months the student will study various modified theories of gravity. These will contain models which are not locally Lorentz invariant or not invariant under coordinate transformations.In year 2 the student will derive the equations of motion for some of these theories and for models not previously studied. An important milestone will be to show that these equations reduce to previously studied models in a certain limit.Next, these equations will be studied in the context of spherical symmetry and in cosmology. The student will work on constructing some explicit solutions to these equations and their properties will be investigated.This investigation leads to the following interesting questions the student can study: First, the existence of regular cosmological models where the initial singularity is removed. Second, the existence of regular black hole solution.