Lorentz invariance and diffeomorphism invariance in modified gravity

研究方向：数学物理近十年来，人们提出和研究了大量修正的引力理论。其中包括改进的遥平行重力模型。在一个适当的限制下，这些理论与爱因斯坦的广义相对论（GR）等效，在这种情况下，人们会说到广义相对论的远平行等效（TEGR）。通过构造，GR在局部洛伦兹变换下是不变的，而TEGR只是拟不变的，即它的作用仅在一个边界项之前是局部洛伦兹不变的。这并不影响理论的场方程，它与gr的场方程等效。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重力模型中的作用。在前6-12个月，学生将学习各种修正的引力理论。这些将包含局部洛伦兹不变量或坐标变换下不变量的模型。在第二年，学生将为这些理论和以前没有研究过的模型推导运动方程。一个重要的里程碑将是表明这些方程在一定限度内简化为先前研究的模型。接下来，我们将在球对称和宇宙学的背景下研究这些方程。学生将构建这些方程的显式解，并研究它们的性质。这项研究引出了学生可以研究的以下有趣问题：首先，移除初始奇点的规则宇宙学模型的存在性。第二，黑洞正则解的存在性。

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.