Landau-Lifshitz-case-study

In this repository, I do a case study of the Landau-Lifshitz or the densitized formalism in General Relativity and modified theories of gravity using the xAct module of Mathematica. The field equations of General Relativity are something people have struggled with to find solutions of. Only in certain symmetric and simple situations, exact solutions have been found. Another way to write down the field equations of GR was found by Landau and Lifshitz using a field redefinition, in which the dynamical variable was redefined from the metric tensor, to a metric tensor density of weight = 1. The redefined field density is commonly known as the gothic metric in the literature of GR. In its usual sense, the Landau-Lifshitz formalism is derived at the action level. However, the formalism can also be derived on-shell, or starting from the field equations of GR. In the first notebook I will be pushing here, the derivation will be done starting from the field equations of GR. The advantages of the Landau-Lifshitz or the densitized formalism is that perturbative solutions about a fully isotropic and homogeneous space-time can be found in a class of coordinate systems known as the harmonic gauge. The field equations of GR in the redefined form, and in such a gauge becomes simpler in form, and resemble a wave equation in flat space-times.