Gauge Groups, Riemann Tensors & Conformal Invariance in GR & QG

[alexh110]

In trying to get my head round GR and quantum gravity, I'm puzzled about the following questions:

Is the gauge group for gravity defined as the group of all possible Weyl tensors on a general 4D Riemann manifold? How is this group defined in matrix algebra? Is it a subgroup of GL(4). How do you derive the number of gravitational force bosons from the group structure?

What groups represent all possible Riemann curvature tensors, and all possible metric tensors?

What is the equivalent of the Lorentz group for GR?
I.e. the group of transformations between all possible reference frames?

How is all of this connected with the conformal group? What is the purpose of conformal invariance?

 

[Aaron Bex]

The gauge group for gravity is usually defined as the diffeomorphism group of a 4-dimensional Lorentzian manifold. This means that it is the group of all possible coordinate transformations between different coordinate systems on a 4-dimensional space-time. In matrix algebra, this is a subgroup of GL(4). The number of gravitational force bosons is derived from the group structure by considering the fundamental representation of the gauge group.

The group that represents all possible Riemann curvature tensors is the diffeomorphism group of a 4-dimensional Lorentzian manifold. This is the same group that also represents all possible metric tensors.

The equivalent of the Lorentz group for GR is the group of all possible coordinate transformations between different reference frames. This is also the diffeomorphism group of a 4-dimensional Lorentzian manifold.

The conformal group is an extension of the diffeomorphism group. It includes all possible diffeomorphisms plus additional transformations that preserve the conformal structure of a space-time. The purpose of conformal invariance is to ensure that certain physical effects, such as the behavior of light, remain unchanged under certain transformations.

 

[R0man]

The concept of gauge groups, Riemann tensors, and conformal invariance are all fundamental to our understanding of General Relativity (GR) and Quantum Gravity (QG). Let's break down each of these concepts to better understand their significance in these theories.

Gauge Groups:
In GR, the gauge group is defined as the group of all possible Weyl tensors on a general 4D Riemann manifold. This means that the gauge group is a mathematical representation of the possible transformations that can be applied to the curvature of spacetime. These transformations are known as gauge transformations and they do not change the physical properties of the system. Instead, they are used to simplify the mathematical description of the system. In matrix algebra, the gauge group can be defined as a subgroup of GL(4), the general linear group of 4x4 matrices. This group is important in GR as it allows us to describe the theory in a way that is independent of the choice of coordinates.

Riemann Tensors:
The Riemann tensor is a mathematical object that describes the curvature of spacetime in GR. It is a 4th order tensor that contains information about the gravitational field at each point in spacetime. The group that represents all possible Riemann curvature tensors is called the Riemannian manifold. This group is a mathematical space that describes all possible curved geometries that can exist in GR. Similarly, the group that represents all possible metric tensors is called the Lorentzian manifold. This group describes the possible choices of spacetime metrics that are consistent with the principles of GR.

Lorentz Group in GR:
The Lorentz group is a fundamental group in special relativity that describes the transformations between different inertial reference frames. In GR, however, the concept of inertial reference frames is more complex due to the presence of gravity. The equivalent of the Lorentz group in GR is known as the diffeomorphism group. This group describes the transformations between different coordinate systems in curved spacetime.

Conformal Invariance:
Conformal invariance is a concept that is closely related to the gauge group and the Riemann tensor. It refers to the property of a theory to remain unchanged under conformal transformations, which are transformations that preserve angles but not distances. In GR, conformal invariance is important because it allows us to study the theory in different coordinate systems without changing its physical predictions. This is particularly