# Gravity II — Field Equations

• Chris Doran
• Anthony Lasenby
• Stephen Gull
Conference paper

## Abstract

In Lecture I we introduced the $${\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{h} }$$ and Ω(a) gauge fields. Their transformation properties are summarised in Table 13.1. Prom these, we need to construct ‘covariant’ quantities which transform in the same way that physical fields do. We start by defining the field strength R(ab) by
$$\frac{1} {2}R(a \wedge b)\psi \equiv \left[ {D_a ,D_b } \right]\psi ,$$
(13.1)
$$\Rightarrow R(a \wedge b) = a \cdot \nabla \Omega (b) = b \cdot \nabla \Omega (b) + \Omega (a) \times \Omega (b.)$$
(13.2)
R(ab) is a bivector-valued linear function of its bivector argument ab. This extends by linearity to the function R(B), where B is an arbitrary bivector. Where necessary, the position dependence of R(B) is made explicit by writing R(B,x) or R x (B).

## Keywords

Gauge Theory Field Equation Bianchi Identity Ricci Scalar Weyl Tensor
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.