Change of frame

In this section we will introduce a number of quantities that depend on a choice of frame. We can then consider a change of frame, a linear transformation of the basis \({e_{\mu}}\) of each tangent space on the manifold. Since an arbitrary manifold is not necessarily parallelizable, a frame is already a local construct; therefore we assume a change of frame preserves orientation, and define it to be a tensor field \({(\gamma^{-1})^{\nu}{}_{\mu}}\) of elements in \({GL(n,\mathbb{R})^{e}}\) smoothly defined in the region \({U\subset M}\) where the frame is defined. We write \({\gamma^{-1}}\) for the change of frame so that the components of a vector field \({w^{\mu}}\) transform according to \({\left(w^{\nu}\right)^{\prime}=\gamma^{\nu}{}_{\mu}w^{\mu}}\).

Δ Note that the vector field \({w}\) is an intrinsic object that is unaffected by a change of frame; it is only the components \({w^{\mu}}\) that transform. It is common to use \({\gamma}\) to denote a matrix and then write \({w^{\prime}=\gamma w}\) or \({e^{\prime}=e\gamma^{-1}}\), where \({w}\) is understood to be a column vector of components and \({e}\) is understood to be a row matrix of basis vectors. We will attempt to always explicitly mention whether variables are matrices, and will in most cases show indices when referring to components to avoid confusion with the intrinsic vector field.

Below we summarize how some common objects transform under a change of frame \({(\gamma^{-1})^{\nu}{}_{\mu}}\).

ConstructIn the original frameIn the transformed frame
Dual frame\({\beta^{\mu}}\)\({\beta^{\prime\nu}=\gamma^{\nu}{}_{\mu}\beta^{\mu}}\)
Vector field components\({w^{\mu}}\)\({\left(w^{\nu}\right)^{\prime}=\gamma^{\nu}{}_{\mu}w^{\mu}}\)
1-form components\({\varphi_{\mu}}\)\({\left(\varphi_{\nu}\right)^{\prime}=(\gamma^{-1})^{\mu}{}_{\nu}\varphi_{\mu}}\)
Linear transformation\({\Theta^{\mu}{}_{\nu}}\)\({\left(\Theta^{\lambda}{}_{\sigma}\right)^{\prime}=\gamma^{\lambda}{}_{\mu}\Theta^{\mu}{}_{\nu}(\gamma^{-1})^{\nu}{}_{\sigma}}\)

An Illustrated Handbook