Connection path transport

Assume the ${\mathrm{A}\mathrm{C}}_{\omega }$ convention for the local tetrad supplier. Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, let $U\subseteq M$ be nonempty and open, and let $E=(E_0,E_1,E_2,E_3)$ be a smooth future-directed orthonormal tetrad on $U$. Choose a smooth metric-compatible affine connection $\mathrm{\nabla }$ on $TM{\mathrm{\mid }}_U$. Its Lorentz-connection matrix $\omega =({\omega }^I{}_J)$ in $E$ is specified by ${\mathrm{\nabla }}_X{E}_J={\sum }_I{\omega }^I{}_J(X)E_I$; the choice of $\mathrm{\nabla }$ is part of the data.

A piecewise-smooth path $\gamma :[0,1]\to U$ is continuous and has a finite partition $0=t_0<\cdots <t_m=1$ on whose closed subintervals its restrictions are smooth up to the boundary. On each such piece let $A_{\gamma ,j}(t)={\omega }_{\gamma (t)}(\dot{\gamma }(t))$, using that piece's one-sided tangent at its endpoints. Each $A_{\gamma ,j}:[t_{j-1},t_j]\to M_4(\mathbb{R})$ is continuous. The piecewise matrix-ODE existence proposition gives a unique continuous piecewise-$C^1$ solution

${\dot{U}}_{\gamma }(t)=-A_{\gamma ,j}(t)U_{\gamma }(t),U_{\gamma }(0)=I_4\mathrm{.}$

The equation holds in each piece's interior with the corresponding one-sided derivatives at its endpoints; uniqueness also makes the solution independent of subdivision. The endpoint matrix $U_{\gamma }(1)$ is the connection path transport in the chosen tetrad trivialization. It maps vector components at $\gamma (0)$ to components at $\gamma (1)$. A smooth local Lorentz frame change transforms this matrix by its endpoint frame matrices; it is a gauge transformation in the principal-bundle sense only if it extends to an automorphism over the identity.

Piecewise-smooth paths and thin homotopies

Local Lorentz tetrad and connection

Existence and uniqueness of transport along a piecewise-smooth path

Gauge group of a smooth principal bundle

This is transport of tangent-vector components in a local tetrad, not holonomy of a foliation or a claim that a global tetrad or spin structure exists. Reversing a path reverses its parameter; path ordering is fixed by the displayed left-multiplication ODE.