Wavefront and frequency conventions

All wavefront sets below are ordinary, unprimed wavefront sets:

means the covector of the bidistribution on $M\times M$. A conventional canonical-relation notation would replace ${\kappa }^{\prime }$ by $-{\kappa }^{\prime }$.

We use the local Fourier convention

The factor $e^{-\mathrm{i}\omega (t-t^{\prime })}$, $\omega >0$, therefore has first temporal covector ${\kappa }_t=-\omega <0$. For signature $(-+++)$, such a null covector is future directed after raising its index. This is our positive-frequency orientation.