  We study scattering rigidity in Lorentzian geometry: recovery of a Lorentzian
metric from the scattering relation $\mathcal{S}^\sharp$ known on a lateral
boundary. We show that, under a non-conjugacy assumption, every defining
function $r(x,y)$ of pairs of boundary points which can be connected by a
lightlike geodesic plays the role of the boundary distance function in the
Riemannian case in the following sense. Its linearization is the light ray
transform of tensor fields of order two which are the perturbations of the
metric. Next, we study scattering rigidity of stationary metrics in time-space
cylinders and show that it can be reduced to boundary rigidity of magnetic
systems on the base; a problem studied previously. This implies several
scattering rigidity results for stationary metrics.
