  We prove solenoidal injectivity for the geodesic X-ray transform of tensor
fields on simple Riemannian manifolds with $C^{1,1}$ metrics and non-positive
sectional curvature. The proof of the result rests on Pestov energy estimates
for a transport equation on the non-smooth unit sphere bundle of the manifold.
  Our low regularity setting requires keeping track of regularity and making
use of many functions on the sphere bundle having more vertical than horizontal
regularity. Some of the methods, such as boundary determination up to gauge and
regularity estimates for the integral function, have to be changed
substantially from the smooth proof. The natural differential operators such as
covariant derivatives are not smooth.
