This isn't true. Given a manifold that satisfies the vacuum Einstein equations and given a world line in that manifold, it can be shown that the world line is undergoing acceleration using only local calculations/measurements. The worldline of a particle on the edge of a gyroscope will have a non-vanishing acceleration and this can be computed using local calculations/measurements.

You can describe the manifold using a coordinate system corotating with the gyroscope, the worldlines will still have a non-zero acceleration. There is absolutely no need for any matter field satisfying hyperbolic PDE's with faster than light characteristics.