An observer principle for general relativity
  We give a mathematical uniqueness theorem which in particular shows that symmetric tensors in general relativity are uniquely determined by their monomial functions on the light cone. Thus, for an observer to observe a tensor at an event in general relativity is to contract with the velocity vector of the observer, repeatedly to the rank of the tensor. Thus two symmetric tensors observed to be equal by all observers at a specific event are necessarily equal at that event. 
Maurice J. Dupre
gr-qc