We see that the only term with an ambiguous sign in the equation 
is the one with 
in it. This is solved by the energy condition.
One can ask what is the link between the curvature and the energy.
In fact the curvature tensor is controlled by the energy-momentum tensor by the formula
There are several energy conditions, all different in their meaning. There is no hierarchy between them.
For the Topological Censorship we are interested in the averaged null energy condition but by assuming a globally hyperbolic spacetime we assure the dominant energy condition. Here we will recall what are these conditions and some of their physical features.
First the weak energy condition:
It is physically acceptable; 
appears to be the local energy density so this condition means that any observer see a non-negative energy density. This seems to be the case for all known kinds of matter.
The dominant energy condition is a slightly stronger condition but still physically acceptable.
It ensures the fact that you cannot travel faster than light. If it is violated there is a causality problem .
The strong energy condition is stricter than the weak condition but still reasonable.
There are other types of energy condition, based on the null vectors instead of the timelike vectors. It is the null energy condition.
A weaker condition is the null averaged energy condition which allows some small region to have negative energy but overall has to be positive on average.
These conditions are used but one aim research a weaker condition than the ANEC because some theorems, like the Topological Censorship, seems to be possible with a weaker condition. This is to do...
