Some Observable Quantities
In order to understand the statements of the various energy conditions, one must be familiar with the physical interpretation of some scalar and vector quantities constructed from arbitrary timelike or null vectors and the matter tensor.
First, a unit timelike vector field can be interpreted as defining the world lines of some family of (possibly noninertial) ideal observers. Then the scalar field
can be interpreted as the total mass-energy density (matter plus field energy of any nongravitational fields) measured by the observer from our family (at each event on his world line). Similarly, the vector field with components represents (after a projection) the momentum measured by our observers.
Second, given an arbitrary null vector field, the scalar field
can be considered a kind of limiting case of the mass-energy density.
Third, in the case of general relativity, given an arbitrary timelike vector field, again interpreted as describing the motion of a family of ideal observers, the Raychaudhuri scalar is the scalar field obtained by taking the trace of the tidal tensor corresponding to those observers at each event:
This quantity plays a crucial role in Raychaudhuri's equation. Then from Einstein field equation we immediately obtain
where is the trace of the matter tensor.
Read more about this topic: Energy Condition
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