Significances of energy conditions I know about these different energy conditions in GR, namely strong, weak and null, but never really understood the full physical significance of them or for example how to 'derive' them or how compelling is a certain condition (for that particular one to be applied). I know that they exist because you need some constraints for $T_{\mu\nu}$ to satisfy as otherwise its completely unconstrained. Can someone please elaborate?
 A: The energy conditions are just various possible generalizations of the statement that "energy density cannot be negative" to the whole stress-energy tensor. The reason why energy density cannot be allowed to be negative - at least in some sense - is that if arbitrary positive- and negative-energy regions were allowed, the empty vacuum would become unstable: it could spontaneously change to regions with positive energy and regions with negative energy, the the "energy density gap" would keep on growing.
We know that a condition of this kind has to hold and prevent the Universe from developing regions of a large negative energy density but we are somewhat uncertain what the precise condition satisfied by the Universe is. As I mentioned, there are many ways how to generalize the condition $T_{00}\geq 0$ to the whole tensor - one that also includes the momentum density and the fluxes of energy and momentum (the latter is the very "stress"). Do we require the energy density to be greater than zero or even the absolute value of the pressure? And so on...
The most important conditions are


*

*null energy condition

*dominant energy condition

*weak energy condition 

*strong energy condition


See their mathematical conditions at:

http://en.wikipedia.org/wiki/Energy_condition#Mathematical_statement

It's somewhat likely that the null energy condition is indeed satisfied in the whole Universe - any macroscopic region of it - and the same may be true for the weak and dominant one (dominant one is strictly stronger than the weak one). However, one may write down models of somewhat exotic matter where the strong energy condition is violated. Although lots of "data" are known, it remains somewhat uncertain what is the precise condition that follows from the fundamental theory that knows all about the allowed types of matter - string theory. I forgot to say that all these conditions may be modified by a special treatment of the vacuum energy density coming from the cosmological constant.
A: The energy momentum tensor at a point $p~\in~M$ in the weak energy condition obeys the condition $T_{ab}U^aU^b~\ge~0$, for $U^a$ a tangent vector in $T_pM$.  The tensor then has components $T^{00}~=~\rho$ and $T^{ii}~=~p_i$ as energy density and pressure along the geodesic through the point $p$.  The dominant energy condition is that for $T_{ab}U^a$ a non-spacelike vector $T_{ab}U^aU^b~\ge~0$.  The strong energy condition is $T_{ab}U^aU^b~\ge~(1/2){\it tr}T U^aU_a$.
These conditions are not derived, but imposed.  They do give conditions on causality and Cuachy conditions.  Violations of them remove these conditions, which permit the existence of closed time-like curves.  The averaged weak energy condition is $T^{00}~\ge~0$.  This condition exists for some quantum field, where we have $\langle T^{00}\rangle$.  Assume this momentum-energy tensor is defined by a scalar field.  We then have
$$
T_{ab}~=~\phi_{;a}\phi_{;b}~-~\frac{1}{2}(\phi_{;c}\phi_{;d}g^{cd}~+~m^2\phi^2)
$$
If we assume it is massless and we just consider $T^{00}~=~(1/2)\phi_{;t}\phi_{;t}$.  The field is expanded into operators $a,~a^\dagger$, and if we ignore for the sake of argument the covariant derivative, energy is $\sim~a^\dagger a$.  We all know that quantum systems have a vacuum, some bounded minimal energy value.  Yet if the averaged weak energy condition is violated then this operator always returns a negative value, unbounded below.  That is a disaster, for it means the quantum system can cascade down an endless ladder of states and emit a vast amount of radiation.  For this reason it is generally thought the weak energy condition holds.  However, this has not been proven to be a consequence of more fundamental physics.
A: It might also help to have a sense of the physical meaning of each energy condition:
Trace EC:  Density must at least equal the sum of the principal pressures
Strong EC: Matter must gravitate toward matter
Dominant EC:  Energy must not flow faster than light
Weak EC: Local mass-energy density must not be negative
Null EC: Stress-energy experienced by a light ray must not be negative
A: An energy condition is merely a restriction on the stress-energy tensor conjectured to hold for all physically reasonable matter.  In other words, it’s just a guess at the precise definition of “physical reasonableness”.   As such, energy conditions are not derived.
Interestingly, every energy condition yet proposed has failed.  This means that for each known condition an example exists of matter considered physically reasonable that violates it.   These examples usually stem from quantum field theory, although there are classical examples based on the existence of (an as yet undetected) fundamental scalar field.
Averaged energy condition are also known to be violated by examples considered physically reasonable.  See for example http://arxiv.org/abs/0910.5925.
