# Null Energy Condition Violation in QFT and Area Theorem

The Area Theorem by Hawking assumes the null-energy condition (NEC) holds, however, it seems QFT in curved space-time violates NEC. I tried to explicitly observe the NEC is indeed violated when there is a scalar field in the Schwarzschild background. For a scalar field in an arbitrary background, the energy-momentum tensor is:

$$T_{ab}=\partial_{a}\phi\partial_{b}\phi-\frac{1}{2}g_{ab}(\partial_{c}\phi\partial^{c}\phi).$$

It is straightforward to see $j^a=-T^{a}_{b}V^{b}$ for every $V$, a future-directed time-like vector, $j^a$ is future-directed causal. So the dominant energy holds and therefore, NEC also holds. I am wondering to know how QFT can actually violate this NEC since if it is not the case, the area theorem has to hold in this case. My question is this:

Is it in some senses true that QFT generically violates NEC?

As a consequence of Hawking radiation, it seemed to be true for me because it seems that Hawking radiation occurs once a quantum field turns on and the only assumption in the Area theorem that can be violated is the NEC.

• I suggest you have a look at quantum inequalities, for example from 'Lectures on QEI' by Fewster. May 30, 2018 at 13:18
• @A.V.S. Thanks. It helped me to understand the local energy-momentum operator cannot be positive in a QFT. I didn't understand how it is related to NEC May 30, 2018 at 17:05
• For NEC in QFT see the paper by Fewster & Roman. May 30, 2018 at 20:24

More precisely, if you consider a Schwarzchild black hole in an asymptotically Minkowski spacetime, then the Unruh vacuum has an outward positive flux of energy at $r = \infty$ and a negative flux into the horizon.
For your particular case, Candelas computed the stress-energy tensor at the horizon and at $r = \infty$. The flux through the horizon measured along a future-directed null curve was found to be negative. Therefore the null energy condition is violated.