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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.

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    $\begingroup$ I suggest you have a look at quantum inequalities, for example from 'Lectures on QEI' by Fewster. $\endgroup$
    – A.V.S.
    May 30, 2018 at 13:18
  • $\begingroup$ @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 $\endgroup$ May 30, 2018 at 17:05
  • $\begingroup$ For NEC in QFT see the paper by Fewster & Roman. $\endgroup$
    – A.V.S.
    May 30, 2018 at 20:24

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For the case of Hawking radiation, the null energy condition is violated due to quantum effects, and so the derivation of area law obeying classical null energy condition doesn't hold true.

Recall that in case of Hawking radiation, at the horizon, particles are pair created with positive and negative energy. The positive energy particles escape to infinity while the negative energy particles fall into the black hole thereby reducing its mass.

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.

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