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So as I understand it, Hawking radiation occurs when virtual antiparticle-particle pair are created near the event horizon of a black hole due to vacuum fluctuations because of Heisenberg uncertainty and one particle flies away from the black hole and the other falls in. The black hole loses mass because the one that falls in has negative energy measured by a far away observer. My question is how can a particle have negative energy in a certain reference frame? Be as techincal as possible in your answer, thank you.

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    $\begingroup$ Possible duplicate of Black holes and positive/negative-energy particles $\endgroup$ – ACuriousMind Jul 24 '16 at 1:17
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    $\begingroup$ Answered by @Renney at physics.stackexchange.com/q/30597, and further expounded on the physics and the Bogoliubov transformation at math.ucr.edu/home/baez/physics/Relativity/BlackHoles/…. However neither really covered the math to make it a little more credible. I suggest leaving this open till a good mathy explanation is provided. $\endgroup$ – Bob Bee Jul 24 '16 at 1:35
  • $\begingroup$ Is it possible for an observer outside of the black hole to measure the energy of a photon which has fallen into the black hole? $\endgroup$ – sammy gerbil Jul 24 '16 at 2:14
  • $\begingroup$ @Jimmy360 : You mean decrease in mass? Caused by a single photon? $\endgroup$ – sammy gerbil Jul 24 '16 at 3:15
  • $\begingroup$ @Jimmy360 : Yes, a negative-energy photon, which will reduce the mass of the black hole - as the question states. $\endgroup$ – sammy gerbil Jul 24 '16 at 3:28
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Why not negative energy? Energy is a scalar quantity that is conserved. Negative energy means that there exists a potential well which for bound quantum mechanical states has negative energy, and for free states positive. A matter of definition. Look at the hydrogen atom states

hydrogen energy levels

Note the - sign for the energy levels

In the case of the black hole the potential well is the gravitational well of the black hole. In an analogous way that an electron falling from 0(free state) to the first energy level it releases a photon of that energy and sticks at that level, the gravitational well will lose the analogous energy to let half of the pair to go free.

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