I have always wondered about this, and since I see there are people around here who probably can explain it to me, here is the question.
As far as I understand, (please correct me if I’m wrong) the current thinking on vacuum fluctuations is that the virtual pair particles that are created/annihilated both have positive energy that is borrowed from the “vacuum energy” and returned to it.
Hawking radiation - the explanation for it is that if such a virtual pair is created right on the event horizon boundary, one particle could fall in while the other escaped, and thus the black hole would “radiate” a particle.
Here is what I don’t understand. If both of these particles in the pair have positive energy, then when such a capture of one of the particles happens, the net result is that vacuum energy on the border is diminished, one particle is radiated and the other is consumed, increasing the total black hole mass. But the radiation is supposed to diminish black hole’s mass. Also, I don’t quite understand how the vacuum energy can go below its minimum level on non-Planck time scales (which is what happens in the mechanism above).
If one of the particles in the pair has negative energy and the other positive, then there is no vacuum energy problem, and when the positive particle escapes, then yes, the black hole mass is decreased. But then the question becomes - how come it is always the negative energy particle that is trapped and always the positive energy one that escapes and not vice versa?
It has negative energy by virtue of being the one that’s trapped. You need to look at the entire energy of the system, including the gravitational potential energy.
It’s a good question, but perhaps the best thing to say is that the idea of virtual particle pairs being created and one partner falling in is jsut a simplifcation of what’s going on.
Rather than considering virtual particles i think it’s better to start with the Unruh effect.
In quantum field theory transforming from non-accelerating to accelerating cooridnates affects the mathematical entities which keep track of how many particles there are in a system with the end result that more particles appear in the accelerated coordinates than the non-accelerated coordinates despite the fact that the two coodirnates represent the same physical system. This is called the Unruh effect.
In general relativity you have the equivalence principle, one of the results of which is that, locally, an observer hoovering just above the event horizon of a black hole is locally equivalent to an accelerated observer. This means you locally get the Unruh effect and some of these extra particles will appear to radiate out from the black hole as Hawking radiation.
Now as to how Hawking radiation affects the mass of a black hole, AFAIK there isn’t a completely firm theoretical basis for saying that it effects the mass of the black hole at all. Hawking radiation is a theory in semi-classical gravity which means that the black hole is essentially just a background and it’s parameter ‘M’ (mass) is completely unaffected in the model by the Hawking radiation. Presumably though Hawking radiation does affect the mass of the black hole. However as Hawking radiation is generally modelled in situations where the conservation of energy does apply and energy is radiating outward from the black hole,it’s no big leap to suppose that M must decrease due to Hawking radiation.
My problem with that explanation is that the Hawking radiation is not only supposed to be observable by someone “hovering just above the event horizon” but in any reference frame, including non-accelerating one far removed from the horizon.
That seems to be a tautological explanation - “the energy is conserved because it is supposed to be conserved”. Or, as Wikipedia put it in its “Hawking Radiation” entry -
“One of the pair falls into the black hole whilst the other escapes. In order to preserve total energy, the particle that fell into the black hole must have had a negative energy (with respect to an observer far away from the black hole).”
So, is that stuff the mystical “negative matter” (not to be confused with antimatter) we need to prop wormholes open with?
Which is to say, if it is not antimatter but negative matter, can we suppose that stuff really exists even though we’ve never seen it? I had always assumed negative matter was in the realm of fairy dust.
I assume antimatter would actually add to the mass of the black hole. Sure it might annihilate with other matter in there into energy but E=MC[sup]2[/sup] suggests that to those of us on the outside of the black hole would see its mass increase so in order to “evaporate” the BH guessing it has to be negative matter. Hell, can matter even exist in a singularity?
IIRC the Hawking Radiation from any stellar mass (or bigger) black hole is below the Cosmic Microwave Background. So, while a black hole “radiates” (sort of) you will not see it. The process of “evaporating” a black hole by these means is insanely slow. Tens or hundreds of billions of years. Far longer than the current age of the universe.
The smaller the black hole the more dramatic the HR becomes. For a microscopic black hole, such as might be formed in a particle accelerator, the HR would cause it to wink out of existence in a fraction of a second.
True, but for tiny ones that are not microscopic the radiation is significant enough to be easily observable in any reference frame. Not that anyone has directly “seen” such a black hole. Or a big one for that matter.
My question is basically how does the Hawking Radiation mechanism explain (without just saying “Law of Conservation of Energy says so, thus it is so”) the supposed reduction in the black hole mass as a result of such radiation?
I think it is the, “Law of Conservation of Energy says so, thus it is so.”
The universe’s balance sheet needs to be kept in order.
If a virtual particle pair pop into existence they usually cancel each other out. If one of the pair falls into a BH then the other particle becomes “real” and adds to the universe. To balance the ledger the other becomes negative matter and thus reduces the mass of the BH.
We see entanglement in other particles so supposing the virtual pair “knows” what happened to the other is not science fiction. It is weird but plausible.
Neither particle has negative energy except in terms of potential energy with respect to the BH. There is no known material that has intrinsic negative energy.
The hole expends enough energy to create two real particles but only gets the energy of one back. Thus it loses mass.
Well no actually Hawking radiation is not (locally) observable by all observers, somoene falling in to the black hole will not locally observe any Hawking radiation. However the Hawking radiation that is ‘produced’ will be observable to non-local observers and they will observe the radiation (or the lion’s share at least) being produced just above the event horizon.
I’ve tried to explain: the physics used, semiclassical gravity, cannot directly model the feedback between Hawking radiation and the black hole. That’s it’s limitation, what we need is full theory of quantum gravity to do that (as an aside from what I understand the best quantum gravity candidate, LQG, does predict Hawking-like radiation). However there are still symmetries in play and thee’s no reason to suppose that Hawking radiation violates these symmetries and in order not to violate them the black hole’s mass must be depeleted.
Well, at bottom, it doesn’t. If conservation of energy were wrong, then Hawking radiation would not necessarily lead to a black hole mass reduction. On the other hand, if energy conservation were wrong, then we wouldn’t have any theory for the existence of black holes (since general relativity predicts at least local energy conservation), nor would we have a theory for particle creation (since quantum theory is built upon energy conservation). So if we’re talking about black holes and particles within the framework in which these entities are defined, it seems that we ought to have energy conservation, as well; otherwise, I’m not sure how to sensibly talk about these things…
When we look at gravitational potential energy and see it decrease as the bodies get closer to each other, we don’t just say “well, there is a law of conservation of energy, so it is conserved”. We have the mechanism that explains HOW it is conserved - the speed increases, thus kinetic energy increases, thus the sum is constant. The same with all other examples of the conservation of energy. Why not this one?
Your intial question was a good one, but I don’t think you’re completely comphrehending the answer.
If you like the details of how Hawking radiation affects the mass of a black hole aren’t known. In the basic semiclassical gravity description, Hawkign radiation doesn’t affect the mass of a black hole because semiclassical gravity doesn’t describe how quantum fields affect black holes (or any other type of genera relatvistic spacetime), it only describes how black holes affect quantum field. How a black holes mass is affected by Hawkign radiation is simply beyond the scope of semiclassical gravity.
However there are good arguments, which come from the global symmetries of the certain spacetimes that we choose to model Hawking radiation in that Hawkign radiation should affect the mass of the black hole. What this basically boils down to is the conservation of energy (note that the conservation of energy in general relativity is a fairly sticky concept, so it’s not just a lazy “energy is always conserved” argument).
The details on how it does that? Well we can only guess until we have a complete theory of quantum gravity. Thoguh we can’t exclude the possibilties that Hawkign radiation will go out the window once we have a QGT or that a QGT will predict black holes evaporate, all we have at the moment are good arguments as opposed to a sound theoreticla framework.
Lastly, thoguh I’d like to add, not all proofs in physics are constructive, so even if we did have QGT and even if we did show it predicted Hawking radiation and that Hawking radiation affected the mass of the black hole there’s no absolute guarentee that it would allow us to instantly identify a nice little ‘mechanism’ for it to do that.
It’s just that, when we’re talking about black holes and particles, we’re already working within a framework (quantum field theory in curved spacetime) where energy is known to be conserved (locally), so in that sense there’s no way for energy not to be conserved when we talk about these things.
Fundamentally, conservation of energy follows from time-translation invariance – provided physics is the same today as it is tomorrow, and was yesterday, energy must be conserved – this is a result of Noether’s theorem, which is unfortunately somewhat abstract.
And from another (equivalent) point of view, conservation of energy is exactly what one uses to derive the change in kinetic energy with the change of potential – potential energy is exchanged for kinetic energy when something falls, for example; if their sum weren’t constant, then the kinetic energy of the falling object could take any value at all.
Half Man Half Wit,I’m welcome to be corrected here but I don’t think the local conservation of energy is really the overriding principle here, I believe that in the local frames where local energy is conserved then local Hawkign radiation is not observed, it’s only in the lcoal frames that do not conserve energy that it appears. However if you assume that the spacetime is spherically symmetric (and asympotically flat) then the global symmetry of the spacetime must mean that the Hawking radiation moving out from the black hole must be accounted for in the reduction fo the mass of the black hole.
Bests in that it’s favoured by the experts and from what i understand as pretty much blown other QG theories of the water in terms of encouraging results.
I’m not an expert thoughand couldrpobably write in oen or two paragraphs what I know about LQG, so you’d have to seek out one and ask them about LQG.
I’m not sure, but isn’t generally energy conserved in asymptotically flat/static spacetimes? So we ought to have both Hawking radiation and CoE in the Schwarzschild case, no? Or am I misreading you?
