How are general relativity and quantum mechanics united in mainstream theory?
How are general relativity and quantum mechanics united in mainstream theory?
They are united by the Grand Unified Field Theory... which doesn't exist yet.
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They are also united by Quantum Electrodynamics (QED) - which I don't even begin to understand at a level beyond the Feynman book of that title.
Will there be a theory of quantum gravity within the next 50 years ? If no, why not ?
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I wouldn't say the efforts would go to waste, though it would be very disappointing. String theory's AdS/CFT correspondence is applicable to parts of condensed matter physics, including superconductors, regardless of whether it is the correct path to quantum gravity.
For the record, Leonov's initial post in this thread was a duplicate of one he made elsewhere which was moved to ATM. I have moved his post here, tusenfem's admonishment, and Leonov's response to that thread in ATM.
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I thought Hawking radiation was an example if GR and QM. Basically at the Schwarzchild radius there is a probability of an annihilating pair being created, separating and recombining within the Heisenberg limit. Except one may get captured by the black hole (or tunnel beyond the event horizon) and the other then escapes. Not sure if this unites them but they co-exist quite nicely if true.
That's true. However, that's a case of a quantum field on a classical spacetime background. In other words, it's a semi-classical treatment of the problem. There's a difference between that and a true unification, in which spacetime itself should be (somehow) be described by quantum mechanics.
The maths for the Unruh radiation do hold up. I'm not nearly as sure about Hawking radiation, but since near the horizon the Schwarzschild black hole is locally very well-approximated by a Rindler chart on Minkowski spacetime, I wouldn't expect there to be any fundamental problems (this does make the comparison of Hawking to Unruh valid, though they're still different effects). The captured-particles picture is a crude description of the field(s), but it isn't wrong.
For a Schwarzschild black hole, ∂t is a Killing field corresponding to conservation of energy outside the horizon, but inside, it is spacelike and thus locally corresponds to momentum instead. Energy conservation requires a virtual particle-antiparticle pair to annihilate again, but if they're separated by a horizon, then this 'energy conservation' actually adds energy of one particle and the momentum of another, and there's no requirement that real particles have positive momentum (this gets interpreted as negative energy by observers outside the horizon). So GR can explain why it's possible, but QM is still needed to make it go.
OK that was not my understanding. Can you explain how the particle emissions approximate a thermal emission spectrum consistent with a temperature? There seems to be no reason that you could characterise the BH to have a temperature in the QM case.
As for the maths - I meant with the QM version. Unruh holds up very well, I thought that the basic mechanism of particle capture was rather hand-waving. I may be wrong - but I was always taught that the ripping apart of virtual pairs was a convenient 'this is how a horizon could emit something' without being rigorous.
Is it your understanding that Hawking radiation and Unruh radiation are different phenomena with respect the a BH?
An intuitive way to think about it if one accepts the Unruh effect as a given is that static observers are accelerated, so if the freefalling observers observes a vacuum of no temperature, the static one should measure a positive temperature. Making this slightly more exact, a Schwarzschild black hole can be approximated by a Rindler chart of Minkowksi spacetime under the substitution , , , which shows that the black hole has surface gravity of 1/4M and acts locally just like the Rindler horizon of an accelerated observer in flat spacetime. Therefore, Hawking temperature of a black hole should match the Unruh temperature with acceleration replaced by this surface gravity. And since redshifting thermal radiation just gives a thermal radiation with different temperature, the end result is thermal as well. This is the correct conclusion that Hawking derived by more rigorous means.
Particle capture is just a heuristic way of describing the mechanism. The particles are pulled apart by tidal forces, and the horizon is one-way. To make this more precise, one would have to propagate both of them in the curved spacetime, and energy conservation can be fulfilled without annihilation if one of them goes inside the horizon (see previous post for basic reason as to why). That's the key point here that makes 'particle capture' a working description. It's certainly not the only heuristic description, but it is a valid one. Alternatively, one can view the particles as tunneling out of the event horizon.
The question is as follows, if the distinction between Hawking and Unruh radiation is due to them being associated with an absolute and an apparent horizon (as seems to be suggested by your "because it is a real horizon"), how can we calculate what happens when the black hole is perturbed and we can't use schwarzschild's apparent horizon as a substitute for the absolute horizon? How can we derive Hawking radiation in this general case? Does it not present a problem that we need to know the entire future evolution of our entire spacetime, and that it must be asymptotically flat, in order to locate the absolute horizon and thus derive any Hawking radiation? If that is not a problem, does it not present a problem that in, what seems to be, our FLRW spacetime we don't even have a future null infinity to define our absolute horizon with?
If the above is not correct, and we're still defining Hawking radiation wrt Schwarzschild's apparent horizon, in what way exactly is it different from Unruh radiation?
I'm far from clear on the quantum-mechanical details of what happens when there is no Killing field in general (in some cases, the distinction simply loses meaning: e.g., in stellar collapse, the scales of time and energy of the process are such that they skirt the uncertainty principle limit anyway) or when it's not everywhere timelike even outside the horizon, but there's a classical approximation that I understand models the situation fairly well, at least for slowly rotating black holes and weak perturbations.
It builds yet another horizon, on the order of a Planck length over the true one. The stretched horizon acts like a membrane of viscous fluid that's electrically charged and conductive, with finite uniform temperature and entropy, but no heat conduction. As matter falls onto this membrane, it grows correspondingly. Pretty much any response to an external electromagnetic field or gravitational perturbation can be analyzed in this formalism. And in addition to being a good approximation (or so I'm assured by people who actually work on this stuff), it also naturally enables several completely off-the-wall ideas, like using black holes in circuits or electric motors. As if black holes weren't crazy enough on their own.
Unruh: an accelerated observer measures a positive vacuum temperature where an inertial one does not
Hawking: a black hole produces radiation
So we need a global definition to even make sense of the latter.
Am i then right that the distinction lies in wether the Killing horizon is degenerate or non-degenerate? So by a "real" horizon you mean a non-degenerate Killing horizon?
As an aside, does the existence of a non-degenerate Killing horizon imply the existence of an absolute horizon? I thought this was not necessarily true, and if it isn't, what happens when you have a non-degenerate Killing horizon without an absolute horizon? It seems like you'd get Hawking radiation without having a black hole, so there must still be something missing from my understanding.