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I am currently working on a project involving the information paradox of black holes and the holographic principle, and as I mentioned in a previous post [1], I would like to illuminate those subjects in a series of gemlogs. For one because I think they are interesting and little talked about in the public sphere, so this could be unique content for gemini space. But also because I am myself still struggling with a lot of the concepts, and teaching is the best way to learn. So please keep in mind I am not a professor and have no authoritative knowledge or anything, I'm just trying to digest my studies by giving my best attempt at an informal review for a general audience. So with that out of the way, here is the first part.
The best model we have of the universe today is made up of two theories: The Standard Model of particle physics, which describes the quantum fields that make up all matter and forces, and general relativity, which describes the dynamics of the background on which those fields live. These theories are incredibly successful in their respective regimes: The Higgs Boson was discovered at the LHC almost 50 years after it was predicted, and LIGO managed to detect gravitational waves from the collision of black holes in galaxies far, far away. In fact it is very hard to make progress in theoretical physics today because the theories work almost too well - there is little room for improvement.
And yet we know the model is wrong. The Standard Model is quantum mechanical, while GR is classical - two completely different and incompatible frameworks. One of the clearest examples of a tension between the two is the evaporation of black holes.
In the 1970's Stephen Hawking discovered that black holes are not completely black, they radiate particles. Since this radiation carries energy, the black hole loses mass and shrinks in the process. At some point it has fully evaporated and all that is left is the Hawking radiation. On its own this does not seem to pose a problem. What I haven't told you yet is that the radiation is mostly random - its properties depend only on the mass, charge and angular momentum of the black hole. In a sense this is all it "knows". Therefore there is no way to figure out what was previously inside the black hole once it is gone. You lose information. But quantum mechanics forbids this, for reasons I will elaborate later.
This problem is known as the "black hole information paradox". While a lot of progress has been made, it is still unsolved and that's very exciting. The hope is that working on it will give us a better understanding of the underlying unified theory of quantum gravity.
Another reason why I am so interested in this is that so many parts of physics must come together to make sense of it: general relativity, special relativity, quantum mechanics, thermodynamics and statistical mechanics all play a role. You can see this just by looking at the formula for the temperature of Hawking radiation:
I will try to avoid using too many equations, but here I just want to point out which quantities appear. T is the temperature of that radiation. ħ is the reduced Planck constant, and represents quantum mechanics. c is the speed of light, representing special relativity, and the Newton constant G represents general relativity. The Boltzmann constant k_B shows us that we also need thermodynamics. M is the mass of the black hole. And of course π appears too for good measure. You don't need to understand what this equation means, all I want to emphasize is that, as far as I know, this is the only equation where all of these disciplines are combined together. As such I will first try to cover a variety of topics important to physics, interesting in their own right, because they will be important in order to even just phrase what the problem is precisely.
The journey towards a resolution of the information paradox has been wild so far, and led to very unexpected implications such as the holographic principle. In the posts that follow I hope to convey the core ideas behind the reasoning used. I will inevitably be vague and unclear, so please ask questions! I'll try to answer them as best I can. You can write me an email at pseudoriemann@dira.cc or write a comment below.
Next time I'll start the explanations by introducing a very central concept related to the the information paradox: entropy.
EDIT: It was pointed out to me that I miswrote - k_B is the Boltzmann constant, not the Planck constant! I corrected the error in the main text now.
Original post where I mentioned my plan for this project
Posted 2022-01-17
2022-01-17 | looking forward to the next part
2022-01-17 | Thank you! - Rie
2022-01-27 | Very interesting read, thank you! -Ceik
