/sci/ - Science & Math

>>10511535
>Why would that lead to evaporation of the hole
That virtual pair suddenly becoming real needs energy. That energy came from the quantum fluctuation of the gravity field. Half of that energy flies away. Energy reduced from field = smaller black hole.
Source: my ass, but I believe it anyway.

http://xaonon.dyndns.org/hawking/

[math] \displaystyle
\begin{align*}
\text{Mass} && M \\
\text{Radius} && R &= M \cdot \frac{2G}{c^2} \\
\text{Surface area} && A &= M^2 \cdot \frac{16 \pi G^2}{c^4} \\
\text{Surface gravity} && \kappa &= \frac{1}{M} \cdot \frac{c^4}{4G} \\
\text{Surface tides} && d \kappa_R &= \frac{1}{M^2} \cdot \frac{c^6}{4G^2} \\
\text{Entropy} && S &= M^2 \cdot \frac{4 \pi G }{ \hbar c \; ln10} \\
\text{Temperature} && T &= \frac{1}{M} \cdot \frac{ \hbar c^3 }{8k \pi G} \\
\text{Luminosity} && L &= \frac{1}{M^2} \cdot \frac{ \hbar c^6}{15360 \pi G^2} \\
\text{Lifetime} && t &= M^3 \cdot \frac{5120 \pi G^2}{ \hbar c^4} \\
\end{align*}
[/math]

>>10511535

There’s a popular description of how Hawking radiation works it goes something like this: empty space is filled with activity of pairs of virtual particles, matter, and antimatter, that spontaneously appear, and then annihilate each other, briefly borrowing energy from the vacuum itself. But when this happens near a black hole, one of the pairs will be swallowed by the event horizon leaving the other free to escape and taking its toll and energy. That energy can’t come from nothing and so the black hole itself pays the debt by slowly leaking away its mass.

This is a nice picture but how accurate is it?

If we follow the narrative of Hawking’s original calculation the story sounds rather different. Space is filled with quantum fields, they can oscillate with different frequencies much like the many possible vibrational modes on the guitar string. A particle is like a note on the string, and just like a real guitar note, real particles tend to be comprised of many vibrational modes.

Those underlying vibrational modes are still present in the absence of real particles, they fluctuate energy due to quantum uncertainty, and those fluctuations give us what we think of as virtual particles. Now don’t take the existence of virtual particles too seriously, they’re really just a tool for calculating the infinite ways in which a fluctuating quantum field can behave.

One way the quantum fields are very different to guitar strings is that they can have both positive and negative frequencies, a negative frequency can be thought of as a mode that travels backward in time and can be interpreted as corresponding to antimatter. Now that’s a whole level of weird or on its own, when a quantum field is in a vacuum state there’s a balance between positive and negative frequency modes, which you can crudely think of as a balance between virtual matter and antimatter particles.

>>10511622
G https://en.wikipedia.org/wiki/Gravitational_constant
c https://en.wikipedia.org/wiki/Speed_of_light
ħ https://en.wikipedia.org/wiki/Planck_constant#Value
k https://en.wikipedia.org/wiki/Boltzmann_constant

>>10511624
>by slowly leaking away its mass.
a non-answer if there ever was one

>>10511624

These all virtually annihilate or cancel out so that no real particles exist, this is all fine in flat space but, spatial curvature can mess with the balance of the underlying quantum field modes, by introducing horizons.

Horizons cut off access to certain modes of the quantum fields, disturbing the balance that defines the vacuum.Stephen Hawking knew that black holes with their insane space-time curvature would wreak havoc on quantum fields, but what would the effect be? To answer that properly he would need a full union of general relativity and quantum mechanics, a theory of quantum gravity, a theory of everything. It didn’t exist then and it doesn’t exist yet. Not to be deterred by the impossible, Hawking came up with an ingenious workaround.

The narrative of Hawking’s mathematics goes something like this: he imagined a single space-time path, a Lightspeed trajectory called a null geodisic. It extends from far in the past to far in the future.This a perilous path, it passes through the location of a black hole in the instant before it forms. In fact, it is its very last trajectory, it emerges barely ahead of the forming event horizon.

Hawking imagined a simple quantum field tracing this path, a field that is in a perfect vacuum state before the formation of the black hole. But he found that the close shave with the black hole disturbs the fundamental vibrational modes that define the fluctuations of the vacuum by the time. By the time this trajectory has found its way back out into flat space again. And those fluctuations look like real particles. A distant future observer sees radiation coming from the black hole. Hawking’s imaginary path from the distant past to the distant future was brilliant, it allowed him to compare the state of the vacuum in two regions of flat space, far from the black hole regions where the nature of vacuums, quantum fields, and particles are perfectly well understood.

>>10511645

But to understand the effect of the close encounters with the black hole, he required an uneasy marriage of quantum mechanics and general relativity. In the absence of a theory of quantum gravity, Hawking needed a hack, and that hack was the Bogoliubov transformations. These can be used to approximate the effect of curved space-time on quantum fields, by smoothly connecting regions of flat space, they describe a sort of mixing of the positive and negative frequency vibrational modes that are caused by those curves.

The physical interpretation of this mixing via the above transformations is tricky, in fact, there isn’t just one valid interpretation. Hawking’s calculation talks about scattering. Certain modes of the quantum field are scattered or deflected by the gravitational field of the forming black hole, they have nudged off their narrow escape path and lost behind the forming event horizon. Meanwhile, other modes avoid scattering and continue unscathed, with the loss of certain fundamental modes, the vacuum state must be constructed from the remaining modes. That distorted vacuum looks like it’s full of particles the nature of the lost modes, it tells us what Hawking radiation should look. Like black holes tend to scatter modes with wavelengths similar to their own sizes, the quantum field that emerges is distorted in the same wavelength range, and so it produces wave packets.

It produces particles that also have wavelengths about as large as the event horizon so, the more massive the black hole the longer the wavelength of its radiation. Hawking calculated the frequency distribution of this radiation found something incredible, it should look exactly like thermal radiation.

>>10511667
>>10511645
>>10511624
Stop copying the script and just post the fucking video
https://www.youtube.com/watch?v=qPKj0YnKANw

>>10511667

Black holes would have a heat glow with an apparent temperature that depends on their mass, more directly, it’s proportional to the surface area of the event horizon. Large black holes should appear cold, radiating excruciatingly slowly, but small black holes should appear hot, and the smallest should radiate explosively.

What about the whole picture of particle-antiparticle pairs being pulled apart by the event horizon?

Hawking’s math describes splitting or mixing of these pure positive and negative frequency modes. It’s fair to interpret this mixing as the promotion of what were once virtual particles into reality, and for the escaping modes there exists a corresponding set of modes linked by quantum entanglement, that are trapped behind the event horizon. We can interpret those as corresponding to the swallowed antiparticle partner, so the split matter-antimatter part of the picture is reasonable, but there are reasons to dismiss aspects of this picture.

Firstly, this radiation is not localized. Remember the Hawking radiation has wavelength the size of the event horizon, the size of the entire black hole? Well, these debris wavelengths have created particles, and they tell us that there is an enormous quantum uncertainty in the location of these particles. Hawking radiation must appear to come from the global black hole, not from specific points on the event horizon. In fact, an observer in free fall through the horizon sees nothing to them, space is locally flat, the vacuum should look like a vacuum.

>>10511681
How can I get to the point that I can understand all of PBS space time videos? I want to learn the math and truly grasp the concepts, and I have a lot of free time. Where do I start (wont go to college) and how much time will it take me to get to that point? 10 years of 6 hours each day studying? im willing to

>>10511535
bump

black holes are like space cyclones which tend to be greater as there is cubic potential of air compression. there is a great tare in the miiddle from the dfense amount of mass created by the cloud or air which surround it. It literally travels faster in time.

>>10511785
Even in the age of the internet, college courses are still by far the best way to truly learn and understand topics in math and science. This is because your learning is structured by people who understand the fields, and you are forced to practice challenging problems regularly.
Now that I got that out of the way: it depends on your level of math experience. Elementary algebra skills are the foundation of all higher mathematics, so learning advanced math becomes easier the more comfortable you are with algebraic manipulation. Even if you feel you have a solid grasp on algebra, it's worth going back and practicing with complicated expressions if you haven't done it in a while (factoring, exponentials, logarithms, trig functions, etc.). Then your math progression should roughly go: calculus, statistics, basic differential equations, linear algebra, some more advanced diff eq's, some functional analysis. That's roughly the extent of my math skills, but beyond that you would eventually want more functional analysis, complex analysis, group theory, and if you want to understand general relativity, differential geometry as well. Probably other math areas I'm missing since I didn't major in math or physics. Textbook learning is very important, but there are some YouTube channels I can recommend to help in your progression of math and physics:

>>10511850

https://www.youtube.com/user/EugeneKhutoryansky

https://www.youtube.com/user/viascience

https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

https://www.youtube.com/user/DrPhysicsA

Ideally these should just be supplemental, but they can be a great help.

>>10511785
I'll also add: calculus and basic linear algebra (vectors, linear transformations, change of basis, eigenvectors and eigenvalues) is really most of what you need to get a lot of special relativity and quantum mechanics. There's no easy road to any of general relativity though, you need to be comfortable with differential geometry. And quantum field theory requires Lie groups, Lagrangian mechanics, and advanced analytical methods on top of standard quantum mechanics.

>>10511899
Also Hamiltonian mechanics for QFT as well, and probably other important things I'm missing

>>10511785
>How can I get to the point that I can understand all of PBS space time videos?
If they suggest watching a previous video before you go on, do that.
They do a decent job with baby steps that build up to explain and tackle bigger concepts.

>>10511785
Also very helpful:

For standard physics
https://m.youtube.com/user/ilectureonline/playlists

For general relativity and an overview of the tensor and topology knowledge needed to get it
https://m.youtube.com/user/XylyXylyX

Here's another one
https://m.youtube.com/channel/UCGDanWUzNMbIV11lcNi-yBg/videos

There are a lot of relatively deep and helpful videos on YouTube. If you find yourself in the middle of something you don't understand, just pause your progress in that channel/playlist and look for one that covers the topic you're missing. Depending on your level of experience, you might have to "go down" several layers before you get to something you understand, but eventually you'll work your way up

>>10511535

Your Platonism is getting in the way. You are inheriting stories of your world into the the story of the mathematical world of astronomy.

A black hole is a story of mathematical behavior. No one has "seen" a black hole, but we have deduced a story of a black hole mathematically and we have observed from what other things we can see would do if it were in the story of the black hole that the story of the black hole is valid, which we take for a verification of the story of a black hole.

That said, there is no "behind" a black hole. There is no "inside" a black hole. There is a mathematical discontinuity where nothing is defined past that point.

>>10511964
>>10511914
>>10511899
>>10511850
Thanks!

Hawking radiation is just a practical exercise of second law at relativistic and astronomical levels.

This is convoluted conservation.

>>10511964
>https://m.youtube.com/user/ilectureonline/playlists
Holy fucking shit. Is this guy god? what the fuck? how does he know so fucking much? he's literally godlike.
What the fuck, do humans like this one really exist?? how tf does someone learn so freaking much? why do brainlets get publicly admired while these literally godlike man get ghosted? so people here are at THIS level?
I guess people like me are just genetically inferior. There is something with you guys that the laymen like me dont have. Youre literally godlike what the hell

I'd gladly kill myself just to allow more oxygen so these people can breathe more. I'm just a waste of space. I'd kindly offer my ass to be used as a sperm cube just to please these übermensch.
Pic related me

>>10511535
bump

>>10512109
>I guess people like me are just genetically inferior. There is something with you guys that the laymen like me don't have.
First I'll say that almost nobody on /sci/ is at the level as that guy.

But most people at that level aren't miracle people. As Euclid said, "there is no royal road to geometry,", or any mathematics in general. Some people do have a special talent for proofs and mathematical reasoning, but 99% of it is grunt work and practice. Nobody gets there riding on talent or intelligence. Being interested in it can help you get through that grind, but you can get to that level too with practice and dedication.

>>10512067
I understand that. But isn't this actually too soon to judge - I mean - we expect there can't be no free energy therefore we think black holes evaporate. But maybe we shouldn't be sure about what laws singularity breaks?

>>10511983
>There is no "inside" a black hole
Well yeah. So how can anything radiate out of it or how can once collapsed space/time glue itself back while a hole in it evaporates?

>>10511682
What my tiny brain can grasp of your answer is that particles popping into existence is just a fairytale for brainlets like me and that's pretty much all I understood. But thanks for trying.
Just one question - is that radiation only a hypothesis? Is there any proof?

here's my take on it -- i researched this for a year before changing fields during grad school under a BH expert so i think i know my shit better than "average"

my feeling is that the whole "black hole swallows one of a pair of spontaneous vacuum pair produced particles, the other one flies out, therefore since energy is coming out the BH needs to lose energy/mass" is really confusing and a bad explanation. if it swallowed the antiparticle it would get heavier, not lighter, right?

imo that picture is stupid because virtual particles in QFT don't represent on-shell particles. virtual particles are a perturbation theory way of describing quantum fields in terms of its normal modes, analogous to mathematically taking a function and decomposing it in terms of sine waves in its fourier decomposition. in other words it's bad to take virtual pair production literally, and certainly it fails in the BH evaporation case.

IMO it is much much better to understand why BHes emit energy by reading up on unruh radiation. in QFT something strange happens: if you're in vaccum (no particles whatsoever) then obviously a particle detector sitting in that vacuum spacetime reads out 0 particles. if it's moving linearly, same thing. however -- this is the weird part -- if it is in an _accelerating_ reference frame, then it starts reading out particles. wow. this is because the vacuum state of QFT transforms into a non-vacuum (it has particles in it) state when you transform to an accelerating reference frame.

now if you think about the equivalence principle, the horizon of a black hole is basically an accelerating surface in space (gravity = acceleration). from here i find it more intuitive that BHes emit radiation

>>10511535
Black holes are a meme, a mathematical construct (solution) to a non working model in the observable universe. I too failed for the meme as well.

>>10511535
I'd say, it's because of negative potential energy of gravitational field. If the falling particle transfers energy to free particle and remains at low speed near event horizon, it has a slightly negative total energy, free particle has a slightly positive energy (rest energy + kinetic energy > potential energy). The sum of black hole energy and falling particle energy is less than initial black hole energy.

>>10512562

No empirical proof, but it keeps apearing in the math no matter how you approach it. The problem is that without a full theory of quantum gravity we cant describe the underlying cause of Hawking Radiation.

>>10511983
>No one has "seen" a black hole
The event horizon telescope team said they've got a big pending announcements.

>>10512562
Virtual particles are weird like that. They don't really exist, but they're essential in QM to describe particle interactions so you get the right answer.
That being said there are things you can do to make those unreal virtual particles into actual real photons. Introducing horizons to a system cuts off available frequency modes which means the energy isn't able to cancel itself out anymore and the feedback creates a real particle. In the case of a stellar mass black hole or higher this is a low frequency photon, the smaller the black hole the more energetic the photons created are. This means the smaller yourblack hole the faster it evaporates.

>>10513135
I probably didn't understand shit, but still an interesting read. Thanks.
So not only quantum fields but also relativity is at work here messing with stuff.
Still I don't see a reason why emited radiation must be paid by the black hoe other than "we can't have free stuff".

>>10513642
Are you saying mass can not be so heavily concentrated it pulls back any photons it emits?

>>10513678
An opposite occurance should happen as often - a negative particle falls out and a positive gets sucked in. One should nullify the other anyway.

>>10514403
Okay. Still can't see why BH has to pay up energy even when it eats a particle. Couldn't it be something like harvesting zero point energy? That would mean BHs can only grow.

>>10514554
A particle with negative energy can't leave the gravitational field, because its energy is negative, so it eventually gets sucked too with total zero change to the black hole energy.

>>10514582
>Still can't see why BH has to pay up energy even when it eats a particle.
TANSTAAFL. Can't get something for nothing.
If you really want to hurt your brain look into what happens with entangled particles when you dump one into a black hole.

>>10511681
videos fucking suck, a script is appreciated

>>10514645
Wait, wasn't that negative particle supposed to be the cause of BH losing energy?

>>10514786
Only if the positive particle is not sucked, then it carries the positive energy away.

>>10514645
>>10514870
Ok, this starts to make sense.
What do those virtual particles become? Electron / positron? Does the negative virtual particle actually become a real one before it gets swallowed?

>>10515245

Hawking radiation is going to be photons and other massless particles. To produce particles with mass, the energy of the radiation has to be high enough to cover the rest mass of the particle.

>>10515263
And the negative particle? Another photon? Antiphoton?

>>10513135
I've checked out the unruh radiation. Mind blowing. I love it.
That would also mean that if you were accelerating and seing a cloud of gass in what once was empty space, you would also create the antiparticles and you would be stealing energy from space behind the event horizon.
But I'm still not convinced this is how it goes and that there is actually a reason for shrinking BHs other than us expecting conservation of energy.

>>10516284
>But I'm still not convinced this is how it goes and that there is actually a reason for shrinking BHs other than us expecting conservation of energy.
You've got to be careful not to think about a black hole like its a classical stellar body. It's not really an object.
For example the volume of the black hole doesn't increase cubically like literally every other 3D object, it increases as if it were surface area. This means the more massive your black hole the less dense it gets, get to an SMB at the center of a black hole and the tidal forces at the event horizon are weaker than you'd feel at the beach at high tide.
They're fucking weird.

>>10516962
>SMB at the center of a black hole
Wut? Galaxy?

>>10518653
Yo dawg, we heard you like black holes...

>>10518653
Yeah, they meant galaxy

>>10516962
Would that create some limiting volume after which a black hoe stops sucking... or starts growing insanely fast and eats space at the speed of light?

>>10519447
not on the bigger and bigger side (they get colder and their surface gravity decreases with increasing mass) but small enough black holes emit hotter and hotter radiation, so that very tiny ones (like masses of a milligram or so) quickly burn out with a little burst of mostly quarks and gluons right as they go away

>>10515340
Photons always have positive energy. The rest mass and kinetic energy is covered by potential energy, gravitational potential at the event horizon is negative infinity, so it can cover anything.

Do the light of the stars compensate for the loses due Hawking radiation? Is the current Stelliferous
era too "hot" for any black hole to lose any energy at all?

>>10519797
i did the calculation once and iirc a black hole the mass of the moon gathers more energy from the cosmic microwave background alone than it loses to hawking radiation.

Sooo how big can a black hole get?

>>10519862

At what distance from event horizon is create the antiparticle-particle to catch it?
The average life of a electron-antielectron is 3,22*10e-22 s, if the maximum velocity in the universe is 3*10e8 m/s, it may be happen aprox at 10e-14 m or 10 femtometres.
A infinitesimal possibility i guess, so don't worry about this BH.

>>10519867
That's just nothing.
Can it get as big as a single galaxy? How about a cluster? How about all visible universe? Can it?

>>10520212
>That's just nothing.
bruh that black hole is bigger than you can comprehend
also no black holes cant get as big as a galaxy you numbskull if they could then the whole universe would be full of absolutely massive black holes that would pull all of spacetime together and destroy the universe

>>10520247
>also no black holes cant get as big as a galaxy you numbskull if they could then the whole universe would be full of absolutely massive black holes that would pull all of spacetime together and destroy the universe
false. they very well could exist no problem. it's just that so far we've never been able to find evidence of anything with radius greater than O(100-1000) solar system radii. the mass of a BH that big is mindblowing already

>>10520212
what would stop it

>>10511622
>pi is always multiplied by an even number
really acTivates The almonds

>>10520272
Those laws have multiples of 2pi because of circles and radians, it’s not that profound

>>10511535
the energy of particle/anti particle formation comes from the energy/mass of the black hole itself.

E=mc^2

>>10520264
what if our known universe is inside a black hole? It would explain The Great Attractor and the acceleration of the expansion of the universe, if we're somewhere between the singularity and the event horizon, it would make sense that we don't recognize the black hole

https://youtu.be/QgNDao7m41M

>>10520264
>they very well could exist no problem
Not enough time has passed for one to get that big.

bumping because physics

>>10521305
It's the time, right? The universe is too young?
it's not like BH ate everything around and now it's invisible, because we would still see it curving stuff that lies behind it, yes?
Or there is actually some limiting volume after which the space around BH collapses, glues back itself and the BH stops to be a part of our universe...

>>10524067

We don't really know from where the Supermassive Black Holes come from.

>>10524067
Even assuming they were direct collapse black holes (they didn't bother with the star phase) they still can only feed so fast because the more matter you try to cram down its maw the hotter and more dense it gets which creates an outward pressure that regulates how much infalling mater you have so you can put some boundaries on how much they should have been able to grow.

>>10521045
This is one of those threads I'll keep around bookmarked to re-read when I have more experience in physics, thanks /sci/ for actually having a decent thread

>>10520390
you mean to say [math]\tau[/math]

>>10511624
>>10511645
>>10511667
>>10511682
Based and PBSpilled

>>10521045
Look into some of Suskind's lectures on youtube, one of the things they're talking about now is that behind the event horizon complexity increases.
He's also talked a lot about how spacetime itself may just be an artifact of quantum entanglement

>>10524435
low iq post

Is there a limit of the space /time fabric's plasticity? Can a black hole bend space so much it breaks?