Deriving Hawking's most famous equation: What is the temperature of a black hole?

Black holes are perhaps the most enigmatic objects in the universe. Popularised in movies and science fiction, they evoke the magic and mystery of our universe and provide inspiration for those looking to make their mark in the world of academic physics. But what exactly is a black hole? And how can we study them?

According to Einstein’ theory of general relativity, a black hole is a region of spacetime where gravity is so strong that nothing, not even light can escape. The boundary of this region is known as the event horizon of the black hole, and according to classical relativity, once an object has passed the event horizon, it will never be able to escape the clutches of the black hole. However, when you throw quantum mechanics into the mix, as is often the case, the situation becomes a bit more subtle. In fact, in 1974 Stephen Hawking demonstrated that by combining certain elements of quantum field theory with General relativity, it was possible to show that Black holes do in fact radiate, causing them to slowly evaporate, and eventually disappear.

Now I don’t know about you, but when I first read about black holes as a child I was instantly hooked and desperately wanted to find out more. The only problem is that the physics of black holes, and in particular Hawking’s work, is notoriously difficult and requires an advanced knowledge of Einstein’s theory of general relativity as well as quantum field theory. But is it possible to determine the most exciting and mysterious properties of black holes using only advanced high school mathematics? Well, it turns out the answer is yes, and in this short video I would like to show you how.

References:

Black Holes and Time Warps - Kip Thorne
Gravity from the ground up - Shutz

[WilliamDye-willdye]

I love that the audience here carefully scrutinizes each video for errors. It's basically peer review, complete with occasional re-uploads. It's a shame we lose all the comments on the original video. Sometimes people put a lot of time into comments, but re-uploads are still the right thing to do.

[sphakamisozondi]

This channel is a goldmine for abstract knowledge.

[adberg]

The fact that you made a video in which you explained this all using physics that a 17 year old school student was able to understand (me) is astounding keep it up!

[Phillip-eydp]

I am 60 years old and I was able to enroll in any university in my country as an engineer. However, at the time, I didn't think I was smart enough to understand the math and physics needed to complete the degree, so I chose to become a Technician instead. After watching three of your videos, you have helped me to remember some of the math concepts I learned 40 years ago. If I had a teacher with even half your ability in high school, I would have pursued a degree in engineering. Your ability to explain complex concepts using only high school math is truly remarkable.

[rc]

The video is improved, the graphic representation of Hawking radiation is removed, which was used by Hawking himself but was ultimately just incorrect. Such a good video! This subject is popular with more sophisticated physics learners, but ultimately purely mathematical and not intuitive and our “mental picture” of what we think is happening is unlikely to be real. All my opinions, of course.

[kendallknight]

I've always wondered how all this stuff was connected and how people came up with these relations. Please keep it up

[devincarlson]

These videos are absolutely incredible. I don’t know how you’ve been able to churn them out so fast and consistently without any drop in production value or content quality. Keep them coming! My new favorite physics channel!!

[jadegrace]

After watching this, I don't get how anyone could find anything more beautiful than physics.

[JoimFormula]

Like this content very much!

Add one, Sir.. ;)
1. Include c, when things are fast
2. Include G, when things are massive
3. Include h, when things are quantum
4. Include k, when things are hot
5. Include pi, when things are spherical/circular

Thanks for the nice sharing..

[lukasz]

Dziękujemy. Thank You. Your explanations of modern physics are brilliant !

[ibanix]

Loving these videos, as an undergrad physics major. Keep it up. I also firmly agree with your assessment of dimensional analysis. On my last exam (Modern Physics), I used it to show the professor wrote the question wrong. Whole class got to skip that problem :-)

[junkmail]

72, Retired Electrical Engineer, Remembered "Dimensional Analysis" for some of the practical engineering calculations, (as a checking system) and remember professors alluding that it was potentially a powerful tool, (and to maybe study up on it after graduation), BUT now I see, Doing simultaneous equations on the exponents of the various dimensions involved! I'm now just stunned! I just subbed. (Mind blown) no recovery insight. I just need MORE!!!
I see at this moment there are 17 videos, having accumulated 3.6 million views Wow. QUALITY WORK!!!
There are some special YT's I enjoy following, and so have a pretty full schedule, but I'm just going to have to nestle yours in among the others. I am going to SAVOR yours like a sweet dessert after a fine meal. Thanks for your great efforts!!!

[narfwhals]

"a topic which I hope to return to in a future video" is a much better shortcut than the unclear pair creation. And it gives us something to look forward to! Good work.

[gimmehendrix]

Probably the best explanation of physics on youtube. Keep em coming!

[Joeleon]

It’s incredible how I’ve learned a lot of what was mentioned here, dimensional analysis, equations of gravity and energy, thermodynamics and enthalpy/entropy, and all of it I was half asleep for. Now I’m presented with a 40min video using it and willingly watch, crazy how different it is learning for a class against learning for yourself.

[littlemissmisses]

I’d just like to extend a huge thank you to you for this incredible content. I failed maths gcse 21 years ago, but I am now enrolled at college to retake it, thanks to you. We all learn in different ways, and this channel (despite maths I barely understand) has made these concepts easy for me to understand, and genuinely curious to see how far I can now push myself with mathematics. When people ask me why (because I don’t technically ‘need’ it for any reason), I reply with “my brain is hungry”. :)

[VintageTechFan]

Just found this channel, and I am amazed. Explaining stuff I heard off but never totally got due to complicated explainations in a extremely clear way.

[pengwei]

Mind blowing content. Such a standout format among science videos. Never thought that the “complicated math” that gets abstracted away actually makes things easier to understand

[Evenfiber]

Some of the most important results in vector calculus, complex analysis, and differential geometry fall out of Stokes' theorem on manifolds. It's incredible to see the same key insight revolutionize physics in real time.

[alexandreolle]

Thank you for being this long-awaited but high quality link between us, amateur physicists and the beautiful but complex physics we desperatly want to understand :)
Outstanding work with crystal clear explanations as always.
Keep it up ! And thank you again !