Black Holes and Dimensional Analysis - Sixty Symbols
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- Опубликовано: 30 мар 2023
- Featuring Professor Ed Copeland with a look at dimensional analysis and how it can be used on black holes, among other things... More links and info below ↓ ↓ ↓
Biographical podcast with Ed: • An A-Class Reject (wit...
Long interviews with Ed: bit.ly/CopelandGoesLong
More videos with Ed: bit.ly/EdCopeland
Ed's University of Nottingham page: www.nottingham.ac.uk/physics/...
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Sixty Symbols videos by Brady Haran
www.bradyharanblog.com
Animation by Pete McPartlan
Email list: eepurl.com/YdjL9 
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Professor Copeland's handwriting certainly looks like the handwriting of someone who has been writing on graph paper for their entire life. I love it.
i have a mont blanc too. the paper he's writing on though is too hard for a ball point, and he shouldn't be writing directly onto the desk. sorry, OCD calligraphy student here. you should see my ex-wife's handwriting, she's japanese, my son's is even weirder.
Professor Copeland is an amazing guy... I was never in physics but as an educator he's fantastic.
he's got that infectious enthusiasm that makes it borderline impossible to not be enthused with him lol
Really?
Maybe it's because I'm not a astrophysicist.
"Take a line, call it big N Little n"..? WTF?
2nd year at school? 12 year olds?
Really?
@@MrBollocks10I think whatever you are replying to has been deleted so you might want to do the same
I remember my physics teacher had a custom made red stamp: "non dimensional", she would stamp our papers and wouldn't bother to check anything else if the result was non dimensional. It taught us some really good lesson, because if you get the units right the rest is pretty easy.
Dimensional analysis is by far the most useful thing I learned in physics class. I learned a lot of useful things, but this is the best.
The lack of rigour in engineering is like a broken leg, but dimensional analyses are the crutches every engineer carries.
chemistry class but same, it is an amazing skill to have
Same, I was able to recover the formula for the frequency of a vibrating string the other day, from what I remembered plus dimensional analysis to figure out what goes under the radical.
@@werdwerdus one of my chemistry teachers was always emphasizing how important this was. usually when i'd make mistakes it was because i got units confused. on an exam my teacher would mark my wrong answer and write "UNITS UNITS UNITS!!!" 😂
@@deltalima6703 should carry*
Videos with Professor Copeland are certainly a treat! Love it!
Professor Copeland is such a pleasure to listen to and watch. Blings a smile to my face - and I'm learning at the same time.
Dimensional analysis was not on the A-level syllabus when I was a kid, but we were taught it as a means of checking answers in exams to make sure you were in the right ballpark. I have found it very useful, but care has to be taken when things start rotating, like torque.
Torque has units of joules!
@@IanBLacy That's my point, it hasn't.
We were given two tricks for checking our answers. One was checking units, the other is checking your powers of 10. Run through the calculation again but only using the order of magnitude (10^x) then see if the final result is within one order of magnitude. If it is, chances are you got it right, and it takes a lot less time to count powers of 10 than to run through the full calculation.
I love how you guys are just having a laugh together while Ed explains his point.
You should make more videos with Ed Copeland. I truly like him.
I love Prof Copeland. What a great chap and educator!
Pluto is a fine example of an object in orbit around the Sun.
Funny way of spelling "planet" but OK.
@@N.I.R.A.T.I.A.S.
Whether or not you consider dwarf planets to be planets is irrelevant here.
The moon is a planet and should be respected as such.
End semantical geocentrism! We live in a binary system with our sister planet, Moon.
So yeah, not only Pluto and Ceres, but Caronte, xena, the moon, Europa, Io..
Not Deimos and Phobos tho, they're not welcomed into the club.
Apart from when it orbits around uranus
@@iseriver3982
No more than Ganymede orbits around Io.
Love Professor Copeland's videos, he always explains things so well
I'll never forget the day I was first introduced to Dimensional Analysis from my calculus professor. The lecture started with analyzing the units of Newton's Laws, as Professor Copeland demonstrated in this video. Then after a few steps my professor exclaims something along the lines of "there you have it, Kepler's Laws of motion without doing any hard math whatsoever." What an epiphany that was for me as a young student!
I found it fascinating that one of the standard works on Aerodynamics, "Fundamentals of Flight" by Richard Shevell, also uses dimensional analysis as a tool to come up with how the different properties of air, geometry and motion influence the result.
I like Professor's representation so much. He is always calm and shows those complex things in a simple way to be easily understood.
This is also useful when doing more complex calculations, when carrying around these G and c constants during long derivations can be cumbersome, so when working in General Relativity we can set the units such that c = 1 and G = 1. Essentially, time, mass and distance are all measured in the same units, say meters. To get time, do dimensional analysis: you have time is X meters, you want it in seconds. c has units of meters / seconds, so you divide X by c and get the number of seconds.
Thus, the Schwarzchild radius in this case can be written as R_S = 2 M. To get everything in normal units, figure out how many factors of G and c you need to get meters on one side, kilograms on the other. So R_S = 2 G M / c^2
Because of this, I can always remember the mass of the Sun, it is 1.5km (that is, it's Schwarzchild radius is 3km), and easy number to remember. How many kilograms is that? I leave that as an exercise, I am not going to remember *that*.
I just absolutely love Prof. Copeland, his episodes are always my favorite! Most everyone Brady features are both brilliant and fascinating; Prof. Copeland is of course no exception but his palpable enthusiasm and sincere humility set him apart. He’s just effortlessly engaging, for me at least and has the character of the ideal educator. 🤓
always appreciate more of prof copeland's commentary
Ed is hands down the best presenter on any of Brady's channels.
I don't mind Neil Sloane on numberphile
Professor Copeland is my hero, I wish he was my uncle! Dimensional analysis was one of the most useful things I learned in high school.
"I don't mess around, Brady." 😂 Loved how much fun was had making this
What? That last part I don't understand. If we shrink the Earth to 2cm it should be a black hole based on what you just explained. How can it be black and under Rs and not a black hole at the same time?
Yes, that’s the question I had. We need answers! :D
It's the limit of the strength of matter (protons, neutrons) that prevents such an object to remain stable over time.
The matter of the object would be pulled in so hard by the gravity of the rest of the earth's mass inside that pingpong ball (i.e. very close by) that it would keep shrinking under its own gravity on its own accord. There is no form of matter that can withstand this continued shrinking and resulting increase of the gravitational force. It's a runaway process that inevitably results in the collapse into a black hole, all within a very, very short time. Essentially it would collapse at almost the speed of light, and you wouldn't even be noticing it, because no light or anything else would escape to show anyone outside what happened exactly on the inside of that little event horizon.
I think he was joking that in Newtonian physics this wouldn't be a black hole, just dark star. In reality it would be a black hole, because of general relativity.
@@MarcinSzyniszewski I think that in the 19th century, knowledge about matter was still somewhat lacking. Even the existence of atoms was still a bit speculative. So they couldn't know what happens when matter gets extremely compressed inside a dark star.
In the 20th century, we learned a lot about matter. As far as we know now, matter can't resist the compression inside a dark star and collapses into a singularity. On the other hand, we also think that a singularity can't be the right answer either. So we're still a bit in the dark (pun intended) when it comes to modelling dark stars / black holes.
My favorite bit of dimensional analysis related to black holes is this:
Many people (unfortunately including physicists who should know better) like to say things like "BHs are the densest things in the universe," or "a mass becomes a BH when it shrinks to an extreme density". Some of this confusion may be conflating the density of the BH with the theoretically infinite density of the (hypothetical) central singularity. But consider the following:
1. Rₛ = 2GM/c²; but 2G/c² is a (universal) constant, so
2. Rₛ is proportional to M.
3. On the other hand, density ρ = M/V (where V = volume), so
4. (ignoring constants) ρ is proportional to M/Rₛ³,
5. which in turn (by point 2) is proportional to M/M³ = 1/M².
i.e., the bigger the BH (measured by mass or by radius), the sparser it is.
e.g., the density of a solar-mass BH would be ≈ 20 trillion g/cc, but the density of a 10 million solar mass ("supermassive") BH would be more like 0.2 g/cc, or about ⅕ that of water. (Etc.)
Black holes are not homogeneous balls of radius Rs though. Most of the mass is supposed to be at the singularity.
Mass would probably be distributed on an inner accretion disk of spaghettified falling matter…Definitely not uniform 😅
I never said nor meant to imply that BH density was uniform; average density is still a valid concept. To put it more clearly in dimensional-analysis terms, density is a [mass]/[volume] = [mass]/ [radius]³ property, whereas "blackholeness" is a [mass]/[radius] one. So saying a mass becomes a BH when it reaches sufficient density is misleading at best.
Meanwhile, the interior mass distribution of a BH, while unlikely to be uniform, is a matter of only partially-informed speculation at this point. This is especially true in the case of supermassive BHs, whose history and formation processes are still mysterious (and may always be).
And if the universe mass/size is big enough (but still very small for us in absolute value), our universe could be just one huge black hole.
@@dragonfly.effect "BHs are the densest things in the universe": if you want you can replace by "BHs contain the densest things in the universe". I find it a bit nitpicky to say that the average density can be low.
"a mass becomes a BH when it shrinks to an extreme density": that is just true. If you consider a sphere, a given radius corresponds to given density (with a fixed mass).
19:17 If light can't escape from it, how can it not be a black hole? Isn't that the definition? I don't understand.
its the type of object between neutron star and a black hole, nowhere to be found in nature but might exist
They're joking and being loose with definitions. In a universe with Newtonian gravity, an Earth-mass body of radius 2cm would be a Mitchell/Laplace "black star", because its classical escape velocity is greater than c, so photons (or rather, Newton's "corpuscules" of light) cannot escape. In a universe with Einsteinian gravity, an Earth-mass body of radius 2cm wouldn't be a swartzschild "black hole", because it wouldn't form an event horizon, and so photons can escape.
@@TheShadowOfMars That doesn't make sense to me. They clearly state that by applying GR -> Rs = 2 * (Gm/r^2) we arrive at radius twice as big as than the one calculated with classical physics. So how is it possible that Earth mass with radius smaller than Schwarzschild radius won't result in a black hole?
I think Ed slightly misunderstood where Brady was going, and was just reiterating that a black star in the Newtonian theory doesn’t have an event horizon. If we assume (for fun) that both the Mitchell-Laplace black star and GR black hole can exist, then because the Schwarzschild radius is twice the black star radius, a black hole would not be a black star.
@@raxxer1234 yeah exactly. So with Einstein we come to a critical radius of 4cm. Anything smaller will result in a BH. So how come 2cm won't do the trick?
Soy un estudiante de física en Paraguay, este canal me inspira a continuar en los momentos dificiles. Gracias.
Yes I’ve been waiting for something from professor Copeland he is my absolute favorite professor thank you so much Brady and CO. For all of the amazing great content consistently intriguing and engaging keep up the great work!
I gained my mathematical consciousness when I was taught Dimensional Analysis in engineering school. Before this epiphany, applied mathematics didn't exist for me. They don't effectively teach this concept in elementary school.
Always a joy to see Professor Copeland in a video!!
Excellent episode and explanation by Professor Copeland. I couldn't stop smiling listening to Professor Copeland.
Always delighted to see Professor Copeland diving into something and driving my brain right up to the edge of pain. (Is that a type of event horizon?)
I am however disappointed that we didn't have Dr Merrifield storming in to denounce the inclusion of Pluto.
Ed has the most calming and captivating voice in the world
For those that want to watch more on this topic, I'd recommend watching a video by the channel Physics Explained about black hole entropy.
Watching this just starting out my journey into physics, I just want to understand it all and I just don't yet, but its so much fun to follow and learn something new everyday. And when we have a internet of knowledge there is so much out there to help understand, I want to understand this as best I can before applying to uni.
Some of my favorite dimensional analysis tricks are obligately dimensionless arguments:
You cannot take ``exp(3m)`` or ``sin(8s)``, so in ``s=A*sin(Ωt)``, ``Ω`` must be a frequency. This can even be proven using Taylor expansion, because if the argument weren't dimensionless, you couldn't do ``1 + x + x^2/2 ...``. Or, you could say "the exponential is a function whose derivative is itself" and show that that means it must have a dimensionless argument (or ``d/dx exp(x)`` would have dimension ``[1/x]`` and could not be compared to ``exp(x)``).
More Dr Copeland!! Would watch him every day.
great video and very well explained
Brilliant more videos like this please, more maths and really physical principles
Thanks for a very informative video! My high school physics teacher taught me a little about this. He said that our answers weren't correct unless we had the correct units. Learning this lesson has made it much easier for me to solve problems, and I still use the technique today to convert from "standard" measures to metric.
This is such an interesting, clever, and vibrant video. Prof. Copeland is great as ever!
Thank you for this demonstration of clear thinking!!
Great video! Professor Copeland is a treasure!
Amazing video!
I could listen to Professor Copeland speak for hours!
Ed is so great!
Fantastic. Thank you.
Thank you so much for the video.
If he was my maths teacher I’d probably have gone into maths. What a soft spoken yet inspiring professor.
I could listen to Ed all day :D We demand more!!
I love how much smarter I feel now. Thank you.
this is such a powerful technique!
To be clear, if you yeet something with escape velocity, it does *not* go into orbit around the Earth.
It would go into an independent orbit around the sun.
Also, at the end they discussed that an object smaller than its Swartschild radius would appear black but it would not be a black hole yet. I think they were referring to the use of classical mechanics (Newton) and Special Relativity; by "not be a black hole yet" they mean it is described in General Relativity. In the real world, if you crushed something smaller than that radius it would become a black hole.
Indeed. If light can't escape the surface, your finger (or any other matter) sure won't. Do not touch!
I wish I had a teacher like Prof. Copeland for every class in school
Good one !
amazing.
as usual.
Excellent!
Ed Copeland, you are great!
I believe you forgot to put the link to Matt Strassler's blog in the description.
i love that the units of energy per mass are equal to an accelerating surface J/kg = m²/s². it is so simple and weird
19:16 so if the Earth was crushed down to 3cm the electron degeneracy pressure would be high enough to prevent further collapse... I had never considered this and had always assumed the crushed-down Earth would just become a black-hole... mind blown.
There are some really neat applications of dimensional analysis in biomechanics.
So nice to mention Pluto :-)
Love it! More math please :)
I could listen to Professor Copeland all day
Professor Ed is like a damn meditation.
Ive thought on it a bit, and from a quick think I think that G = (p/t) (d/m1) (d/m2), big G is a constant transfer in momentum over time that depends on the distance between two objects. The lower the distance, the higher the momentum. I think simplifying equations in a way makes you miss some hidden details.
I found dimension analysis indispensable while going physics course work in university for checking complex problems worked out.
Pretty interesting, almost makes me wish I paid more attention in physics
I really liked the editing at 14:05 after Ed says sarcastically “I am known for my rapid speed” hahaha
@1:50 there's supposed to be a link in the description to a blog post by Matt Strassler (SP?) on dimensional analysis, but it is not there.
When I heard that engineers were going to fix Planck's constant, I figured out the "frequency" of one kilogram. Then I proceeded to figure out what its "pitch" would be. I tuned C4 to 256 Hz because 256 is 2 to the power of 8. I came up with F-sharp, 158.5 octaves above C4, 2 to the power of 166 and six-twelfths Hertz. I know not of what use this calculation could be, except maybe as a tone a bathroom scale could generate in order to represent my weight [mass]. Dimensional analysis made me observe that (kilogram times second) is a constant🤔
My first physics professor in college was an arrogant, abusive, drunk. It was tough to get to class and stay in my field of study. After that, I had better instructors who were humble and excited like Professor Copeland.
Looks like we all love mr. Ed
The Schwarzschild radius formula? The most famous formula of all time!
Worth emphasising that black holes in general relativity have nothing to do with escape velocities. Indeed, in Newtonian mechanics, an object with a speed-of-light surface escape velocity would actually still be visible - light would be able to leave the surface and travel any finite distance outwards, it just wouldn't be able to travel out to infinity instead of 'falling back down'. And in Newtonian mechanics, you can escape an object even if you're going much slower than the escape velocity as long as you're being propelled - that's why rockets can leave Earth without going anywhere near 11 km/s.
Neither of these apply to black holes - light cannot travel any finite distance outwards, and you cannot escape even if you're being propelled. The reason the escape velocity calculation works is not because the physics is at all similar, but rather because the fundamental physical constants (G and c) are the same in both cases. The power of dimensional analysis is why the unphysical calculation gets you the right answer to within order of magnitude.
What about "tensive analysis"? (i.e. looking at extensive and intensive properties)
That's an amazing intro title card
Dimensional analysis is a very powerful tool.
Cool professor. Can be a narrator also. His sound is smooth
E=mc2 was independently discovered by other physicists before Einstein, the difference was that Einstein interpreted the equation literally rather than believing it merely represented equality of magnitude, as Henri Poincare and others had
01:50 there is no link to the blog in the description :-(
One very serious question I have about black holes that has been in my head for many years. Do black holes act, in some capacity, as super conductors? Given their amazingly low temperatures, it seems to be an obvious question to ask.
I’m only 6:50 into the video so this may be explained later, but why does [G][M] necessarily correspond to Kepler’s Law that R^3/T^2 ? I didn’t really follow why those two things should have anything to do with each other.
19:35
Schwarzschild's solution: e-λ=еν=1+A/r (the constant A can be defined from the result according to which in a weak gravitational field g00~1+2ф/c2, where ф=-GM/r -Newtonian potential) satisfies this requirement. So, A=-2GM/c2, and consequently, e- λ=eν=1-rG/r, where the gravitational radius (or Schwarzschild radius) is a characteristic radius defined for any physical body with mass: r(G)=2GM/c2 (here vacuuming string).
Consequently: 2E0/rG=Fpl=c4/G=εpl/rpl=ħw2pl/с=4ф2pl/G=4FGpl , where: фplG=(+/-)(1/2)c2=(+/-)1/2)[Għ/с]1/2wpl ; with indicating the mutual quantization of the mass (energy) of "rest" and space-time: m0/mpl=rG/2rpl=n, where n=0,1,2,3..., number of quanta.
From this (generally, from Einstein's equations, where the constant c^4/G=F(pl), and without the need to involve the concept of curvature of space-time), one can obtain a quantum expression (as vibration field) for the gravitational potential: фG=(-1/2)[Għ/с]^1/2w = -[h/4πm(pl)]w.
By the way, to this expression for the gravitational potential: "Containing all information about the gravitational field." (Einstein), you can come according to the classics (G), SR ©, and De Broglie's hypothesis (h), - without GR and QM.
What is the lower limit synchronization threshold
@Sixty Simbols a gravitációs-hullámok interferációs pontjaiban lévő barionos anyag plusz energiát kap a tömeg-energia ekvivalencia miatt s így jön létre a szingularitás! Mit gondol?
"I'm known by my rapid speed." - Professor Ed Copeland.
1:50 The link is never in the description.
Isn't E=mc2 really just the same as E=mv2? While yes, radiation have no mass, but on the other hand it velocity is fixed to exactly where the mass of a object that had any mass would be infinitiy....hence compensating for the lack of mass.
It's not totaly intuitive, but it kind of makes sense.
Question : is there a possibility that there is no singularity but some quantum force resisting the collapse. Similar to electron degenaracy force keeping star from collapsing for small star
Possibly. Regular stars resist collapse from the heat of their thermonuclear reactions. When that does not suffice, they will collapse. If their mass is low, they will stop as a white dwarf due to the electron degeneracy pressure. If their mass is a bit higher, protons will absorb the electrons and then you reach a big ball of neutrons (neutron stars) where the neutron degeneracy pressure prevents further collapse. Both are from the Pauli exclusion principle.
What would prevent further collapse? Nothing is known fur sure. Maybe some form of quark degeneracy pressure? But then, is there a limit? A "solar mass" black hole, around say 100 times the mass of the Sun, might be stopped from becoming a true singularity by that, but what about supermassive black holes? Could they overcome quark degeneracy pressure? What then? There is a lot yet to discover and learn!
Well, yes. That would be Quantum Gravity.
I wish I was taught dimensional analysis this way in univ.
@15:00, you correctly relate Escape Velocity dimensionally to [GM] which is correct with respect to velocity; however, you don't justify that Escape Velocity is that velocity, nor do you identify it as an assumption for your viewers. An exact calculation of Escape Velocity from the Earth based on Conservation of Energy starts by equating the Kinetic Energy of a mass positioned on the surface of the Earth with the Potential Energy of that mass at the same location. By Conservation of Energy, the Escape Velocity of the mass or its Kinetic Energy must be zero at that position when Potential Energy is zero, which is by definition zero at an infinite distance from the gravitating mass, i.e. the Earth. Thus, at the surface of the Earth, the Kinetic Energy must also match the Potential Energy, and it is curious that the Potential Energy of that mass on the surface of the Earth is negative.
At work but can't wait to watch this later. Ed, Mike, Scouse Tony, Grimey... the OG guys are the best
I wonder if a object's mass is just few grams less than escape velocity of light- what would it look like? Because time slows down in extreme gravity- wouldn't it Emmitt very few photons?
Is there any deeper discussion on this channel or elsewhere, into why this technique is so powerful? There is tonnes of stuff out there on the "unreasaonable effectiveness of mathematics" but this has very little mathematics in it, and seems even more "unreasaonably effective" to me.
That last statement is counterintuitive to me… If you cram enough mass in a small enough area such that nothing can escape, isn’t that by definition a black hole? Someone feel free to enlighten me, I don’t see the distinction.
I think maybe the idea might be that classically one would expect that light couldn’t escape at that size, but when one actually does the general relativity one gets that factor of 2 ? But that’s just a guess really...
Alternatively, maybe that’s the radius for when like, light moving sideways wouldn’t leave?
Like there’s a radius of “smallest possible stable orbit” iirc, where the only thing that can orbit (without like using some propellant to stay up) at that radius, would be light? I think?
Gotcha
Hmm. Even though this is "just" dimensional analysis, plugging the speed of light into a formula that was obtained starting with relationships that hold for objects having mass seems questionable to me. (Note: I'm not thinking that the remaining M is the mass of the object going speed c. Obviously it is not. But we started with F=GMm/R^2 and F=ma, both of which assume an object having mass m.)
What did he mean at the end that the earth mass sized black hole wouldn't be a black hole?
They're joking and being loose with definitions. In a universe with Newtonian gravity, an Earth-mass body of radius 2cm would be a Mitchell/Laplace "black star", because its classical escape velocity is greater than c, so photons (or rather, Newton's "corpuscules" of light) cannot escape. In a universe with Einsteinian gravity, an Earth-mass body of radius 2cm wouldn't be a swartzschild "black hole", because it wouldn't form an event horizon, and so photons can escape.
@@TheShadowOfMars I think they may have got it the wrong way round, though. The GR Schwarzschild radius is larger than the “black star” radius.
I agree., that was confusing. If the Earth were the size of Brady's OK sign circle such that light may not escape, it must have collapsed to a black hole. A neutron star dense Earth would be larger and not black.
19:17 In the astrophysical sense, at least, it would be a black hole. The idea of a point mass "singularity" is a popular mathematical idealization that a lot of people can understand, and is I think the idea behind a Schwarzschild black hole, but it's by no means certain that matter or energy can really cross an event horizon (or that any actually has yet), due to infinite time dilation, so it's quite likely that actual astrophysical black holes, or any black holes you could conceivably make, aren't even QUITE as small as their Schwarzschild radii, and it's even less clear if it's even meaningful for them to be smaller than their Scharzschild radii, as the traditional point mass model is.
Kepler must have been a big fan of sushi, also bringing his chop sticks to his portrait painting sessions.