Black Holes and Dimensional Analysis - Sixty Symbols

I'm gonna try and show you how we can obtain the the size of the Event Horizon of a black hole with using no more than School based maths as a theoretical physicist it's what you do when you first think about doing a calculation you don't plunge straight in and start in trying to solve some very very complicated set of differential equations what you tend to do initially is you do what is called a dimensional analysis you look at the units and and you try and understand what what do I expect the answer to look at in terms of units and that often gives you a real handle on what this solution should look like it can be very powerful when you're marking exams actually it's a really good way of finding where a student has made a mistake by looking back and finding where the units don't work out if you have an equation that says x equals y then you don't think of X as being meters and Y as being kilograms you don't say this person's six meters tall they must be the same as this person over here who's six kilograms you've got to have the same units on either side of an equation and it's as simple as that that's the basic idea behind this but you can actually make some real progress and we'll we'll reach the black hole but on the way we'll do Kepler's laws which is one of the things we teach in second year uh classical mechanics course as a solving a rather complicated differential equation we'll kind of get to the the nub of it all right Ed and if I don't like it you know where it's going to end up in the confidential waste oh I'm gonna put it in your confidential waste bin okay all right so where do we start I guess we start with the title dimensional analysis there's a nice blog post that does something along this Lines by a very good theoretical physicist called Matt strassler Link in the description so the idea is the following right we we're going to use units or in particular the consistency of units to try and glean information about a problem and it's likely solution and it's remarkably powerful imagine you've got the Sun and we've got the earth going around the Sun okay and it's got some typical radius R and it's got some Associated period the time for it to go around T and you can ask what's the average velocity what would be the possible equation that relates velocity to the radius and the time well everyone's going this is silly right we know velocity is distance over time and so it's going to be some possible number times the the radius divided by the time and it can't be anything else it can't be r squared divided by T because that would have units of kilometers squared divided by seconds it wouldn't be a velocity so this is the first example of a dimension dimensional analysis giving me some information and it turns out in almost all situations that we deal with this number usually becomes a number of order one so we might not be bang on because the the value will depend you know if this was a circle that would be 2 pi right because the circumference would be 2 pi r and if it was an ellipse it would be something slightly different but this number will typically be something of order one and that's a common thing that will come out it won't be a billion or a billionth but now I want to go and move on to Kepler right Kepler's laws one of his most famous laws relates the radius to the time with this rather weird relationship that the radius cubed is proportional to the time squared or if I divide by the amount it tells me the radius Cube divided by the period squared must be some constant and this is true of all the planets you know whether it's Mercury all the way out to Uranus or Neptune or you didn't go for Pluto I am going for Pluto I didn't know whether you'd let me go for it but I am going for Pluto's in there right this I can't let go of Pluto I know Mike doesn't is a bit more hard-nosed about it and it's true of the moon going around the earth as well so I thought maybe we'll try and show this using what I hope will be standard formula that people will remember so the only thing I want you to assume is our local hero well one of our two Local Heroes we've got George Green who's our really Local Hero but we'll stop Manor is no more than about 30 miles away from here so Newton's hours so we'll assume Newton's Laws if I have two objects let's call it m and Little M and they're a distance R apart Newton told us that the force of attraction between those objects was some constant mu is constant times Big M times Little M over r squared so I can rearrange this it's units that I'm going to be interested in so I can rearrange this right so this is just G is f r squared over big n times little n but there's one more equation what about the what about F Well possibly the most famous equation alongside E equals m c squared is f equals m a we learned that at score mass of the object times acceleration but what's acceleration in terms of units this becomes the mass I'm going to use these square brackets here to say we're talking about units that's the mass and what's acceleration kilometers per second squared for example so it's distance over time squared so now I can ask about the dimension of G the units of G so I put in the units here right so I get the units of G and it becomes that says mass times and then there's a length here but there's also a length squared there from the radius so we'll call that length cubed divided by on the bottom here we've got two masses so it's like a mass squared and in fact one of this this mask actually cancels these two little m's are the same they cancel one of them and that's why we've got that there and then we've got the time squared from here so this is times t squared so this then becomes L cubed over and it's actually this big Mass that's left and it's times t squared and all of the orbits that we're talking about are going around the same object so this object of mass m and so if I bring the mass over to this side I have G times m is actually L cubed over t squared and the mass of the smaller object doesn't even matter doesn't matter the mass of the smaller object doesn't matter and and so you know this is for example the sun of mass m and so this is the the relationship that comes from Kepler's Law there'll be some number here because when you've done the actual calculation there'll be some numbers so what you actually get is you know the the units of G times m is actually going to be some number times an L for us is the radius and the time is the period so for a given M then all the objects have the same value of R cubed over t squared that are going around it what we've effectively done here is we've kind of guessed Kepler's Law without solving some equation which is telling me about when you solve for Kepler's Law you've got to solve for an angle that's going around and it's a differential equation you've got to do changes of variables it's it's quite a complicated thing but you've got the the nub of the problem just just there and so we've effectively guessed Kepler's Law and we've done it from dimensional analysis which I think's nice and in fact we've got this unknown number here and we've got G which we haven't said what its value is but actually it turns out if you're interested in ratios of things then you actually can still make a lot of progress imagine that I've got the following that instead I think of both the earth going around the Sun but I'm also thinking about the moon going around the earth not necessarily got my length scales very good here but the Earth is going around the sun and the moon is going around the earth and in each case then I've got an equation that something like says G times the mass of the sun is equal to some number times the radius of the Earth to the Sun cubed divided by the period of the earth which is a year of course squared and we know what the radius is we've also got G times the mass of the earth is the same number times the radius of the Moon to the Earth cubed divided by the period of the Moon to the Earth with what it's about a month right look what happens if I divide one by the other this unknown number cancels and the g canceled and I'm left with a rather nice formula which says that the mass of the sun divided by the mass of the earth is equal to and it's this ratio it's the radius of the Earth Sun divided by the Moon Earth cubed and then actually it's this is the other way around it's the period of the Moon earth divided by the period of the Earth Sun squared well I know these numbers right I can measure these numbers and when you do it you get that this is a number of order 325 000. in other words I've estimated the mass of the Sun compared to the mass of the Earth without knowing anything about it you just had to know the radius I just needed to know the radiuses and the times okay and and you know I've just used dimensional analysis for it knowing the radiuses was a feat in itself though no yeah I mean I'm being a bit flippant yes it's not not easy to to determine the the radius yeah no it's it's true all right that's Kepler let me just mention one that um is probably the most famous equation of all time but just to remind people that maybe you're pretty liberal with most famous well I'm waiting for my own there are lots of most famous equations of all time what's this most famous this one's equals mc squared okay that is but what I want to um give a flavor for is that maybe people when they first saw it might be really surprised but perhaps they shouldn't have been based on dimensional analysis because I want to just remind people that at school when we're learning about energy we learn what kinetic energy is right we learned that the kinetic energy of an object of mass m moving with a velocity V is equal to a half m v squared most famous equation of all time the most famous equation of all time and but this now tells me immediately tells me right that the units of energy here because it's an equation the units have got to be the same so we pretend we don't know what the units of energy are but we do know what the units of this are we've got the mass so that's got a unit of mass and we've got velocity which has got units of length over time so this has got units of L squared divided by t squared and the question is where the physics comes in is if I want to get this relationship here equals mc squared which of course has those units right that's the point that I'm sort of driving at we shouldn't be surprised that it's of the form a mass times a velocity squared because that is the only thing that will have the correct units now what we can be surprised about is that speed of light and we might be surprised that the number is one that we can't pluck out so why isn't it so the speed of sound or why isn't it you know some other speed like the speed of of a you know a traveling car or something why is that a fundamental object because we think of this as a very fundamental thing and that of course is where Einstein's Brilliance came in I mean if you're gonna guess any speed that it was going to be you surely you would guess the speed of light because it's such a fundamental thing in the universe whereas everything like the cars and the speed of sound a moving feasts yeah you're very good okay all right I mean I would I would have got that equal since I would have figured that in no time I mean you're bang on right I mean Maxwell is the guy who was the one who linked it to electromagnetism right he realized when he was doing his phenomenal equations it was this fundamental speed emerging which was the speed of light of massless particles so it was clear that there was something in there you know the natural thing as you said linking it to light linking it to electromagnetism it was actually Einstein that really pinned down this because he he demonstrated through his theory of a special relativity that that this was a this was a cosmic speed limit that you just couldn't have information going faster than this speed and so it was an upper speed limit that was there a natural one and so it was natural that that Einstein would have this coming out and in fact it's really neat in in his paper it's a very short paper in 1905 was a phenomenal year right for Einstein with some about six papers all of which were Nobel prize winning type things this was a short paper he wrote and and what he basically said was if there is a if you have an object with a mass m that loses radiation the amount of radiation it will lose in light is is this is its mass will decrease by this amount I mean and and in here of course is a phenomenal statement about the the the amount of energy for a small Mass because we know in in units of meters per second the speed of light is like three times ten to the eight ten to the nine so Square it you've got a factor of 10 to the 18 multiplying so the speed of light wasn't quite such a big deal until it equals MC yeah I think that's right I think that I think Maxwell and and others realized there was something special about it through the electromagnetism and it plays such a crucial role but it was Einstein linking it to space and time and the evolution of space and time that that was the big breakthrough are you getting us to Black Ops black holes yeah yeah yeah I don't mess around Brad a bit all right known for my uh rapid speed so here we go black holes right so what am I going to do are we going to do a big course in general relativity no we're going to do it via dimensional analysis and we're going to do it via this skip velocity one of the things we learn at school calculator in the escape Velocity how fast would a projectile have to be sent from the earth in order for it to go into orbit before before we did this video we spent a great few minutes looking at the polo films I guess like everybody else that just arrived or all three of us are plastered to the windows look at them excuse me do you keep that way you are anyway for sure and and that had to have an escape last year right and and so that's what we're going to try and do and and from it end up with a a remarkable close approximation dimensionally perfectly correct to the event horizon of a black hole okay that's where we're going so let's estimate it and we'll estimate it for an object of mass m so that's going to be like your Earth and of a radius R so that's like the Earth orbit remember earlier we saw the units of of Newton of Newton's constant we said it's it's got units of L cubed over mass times t squared and that means that g m If I multiply it by the mass has units of L cubed over t squared you might think what you're doing what you're playing at I'm gonna get us to remember I'm doing dimensional analysis I want to get this to a velocity that's what I want us to get to look if I divide out by say the radius g m over r a radius has got dimensions of length I'm left on the right hand side with something that's got dimensions of L over t squared that's a velocity squared I've got a velocity squared out of these these combinations of the mass of the object the radius of the objects and Newton's constant so this is consistent so so I can I can say oh there's some relationship which I'll call the escape Velocity which is some number which I don't know but I'm estimating it it will something be of order one times by this combination g m over R okay let's look at this just for a moment let's remind ourselves what we're looking at we're looking at an object of some mass m and some radius R and we're asking there's little U on your would-be rocket desperate to get out and up and you will need to know what's your escape velocity to get Alpha now into orbit or out into space one thing to notice is R decreases for a given mass as I ring this object this velocity increases so as R decreases then it leads to ve increases for a fixed that's important right for a fixed mass and a natural question to ask is what size would this object have to be in order for it to not allow light to go instead of you it was light how big would that be and this was and light can only have one velocity right yeah yeah yeah what about light I keep the mass fixed and just shrink it down light's got a velocity that we know so we know that light has a velocity of C so what we can do is we can rearrange this equation here to write down what the radius will be R is equal to some number times g m over c squared and this was first spotted by John Mitchell and independently by LaPlace Pierre Simon Simon is it now plus and the remarkable thing is this was done in 1784 by John Mitchell and 1796. so this is the radius Beyond which light couldn't escape from this object it's not a black hole it's just an object that's you've just found the radius for a given Mass that's too big for anything that want to move at the speed of light to get away it's a black stone if you like or a planet or something and you can sort of put in some numbers right if I put in the mass of the Earth this would give me a radius of around two to three centimeters golf ball type of size if it was the mass of the Sun it would be the radius of around a few kilometers around two kilometers or so and below that size like couldn't escape this remarkably right this GM over um us over c squared is up to a factor of two the size of the Event Horizon of What's called the schwarztel black hole and it was discovered by it's a solution to Einstein's general theory of relativity it's and it's an object that light cannot escape from if if light ends up anything that ends up within that radius of the singularity of the black hole that light from it can't escape you can emit as Photon and it will never be able to escape and the actual size is just a factor of two different so in Einstein's general theory of relativity which you know is about 1915 he found or actually he didn't know about the existence of this until Schwartz child found it that you have a black hole with What's called the schwarzed held radius which is 2 GM over c squared where you know C is our Universal speed limit so Ed if we crunch the Earth down to less than you know two or three centimeters yeah in in size it wouldn't be a black hole yet no not this one no yeah but it would but light wouldn't be able to escape light wouldn't be able to escape you could go up to it and you'd be black be careful wow well I've calmed down a bit now Brady and I think I can begin to talk about it

Channel: Sixty Symbols

Views: 134,319

Keywords: sixtysymbols

Id: MAGdU-G5OZg

Length: 19min 58sec (1198 seconds)

Published: Fri Mar 31 2023