Black Holes - An Introduction

hello today we're going to be looking at black holes and we'll be covering issues such as the escape velocity the Schwarzschild radius the energy the mass the temperature and the entropy and Hawking radiation so get yourself a cup of tea and prepare first a working definition of a black hole is any body from which the escape velocity which I'll describe in a moment is greater than the speed of light the escape velocity means the velocity you have to travel in order to get away from that body and if you have to travel at more than the speed of light then obviously since nothing can travel at more than the speed of light including light itself nothing can get away from that body so it traps everything and you can't see it because light can't escape from it hence it is a black body what do I mean by escape velocity well it's consider the escape velocity of the earth here is the earth here am i standing on the earth and I have a tennis racquet in my hand and my plan is to hit a ball up in the air such that it escapes from the earth now ordinarily if I hit a ball up in the air with a certain velocity it will travel upwards it will slow down all the time affected by gravity until eventually it stops and then it will turn around and fall back to the ground again if I hit it a little harder so that it travels with a greater velocity then it will go up further slowing down all the while because of gravity till eventually it stops and then it will fall back down to the ground again now I want the situation where I hit the ball up into the air and it never stops it continues to get out of the Earth's gravitational field and that is the velocity I need to use to get that to happen is called the escape velocity so what is the escape velocity for the earth well we first just need to remind ourselves of two formulae that we've covered in previous videos the first tells us the force that exists between two masses let's say that they are mass Capital m and little in then the force between those two masses is G the gravitational constant times the first mass Capital m times the second mass little m divided by R squared where R is the distance between them and strictly it is the distance between the two centers of the masses assuming the masses are spherical we can also remind ourselves that the potential energy that exists is equal to G M m so it's exactly the same as for the force but this time it's simply divided by R naught R squared now let's consider the earth from which the tennis ball is going to escape and what we're going to say is it's going to travel all the way up here and it can never stop until it reaches infinity of course it will never reach infinity but conceptually if it were to stop before it gets to infinity then there will be some potential energy because the potential energy never reduces to zero it gets smaller and smaller as R increases R is the distance away from the earth but it's never zero it will only be 0 strictly when R is infinity so what we're saying is that the tennis ball leaves the earth with a certain velocity it will slow down due to gravity all the way along but it won't actually stop until it gets to infinity so at infinity the velocity will be zero so the kinetic energy which is 1/2 MV squared is also 0 because V is 0 and the potential energy is also 0 because if there were any potential energy that would pull the ball back to earth so that means that the total energy which we'll call 'total is the kinetic energy plus the potential energy and that is zero well energy can be neither created nor destroyed so the total energy kinetic energy plus potential energy has been zero all the way along this particular trajectory so if you take a point say roughly halfway what you can say is that the ball will have a kinetic energy associated with a velocity in that direction and it will have a potential energy the the scope to pull the ball back to earth due to its gravitational attraction acting in that direction and the two are exactly equal thus making the total energy zero so what happened right at the beginning when you first hit the ball well when you first hit the ball it would have a kinetic energy which is 1/2 MV squared and that V is the escape velocity it's the velocity you need to get it the velocity you need to give it in order for the ball to go all the way to infinity before it stops it would also have a potential energy and that potential energy would be G times M where m is the mass of the Earth times M where m is the mass of the tennis ball divided by R and R will simply be for these purposes the distance between the tennis ball on the surface of the earth and the center of the earth in other words it will be the radius of the earth and since we've said that at all times these two have to be equal because the sum of the two is zero then we have this equation and we can simply rearrange that to show that V squared is equal to bring this two up here to the MS incidentally cancel so this now becomes 2 GM over R that's the V squared and we can calculate that V squared is equal to 2 well G is the gravitational constant that's six point six seven times 10 to the minus 11 the mass of the earth is six times 10 to the 24 kilograms divided by the radius of the earth which is about 6,400 kilometers so that six point four times ten to the six meters well let's not be too precise about that let's just say that six point four and six point six seven broadly cancel out and then we've got two times six is 12 and then we've got 10 to the six downstairs you bring that upstairs it becomes 10 to the minus 6 times 10 to the minus 11 is 10 to the minus 17 times 10 to the 24 is 10 to the 7 or 120 times 10 to the 6 but that's V squared so V is the square root of that what's the square root of 120 well look it's about 11 11 11 121 let's not worry too much about that what's the square root of 10 to the 6 it's 10 cubed so what we're saying is the escape velocity from the earth is 11 kilometers per second which is approximately equal to 7 miles per second which is approximately equal to 25,000 miles per hour that's the speed you would have to give that tennis ball if it were to escape the clutches of the gravity of the earth now a professional tennis player probably hits a tennis ball at about 160 miles an hour so that's a long way short of 25,000 miles an hour so there's no danger that tennis players will launch their tennis balls into orbit but you'll also notice that 7 miles a second which is the escape velocity for the earth is a long way short of the speed of light which is 3 times 10 to the 8th meters a second or perhaps we should compare this it is 11,000 meters per second 11,000 the speed of light is 3 times 10 to the 8th 300 million meters per second so the escape velocity from the earth is a long way short of the speed of light so the earth is not a black hole because a black hole needs an escape velocity of greater than the speed of light so let's just remind you of that formula for escape velocity again V squared equals to GM over R where m is the mass of the earth but you'll notice that V squared will get greater as R gets smaller so if you could take the earth and squash it retaining all the maths you can't get rid of the mass the mass must remain the same but you just squash the earth down to a smaller size how small do you have to make the earth before the V squared or the velocity the escape velocity becomes equal to the speed of light well let's rearrange this formula and we get that R is equal to 2 GM over V squared but we are saying that we want the escape velocity to be the speed of light so that becomes 2 GM over C squared well we can work that out that means that the radius of the earth if it were to be a black hole that means all the mass of the earth must remain within that new squashed sphere the radius will be 2 times G which is 6 point 6 7 times 10 to the minus 11 times M which is as we said 6 times 10 to the 24 kilograms divided by C squared well C is 3 times 10 to the 8th so C squared is 9 times 10 to the 16 well very roughly again 2 times 6 times 6 is about 72 72 divided by 9 is 8 so that's dealt with the numbers now we need to deal with thee with the orders of magnitude well 10 to the 16 brought upstairs becomes 10 to the minus 16 times 10 to the minus 11 is 10 to the minus 27 10 to the minus 27 times 10 to the 24 is 10 to the minus 3 and so we calculate that you'd need to squash the earth down such that it had a radius of 8 millimeters and then it would be a black hole but only if so we're talking about the size of a marble here we're talking about a diameter of about one point six centimeters so if you could squash the earth down to the size of a marble with radius eight millimeters retaining all the mass because you have to do that retaining all the mass inside that marble the earth would become a black hole now I want to ask the question if you could indeed squash the earth down to a marble whose diameter is one point six centimeters but whose mass is six times ten to twenty four kilograms in other words the full mass of the earth is contained within this marble if you could do that would it stay that size well let's think about the ordinary earth the earth before we squash it and I'm going to put here's a floorboard on the earth and I'm going to put a 1 kilogram weight on that floorboard and I want to know what is the force of gravity acting on that one kilogram weight well we had the formula before the formula is GMM over R squared and in this case G of course is the gravitational constant we know that M is the mass of the earth gets capital M little m is my 1 kilogram weight and our just move that back R is the radius of the earth it's the distance from the mass to the center of the earth so we can work that out F is going to be G which is six point six seven times 10 to the minus 11 times the mass of the earth which is six times 10 to the 24 times the mass of the one kilogram weight well that's just one / r squared well the radius of the earth as we've said before is six point four times ten to the six meters so this is six point four times ten to the six all squared because this is now an R squared term and we can cancel six times six and cancels with the six squared term now you've got a ten to the six squared term which is 10 to the 12 bring it upstairs it becomes 10 to the minus 12 10 to the minus 12 times 10 to the minus 11 is 10 to the minus 23 times 10 to the 24 give you 10 so the force is going to be 10 Newtons 10 Newton's of force acts on the floorboards due to the gravitational attraction why then does that 1 kilogram weight not simply fall through the floorboards why is it that when I stand on the floorboards gravity doesn't pull me through them the answer is that the floorboards are themselves made up of a solid lattice of molecules and atoms and they are held together by their own atomic and molecular forces or bonds as they are called and they actually have a strength that's what makes a solid a solid it's because there are bonds between atoms and bonds between molecules that exert a force to hold them together and those forces are stronger than the 10 Newton force which is applied by gravity to the 1 kilogram weight so the 1 kilogram weight doesn't fall through the floorboards because the molecular structure of the floorboards is capable of withstanding that now let's suppose you do squash the earth down to the size of a marble here's a floor board in theory and there's my 1 kilogram weight and I now want to ask what is the force acting on that one kilogram weight due to the gravity of the earth which is now the size of a marble exactly the same formula is going to apply f it was GMM over R squared and the GMM are not going to change that means the force is going to be six point six seven times ten to the minus eleven times the mass of the earth which remains the same we've squashed the the same mass into the size of a marble so that's six times ten to the twenty four times the mass of this little m times the mass of the one kilogram weight which is one divided by R squared but R now is eight millimeters because we've squashed the earth down to eight millimeters well that's eight times ten to the minus three squared now let's see if we can calculate that we've got six times six divided by eight squared look let's just call that a half we could actually call it one that's not going to be of any consequence at all now we've got 10 to the minus three squared well that's came to the minus six bring that up to the top it becomes 10 to the plus 6 10 to the 6 times 10 to the 24 is 10 to the 30 10 to the 30 times 10 to the minus 11 is 10 to the 19 Newtons the force acting on that one kilogram mass due to the earth if it were the size of a marble is 1/2 well let's forget about that but 10 to the 19 Newtons 10 billion billion Newtons the force acting on the earth when it was the ordinary size was just 10 Newtons and the floorboards were well able to withstand that but when you get to the earth crushed to the size of a marble the force acting on that one kilogram weight is 10 billion billion Newtons and there is no known forced man that wood can withstand that atomic and molecular forces can't even the strong and the weak nuclear forces can't and the electromagnetic force card so you now have an irresistible force acting according to the black hole so consequently if the black hole were one point six centimeters across gravity would be acting on the outside of that mass pulling it in and there would be no force that could stop that from happening so consequently if you could squash the earth down to the size of a marble it wouldn't stop it would carry on contracting under its own gravitational force it would be as it were shrunk still further and nothing could stop it until it reached a dimensionless point which is called a singularity and what we say is all the mass of the earth would then reside in a dimensionless point it can't have any dimensions if it has any width at all the gravity will just pull it in so it keeps pulling it in until it has no finite dimensions and that is what's known in mathematical terms as a singularity there will however be a radius around that singularity of eight millimeters which says that anything inside that radius has an escape velocity of greater than the speed of light but anything outside that radius has an escape velocity of less than the speed of light and that radius is called the Schwarzschild radius so a black hole will always fall into a singularity because the gravitational attraction on the outer parts of the matter making up the black hole will always be so great that nothing can withstand it so the black hole must always shrink to a singularity but there will be a radius around that singularity inside which the escape velocity is greater than the speed of light outside which it is less so the point is that if you travel inside the Schwarzschild radius you are doomed you will never be able to escape but at least outside the Schwarzschild radius there is hope and I just want to clear up one other point which is often a point of misunderstanding if this is the whole uncompressed earth the normal sized earth with a radius of six point four times ten to the six meters we are not saying emphatically not saying that at the center of the earth there is a marble shaped piece of earth that is a black hole there is no black hole at the centre of the earth because in order for it to be a black hole the whole of the mass of the earth would have to reside inside that marble at the center and it clearly doesn't the mass of the earth in the shape of a marble at the center of the earth would be very small only if the whole of the earth has been squashed into that marble shape at the centre of the earth would you have a black hole there is no black hole at the centre of the earth so now the question is how big do black holes have to be and the answer is they can be any size they can be ultra massive and they can be as small as the size of a subatomic particle but what you have to ask yourself is how are you going to create a black hole because the way we've talked about it so far is to take the earth and squash it well actually I don't know how you could do that so that becomes a rather impractical proposition we've simply been looking at it theoretically but suppose instead of this being an earth this is a star and it needs to be a star that is significantly bigger than our Sun what is happening for the bulk of the time that the star is alive is that gravity is acting on the outer edges of the star trying to pull it in words but that is balanced by the pressure that is created through the nuclear fusion reactor that are happening inside the star and so consequently the star remains stable the gravity which is tending to cause it to collapse is balanced by the nuclear fusion which is pushing it outwards but at the end of the star's life the nuclear fusion stops and now there's only gravity consequently the star collapses in on itself and once again the question you have to ask is what's going to stop it now in some cases it may be that the atomic forces will stop a collapse in some cases it may be that the power Li exclusion principle will stop the collapse in other words Polly says no two electrons can occupy the same state and that may stop the collapse in some times gravity is so great that it just knocks Pauli's exclusion principle for six all the electrons and all the protons are squashed together to become neutrons and then you get what's called a neutron star it's very much smaller it contained principally of neutrons and the neutron forces can withstand the pressure of gravity but if the star is massive enough nothing can withstand the gravitational collapse and consequently the star collapses to a dimensionless point and that is a black hole so that is one mechanism for creating a black hole but in theory any mess could become a black hole provided there is a mechanism for getting that mass to live within its shorts chilled radius effectively to become a dimensionless point now let's ask the question does a black hole therefore suck everything into it is everything in its force in vicinity doomed well the answer is no let's consider just the ordinary earth here is the ordinary earth and here is the moon and the moon of course as we know orbits the earth and the reason it orbits the earth at least as far as Newtonian mechanics is concerned is because the earth exerts a force on the moon which is GMM over R squared and that force provides the centripetal force that keeps the moon in orbit and the centripetal forces MV squared over R so that's what's keeping the moon in orbit and R is the distance between the center of the earth and the center of the moon now suppose the earth were suddenly by some mechanism to become a black hole so it just becomes a black hole with all its mats residing in a dimensionless point as far as the moon is concerned it doesn't care the g m and m remain the same the mass of the earth is still Capital m it's just that it's now all in a dimensionless point at the center R squared remains the same the distance between the moon and that Center that singularity remains the same so the moon just carries on orbiting the Earth which has become a black hole there is now no greater gravitational force because these features remain the same so consequently the moon wouldn't care if the earth became a black hole it would simply continue to orbit it this incidentally is a feature of what's called Gauss's law Gauss's law says and this applies both in the electrostatics and also electromagnetism and also in gravity that when you're talking about the effect of a mass it doesn't matter whether that mass is a spherical mass or all the mass is residing in a single point the singularity as long as you're outside the sphere as long as you're here you must be there as long as you're outside the sphere then this formula applies it doesn't care whether M is a spherical mass or a singularity the same conditions apply that's a feature of Gauss's at all but of course we can ask the question what happens if you've got something like a meteor heading towards the earth well what it would have done when the earth was the normal size what it would have done here's the meteor coming towards the earth let's forget about the atmosphere of the earth probably the meteor would burn up in the atmosphere but let's forget the atmosphere it would just continue to accelerate under the force of gravity until eventually it hit the earth and depending on how big it is there might be an explosion well suppose that the earth has become a black hole and the black hole of course will have a Schwarzschild radius around it all that now will happen is that the meteor will continue to be attracted it will cross the Schwarzschild radius where in theory nothing particularly happens although in a moment I'll explain something might and it crashes into the singularity and it is absorbed and consequently the black hole increases its mass by any material or energy that hits it so for example the cosmic microwave background radiation which is coming from all directions will hit the black hole it cannot be really mitad so consequently the black hole increases in energy due to the cosmic microwave background radiation and any other space debris that happens to be passing by and is stupid enough to get caught will fall into the black hole now I said that you could pass the short chilled radius and nothing particularly happens well actually that's not quite true it rather depends remember the gravitational forces at this point are going to be massive if you remember we calculated the gravitational force of a black hole the size of the earth on a 1 kilogram weight were some of the order of 10 billion billion Newtons so there's a massive gravitational attraction that's not the problem the point is that there will be what's called very high tidal forces so the difference between the gravitational attraction at the Schwarzschild radius and say six feet above the Schwarzschild radius will be huge and so if you were falling into the black hole the gravitational attraction and you were feeling falling feet first the gravitational attraction on your feet when you got to the shorts chilled radius would be 10 to the 19 Newtons but the gravitational attraction on your head now here you are falling in feet first the gravitational attraction on your feet is going to be huge the gravitational attraction on your head is going to be significantly less and the consequence is you're just going to get stretched out of all existence you will be torn apart as you cross the shorts your radius if not before for the next part of this video I won't forget about black holes just for a moment because I want to remind you about features of space-time you remember that we draw space-time as time and space now space of course has three dimensions up and down side to side back and forth but I can't easily draw three dimensions of space and one dimension of time so I'm simply going to for simplicity draw one dimension of time because there is only one and one dimension of space we'll forget about the other two dimensions this is a one dimensional space and remember that the principle of space-time is that if you stand still in space you still move through space-time because time is moving so if you're standing still in space you won't move in this direction but standing still in space will result in you doing that because you are moving forward in time when we draw a spacetime diagram conventionally we draw the speed of light at an angle of 45 degrees so that represents the speed of light that is light traveling at a distance in a certain time so this is three times ten to the eighth meters per second consequently I think you'll agree and understand that it is quite impossible for you to travel all like that in space-time because that implies that you have traveled a distance in space but in no time at all you have traveled at infinite speed and that's impossible special theory of relativity says you can't gain more than the speed of light so consequently motion in space-time must be at an angle of greater than 45 degrees that is possible that is not because that would suggest you're going greater than the speed of light that suggests you're going less than the speed of light this line remember says you're not traveling at all in space but you are traveling in time and of course you also agree won't you that it's impossible to do that in space-time because that implies you're going backwards in time okay so now let's think of space-time and we're going to consider an accelerating object in space-time so first of all let's just draw our space-time coordinates time and space and we'll start here at 0 velocity and we're going to accelerate away and we've seen in previous videos that the shape of an acceleration is something like that it just means that for every second that goes by you travel faster and therefore you travel further so consequently this curve is produced that is the curve of an accelerating body constant acceleration but of course continually increasing velocity means that you cover a lot more ground in the same amount of time but in a space time chart we've already said that you can never travel at an angle of grate of less than 45 degrees so once you've got to and here is the angle of 45 degrees is about here the acceleration does not allow you to do that so strictly if we're going to draw a proper space-time chart because you're not allowed by special relativity to go at greater than the speed of light if that's the speed of light and you start out here accelerating what you actually do is to asymptotically approach that line but you'll never get to it that's the point you never get to it and well you do when you reach infinity but not before so you accelerate you get progressively faster but you never reach the speed of light you continue to get faster and faster by infinitesimal amounts and your Assam topically asymptotically approaching the speed of light and this curve here I could for example draw the other side of it this curve is called a hyperbola this is a hyperbolic function and it shows how you would accelerate in space-time of course we're talking about acceleration that is getting you up too close to the speed of light ordinarily if you were if you accelerate on a motorway you probably can do this but of course the acceleration would not take you anywhere near the speed of light but when you're accelerating such that you're getting close to the speed of light that's when you get asymptotically closer to the speed of light and your acceleration curve becomes a hyperbola okay so let's draw the space-time chart again this time we're going to imagine that we've got there's the speed of light here is the acceleration which pipe which is a hyperbola and asymptotically approaches the speed of light and we're going to consider this is a rocket and the rocket contains Alice and Bob it's always Alice and Bob and when the rocket gets to this point here at Bobby is going to do something very unbalanced he's going to actually put Alice outside the rocket the rocket is accelerating constantly and that's its path through space-time but of course as soon as Alice is put outside the rocket she is no longer got any separate propulsion she's got no force Newton's first law kicks in and she will simply continue to travel at the velocity was traveling when she was put outside the rocket by Bob and since the velocity she was traveling in is that direction she will proceed through space time now in a straight line traveling at a constant velocity now can Alice this is Alice traveling and this is Bob still in the rocket can Alice see Bob and I'm talking about visual communication or any other form frankly of electromagnetic radiation she can communicate by radar waves if you wish well remember that the speed of light can only travel at 45 degrees so it can either go in that direction or it can travel in that direction those are the only options that you've got for light traveling through space-time according to the rules that we have said about the speed of light so yeah Alice has no problem seeing Bob because light from Bob traveling at 45 degree mark will continue to reach her wherever she is so what she's actually seeing of course what this space-time diagram is telling you is that she sees Bob getting progressively further away represented by the increasing length of these lines and that's perfectly fair what does he fective lee happened is that bob has dropped her off traveling at 30 miles an hour whilst he accelerates away consequently she will always see him getting further and further away that's fine what does Bob see when he looks for Alice well I think the first thing you can recognize is that once Alice crosses this point here in other words once she crosses this line which represents the speed of light Bob will never be able to see her why because light which leaves Alice from this point on can never get back to Bob to do so it would need to travel at a angle that's less then 45 degrees and that means it would have to travel it more than the speed of light and light travels at the speed of light so light from Alice goes like that like that like that like that like that but it's never going to reach Bob because Bob is always on this side of the speed of light line whereas the light coming from Alice which can go up in either direction of course it can go that way at 45 degrees or this way at 45 degrees never gets to Bowl so you might conclude from that that what Bob will see and it's okay when when Alice is here Bob can see her then because light from Alice will indeed reach Bob but you might think that what Bob will see is Alice drifting towards this point here and then disappearing from view but that's not actually what happens I'm just going to blow up this particular part of the diagram so what we've actually got is here is Alice's trajectory here is the speed of light and here is Bob's trajectory what will actually happen is that light from Alice will reach Bob traveling at 45 degrees you know the speed of light but it will take an increasingly long time for that light to reach Bob because Bob is travelling further and further away and remember he's going very fast he's almost going at the speed of light so light itself is going to take a long time to catch him up it will do because the light is traveling at the speed of light and Bob is traveling at slightly less than the speed of light but this is just going to continually go like this and what is actually going to happen is that Bob looking back will perceive that Alice is traveling slower and slower towards this point here but never crossing it now that is just a pure perception in fact what will happen of course is that Alice traveling at the velocity she was traveling when she was dumped off the rocket or the spacecraft will just continue to go through that line and carry on no problem but Bob's perception is quite different he perceives that she goes slower and slower and slower and slower and slower and slower concern but never crosses that line is the difference between reality and perception and the reason I went through all of that we can now go back to black holes is that that is exactly what happens here's a black hole a dimensionless singularity here's the Schwarzschild radius around it that is the radius remember within that radius the escape velocity is greater than the speed of light outside it's less if you've got somebody traveling towards the black hole to certain doom and you've got somebody observing from the outside but safe maybe they're in orbit around the black hole so they're safe the people traveling towards the black hole simply recognize that they are accelerating they're getting faster and faster and faster until eventually they hit black hole and Calamity but the observer observing those people will actually see the same kind of phenomena as ball what they will see is the rocket getting slower and slower and slower and slower and slower and slower and never actually reaching the short children radius so actually it's a bit of a false hope because the observer thinks that they haven't yet met their doom and there's still time to save them the reality is they have long since crashed into the black hole and been completely annihilated and the reason for this is that the gravitational forces here are so strong that light is having an enormous difficulty traveling away and consequently the observer simply sees this spacecraft apparently taking longer and longer and longer and traveling slower and slower and slower to get to the short chill radius but never crossing it and there is a mathematical equation which straights this just to remind you from special relativity we have shown that there is an invariant quantity called the proper time the point about special relativity is it says that all observers if they are measuring a particular distance or they're measuring a particular time depending on their relative velocities they will measure different distances and different times so nobody agrees on the measurements but there is a quantity called proper time which everybody can agree on and the way you can calculate proper time is to say that C squared which is the speed of light times tau squared where tau is the so called proper time the time that everybody can agree on is equal to C squared T squared which is the time that any observer measures minus x squared where X is the distance that anybody measures so if you measure a time and distance and somebody else measures a time and a distance then they will get different answers but if you plug it into this formula both parties will get the same tau that's just from special relativity and you can see my videos on special relativity for more information about that but i do it to show that there is a similarity to this new equation which i'm going to show you which is called the Schwarzschild metric and that is derived from Einsteins field equations of one day I will work out a way of explaining that with sufficient simplicity that I can do it in the style that I like but basically what the Schwarzschild metric says is that C squared tau squared is and I should say this now has to be done in a polar coordinate so we have where instead of using x y&z Cartesian coordinates we use or sorry ordinary Cartesian coordinates we use R theta and Phi and it becomes 1 minus R s over R where R s is the Schwarzschild radius times C squared T squared minus R squared over 1 minus R s over R minus R squared D Omega squared where D Omega is this is shorthand it's a combination of theatres and thighs which I don't need to go into now because that's not going to be relevant to what I'm going to show you but you can see a kind of similarity with this term and this term you've got C squared Tao squared in both you've got a C squared T squared you've got an R squared instead of an x squared but that's because we're using polar coordinates but you've got this peculiar term here 1 minus RS over R and we said that RS is two mg over C squared that is the formula for the Schwarzschild radius that we calculated earlier in this video so now you can see that two things happen there are peculiar things happening when R is equal to RS this term becomes 0 because RS over RS is 1 and 1 minus 1 is 0 you also get funny things happening when R itself is 0 in other words when you hit the black hole when you hit the black hole RS divided by 0 becomes infinity and so you've got an infinity term here you've also got an infinity term here and effectively the whole thing blows up and the singularity is not one that obeys the speed Schwarzschild metric anymore indeed the whole point about the singularity is that the laws of physics generally don't apply anymore but there's also the interesting question what happens when R is RS in other words when you are at the Schwarzschild radius in those circumstances is our s over art is 1 1 minus 1 is 0 0 times C squared T squared is 0 minus R squared over 1 minus artists over R well 1 minus R is over re 0 because we are at the Schwarzschild radius so R equals R s so this term blows up to infinity and you can forget about that one for these purposes and what that is showing is remember proper time is the time that is read by the person traveling so if you are traveling in the rocket or in the rocket is run out of fuel if you are traveling in that rocket heading towards the black hole to your certain doom and you've got a clock inside that rocket with you proper time is the time you measure on the clock because you're traveling with the clock the times that other people measure are the times that they observe of you and what this formula essentially is showing is that for an infinitesimally small amount of time on the rocket when the proper time as measured by the person on the clock on the rock on the rocket is an infinitesimally small amount of time the time measured by the observer will be infinite and that's why the rocket the rocket appears to travel ever slower ever slower ever slower without ever crossing the Schwarzschild metric because what is actually happening is that the observer is seeing the rocket over a very brief instant of time but that instant of time appears to be an infinity of time in practice the rocket of course has accelerated accelerated accelerated and smashed into the black hole but the observer according to this metric here will see time expand enormous Li so that in fact what seems to be a fraction of a second on the rocket becomes an eternity of time for the observer and that's why they never see them cross the Schwarzschild metric of the Schwarzschild radius I now want to look at the energy and entropy of the black hole so there is the dimensionless black hole and there is the Schwarzschild radius around it and I want to add one unit of energy to the black hole so what I'm going to do is I'm going to allow one photon to fall into the black hole and we know that the energy of a photon we've done this many times before is H nu where nu is the frequency of the light of which the photon is apart and that is the same as HC over lambda H is Planck's constant and we've done this in many other videos now if we assume that the wavelength of the light is of the order of the Schwarzschild radius which it needs to be if lambda is very much bigger than the short short radius then the light might just pass it by with never ever seeing it but if lambda is broadly of the order of the Schwarzschild radius then you can have this situation we take the famous Einstein formula e equals MC squared which means that a change in mass is equal to a change in energy divided by C squared that just comes from this formula here but Delta e well we can say for one photon Delta e is HC over lambda so that's HC over lambda that's Delta e times C squared and the C cancels so that now becomes H over lambda C so that is the change in malice as a consequence of one photon falling into the black hole now we have said that lambda is our s and we know that our s is equal to two mg over c-squared because we calculated that earlier in this video and if we take this formula here and ask how does the radius vary according to increasing mass then you would simply say that Delta R an increase in radius will equal to Delta mg over C squared because that just falls out of that a small increase in mass leads to a small increase in the short chilled radius but we can substitute for Delta M here what we what we calculated here with lambda equal RS so now we can say that Delta R is equal to Rho 2 G over C squared times Delta M which is H over lambda C but lambda we're going to say is comparable to the Schwarzschild radius that's 2 G not 29 up here and if I bring that RS term up here I get that RS Delta R is equal to 2 G divided by C squared times H over C from here I've just brought that RS term up on this side of the equation and that's 2 G H over C cubed and that's a constant sorry 2 G H over C cubed is a constant now the surface area associated with the Schwarzschild radius is going to be 4 pi times the sportster radius squared because that is the surface area of a sin and if I take da by D R that is just going to be 8 PI R s because you're differentiating with respect to R you get 8 pi r s and that means that da is equal to 8 pi RS D are just multiplying by dr but we know what our SDR is because we calculated that up here so now we can substitute that da is equal to 8 pi times our SDR which is 2gh OC cubed and that if you like is saying that the area increases by an amount da as a consequence of putting one photon of energy also one element of entropy because we'll regard a photon as being the basic unit of entropy and that is that formula if you put in several units of entropy several photons then the increase in the surface area will be 8 pi 2 gh over C cubed times D s where D s is the number the amount of entropy and if we rearrange this formula we get that and now have SN a we now get that s is equal to a C cubed divided by a set of constants times G H in other words we're going to take the 16 pi and just make it a constant and that is known as the bekenstein formula it shows you how the entropy of the black hole changes as its area increases that's the area of the surface area associated with the Schwarzschild radius we can also consider the temperature the black hole in our thermodynamics video we derived the idea that the change in energy is equal to the temperature times the change in entropy now for one you photon of energy we have said above that the change in energy you is HC over lambda that was what we derive right what we can derive it we we stated it because we've derived it in or shown it in other videos so the change in energy from one photon is HC over lambda so now we can say that for one photon where D s essentially is one because we're saying one photon is one unit of entropy so for one photon we've got that de equals T because D s is one and D is also HC over lambda but remember we're also saying that lambda is broadly of the order of these shorts shield radius so that's HC over R s and R s is two mg over C squared so that's HC over two mg over C squared and you can see that H and C and C squared and G are all constants which means that you now have that T is proportional to one over m the only variable in this term here is the mass M so this is an interesting and surprising result that as the mass of the black hole increases its temperature decreases and as the mass decreases the temperature increases so a small black hole will be hotter than a big black hole that's contrary to usual concepts because we reckon that as mass increases energy increases because e equals MC squared so an increase in mass equals increasing energy and of course we usually associate increasing energy with increasing temperature but black holes it's quite the reverse and finally professor Stephen Hawking has proposed that black holes could lose energy and indeed in some circumstances evaporate away by radiating out energy and you might say how can that possibly be given that we've just spent most of this video showing that once material or radiation it's into a black hole it can never escape because the escape velocity is greater than the speed of light so how can anything ever get away from a black hole and the answer is that it actually doesn't the principle is all based on what's called quantum fluctuations the argument is that in space where you have nothing from absolutely nothing at all where energy is nothing quantum fluctuations allow particle pairs to be created usually matter and antimatter they then recombine and take you back to energy is nothing so the books balance you start with nothing you end up with nothing ordinarily you would say nothing comes from nothing but in quantum mechanics that's not true from nothing something can appear as long as the books balance and you go back to nothingness again you certainly can't create energy out of nothing but for short time you get these quantum fluctuations where things materialize and then D materialized it's a fundamental principle of quantum mechanics now suppose that happens here is my black hole it's the singularity and here's the Schwarzschild radius around the singularity and suppose a quantum fluctuation happens here outside the Schwarzschild radius and two particles appear matter and antimatter and this particle flies off in this direction and this particle flies off in this direction now they if this is a capable of escaping because it's outside the shorts to radius so there's a theoretical possibility that it can escape the two will never remit in order to annihilate consequently the argument goes that the universe which is the universe out here has got additional energy from this particle from nothing this particle was created together with this particle if this particle survives and goes off into the wider universe the wider universe has gained energy the books must balance so this particle must effectively represent negative energy and negative energy going into the black hole will reduce the amount of energy of the black hole consequently you can have a situation where the black hole will evaporate because it is getting negative energy and therefore reduction in its total energy in practical terms of course if you take massive black holes that doesn't happen because the amount of material going into the black hole the cosmic weight microwave background radiation the debris of space that may be attracted to the black hole that input into the black hole is the black hole with all the material going into it radiation and matter which is adding to the mass of the black hole against that you've got some fairly feeble reductions in the energy in the black hole due to this process of payer production and as a result of quantum fluctuations outside the Schwarzschild radiation outside this Schwarzschild radius the evaporation is nowhere near as great as the amount of material coming in so the net effect is the black hole just continues to grow but if you have a very very small black hole that could evaporate away almost instantaneously and that's possibly the reason we don't see small black holes because the capacity for them to evaporate is very high and they just disappear altogether and that is a brief introduction to black holes

Channel: DrPhysicsA

Views: 411,415

Keywords: black hole, Schwarzschild radius, energy, temperature, quantum mechanics, hawking radiation, cosmology, entropy, Schwarzschild metric, Einstein's Field Equations, Escape velocity, Force, Potential Energy, Kinetic Energy, Physics

Id: 9WbrujNXSw8

Length: 61min 46sec (3706 seconds)

Published: Tue Nov 20 2012