Norton Paradox Explained | I Find Out It isn't a Paradox!

追加: 2026-05-09

[00:00:00]Welcome back to my channel. So today I will make a faster versus to prove the y not on assumption for the part of uh the non dom product was wrong. So first of all we can see that uh we can derive the differential equation r prime mt which is the acceleration which is equal to the square root of the distance r when it distance or it actually aistic from the top of the dom to the bottom of the dom and followed by the equation a = 2 + g * r to the power over two. So I can tell it by using the oiler electron equation to derive even though people like to derive with geronomy but uh I will say that the kinetic energy minus the potential energy which I find is where how I can find it because I just plugging the equation mga where I replace a with this quantity. So I can finally find out that well after I derive that r prime equal to negative square root of r. So what is this negative here? Why did not don't have this negative or actually because uh the a here it actually reduction in my assumption but in hit assumption that this this a actually increases. So that's why that solution doesn't have a negative. So I will come straight to the problem. First of all the oiler electron equation I derive what correct. So um this equation we can see that it related to the equation mg8. So we can plot it in and we check out that's energy conserve which mean I I can use the assumption where energy conserve in this uh solution of the non and what is the non paddle actually are well they say that the first solution say that when you put a point m on the top of the dome because there's no slope it mean it like locally flat surface so therefore only the for pushing it down it gravity and the normal for pushing it up this make the acceleration equal to zero which is saying that the object can be only stationary or moving at constant speed.

[00:02:04]But because the non using the initial condition like this the initial condition is that the velocity equal to zero at time equal to zero and the location also zero at time equal to zero. Well this mean that the ball should not moving and this first solution make total s because it should not moving. level r equals z. But the second solution is that he had a quantity big t here called the spontaneous time which mean that we can put this and after the spontaneous time the ball will start moving and he said it had to do with the Newton uh determinism problem. He said that Newtonian physic cannot give determinism. So what about this? Turn out the mystery of the second solution come from the special point at the top of the dump where the slope suddenly become zero and the slope is not zero at any time at any r given that it for example if r equ= to 0 01 is already not zero so that act like kind of the singularity kind of like the problem of general relativity which I will explain later. So look at the another classic problem is that the mass on spring problem which is uh the simple harmonic motion we need to say the for equal to negative k time x and notice when I put the initial condition is that the the mass is on the location where the for acting is zero here's no zero force here so the velocity at the initial time equals zero is also zero I have only one solution that x equal zero. So that tell you something. So at this turn out so I know this statement is true in a force field for potential energy to transfer the kinetic energy at a point the for zero must require a nonzero initial velocity otherwise the total energy will equal the potential energy which always stay constant because when a system when there is no external force from outside the system only the system within the energy is conserved there's always potential energy coming along with associate to the four mean that the derivative of potential energy with respect to rotation can tell you the magnitude of the four which can easily see in the radian uh equation. So in that special point where the four is equal to zero which mean that this location the four is zero. If we set an initial condition equal velocity equal to zero, it mean that potential energy no longer able to convert to kinetic energy. And this is a very strong state.

[00:04:51]Then imagine if you dig the hole through the entire earth assuming no air resistant, you can ch and you will accelerate and then bass the center and come to the other side. But then you will continue to fall back and you will al isolate it like this. But if you start at the center at the initial velocity, you will stuck there forever because the force is zero and you require an initial velocity to overcome that right. So that mean the equation here is uh similar. We get only one solution x equal to zero. So this should tell you something. This is exactly the same for the situation of the dom. So we have a force field that can input the location of r and tell you the four is zero at r equal to zero for the dom. So this part of initial condition which is not on you to divide heat equation which already tell you that the kinetic energy is zero. So the potential energy must equal to the total energy and because the initial position is also zero. So it mean that the force on this location is zero here. So it mean that the object no longer able translate potential energy to the kinetic energy because the force acting on the objects it inside the system unless if we push the objects then it can fall. So there's an agreement that you can uh slide at a a table on this and you can make this solution valid by put this table slightly at the uh at time equal to big t here. So it can fall but it will violate the conservation of energy. That is why I don't like that equation because when the fall come from outside it mean that you act the energy into the system which is make this uh potential energy no longer valid to you but that just mean that the force come from outside the system and it's pro Newton is correct because mass time acceleration equal to the force upon it.

[00:06:46]So if an object before it is standing still and after that is moving at a constant velocity it prove there must be a for acting on it on the path. People might like wait a second why do I bring the conservation of energy here but if not just um challenge me that though solve the problem only doing with Newtonian physics but conservation of energy doesn't seem so well physic right so physic depend on absolute quantity that we all read about for example in relativity that's quantity is called the constantity of speed of light so speed of light is always constant due to divide the entire theory of special and general relativity. So the not theorem telling us that uh when respect to the symmetry of the universe always arrive conservation laws and one of that is the conservation of energy which is a loss of nature and that must make a clue for us to telling what is happening. So sorry not northern I had to explain further than Newtonian physics to explain that but in the end you got your answer and I will explain further this had to do with the math rather than uh the issue with determinism. For example, if I have one this ball here sitting on the 1D table that's infinitely long we can say there's the for gravity putting it down and the normal for pushing it up. So the total for zero which mean that the acceleration equal to zero. If you rewrite this differential equation, you get this solution which either tell you two things. One, the ball is stationary. Two is that the ball moving at a constant speed because why? Because uh Newton already said that when no acting object, the object can move at a constant speed uh or standing still. But even that you give two outcome, not one outcome. That's just the nature of differential equation. But it didn't seem like I get to the what not talking about. Noon talking about the special condition at the top of the dome where we had this weird solution. Well, this is kind of like the singularity problem in the general relativity. Now I come back to that problem. So when structure uh derive a equation from the Einstein field equation his solution telling us that we had a kind of like sphere spherical coordinate called the structure coordinate. But when the coordinate r the radial coordinate r come at some point equal to a quantity called the structure radius you get infinity. So that isn't come to the center of the black hole. Yes. So that got the first singularity but then many mathematician and physicist trying to change the coordinate to the different coordinate that does that obstruct you and turn out that's that infinity go away.

[00:09:42]So that mean that the problem come with the coordinate that is right the situation not the situation itself and for the dom here we had that uh exactly issue here the issue come from the top of the dome where mathematician say it doesn't violate the coppit equation uh sorry the cosepit condition where the it require a units net to solve a qual equation but in the end I can easily prove to you using the conservation of energy total energy in the system and say that at a point of 4= Z in order to potential energy to convert to kinetic energy an initial velocity is required.

[00:10:20]If it is zero the object no longer able to convert potential energy to kinetic energy. So the object must be stationary. That's only one outcome. So, thank you for watching.