The Dome Paradox: A Loophole in Newton's Laws

I've got a question for you what will happen if I place this ball right at the Apex of this Dome and let go well you probably guessed that but now imagine we're in a theoretical world where this Dome is perfectly smooth no friction or bumps there are no disturbances or forces acting other than gravity and I can balance this ball perfectly perfectly perfectly at the Apex now what do you think will happen when I let go if that was your guess you're right Newton's first law says that an object at rest will remain at rest

unless acted upon by an external force or so we thought this is a paper by an Australian philosopher of physics named John Norton it claims that Newton's Laws also say that at some random time the ball will spontaneously start rolling off the Dome Without a Cause no quantum mechanics involved just good old Newtonian physics why am I making a video about this you ask there's obviously some mistake a ball can't just start rolling on its own Without a Cause but here's the thing this paper was published in 2008 it's been 16 years since Norton published

this result and no one can point out what's wrong with it and it's not for like of trying this is a map that shows other research papers that have talked about the Dome there are rebuttal rebuttal of those rebuttals outright denials and a lot of heated arguments some scientists think that this result might even reveal a fourth law of motion and others think it's a proof that humans have free will I'm responsible for lots of provocative ideas and philosophy that gets me into trouble now when I first read this paper I thought surely this is

just some kind of light of hand math trick but after weeks of investigating I realized just how deep this Rabbit Hole goes It goes beyond hey a ball is moving weirdly in this one specific scenario it challenges the entire mathematical Foundation of Newtonian mechanics the most widely used and well understood framework in modern physics Humanity would not be where it was if it wasn't for Newton's laws and the math derived from them while they can't quite describe the inner workings of an atom or when things are moving at near light speeds they describe pretty much

everything in between so how does this little Dome challenge all of that and what does it mean for physics before we dive in this is an extremely subtle topic and I want you to understand what this video is and is not about it's not about whether a physical ball will really move off the top of a physical dome in the real world in reality you can never place a ball right at the Apex without it rolling down as you can't create the the ideal conditions and even if you could quantum mechanics would interfere at some

point this video is about what the mathematical theories we use to understand reality tell us now I know some of you are thinking what's the point of that if this is purely theoretical and doesn't apply to the real world who cares well there are two reasons one sure we might be working in an Ideal World Without friction or disturbances but anyone who's done even high school physics would know that that's basically how we deal with the real world in physics to make problems simple enough to solve we make assumptions like there's no air friction objects

are Point masses and no energy is lost in collisions so this idealized framework is pretty standard in physics and two imagine two gigantic numbers numbers so big that that many things couldn't physically fit into our universe now imagine that if we found out that our rules of addition didn't work for them when we add them together we get something weird and unexplainable it wouldn't physically affect anything in the real world but it would mean that our theory of addition doesn't work all the time and that's something worth knowing that's exactly the point of the Dome

you'll see limits in Newton's theory that we weren't aware of and how even the most wellestablished theories can still surprise us now we're actually going to ease into this with a not too technical handwavy explanation to give you some intuition for why this works in Newtonian mechanics the laws of physics work the same way forward and backwards in time if you record a physical process and then play the recording backwards the Reversed sequence should still obey the same physical laws this is called time reversal symmetry now if you accept that it's possible to nudge the

ball with just the right amount of force so that it rolls up to the Apex and stops there by time reversal symmetry it's a valid solution for the ball to rest at the top of the Apex for a while and then roll down the starting point of this is John man's book on called a primer on determinism that came out I think sometime in the uh in the 1980s and John just alerted us to the fact that there were all sorts of cases of indeterminism all the way through physics this is super important let's break

it down Newton's flws are awesome at making predictions they were even used to predict the existence of Neptune before it was ever observed if we're given the position and velocity of something at the present time we can calculate its position and velocity for any future time said simply the future state of things depends on their present State and importantly there is only one possible future state for any present state if there were more than one possible future State well that would throw our powers of prediction out the window the idea that any present state has

only one future state is called determinism determinism is a pretty huge part of the philosophy of Newtonian mechanics pretty much since its conception physicists have viewed Newtonian mechanics as a deterministic theory the way determinism manifests itself in the math of new's theory is like this you might have heard that maths is the language of the universe in that case the dialect of Newtonian mechanics is differential equations the solutions to differential equations tell you how things change over time you can imagine that would be pretty useful as all of motion is pretty much how an object's

position changes over time the way determinism is expressed mathematically is that each differential equation has only one solution one possible trajectory this is called the uniqueness theorem if a Newtonian equation had more than one solution for the same initial conditions that would mean that a present state had more than one possible future State and that would break determinism and that is exactly the problem with this Dome the equation of motion that describes how the ball moves when it's placed at the Apex has more than one solution it's like one of those Choose Your Own Adventure

books if you start at the beginning the initial conditions you can end up reading entirely different stories but determinism says that Newtonian mechanics is like a regular book when you start at the beginning or when you're given one set of initial conditions there should only be one story so how did Norton come up with a scenario that defies centuries of classical physics it must have taken years of hard work and toil I mean this is really the work of just a short afternoon there wasn't terribly much to to do it oh so here are the

solutions to the equation of motion see more than one this solution describes the ball sitting at the Apex forever exactly what we would expect and this solution says that at some random time called the excitation time the ball just starts rolling down the Dome on its own so my first thought when I read this was I do not understand where did this come from why did this come from how does it what sure we can plug the solutions into the equation of motion and see that they both work but that's just like yeah okay it

works because math obviously those thoughts aren't what you want in an educational video so I thought you know what the only way we're going to understand this is if we just figure it out ourselves from scratch right now so that's exactly what we're going to do imagine you hate Newtonian determinism and you want to come up with some situation that breaks it what's the first thing you would do well you might first want to figure out how to break determin ISM in other words how to come up with a differential equation that has more than

one solution there are conditions that tell you uh when a differential equation will have a unique solution and I knew one of those conditions was the LIF shits condition and it was just it's just standard in in mathematics books that that they'll say well if you want to have a unique solution you need special conditions the lipshits condition is one that will give you the unique solution if you don't have it here's how uniqueness might fail thanks Nan okay lipshits condition in the theory of differential equations lipshits continuity is the central condition of the Picard

Lindelof theorem which guarantees the existence and uniqueness of the solution to an initial value problem okay that seems promising H so it turns out there's this really important theorem in math called the Picard Lindelof theorem it gives us set of conditions for an equation to have a unique solution if all these conditions are met this guarantees uniqueness but if we don't meet all these conditions an equation can have multiple Solutions that's good that's exactly what we want an equation that has more than one solution so we can break determinism which we hate the central condition

of the bicard Lindelof theorem is the lipshits condition or lipshits continuity if we don't have lipshits continuity this can lead to splitting and branching of a solution so guys we just need to break lip's continuity how do we do that well the first step might be finding out what it is H yes interesting what could it be H basically the lip shits condition makes sure a function doesn't change too abruptly that its slope doesn't get too big too quickly or explode if you will it definitely should not be vertical in math talk it needs to

be increasing by a finite amount not an infinite amount there are actually a lot of functions whose slopes explode like this at some point a simple one is y equals the < TK of X it doesn't look too crazy but let's zoom in right around xal 0 you can see the slope becoming more and more vertical and if we zoom in far enough we can see the slope blow up to Infinity breaking lipshits continuity guys we've made good progress we've found a function that breaks lip shit's continuity which leads to multiple Solutions which breaks determinism

which we hate okay now what this isn't a Newtonian system this is just a function an abstract relationship between X and Y we're trying to break newtonium determinism which means we need to break determinism within a Newtonian system okay so next question what's a Newtonian system basically any physical scenario that's described by Newton's Laws objects moving under Gravity tension friction or any other force that can be described by Newton's Laws so how can we use our function and get it to describe some kind of Newtonian system well imagine you've been given a job to build

the slide at your local playground you can choose how steep and curvy it is which directly affects how fast the kids will accelerate at each point but you've built it too Steep and too curvy now you've been asked to make sure the slope and curves are such that the kids stay on the slide and aren't going so fast they get scared so you figure out the right acceleration and you work backwards to build the shape of the slide that that ensures this acceleration so just as a shape influences the motion of an object on it

if you start with what kind of motion you want you can reverse engineer the right shape similarly let's say we want an object like a ball to accelerate according to this relationship the simplest way to get a ball to accelerate is to put it on a slope and add gravity a ball accelerating under Gravity that sounds like a Newtonian system to me let let X be the position of the ball and Y be the acceleration of the ball when the ball is at position zero its acceleration is also zero what could that physically correspond to

well it could be a ball sitting still on a flat surface now as the position of the ball increases so does its acceleration what can we do to our surface to make the ball move like this well we could make it get steeper as position increases we end up with a kind of ramp and if we make it symmetrical about the origin we get a dome in fact if we do the math we end up with exactly this Dome so that's how the Dome breaks determinism as I mentioned it sparked a lot of controversy a

lot of people outright hated the Dome and tried to invalidate it scientists and philosophers had plenty of problems with it but no one could agree on a specific definitive FLW that complet completely invalidates it nothing concrete enough to point at and say this is the reason the Dome doesn't work one of the biggest points of contention and what was my biggest problem with it is this whole Ball moving by itself thing surely this is just wrong so why are we even entertaining it as a valid solution even theoretically we discard Solutions all the time in

physics so why not just Chuck it out physics cares about math but math doesn't care about physics it's up to us to determine what makes sense and discard what doesn't but this raises the question how do we decide what makes sense how do we know when to discard a solution well a good indication is when it doesn't make physical sense or if it violates a known law of physics take the example of a ball being rolled against a wall viewing this situation through the conservation of kinetic energy we can say that the square of the

initial speed used s is equal to the square of the final speed after it's hit the wall v^2 solving this equation we get two solutions instinctively this is the correct one as it corresponds to the ball rebounding Off the Wall this one corresponds to the ball going through the wall which from experience doesn't make any physical sense we know balls can't travel through walls this solution both doesn't make physical sense and it violates Newton's third law of motion when two objects Collide they exert an equal and opposite force on each other so we can safely

check it out so can we Chuck out this problematic Dome solution well it seems pretty unphysical to me that a ball can just start suddenly moving by itself but here Norton's got an answer for us he says that the whole unphysical argument doesn't apply here because the Dome is a mathematical creation within Newtonian Theory therefore we have no prior knowledge about whether the ball should stay there forever or whether it could spontaneously move off the fact that we get an inmic solution is not impossible it's just unexpected okay but what about the fact that this

result clearly breaks Newton's first law of motion which stated in its entirety is in the absence of a net external force a body remains at rest or in a state of uniform motion in a straight line but here nordan's got an answer too he argues that we're used to thinking of uniform motion in a straight line over some time interval but in this context we need to apply the law to a single instant of time the law then becomes in the absence of a net external force a body should have zero acceleration in other words

zero net force equals zero acceleration so is there a moment where this instantaneous version of Newton's first law breaks well before the ball moves or as Norton calls it before the excitation time the masse is sitting still at the Apex there's no net force acting on it as gravity is balanced out perfectly by the normal force and it's not accelerating this meets our conditions after the excitation time the mass is accelerating down the dome but it's also got a net force acting on it gravity it accelerates in accord with FAL Ma so this situation checks

out too and at the exact moment of the excitation time well this is obviously the point of Interest now I don't generally like doing this but the only way I could think to explain this part is through the math I don't like doing that because I don't think it gives great intuition but I honestly don't think there's an intuitive explanation for this if you take the solution where the ball spontaneously rolls off the Dome you can get its acceleration by differentiating twice or just looking at the paper the exact moment the ball moves is when

time small T is equal to the excitation time Big T so when we plug that into our equation this term becomes zero so the entire thing is zero no acceleration along with no net force just as Newton's first law says so there is no actual moment we can pinpoint where this instantaneous version of Newton's law is broken so why does the mass move well this is exactly Norton's claim Newtonian physics is indeterministic there doesn't need to be a why we're just so used to thinking about this in a causal way he says our natural causal

instinct is to seek the first instant at which the mass moves and then look for the cause of the motion at that instant we're tempted to think of the excitation time as the first instant at which the mass moves but that is not so it is the last instant at which the mass does not move there is no first instant at which the mass moves the mass moves during the interval after the excitation time and this interval has no first instant so there is no first instant of motion and thus no first instant at which

to seek the initiating cause but I'm curious to know what you guys think are you convinced by Norton's arguments leave your opinions in the comments below so now the burning question what does this all mean for physics well in the most extreme case it could mean that Newtonian mechanics is not a deterministic theory but it feels a bit strange to say that an entire theory is not deterministic because of a few edge cases that break it more likely is that Newtonian theory is just not as clearcut as we thought and we don't understand it as

well as we thought we did some philosophers go so far as to say there isn't a single conception of Newtonian physics there is not one true version but many versions which are all correct even if they're all at odd with each other this journey into the Dome has been so cool because it revealed that something that we thought was kind of pure and unified and whole and complete could actually be this fragmented thing that has many versions it's surprising to think that after all this time we're still not sure how to interpret one of the

oldest models of physics isn't that surprising though you might be shocked at what kind of questions you'd think are pretty basic that still puzzle scientists especially to do with something you'd think we'd have figured out by now us humans scientists still can't agree on things like how we first started speaking was it a slow gradual process or was it like a linguistic Big Bang why do we sweat while other animals don't what were the moments that made us human it's a really fascinating topic and my friends over at real science made an entire series about

it called becoming human which I highly recommend it's a deep dive into the Define ing moments of our Evolution the different theories put forward by scientists and it's one of my favorite things on nebula a platform that's quickly becoming the go-to space for the best educational content on the internet nebula is built and run by creators with the mission to be the best place for us to make the work that we couldn't make anywhere else I've always wanted to take my videos to the next level and nebula is enabling that for our entire list of

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