Vashta Nerada

You got the point 😉

Agreed, all of that IS YOU. But what is exactly "you"? And also, the hole point here is to understand if you are a conscious entity or just an unconscious reproducer "machine" of information. And that second one for me don't feel like a free will. I mean... If free will is just the fact that your brain is capable of process data that was conceived by organic sensors and then generate a response to that, then sure, we all have free will, there's no doubt that the brain do that. Now, for me, free will were something more like the ability of having a master conscious that allow you decide whatever you want, independent of the chaos surrounding you. And that for me is not true. In that case we would be creatures without free will. Yeah, that enters in what I said of "Maybe that's where our Free Will is 😋."

No, Perhaps I did not make myself clear. What I tried to say is that always when you have to make a decision, since you had a specific and unique set of inputs for your whole life until now, then there's only one possible way that you could response to that, you don't have a "choice", your response is just a resultant of the chaos of the universe. You think that your response was created and elaborated by you as if you were a conscious entity, but the fact is that you were always doomed to response exactly like you did. Simplifying, think in an isolated box full of particles with random positions and directions speeds. Then at a given time one particle hits another particle and changes it's direction and speed. Did that particle changed it's direction speed because it wanted to? No, it was just a causality given by the previous state of the system, and it could be easily predicted once knowing that state. Same idea for our brains, but in a much greater and complex scale.

I think we don't. Thinking in a biological perspective, I think (and that's only my opinion) that we are just a very complex biological machine that can process information and responds to it. Information that was captured by our biological sensors that are all around our body. What can explain why different people receive and interprete the signals in different ways, because each one is unique, and have a unique set of biological sensors. And also no one has gone through the same experiences as the other. Our brain is a very incredible machine that can store incalculable Terabytes of memories and experiences. So I like to think that we as humans are only a result of all the experiences that we had in our lives added to the characteristics of our tools that externalize that processed information. So, no... I don't think we have free will at all, I think that all decisions that you take is just a result of your past and the way you process the information. It's like the idea that if you know the status of all particles in the universe (such as position, energy, momentum, velocity, etc.), you would be able to "predict the future", because you would "just" have to calculate all the interactions between all those particles to know their future interactions and so on... So, why couldn't this idea be applied to our brains too? Now, just one thing that makes me consider this idea is the fact that we can't predict at all the interactions of particles in a quantum level. For us (at least so far) these interactions are literally random and unpredictable. Maybe that's where our Free Will is 😋.

Okay, so, can you clarify to us what is a Virtual Number, please? Yeah, but actually he didn't mention the imaginary numbers. Imaginary is different from Virtual.

First of all, zero is not a "virtual number" or whatever it means, zero is so real as any other positive or negative number. Also, zero is an even number too, making it an integer. It follows all the "even number rules": Its between two odd numbers (-1 and 1), it can be divided by two... And about infinity, different from zero, it's not a number, infinity is an idea. You cannot consider it as a number because it can literally have many sizes. For instance, think in the quantity of all positive integers, it's infinity, right? Now think in the quantity of all integers (both positive and negative), it's also infinity, but somehow it's twice as big as the previous infinity that we thought. Now think in all the real numbers, somehow it's something like infinitely times the others infinities before. But anyway, we treat it all as just infinity... So it is definitively not a number. And about infinity being the "polar opposite" of zero I hope that you're not talking about the inverse, because the inverse of zero is not infinity. What we can say is that the limit of [1/x] as x approaches zero from the right side is equal to infinity, and if we imagine this same limit as x approaches zero from the left side it leads us to negative infinity, this duality prevents us from saying that 1/0 is infinity. Now if you're talking about the diametrically opposite point of a circle then I imagine that you're taking the number line and bending it in a circular shape, connecting the two infinity extremes, but it makes no sense to me, and I don't see a correlation with the Big Bang.

Because I believed that the /r² term was a consequence of our 3D universe, so if we try to adapts that to a 2D universe it would change the term to /r. But as you said, it's just an assumption so it doesn't matter, there's not a right way to do. It could be 2D universes where indeed this assumption is very right (thus the PE is weird), but also others where it doesn't, leading us to infinite possibilities. So I think the right answer is: Undetermined 😅

Exciting? Hell yeah! Realistic? Absolutely not. Since our conception of reality is limited in only one "time dimension". I mean... We can't even insure that time is a dimension, I love the idea that it might be, but in fact it can also be other thing totally different of a dimension aswell. One thing that makes me believe more that time could be a dimension is the strong space-time curvature inside the Event Horizon of a Black Hole, at this region of space (or this moment in time :P) the curvature is so strong that space starts to become time-like and time starts to become space-like, the Event Horizon is no longer a place where you where but a moment at your past, and the singularity is no longer a place aswell, it is a moment at your future, that you cannot avoid. I like that because time and space literally switch, leading us to assume that they could be the same thing. For me, the only truth of time is that: Time is nothing but a stubborn illusion... 😅 But anyway... As @studiot said, we're trying to (at least I insinuated that at the first topic) think in an universe like our universe, as it is, but with the only exception of taking out one space dimension. And also trying to maintain or readapt the physical properties, but that's the tricky part.

Hi, sorry if I didn't answered your questions, guys. It's been a rush day at job. So, first to @studiot: Yeah, energy density seems a new good approach to my supposition, I'll take a look on that. And about the vector curl, well... being honest I'm not very familiar with this concept, but I'll surely take a time to study that later. Now, @Genady. You're talking about the integration of the force in 3 dimensions or in 2? Because in 3 dimensions is just fine, works perfectly. I even deduced the integration step-by-step at the OP, leading to: PE(x) = -G.M.m/x But when we go to the 2 dimension situation, is where the scenario goes weird, and I've been saying it several times along our discussion. Taking the G.M.m.ln(x) solution as true does not fits on the second condition, the limit: x->∞ would leads us to infinity. That would imply that objects very very far apart have already a very strong PE associated, that sounds wrong. This universe would quickly collapses in itself.

Yes like the Quantum Wave Equation, right? I thought that too, but now the things starts to be more complex. I was hopeful that we would be able to obtain the PE equation only with Classical Mechanics.

The idea of Newton's Law is right (I think). What's wrong is when we try to integrate to obtain the PE.

Actually, there's not only one way to derive an equation, we have like infinite ways to do that. Newton did it by the Kepler's Law because that was what he had at his time, and with that he just invented the hole thing (Genius). But now we are in 2022, and we have Newton Classical Mechanics, and also all the other contribution of scientists after him. So we have the knowledge in our favor. Also, remember: Kepler's Law was obtained mostly by observation. We don't have a 2D universe to observe. But if the idea of field "dilution" in 3D is right (I even don't know if it is) then we could apply it to a 2D model and we should get a similar result. At least that was I wanted to...

You miss my point, I don't want to think in an equation that satisfies these 4 statements, for that we have infinite solutions. What I want is to get the right one, by taking the right steps, understanding the process. Like Newton did. But for that, I know that when I found a solution, it must satisfies these 4 statements. You get what I mean?

Exactly.. perfect. Is that what I want... I want to find a solution similar to -G.M.m/x² (obviously not the same), but similar. And with similar I mean have 4 things in common: The graph needs to tend to negative infinity when the distance tends to zero; The graph needs to tend to zero when the distance tends to infinity; As we increase the distance the value must increase too (the derivative must have only positive values); The integral from any point (greater then zero) to infinity must be a finite value. If the graph founded violate any of those 4 statements, then we have a problem. And the solution G.M.m.ln(x) violates the statements 2 and 4. you're talking about what case? the G.M.m.ln(d) one?

Yeah, agreed. Maybe I didn't detailed, but since the beginning the whole point was to calculate the Energy from the interaction between the two bodies. For example the Earth (that is "stationary") and a smaller object (like an apple). Yes, the graph of the PE makes perfectly sense and it's beautiful, it starts as zero at an infinite distance and then starts to grow (negatively) as the bodies approach, tending to a negative infinite value (in the hypothetical case where the distance is zero). And also, to have a good understanding it's nice to perceive why is that Energy always negative: is just linked to the attractive nature of the Gravitational Force. The two bodies wouldn't spontaneously accelerate towards each other if that was a case of greater energy (since the universe always tend to a less energy state), but since they do approach, we might assume that the closer they are, the less would be the PE. It's simple, just think that to separate two bodies we have to apply a force within a distance, that means, apply an Energy to the system in order to increase the distance of the bodies. And if we keep doing that, applying more and more Energy to separate more and more the bodies, someday they will be at an infinite distance apart, where the PE would be zero. But how could it get into zero if I was increasing the Energy? Simple... It was negative before. That's also why I don't buy the idea that the PE equation for a 2D universe would be G.M.m.ln(d) , because that graph is negative for distances less then 1 and positive for distances greater than 1, tending to infinity. It's like the bodies would always approach each other until they reach 1 meter apart, then they starts to repulse each other. Don't make any sense to me.

Hi, Genady! Sorry, I actually saw your comment, but I'm still trying to convince myself that this equation may be right, but the truth is I kind of don't think so. It just don't feel right. Because if EPG(x) = G.M.m.ln(x) , then the Energy at the infinity would be: EPG(∞) = G.M.m.ln(∞) = ∞ And I don't kow... That makes no sense for me... How could the Energy of two bodies that don't interact with each other be Infinite?

Hi, Studiot, Yeah, I actually already done calculus and I'm well aware of the difference between a definite and an indefinite integral. But the thought in question was actually in combine three things: Classical Physics, Calculus and The fantasy of 2D universes. Here I didn't understand why you reinforced that "for energy we must use a definite integral" because it was exactly what I did... I made an integral going from Infinity to "d"... It is in the first post. And actually, We can technically use a Indefinite Integral, but for that we must use a boundary condition to found the value of C, like that: EPG(x) = Integral(G.M.m.dx/x²) = -G.M.m/x + C Boundary Condition: EPG(∞) = 0, so: 0 = -G.M.m/∞ + C -> 0 = 0 + C -> C = 0 Finally: EPG(x) = -G.M.m/x So the question was all the time in how would it be the EPG for a 2D universe since the Energy (Area below the Force graph) tends to infinity in that case, and for that I got no answer.

Yeah... Just like @Genady said, 2D World is (yet) only a fantasy that we create. We don't know if it exists or if it's even possible to exist. However, it makes sense. An analogy that I like to make is to just think in a Universe with 4 dimensions, the intelligent forms of life at that universe would be so more complex than we are that from their perspective, a 3D universe would seem easily impossible or unlikely to exists (just like 2D are from our 3D world), but here are us . We exists. So maybe it's very possible to exist a 2D universe, but all the things would be total different, particles would be total flat, without thickness, mass is now something attached to the Area of things, and not the Volume. Remember... there's no Volume in 2D. You can't imagine a functional 2D world without letting go of our models of the universe. But maybe some concepts of physics is similar to ours, who knows... That's why I started this topic.

Just like I wrote at the beginning... Actually, I think Newton derived using Kepler 3th law (T²/r³ = constant). But anyway... If we think, let's say, the light intensity of a point source spreading in the space (in our 3D universe), the formula would be simply: B(x) = P0/4.π.x² given in [W/m²]. It's obvious to see that the power is diluted in the surface of a sphere of radius x. So it is inversely proportional to the distance squared because it is spreading in an area. So it's reasonable that we imagine that gravity would kind of do the same. It's intensity would spread in the universe inversely proportional to the distance squared. Now, for two dimensions, the power will expand in a circle shape, being diluted in the linear circumference of radius x. So the formula is: B(x) = P0/2.π.x given in [W/m] The same idea for gravity... For two dimensions it would be inversely proportional to the distance, not the distance squared.

Ok. But I thought that we consider the force "starting" at the infinity because at that imaginary distance the two bodies wouldn't experience any force or energy at all, It's like there's never been an interaction between them, because they have never even been close. At the infinity, the body A doesn't exerce Gravitational Force in body B (consequently they have no EPG) and vice versa. If the bodies are from a finite distance apart, that mean that they already had a Gravitational Force applied, so they already have EPG associated (not zero). So, for me it feels very wrong to define the EPG being 0 at a finite distance L because that would imply that for distances greater then L the energy would be inverted in signal, that's madness.

Okay, but. Why in 3D we can get a general equation that gives us the absolute value for any distance but in 2D we can't?

Hello everyone! I was trying to imagine how some properties of physics (like energy and forces) would be in a 2D Universe. But I found some irregularities... Thinking in 3D terms and considering that gravity exists, we can easily deduce the equation for the Gravitational Force (like Newton did): It must be proportional to both two-body masses; Since we live in a 3D universe we can imagine that the gravity extends by a distance in 1 dimension, remaining the other 2 dimensions for it to spread, forming a spherical propagation. So it must be inversely proportional to the distance squared; We need a constant to fix the units and values. So, the equation must be: FG = G.M.m/d2 Now, to derive the Gravitational Potential Energy formula we just need to integrate the force with respect to the distance that it was applied (Energy = force x distance). We also need to think that this force was applied since an infinite distance all the way through a distance "d" (the distance where the two bodies are now appart), these will be our integration limits. So: EPG = Integral[∞->d](G.M.m.dx/x2) = -G.M.m.Integral[d->∞](dx/x2) = -G.M.m.(-1/x)[d->∞] = G.M.m/x[d->∞] = G.M.m/∞ - G.M.m/d = 0 - G.M.m/d = -G.M.m/d Ok, all good. But now, if we think in a 2D universe, we can deduce that the Gravitational Force would be inversely proportional to the distance, since the gravity would spread like a circle. So, the equation must be: FG = G.M.m/d (of course, the G constant will have different units) Now, the irregularity that I found was in trying to derive the Gravitational Potential Energy, because following the same idea, we have: EPG = Integral[∞->d](G.M.m.dx/x) = -G.M.m.Integral[d->∞](dx/x) = -G.M.m.ln(x)[d->∞] = -G.M.m(ln(∞) - ln(d)) = G.M.m.ln(d) - ∞ :( That was the irregularity... this energy makes no sense... So it must exist some wrong definition that I made... I don't know if it was in the 2D Gravitational Force or in the concept of Energy in a 2D universe... But if someone know, please share with me the knowlage. Thanks!