Gravitational Potential Energy in a 2 dimensional Universe

[Vashta Nerada]

47 minutes ago, studiot said:

Gravitational potential is not 'energy'

The PE results from a potential difference

A potential is not the same as a potential difference, although they will have the same units.

 

Take an isolated single body, possessing mass.

There exists a gravitational potential around it.

But there is no energy, potential or otherwise involved.

 

Now introduce a second massive body.

Potential energy now arises from the configurational interaction of the two bodies and perhaps also kinetic energy.

This is the same for electric and magnetic fields.

It is also worth noting that the potential energy is conventionally set to negative infinity at the source, reducing to zero at infinite distance.

I repeat it is easiest to try these questions out in one dimension first.

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.

[studiot]

5 minutes ago, Vashta Nerada said:

(since the universe always tend to a less energy state),

 

The universes in this case cannot tend to a state of lesser energy.

The total mechanical energy of an isolated system is constant.

[Genady]

5 minutes ago, Vashta Nerada said:

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.

This is incorrect. Force is a gradient of potential energy and its direction is the direction of the gradient. Does not matter what a sign or even the value of the potential energy is, only its gradient.

6 minutes ago, Vashta Nerada said:

It's like the bodies would always approach each other until they reach 1 meter apart, then they starts to repulse each other.

Again, this is incorrect. Since the force is a gradient it would change direction only if the gradient changes direction. Changing the sign of the potential energy doesn't change the direction of force.

[Vashta Nerada]

30 minutes ago, studiot said:

If that were the case then the gravitational field would contain an infinite amount of energy.

The point that I already made is that whilst the length of the bounding curve is infinite, the area under it remains finite.

https://en.wikipedia.org/wiki/Improper_integral

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:

1. The graph needs to tend to negative infinity when the distance tends to zero;
2. The graph needs to tend to zero when the distance tends to infinity;
3. As we increase the distance the value must increase too (the derivative must have only positive values);
4. 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.

12 minutes ago, studiot said:

 

The universes in this case cannot tend to a state of lesser energy.

The total mechanical energy of an isolated system is constant.

you're talking about what case? the G.M.m.ln(d) one?

Edited February 17, 2022 by Vashta Nerada

[Genady]

11 minutes ago, Vashta Nerada said:

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:

1. The graph needs to tend to negative infinity when the distance tends to zero;
2. The graph needs to tend to zero when the distance tends to infinity;
3. As we increase the distance the value must increase too (the derivative must have only positive values);
4. The integral from any point (greater then zero) to infinity must be a finite value.

How many solutions like this do you want? Here is one, for example, -G.M.m/(e^x-1). Or, -G.M.m/x⁴

Edited February 17, 2022 by Genady

[Vashta Nerada]

7 minutes ago, Genady said:

How many solutions like this do you want? Here is one, for example, -G.M.m/(e^x-1)

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?

Edited February 17, 2022 by Vashta Nerada

[Genady]

Just now, Vashta Nerada said:

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, as you and @joigus said, Newton did it from the Kepler's law, not from the idea of field "dilution" that you've described in the OP. If you want to do it like Newton did, you need to start with a 2D version of the Kepler's law.

[Vashta Nerada]

1 minute ago, Genady said:

But, as you and @joigus said, Newton did it from the Kepler's law, not from the idea of field "dilution" that you've described in the OP. If you want to do it like Newton did, you need to start with a 2D version of the Kepler's law.

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...

[Genady]

25 minutes ago, Vashta Nerada said:

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...

in this case, for your 4 statements to hold, you have proved that the idea of field "dilution"  can't hold. But what is wrong with the idea that the Newton's law holds as is?

[Vashta Nerada]

1 hour ago, Genady said:

in this case, for your 4 statements to hold, you have proved that the idea of field "dilution"  can't hold. But what is wrong with the idea that the Newton's law holds as is?

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

[studiot]

1 hour ago, Vashta Nerada said:

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...

I'm not sure what your idea of field dilution is or entails.

However I have some suggestions.

If you want to work in terms of what I suspect 'dilution' to mean, I suggest you work in terms of energy density not energy.
This will also work more easily in any number of dimensions. This would be energy per unit length, area or volume as appropriate.

I attempted to address to your idea of 'leakage' by pointing to the vector curl, but you have not responded to this.
Perhaps I have misunderstood your explanation of this issue ?

[Genady]

8 minutes ago, Vashta Nerada said:

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

What is wrong when we try to integrate?

[Vashta Nerada]

2 hours ago, studiot said:

If you want to work in terms of what I suspect 'dilution' to mean, I suggest you work in terms of energy density not energy.
This will also work more easily in any number of dimensions. This would be energy per unit length, area or volume as appropriate.

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.

[studiot]

5 minutes ago, Vashta Nerada said:

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.

No, I was keeping in mind this is a discussion about Classical Physics.

Working in terms of densities -  something per metre, something per square metre, something per cubic metre is a very common technique.

Of course if you go to two or three dimensions your integrals become area or volume integrals, which is why I keep recommending using the one dimensional case for starters.

 

How about answering my questions, they are designed to help ?

[Genady]

And, answering my question would be nice, too. Here it is again: what is 'wrong' when you try to integrate the Newton's gravitational force? it seems to obey all your conditions.

[Vashta Nerada]

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.

Edited February 17, 2022 by Vashta Nerada

[Genady]

2 minutes ago, Vashta Nerada said:

Taking the G.M.m.ln(x) solution as true does not fits on the second condition, ...

No, the G.M.m.ln(x) is NOT a solution for the Newton's law. The -G.M.m/x is. My question is, what is wrong with it? In 2D, I mean. This is what I mean by "the idea that the Newton's law holds as is."

[studiot]

24 minutes ago, Vashta Nerada said:

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.

 

15 minutes ago, Genady said:

No, the G.M.m.ln(x) is NOT a solution for the Newton's law. The -G.M.m/x is. My question is, what is wrong with it? In 2D, I mean. This is what I mean by "the idea that the Newton's law holds as is."

 

Actually none of this is a solution to Newton's Law of Gravity.

Newton did not use potential theory  -  It had not been invented in his day.

In modern notation Newton's Law of gravity is

[math]F = G\frac{{Mm}}{{{r^2}}}[/math]

 

This is a straighforward algebraic expression and a solution means that you have a value for all the variables, except one so you substitute them into the expression to obtain the unknown quantity.

So instead of arguing at cross purposes, Vashta how about posting your maths working to obtain whatever you have obtained ?

Then we can help you tighten up your maths.

 

As regards the curl, do you understand the concept of a vector that represents a turning moment ?

I am convinced that this is the key to your difficulty reconciling 2D and 3D.

 

 

 

Actually

[Markus Hanke]

If you want to know the gravitational potential for 2D, just solve Laplace’s equation - you’ll find that the potential is logarithmic, as it needs to be, since the force now follows a 1/r law.

However, the story is actually more complicated - the more general law of gravity isn’t Newton, but GR. If you apply GR two a 2D universe, you find that the Weyl tensor identically vanishes; in vacuum, the Ricci tensor vanishes as well, as per the Einstein equation. Since the Riemann tensor decomposes into Weyl and Ricci, the result is that in 2D there is no gravity in vacuum at all, outside a mass distribution. You only have gravity in the interior of masses.

[joigus]

1 hour ago, Markus Hanke said:

If you want to know the gravitational potential for 2D, just solve Laplace’s equation - you’ll find that the potential is logarithmic, as it needs to be, since the force now follows a 1/r law.

However, the story is actually more complicated - the more general law of gravity isn’t Newton, but GR. If you apply GR two a 2D universe, you find that the Weyl tensor identically vanishes; in vacuum, the Ricci tensor vanishes as well, as per the Einstein equation. Since the Riemann tensor decomposes into Weyl and Ricci, the result is that in 2D there is no gravity in vacuum at all, outside a mass distribution. You only have gravity in the interior of masses.

Indeed. 1+2 gravity has no local degrees of freedom. Yet people still use it as a topological theory.

https://en.wikipedia.org/wiki/(2%2B1)-dimensional_topological_gravity

Just a disclaimer:

I wasn't thinking of GR when I asked my questions, even though we know it to be the right framework. I was thinking under the premise "if Newton's gravity were correct..."

And I stand by what I said. Namely: we can't deduce any of that. We can't deduce Laplace's equation either. We can guess at it from symmetry properties. Also, I think it would be interesting, like @studiot was trying to do, to find out at what level the OP wishes the question to be answered. Maybe they're not familiar with GR.

[Markus Hanke]

2 hours ago, joigus said:

Namely: we can't deduce any of that.

Sure, we can’t deduce the exact law, in particular not the constants. However, I think we can deduce the general form it needs to have - that is just a consequence of the generalised Stokes Theorem. 

[joigus]

8 hours ago, Markus Hanke said:

Sure, we can’t deduce the exact law, in particular not the constants. However, I think we can deduce the general form it needs to have - that is just a consequence of the generalised Stokes Theorem. 

Agreed. If we adopt the basic assumption that the equation for the gravitational potential be the topologically simplest assumption (no field lines start out from the vacuum), then Laplace's equation is the rotationally-invariant version of this assumption. From the initial assumption, gravitation can be deduced now.

If we go to GR, as you said, it's a different matter. I do remember that the binding of a small particle coupled to a strong spherically-symmetric gravitational field has higher-order inverse powers of the distance than 1/r².

[Genady]

1 hour ago, joigus said:

Agreed. If we adopt the basic assumption that the equation for the gravitational potential be the topologically simplest assumption (no field lines start out from the vacuum), then Laplace's equation is the rotationally-invariant version of this assumption. From the initial assumption, gravitation can be deduced now.

If we go to GR, as you said, it's a different matter. I do remember that the binding of a small particle coupled to a strong spherically-symmetric gravitational field has higher-order inverse powers of the distance than 1/r².

Agreed. "From the initial assumption" is a key phrase. We can make ANY physical assumption about a 2D universe. What would stop us?

I wonder, why 2 dimensions in space. Wouldn't it be much more exciting to consider a universe with 2 dimensions of time?

Edited February 18, 2022 by Genady

[joigus]

5 minutes ago, Genady said:

Agreed. "From the initial assumption" is a key phrase. We can make ANY physical assumption about a 2D universe. What would stop us?

I wonder, why 2 dimensions in space. Wouldn't it be much more exciting to consider a universe with 2 dimensions of time?

You just touched my soft spot.  

It is an interesting possibility, that I've tackled before in the way of a suggestion: That it's just possible that both time and space are multidimensional, but we (and other physical systems) may just be constrained to perceive it as 1+3-dimensional or, IOW, 1+a bundle of the rest, that we perceive as spatial+internal. It's just a speculation, of course, but a speculation that I think can be argued in favour of within the limits of mainstream, widely-accepted science.

I'm not sure if anybody has proposed anything like this. My intuition is that somebody must have, because almost any idea you can conceive of has previously been considered by someone.

[studiot]

9 minutes ago, Genady said:

We can make ANY physical assumption about a 2D universe. What would stop us?

You have made some very valid points in this thread, but I believe Vashta is talking about our Universe.

Further the rules of SF require our Universe as this was posted in classical Physics.

 

12 hours ago, joigus said:

And I stand by what I said. Namely: we can't deduce any of that. We can't deduce Laplace's equation either. We can guess at it from symmetry properties. Also, I think it would be interesting, like @studiot was trying to do, to find out at what level the OP wishes the question to be answered. Maybe they're not familiar with GR.

I agree, but would go further and suggest that GR is inappropriate in this thread, except as a passing mention. +1

Incidentally Markus said solve Laplace, not deduce it.

I have plenty of expositions of Newtonian gravity involving Laplace, I just want to help Vashta find the appropriate format.

Of course solving Laplace will not get us the potential. It will get us the potential function, which is different.

It is confusing to those just studying this subject that the word 'potential' is used in several different ways.

I actually must now apologise and correct an incorrect statement I made earlier about the units.

I said that potential energy and potential (difference) have the same units.

This is not quite correct.

PE has units of energy, potential by itself or PD has units energy per unit mass.