Gravitational Potential Energy in a 2 dimensional Universe

[Vashta Nerada]

1 hour ago, Genady said:

Wouldn't it be much more exciting to consider a universe with 2 dimensions of time?

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.

[Genady]

7 minutes ago, Vashta Nerada said:

... 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, ...

Then, why do you need to change the Newton's law at all rather than hold it as it is, i.e. GMm/r²? See what the consequences are and change something else if needed.

[Vashta Nerada]

21 minutes ago, Genady said:

Then, why do you need to change the Newton's law at all rather than hold it as it is, i.e. GMm/r²? See what the consequences are and change something else if needed.

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 😅

Edited February 18, 2022 by Vashta Nerada

[joigus]

2 hours ago, studiot said:

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

I couldn't agree more. While the GR discussion, as well as alternatives/generalisations/toy-models is very interesting, perhaps @Vashta Nerada is interested in another level of discussion.

Markus did say "deduce,"

13 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. 

but of course we all understand what he meant. I've crossed paths with him at least once in relation to this precise matter. And it hasn't scaped my attention that his signature displays Stokes' theorem in its most general differential-form form.  

[studiot]

31 minutes ago, joigus said:

Markus did say "deduce,"

I think you need more black coffee.

17 hours 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.

 

16 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.

 

14 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.

Was he not only following your lead?

 

1 hour ago, Vashta Nerada said:

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.

 

Since you don't want to discuss the basics with me I will leave you with this 1950s University Physics textbook extract, which directly addresses what seem to be you basic misunderstandings.

In particular it provides a simple mathematical derivation of the 1/r relationship from the 1/r² Law.

Please be aware this applies in any number of dimensions for Newtonian gravity.

Also note the explanation about infinity and the negativity of PE and the difference between PE and potential.

As regards the reasons for our universe being a 3 + 1 universe:

Vector statics is complete in a 3 spatial universe and vector dynamics complete in a 3 +1 universe, but not a 2 spatial universe or a 2 + 1 universe.
As I have been trying to lead you towards.

In the early days of Einstinian Relativity, there was much discussion about the possibility of other universes with A spatial and B time dimensions and the consequences if these were assumed.
Eddington provides a good mathematical reasoning to discount these for our own in his book 'The Mathematical Theory of Relativity'.

[joigus]

1 hour ago, studiot said:

I think you need more black coffee.

I missed that nuance. Thanks!!

[J.C.MacSwell]

On 2/15/2022 at 4:26 PM, Vashta Nerada said:

Now, the irregularity that I found was in trying to derive the Gravitational Potential Energy, because following the same idea, we have:

E_PG = 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!

Does it not go to infinite in a 3D universe also? (obviously not in this one, but we just don't know how the formula we know well should  change at small scales)

Edited February 19, 2022 by J.C.MacSwell