Mass in 2D: Understanding the Concept

[Stranger]

I was wondering what mass means in a 2 dimensional world...

 

[lethe]

Originally posted by Stranger
I was wondering what mass means in a 2 dimensional world...

mass is the invariant length of momentum. mass is the coefficient of the "potential" term of the particle field lagrangian. mass is a particle s resistance to acceleration.

none of these definitions makes any reference to the dimensionality, therefore mass means in 2 dimensions exactly the same thing as it means in 3 dimensions.

 

[Stranger]

mass is the invariant length of momentum. mass is the coefficient of the "potential" term of the particle field lagrangian. mass is a particle s resistance to acceleration

Can you please elaborate on this...thanks

 

[Stranger]

I'm sorry...i know the last one ...and quiet a bit about the 1st one...but the second one...

 

[FZ+]

Originally posted by lethe
mass is the invariant length of momentum.

No. With relativity, mass is most definitely not invariant.

 

[lethe]

Originally posted by FZ+
No. With relativity, mass is most definitely not invariant.

uhh...? are you sure? you should double-check your textbook, eh?

so what exactly do you think the invariant length of the momentum four vector is, if you don t think it is mass?

did it escape your attention that the invariant length of any four vector is invariant? why do you think it s called "invariant length"?

i hope you re not going to start using the 1960s definition of mass, that debate over semantics is completely boring to me.

 

[FZ+]

Ah sorry. Thought you were referencing relativistic mass or matter <-> energy conversions.

 

[lethe]

Originally posted by FZ+
Ah sorry. Thought you were referencing relativistic mass or matter <-> energy conversions.

relativistic mass

Of the two, the definition of invariant mass is much preferred over the definition of relativistic mass. These days, when physicists talk about mass in their research, they always mean invariant mass. The symbol m for invariant mass is used without the subscript 0. Although the idea of relativistic mass is not wrong, it often leads to confusion, and is less useful in advanced applications such as quantum field theory and general relativity. Using the word "mass" unqualified to mean relativistic mass is wrong because the word on its own will usually be taken to mean invariant mass. For example, when physicists quote a value for "the mass of the electron" they mean its invariant mass.

"Ouch! The concept of `relativistic mass' is subject to misunderstanding. That's why we don't use it. First, it applies the name mass--belonging to the magnitude of a four-vector--to a very different concept, the time component of a four-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of space-time itself."

-wheeler

"It is not good to introduce the concept of the mass M = m/(1-v2/c2)1/2 of a body for which no clear definition can be given. It is better to introduce no other mass than `the rest mass' m. Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion."

-einstein

you only find the concept of relativistic mass used in popular science books these days, and on internet physics boards or newsgroups. in real science textbooks, it is simply not found at all.

 

[lethe]

Originally posted by Stranger
I'm sorry...i know the last one ...and quiet a bit about the 1st one...but the second one...

well, fields act a lot like harmonic oscillators, that is to say, they act like springs. springs obey hookes law, which states that the restoring force is proportional to the displacement. in other words, F=-kx. it s easy enough to see from this definition that the potential energy of a spring is 1/2kx². the lagrangian is then, by definition, T - V = 1/2mv² - 1/2kx².

it turns out that most fields are a lot like harmonic oscillators. their lagrangians look something like &part;&phi;² - k&phi;². the first term is a derivative of the field squared, and is called the kinetic term, to make the parallel to the harmonic oscillator lagrangian (v is the derivative of x). the second term contains the field squared, and to make a parallel with the harmonic oscillator, it is called the potential term. it shows how strong the restoring force of the field is, although i don t know how far one can take that analogy.

point of the story is, the coefficient of the potential term turns out to be the rest energy of a single quantum of the field. thus the mass of an electron is just a measure of how tightly the electron/positron field oscillates: how much energy it has! this is easily understandable in terms of the equivalence of mass and energy. a tighter spring has more potential energy, and therefore weighs more.

 

[Stranger]

another question...if there is a circle in a 2 dimensional world, where doest the gravity act from...centre?

 

[lethe]

your question is a little vague.

massive bodies gravitate. are you asking about a circle in a 2 dimensional world that has mass? if you are outside of the circle, you feel the gravitational attraction as though it were at the circle, yes. if you are inside the circle, there is no gravitational attraction from the circle. the attraction from the circle cancels out in every direction.

 

[Stranger]

the attraction from the circle cancels out in every direction.

How? Is the thickness a 2 dimensional object equal to that of plank length...

 

[lethe]

Originally posted by Stranger
How? Is the thickness a 2 dimensional object equal to that of plank length...

a 2 dimensional object has 2 dimensions. if you call one length and one width, and you want to call thickness the measure in the third or any higher dimension, then the thickness of any 2 dimensional object is 0. i m still not really sure what you re trying to ask, but i can assure you, the Planck length has nothing to do with geometry, or classical gravitation.

 

[Stranger]

i m still not really sure what you re trying to ask, but i can assure you, the Planck length has nothing to do with geometry, or classical gravitation.

No...I was just wondering if there really was no thickness...anyway...its mathematical...so the thought that a 3 dimensional being can rip the two dimensional being or object from its 2 dimensional world doesn't seem to work...because he will have nothing in his hand...if he does then he can tell that it has some thickness...

 

[lethe]

Originally posted by Stranger
No...I was just wondering if there really was no thickness...anyway...its mathematical...so the thought that a 3 dimensional being can rip the two dimensional being or object from its 2 dimensional world doesn't seem to work...because he will have nothing in his hand...if he does then he can tell that it has some thickness...

i am not really sure how a three dimensional object would interact with a dimensional object.

in all likelihood, if the 2D object and the 3D object were to meet in any realistic universe, then that would imply that the 2D object is only approximately 2D, and actually has some thickness.

i can t imagine a universe that contains both 2D fields and 3D fields. there would have to be a discontinuity in the in the spacetime.

 

[chroot]

Wow, lethe, it looks like you're being particularly patient with this one. I would have thumbed my nose and referenced Halliday and Resnick by now.

- Warren

 

[Stranger]

2D object is only approximately 2D, and actually has some thickness.

Yes...thats what I was thinking of...Its kinda hard to imagine something without a thickness...so I think it can also be that our 3D space also has a little extensin in 4D space...every 3 D object...

 

[Stranger]

Wow, lethe, it looks like you're being particularly patient with this one. I would have thumbed my nose and referenced Halliday and Resnick by now.

You mean he is being patient with me...lethe is just helping me out..