Where do we use infinity in physics, and why?

[FredLau]

Hello!
I'm doing a school project, where I am writing about Infinity in Math and Physics. I've got the math part settled, but it's the physics part that has begun to bother me.
One part of my task is to write about some of the mathematical expressions in physics, where we use infinity - but the only real thing I've been able to find about infinity in physics, is when we are talking mass and density of black holes, which is fine, I think I have a pretty basic understanding of it, but can anyone tell me about other things in physics where we have something that gets "infinitely big", or where we use infinity? If you've got some sources for it, I would be forever thankful!

 

[DrClaude]

What is your level? You marked the thread as "I" (undergraduate), but from your question I would assume that "B" (high school) would be the correct level. (In other words, if you really are an undergraduate, you surely have seen ∞ put up all over the place )

 

[PeroK]

Hello!
I'm doing a school project, where I am writing about Infinity in Math and Physics. I've got the math part settled, but it's the physics part that has begun to bother me.
One part of my task is to write about some of the mathematical expressions in physics, where we use infinity - but the only real thing I've been able to find about infinity in physics, is when we are talking mass and density of black holes, which is fine, I think I have a pretty basic understanding of it, but can anyone tell me about other things in physics where we have something that gets "infinitely big", or where we use infinity? If you've got some sources for it, I would be forever thankful!

You could make a case that infinity has no place in Physics, only in the mathematics.

A common idea is to consider "a point at infinity", but that really means "a point far enough away that some quantity may be neglected or taken to be 0". E.g in a gravitational field, "a point at infinity" is far enough away that the gravitational potential energy is effectively at a maximum.

The current theory is that the universe may be infinite in spatial extent, so that's an exception of sorts; although it might be interesting to consider what that might actually mean.

You can also look at infinity as the number of possible positions or states that something may be in. If a particle moves from A to B in a given time, then we generally think that the particle is at each of the infinitely many points between A and B at one of the infinitely many instants of time. In fact, this is what is called an "uncountable" infinity, if you have come across that in your mathematics.

 

[sophiecentaur]

It strikes me that, in Physics, there are two types of Infinity. As the distance from a celestial object increases, there is no 'limit' - it's just further and further away and we approach that sort 'Infinity' in a graceful manner. But the Infinity that's associated with, say, the forces involved with the Inverse Square law, as objects get closer and closer, is more problematic. You get the problem of the 'Infinity' that you get with 1/x² as x approaches zero. That's far harder to deal with unless you acknowledge that the inverse square law has to stop operating for a small enough distance. You can't just say "1/0=∞" ; it's far too crude. We don't actually use the word Infinity for that situation, we tend to say 'Indeterminate', instead. We use the word Singularity and the Maths has to change gear in order to get around it.
The problems we have with Infinity are only the tip of the iceberg, though. It is a dramatic reminder that Maths is only a Model of what goes on in Physics and any mathematical result needs to be scrutinised to see if it is really meaningful. An obvious example of this is when we take a simple square root when working our the length of the walls of a square room of a given area; there are two possible answers there, one positive and one negative.

 

[Nugatory]

...but can anyone tell me about other things in physics where we have something that gets "infinitely big", or where we use infinity? If you've got some sources for it, I would be forever thankful!

A few standard examples of infinity being used in problems from high school physics:
- Look at Newton's law of gravity: $F=Gm_1m_2/r^2$. What happens to the force as the two gravitating objects move closer together ($r$ becomes closer to zero)? Conversely, under what conditions will the force be small enough that we can ignore it?
- When I bounce a ball off the ground it starts off going down at speed $v$, ends up moving up at the same speed. How does this not violate conservation of momentum?

 

[Stephen Tashi]

The idea of a continuum involves the idea of "infinitely divisible". Which things are a continuum and which thing are not is a debate conducted in physics.

 

[FredLau]

A few standard examples of infinity being used in problems from high school physics:
- Look at Newton's law of gravity: $F=Gm_1m_2/r^2$. What happens to the force as the two gravitating objects move closer together ($r$ becomes closer to zero)? Conversely, under what conditions will the force be small enough that we can ignore it?
- When I bounce a ball off the ground it starts off going down at speed $v$, ends up moving up at the same speed. How does this not violate conservation of momentum?

Okay, so I get that if you let r -> 0, the force of gravity will by definition become infinite. My question here is, what does it mean when the force of gravity becomes infinite in this sense?

 

[Drakkith]

Okay, so I get that if you let r -> 0, the force of gravity will by definition become infinite. My question here is, what does it mean when the force of gravity becomes infinite in this sense?

It means you've reached a breakdown in your model, as an infinite force makes no sense. Force example, if you try to calculate the force felt by a proton and an electron as they get closer to each other you will hit infinity at r=0. However, the reality is that this simple model, one where the electron and proton approach and r becomes 0, is not accurate. Instead the two bond together to form a hydrogen atom and you have to use quantum physics to model them. In this new model, the laws are very different and you don't have the r=0 issue.

 

[FredLau]

It means you've reached a breakdown in your model, as an infinite force makes no sense. Force example, if you try to calculate the force felt by a proton and an electron as they get closer to each other you will hit infinity at r=0. However, the reality is that this simple model, one where the electron and proton approach and r becomes 0, is not accurate. Instead the two bond together to form a hydrogen atom and you have to use quantum physics to model them. In this new model, the laws are very different and you don't have the r=0 issue.

Okay, that makes somewhat more sense! Thank you.

 

[Cutter Ketch]

Speaking of gravity (or electrostatic force too) physicist often use infinity as a reference point for the potential energy, i.e. one of the bounds of a potential integral. We integrate the force in from infinity where the force is 0 to a given finite distance to get the depth of the potential well. Come to think of it we often use infinity as a limit of integration for example integrating over a wave function in quantum mechanics or similar. In these cases we are adding up all the values of a function from "way over there" where we know the value is zero through the local region where the function has finite value to far away where once again the value has fallen to zero.