Does the existence of a potential render a space inhomogeneous?

[MuIotaTau]

A common definition of an inertial frame is that it is a reference frame in which space and time are homogeneous and isotropic; see, for instance, Landau and Lifshitz's Classical Mechanics. L&L also use homogeneity and isotropy to justify the functional form of the Lagrangian. But intuitively, it seems like the introduction of a potential renders a field inhomogeneous. Is this true?

 

[Simon Bridge]

Not by itself - no - but, if the gradient of the potential is not zero, and nothing else happens, then the object the frame is attached to is accelerating - making the frame non-inertial.

Can you provide a reference for the "common definition" of an inertial frame?
The vast majority of resources I have here just define it in terms of Newton's Laws.
i.e. Robert Resnick (1968) "Introduction to Special Relativity".

 

[MuIotaTau]

Thinking about it, I'm not actually sure that I know of any other pedagogical text that uses that definition, so it's possible that I've only really seen people reference that definition. Wikipedia uses it, but that's not very satisfactory. Thanks for the answer!

 

[Simon Bridge]

You'll also see that a reference frame can be non-inertial but space itself is homogeneous and isotropic.

Wikipedia says that time and space are described in an isotropic and homogeneous manner - the potential, in this description, is something that gets imposed over space. Having a potential does not make the space itself any different.

Even so - a characteristic of a non-inertial frame is the existence of pseudoforces ... i.e., it is as if the observer were in an inertial frame with an applied force-field.
Make more sense now?

 

[MuIotaTau]

You'll also see that a reference frame can be non-inertial but space itself is homogeneous and isotropic.

Wikipedia says that time and space are described in an isotropic and homogeneous manner - the potential, in this description, is something that gets imposed over space. Having a potential does not make the space itself any different.

Okay, I was beginning to think it was something like this, thank you. So what exactly would it look like if space itself were inhomogeneous?

Even so - a characteristic of a non-inertial frame is the existence of pseudoforces ... i.e., it is as if the observer were in an inertial frame with an applied force-field.
Make more sense now?

Ohhh, so a potential, which would produce a force field (right?) would create pseudoforces and make a frame non-inertial? So if I were in an accelerating reference frame, for instance, it would be as if there were a gravitation force field. But this wouldn't make space itself inhomogeneous? Am I understanding things correctly or am I still mixing something up?

 

[Simon Bridge]

So what exactly would it look like if space itself were inhomogeneous?

On cosmological scales, you are looking at it. It is a space in which classical mechanics does not work.
http://en.wikipedia.org/wiki/Homogeneity_(physics)#Homogeneous_cosmology

Note: homogeneity in space means conservation of momentum, and homogeneity in time means conservation of energy. If space were not so arranged, then these laws would not hold everywhere.

Ohhh, so a potential, which would produce a force field (right?) would create pseudoforces and make a frame non-inertial? So if I were in an accelerating reference frame, for instance, it would be as if there were a gravitation force field. But this wouldn't make space itself inhomogeneous? Am I understanding things correctly or am I still mixing something up?

I think you are close enough to make sense of what you read - for now. ;)

In the context of classical mechanics ... we would consider, say, we can use an inertial observer to study the effects of, say, an electromagnetic potential well.

Newtonian gravity compares with being in a accelerating reference frame is a nice example.
But you do need to be careful about what it is about space that you are calling homogeneous.

It's a heady subject - leads to gauge theories and descriptions based on symmetry.
Take it slow.