Does Gauss' law imply that the universe isn't compactified?

[Rocky Raccoon]

The first Maxwell equation, Gauss' law makes a clear distinction between "inside" and "outside". But such a distinction can't be made in a compactified space (e.g. circle). Does that mean that the universe isn't compactified in a sense that if one was to move in a "straight" line one would never return to the starting place?

 

[Vanadium 50]

I am not sure what you mean by circles not having an inside and outside.

The Maxwell Equations in differential form don't make any inside/outside distinction. If you'd like to integrate them over a larger space, I suppose that you could say something about the volumes that you can integrate over, but this is a statement about integration, not about electromagnetism.

 

[AlephZero]

I thnk the OP means a "curved" space, as in the pop-sci analogies of the circumference of a circle or the surface of a sphere .

But in both those examples, you can define a sub-region of space which has a closed boundary, and any two points are either on the "same side" of the boundary or on "opposite sides", so I'm not sure exactly what the OP's question is.

 

[Rocky Raccoon]

What I meant to ask was:

Imagine 1+1 dimensional world with space compactified in a circle. If you go straight ahead for enough time, you reach the starting point. Put a point charge Q somewhere on the circle. Choose two point A and B so that Q is "inside" [A,B]. But then Q is "outside" [B,A]. What would Gauss' law give for an electric field in this case?

 

[coelho]

What I meant to ask was:

Imagine 1+1 dimensional world with space compactified in a circle. If you go straight ahead for enough time, you reach the starting point. Put a point charge Q somewhere on the circle. Choose two point A and B so that Q is "inside" [A,B]. But then Q is "outside" [B,A]. What would Gauss' law give for an electric field in this case?

I think i understood what you mean. I may be talking nonsense, but i would risk saying that Gauss' law implies that the total electric charge of any space of this kind is zero, as the only way to avoid inconsistencies.

Like if there is a +Q charge at [A,B], there must be a -Q charge at [B,A] so the field lines that were diverging from +Q can converge to -Q. If there were not this -Q charge, the field lines that were diverging from +Q would converge somewhere without there being any charge there, and that would be inconsistent.

 

[Rocky Raccoon]

I think i understood what you mean. I may be talking nonsense, but i would risk saying that Gauss' law implies that the total electric charge of any space of this kind is zero, as the only way to avoid inconsistencies.

Like if there is a +Q charge at [A,B], there must be a -Q charge at [B,A] so the field lines that were diverging from +Q can converge to -Q. If there were not this -Q charge, the field lines that were diverging from +Q would converge somewhere without there being any charge there, and that would be inconsistent.

I agree with you. So, in a closed universe, electric field lines would also have to be closed. In that case it doesn't matter which side is in or out since Gauss' law will always produce the same result.