Why homogeneous?

Question

Why in the conservation laws ( I mean energy, linear momentum and angular momentum conservations) we consider time or space to be homogeneous _as the characteristic of inertial frame? Can anyone please tell me what the reason for this is in mathematics and physics? what does it mean exactly to be homogeneous?

Answer 1

The basics of this is Noether's theorem. That theorem says that whenever there is a symmetry in your framework, there is a corresponding conserved quantity. The proof usually relies on Lagrangian mechanics.

(Newtonian mechanics tells you how an object moves by analysing the forces that act upon it, which tells you its acceleration. Lagrangian mechanics tells you how things move by declaring that any object follows the path of "least action" (the difference between potential energy and kinetic energy is, on average, as low as possible). They give the same physics, but some results, like Noether's theorem, are easier to deduce in the Lagrangian framework than in the Newtonian.)

For instance, the fact that physics (as far as we know) is the same here is it is a lightyear to the right, and everywhere else (translational symmetry), directly implies that momentum is preserved. If physics had been different in the Andromeda galaxy, then you could increase or decrease the total momentum of the universe just by sending something back and forth between here and there.

As far as we can tell, there is no evidence that this happens, so we assume that it doesn't.