# Expanding universe and Time-translation invariance

My understanding is that there are 5 implicit assumptions when introducing the Lagrangian formulation.

Assumptions:

1. We appear to know that the principle of stationary action is true for the universe
2. We define Kinetic energy of the system to be to be $$T = \sum f(\mathcal{P}_n(\dot{q}))$$ where $$\mathcal{P}_n$$ is a polynomial of some degree
3. We define $$V(q)$$ to be the potential energy of the system
4. We assume the system is time-translation invariant
5. We define Lagrangian to be $$L(\dot{q}, q) = T(\dot{q})-V(q)$$

Question:

If our system is the universe, how do we know the universe is time-translation invariant? I am not fully convinced with the "Hamiltonian is a generator of time transformation", because it reasoning appears a little circular to me, given that's how the Hamiltonian is defined in the first place. Or maybe I have a fundamental mis-understanding of the Hamiltonian?

Also, how do we know that the duration of 1 second, say 50 years ago, is the same as the duration of 1 second today? Can we guarantee that it'll remain the same 50 years in the future from now?

From Noether's theorem, we know that we can connect symmetries with conserved quantities, and as you said energy conservation is connected to time-translation invariance. However, universal expansion breaks time-translation invariance, and hence energy is not conserved. (At least in the standard cosmological models).

For the second part, I think you have to be more specific when you think about time measurement and specify who observes and what exactly. For example, a comoving observer in his clock would measure a constant duration of 1 sec always.

The lagrangian formalism doesn't require that the state of the system be time-translation invariant. It only requires that the differential equations governing the system be so.

The universe is not time-translation invariant, and nobody has ever imagined that it was.

Also, how do we know that the duration of 1 second, say 50 years ago, is the same as the duration of 1 second today? Can we guarantee that it'll remain the same 50 years in the future from now?

Relativity says that time is what a clock measures. Therefore this question is meaningless. Relativity expresses gravity as curvature of spacetime, not as a time-dependent dilation of time.

• Thank you for the answer. I accepted @MariosN3's answer because his answer was first, even though it's very similar to yours. Commented Apr 30, 2020 at 12:52