Noether's theorem and energy in four-momentum

Question

In Newtonian physics, momentum and energy are often treated as distinct entities, which happen to be separately conserved. In relativity, energy is regarded as the "time" component of the four-momentum. Energy and momentum are still separately conserved, but they mix under Lorentz transformations in a way that identifies energy with time and momentum with space.

Meanwhile, there is an analogous relationship energy:time::momentum:space in Newtonian physics, which becomes apparent in the Hamiltonian formulation. The Hamiltonian (energy) is the generator of time translations, whereas momentum is the generator of space translations.

What's the "secret" behind this? As pointed out in the comments, Noether's theorem relates energy and time via time translation symmetry in Newtonian mechanics, so this symmetry is probably also at play in relativity. But how? Is there a clear line of reasoning that goes from Noether's theorem + spacetime translation symmetry in relativity to "energy and momentum combine as a four-vector"?

Answer 1

You pretty much answered the question on your own. One can understand the very definition of energy (momentum) as "The Noether charge associated with time (space) translations".

Consider, for example, the Lagrangian for a free relativistic particle, $$L = \frac{1}{2} U^\mu U_\mu,$$ where $U^\mu$ is the four-velocity. If we make a transformation of the coordinates $x^\mu \to x^\mu + \epsilon\psi^\mu$, the Lagrangian stays invariant (the coordinates are all cyclic), and hence there is some associated conserved quantity. Picking $\psi^\nu = \delta^{\nu\mu}$ for different choices of $\nu$ and using Noether's theorem, we find that $$- \sum_{\nu} \frac{\partial L}{\partial U^\nu} \psi^\nu = - \frac{\partial L}{\partial U^\mu} = - m U_\mu$$ are constants of motion. We see we can arrange these quantities in a four vector by defining $p^\mu = m U^\mu$ and analysing the explicit expressions for each component we'll find out that these turn out to be precisely the energy (which was indeed associated with time translations in non-relativistic Physics) and the momenta (associated with spatial translations).