The problem of energy appearing out of nowhere

[DanteKennedy]

TL;DR

Is energy conservation law fundamentally a consequence of the universe preserving relativity principle? In other words, can we explain why time translation symmetry exist?

To help with the idea, imagine a box sits in lab in frame S. At some moment, it somehow spontaneously creates 10 J of energy from nothing, without any push, so its momentum doesn't change: ΔE = 10 J, Δp = 0.
Observer S' moves past at v = 0.6c (so γ = 1.25).
Does this energy-creation event look the same to both observers? If it's not, does it mean that theoretically, no self-consistent universe could hold the principle of relativity while also permitting the arbitrary creation and destruction of energy out of nothing?

 

[Dale]

You could have a universe that obeyed relativity and did not respect the conservation of energy. Such a universe would also violate the conservation of momentum.

What you could not have is a relativistic universe that had conservation of energy but not conservation of momentum. Or vice versa

 

[PeroK]

TL;DR: Is energy conservation law fundamentally a consequence of the universe preserving relativity principle? In other words, can we explain why time translation symmetry exist?

To help with the idea, imagine a box sits in lab in frame S. At some moment, it somehow spontaneously creates 10 J of energy from nothing, without any push, so its momentum doesn't change: ΔE = 10 J, Δp = 0.
Observer S' moves past at v = 0.6c (so γ = 1.25).
Does this energy-creation event look the same to both observers? If it's not, does it mean that theoretically, no self-consistent universe could hold the principle of relativity while also permitting the arbitrary creation and destruction of energy out of nothing?

You may want to ask yourself the following questions:

Can a quantity be conserved but not invariant?

Can a quantity be invariant but not conserved?

If relativistic momentum ($p = \gamma mv$) is conserved in one inertial reference frame, then is it conserved in them all (under Lorentz transformations)?

What about classical momentum ($p = mv$). If that is conserved in one inertial reference frame, is it conserved in them all (under Lorentz transformations)?

 

[PeterDonis]

At some moment, it somehow spontaneously creates 10 J of energy from nothing

In order to even try to answer what happens in such a scenario, you have to have some set of physical laws that you're going to apply.

But no one here has given a set of physical laws that (a) has a well-defined transformation between inertial frames, and (b) does not conserve energy.

So your question is unanswerable because we don't know what laws to use to predict what will happen.

 

[PeterDonis]

You could have a universe that obeyed relativity and did not respect the conservation of energy.

Has anyone ever proposed a consistent set of physical laws that has this property? I'm not aware of any.