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.

 

[Dale]

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

I would think that you could take a relativistic Lagrangian and just add an explicit symmetry violating term. That should be self consistent, no?

 

[Matterwave]

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

Do you define GR to "obey relativity" (or do you read "obey relativity" as global Lorentz invariance)? Depending on the reading of the words... one could argue that GR would fit. It's locally Lorentz and many of its solutions do not admit a time-like Killing vector field (e.g. FLRW) --> no time translation symmetry --> no (in a simple sense at least) conservation of energy.

 

[PeterDonis]

Do you define GR to "obey relativity"

Yes.

one could argue that GR would fit.

No, it doesn't, because in GR the local conservation of stress-energy is enforced by the Einstein Field Equation. So it's impossible for energy to appear out of nowhere.

 

[PeterDonis]

no time translation symmetry --> no (in a simple sense at least) conservation of energy.

No "energy" in the sense of Noether's Theorem; but local conservation of stress-energy still holds, since the EFE always holds in GR. So even in spacetimes that don't have a timelike KVF, it's still impossible for energy to "appear out of nowhere".

Indeed, the notion of "energy" given by Noether's Theorem, in general, has to be interpreted physically very carefully. In asymptotically flat spacetimes, for example, (e.g., the Kerr-Newman family of spacetimes), it is usually called "energy at infinity" and does not correspond to the energy that would be locally measured at any finite location in the spacetime.

 

[PeterDonis]

I would think that you could take a relativistic Lagrangian and just add an explicit symmetry violating term. That should be self consistent, no?

If there's an example in the literature, I'd be interested in seeing it.

Note that in GR, for example, spacetimes that do not have a timelike KVF are not generated by adding anything to the Lagrangian. The Lagrangian in GR is just the Einstein-Hilbert Lagrangian, $R \sqrt{-g}$ (plus a cosmological constant term if you want to include it). But this Lagrangian has many different possible solutions, some of which have a timelike KVF and some of which don't. So that's not something that is put in by hand, so to speak; it just pops out of the family of solutions that have it.