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.

 

[Roberto Pavani]

My 2 cent:
A standard example:
In any expanding universe, photon energy decreases (cosmological redshift) without being transferred anywhere.
In a contracting universe, photons would blueshift and gain energy. Local conservation (∇_μ T^μν = 0) holds, but no globally conserved energy quantity exists in the absence of a timelike Killing vector.

 

[PeterDonis]

In any expanding universe, photon energy decreases (cosmological redshift) without being transferred anywhere.

More precisely: a photon emitted by a comoving observer and measured to have a certain energy at emission by that comoving observer, will be measured to have decreasing energy by future comoving observers as it passes them.

However, the stress-energy contained in the photon is still locally conserved, per what I said before about the Einstein Field Equation. The photon energy measured by comoving observers is due to the effect of the expansion on their 4-velocities. It does not reflect any change in the photon itself: indeed, in order to derive the result, one has to treat the 4-momentum of the photon itself as being determined at emission.

no globally conserved energy quantity exists in the absence of a timelike Killing vector.

Yes, this is true. But the photon energy measured by comoving observers is not a "global" quantity in any case. It's a local quantity at each measurement event.

 

[Dale]

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

I’m sure I can google up a reference for a relativistic damped harmonic oscillator for you. I will try to do so tonight

 

[Matterwave]

Yes.


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.

This is quite a loose definition of conservation of energy imo. Yeah, the OP's original grossly energy violating "energy just pops out of thin air" doesn't make sense, but e.g. in FLRW, light red-shifts while there's no easy way to define an "energy of the gravitational field" to which that lost energy can be absorbed into (there is no asymptotic flatness in FLRW so global energy / "energy of the gravitational field" is pretty hard to define here). I think by pretty general physical senses -- that would violate conservation of energy.

Btw, I don't think we disagree on *any technical points*. But my point was just trying to read Dale's statement:

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

In kind of a plain English reading. I would say GR "has a local *sense* of conservation of energy due to $\nabla_a T^{ab}=0$ but this definition is not generalizable in a way that would fit my definition of "conservation of energy" for all spacetimes (specifically not ones without a timelike KVF).

 

[Matterwave]

Maybe to the OP I can make the following (I believe) fully non-controversial statement:

GR "obeys relativity" and yet admits solutions which are not time-translation invariant.

 

[PeterDonis]

This is quite a loose definition of conservation of energy imo.

It's the standard one accepted by physicists who work on GR, and has been ever since the Bianchi identities satisfied by the EFE were understood. Also, as you appear to agree, it captures the intuitive notion of "energy can't appear out of nowhere" (or disappear into nowhere).

in FLRW, light red-shifts while there's no easy way to define an "energy of the gravitational field" to which that lost energy can be absorbed into

Nor does there need to be, because there's no need to interpret the redshift as a "loss of energy" in any case. See my post #12.

this definition is not generalizable in a way that would fit my definition of "conservation of energy" for all spacetimes (specifically not ones without a timelike KVF).

I'm not sure what you mean. As I've already pointed out, $\nabla_a T^{ab} = 0$ holds in any spacetime whatever. That's as "generalizable" as it gets.

What is "not generalizable" is the notion of energy that depends on a timelike KVF, since obviously no such notion exists in spacetimes with no timelike KVF. But IMO the best response to that is to accept that that notion of energy has limitations, not to try to conjure up a way of interpreting things that happen in a spacetime with no timelike KVF as "non-conservation of energy". Particularly if, as you appear to agree, there is no invariant way to conjure up such things anyway.

 

[Matterwave]

Fair, if the consensus is to treat $\nabla_a T^{ab}$ as a general statement of conservation of energy in GR then that's fine with me~

At any rate, I don't think we are arguing anything technical. Very quickly we will get into arguing semantics so I'll leave it here. :)

 

[Matterwave]

Ah, found the statement from Wald that I was trying to look up. I should have just quoted him instead. He is far more rigorous than me -- this I will concede:

However, although the condition $\nabla^a T_{ab}=0$ may be interpreted as expressing local conservation of the energy-momentum of matter, this condition does not, in general, lead to a global conservation law, i.e. a law which states that the total energy (expressed as an integral involving $T_{ab}$ over a spacelike hypersurface) is conserved.

(Not trying to bring up any additional argumentation, just trying to give the rigorous statement to the OP)

 

[Dale]

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

Here is a relativistic damped oscillator:
https://www.jstor.org/stable/2098796

It was not constructed as I had thought, by adding an energy-violating term to a Lagrangian. Instead it was derived with the relativistic version of Newton's laws.

On the Relativistic Damped Oscillator
Author(s): Thomas P. Mitchell and Daniel L. Pope
Source: Journal of the Society for Industrial and Applied Mathematics, Vol. 10, No. 1
(Mar., 1962), pp. 49-61
Published by: Society for Industrial and Applied Mathematics

 

[PeterDonis]

Here is a relativistic damped oscillator:
https://www.jstor.org/stable/2098796

It appears to be paywalled, unfortunately.

It was not constructed as I had thought, by adding an energy-violating term to a Lagrangian. Instead it was derived with the relativistic version of Newton's laws.

Yes, that makes sense. A damping term in the relativistic force equation would result in energy loss if you're only modeling the oscillator's motion and not including heat or other energy sinks.

 

[Dale]

It appears to be paywalled, unfortunately.

Sorry about that. There doesn’t appear to be much in the way of later follow-up.

 

[DanteKennedy]

GR "obeys relativity" and yet admits solutions which are not time-translation invariant.

So according to current understanding in physics, we cannot have any type of universe that obeys relativity while not holding energy-momentum conservation locally? Which means local conservation is a consequence of relativity in a mathematical level?

 

[PeterDonis]

according to current understanding in physics, we cannot have any type of universe that obeys relativity while not holding energy-momentum conservation locally?

That's correct.

Which means local conservation is a consequence of relativity in a mathematical level?

It's a consequence of the Einstein Field Equation, plus including all forms of local stress-energy in your model.

In a case like the relativistic damped oscillator that @Dale posted a reference for, the appearance of energy being lost in the oscillator arises from the fact that the model only includes the oscillator's center of mass motion. It does not include heat stored in the oscillator, which is where the energy that is taken away from the center of mass motion by damping goes (at least in the simplest case). A complete model that includes all of the local stress-energy will have to include that heat, and with it included, local conservation of energy holds.

 

[DanteKennedy]

That's correct.


It's a consequence of the Einstein Field Equation, plus including all forms of local stress-energy in your model.

How about the idea that the universe could give an illusion of energy appearing or disappearing out of nowhere by having extra invisible field that can absorb and give energy to matters? Not a conventional physics but could be a fun thought experiment

 

[PeterDonis]

How about the idea that the universe could give an illusion of energy appearing or disappearing out of nowhere by having extra invisible field that can absorb and give energy to matters? Not a conventional physics but could be a fun thought experiment

This is personal speculation and is off limits here.

 

[PeterDonis]

Not a conventional physics

Actually, "conventional physics" does have a concept something like what you describe--indeed, that's how the concept of a "field" arose in physics in the first place, in order to explain how charged objects could be accelerated without anything visible appearing to push on them. The electromagnetic field, which can store energy and can transfer it to and from matter, was introduced in order to deal with such phenomena (and other phenomena in electricity and magnetism which have similar properties). But any such field still has to obey physical laws, and still has to be included in the local stress-energy that is conserved.

 

[Matterwave]

So according to current understanding in physics, we cannot have any type of universe that obeys relativity while not holding energy-momentum conservation locally? Which means local conservation is a consequence of relativity in a mathematical level?

Local energy momentum conservation also happens in Newtonian physics (and indeed there, global energy momentum is well defined and conserved).

I think your first point is true. I just don't really follow why that would lead to the second point and why you feel such a strong connection between relativity and energy conservation. There is a deep connection between time translation symmetry and energy conservation. It seems you understand that fact. Why you insist on making it about relativity eludes me.

Energy conservation is baked in to all sorts of physics.

 

[Dale]

So according to current understanding in physics, we cannot have any type of universe that obeys relativity while not holding energy-momentum conservation locally?

That is the opposite of what I said above.

The discussion that others were having is whether or not the FLRW spacetime is a counter example. But even if that one specific spacetime is not a counter example does not imply that there is no other counter example.

I gave the counter example of a relativistic damped oscillator.

I would also suggest that lack of a global conservation could reasonably be taken to constitute “creation and destruction of energy out of nothing” even if it is not localized. So personally I would consider FLRW to be a counter example.

 

[Dale]

A complete model that includes all of the local stress-energy will have to include that heat, and with it included, local conservation of energy holds.

A complete model in this universe, yes. The question was if an alternative universe without that feature is self consistent. And it clearly is.

So our universe has both relativity and conservation. But that doesn’t mean “no self-consistent universe could hold the principle of relativity while also permitting the arbitrary creation and destruction of energy out of nothing”.

 

[PeterDonis]

I would also suggest that lack of a global conservation could reasonably be taken to constitute “creation and destruction of energy out of nothing” even if it is not localized.

Lack of global conservation of what?

In FLRW spacetime, the issue is not that there is a global invariant that is not conserved. The issue is that, since there is no timelike KVF, there is no such global invariant to begin with. It doesn't make sense to say that something that doesn't even exist is not conserved.

To put it another way: Noether's Theorem does not say that some pre-existing global "energy" is conserved in a spacetime with a timelike KVF, which would not be conserved in a spacetime without one. Noether's Theorem uses the timelike KVF to define a global "energy" that is conserved. So in a spacetime without a timelike KVF, where Noether's Theorem doesn't apply, you can't even define a global "energy" to begin with. So there is nothing to not be conserved, so to speak.

personally I would consider FLRW to be a counter example.

I disagree, for the reasons given above.