When Conservation of Energy FAILS! (Noether's Theorem)

[John Clark]

When Conservation of Energy FAILS! (Noether's Theorem)

John K Clark

[Brent Meeker]

Sean Carroll and Jackie Lodman published a paper on non-conservation of energy in quantum measurements. 

https://www.preposterousuniverse.com/blog/2021/01/28/energy-conservation-and-non-conservation-in-quantum-mechanics/

They're proponents of MWI and it's still conserved across the multiverse and on average in each universe.  But locally in a single measurement it can be violated and they propose an experiment (which is probably impractical) to demonstrate this.

John Baez and others have long pointed out that total energy is not well defined in an expanding universe and Noether's theorem doesn't apply because there's no Killing field to provide the symmetry. 

https://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Phillip Gibbs has written several papers discussing this limitation and advocating for a conserved energy based on a generalization of Noether's theorem and/or pseudo-tensors.  I think Gibbs has some good points but he's not really contradicting Baez et al.

Brent

On 4/3/2022 12:18 PM, John Clark wrote:

When Conservation of Energy FAILS! (Noether's Theorem)

John K Clark

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[Samiya Illias]

And the heaven We constructed it with strength, and indeed, We (are) surely (its) Expanders. 

https://www.islamawakened.com/quran/51/47/default.htm 

On 04-Apr-2022, at 12:18 AM, John Clark <johnk...@gmail.com> wrote:



When Conservation of Energy FAILS! (Noether's Theorem)

John K Clark

[Lawrence Crowell]

On Sunday, April 3, 2022 at 2:18:49 PM UTC-5 johnk...@gmail.com wrote:

When Conservation of Energy FAILS! (Noether's Theorem)

John K Clark

 I disagree with his statement about the CMB, for the most part. The universe early on when it was radiation dominate was gravitation dominated by radiation. The radiation was red shifted and that energy in a sense went into gravitational potential energy.

The issue with dark energy is more relevant. If the spatial surface of the universe is an ℝ^3 then the expansion of the space makes no difference. There is an infinite vacuum energy and it just keeps expanding into itself. Infinity has that property that multiplied by anything give infinity back. If space is a 3-sphere then there is something wrong with energy conservation.

Noether’s theorem is easy to derive, and in our mechanics course we have Noether’s theorem in our hand and throw it away. A Lagrangian L(q,q’) (‘ = time derivative) defines the action S = ∫L(q,q’)dt. The variation δS = 0 means that

δ S = ∫δL(q,q’)dt = ∫[(∂L/∂q)δq + (∂L/∂q’)δq’]dt

and by chain rule

δ S = ∫[(∂L/∂q) - d/dt(∂L/∂q’)]δqdt + d/dt∫(∂L/∂q’)δqdt.

The first integral is the Euler-Lagrange equation. The second is the boundary term that is zero, which means the integrand (∂L/∂q’)δq = pδq is a constant. What this tells us is that momentum is conserved under translation by coordinates. This translation of coordinates means space is homogeneous. This symmetry of space conserves momentum. That is Noether’s theorem in a nutshell.

With general relativity most spacetimes have Killing vectors K_a that act on momenta p^a so that K_ap^a = constant. For type D solutions these are the K_t = √(g_{tt})∂_t and angular Killing vectors. One signature of a Killing vector is that the metric K_t for instance is such that the metric terms do not depend on t. The Petrov types D, to II to III to N, black hole near field out to gravitational waves as far field have Killing vectors. So energy conservation does work. This is why my interest is primarily on these. A theory of quantum gravitation is possible with this more local type of spacetimes, spacetime solutions with an asymptotic limit.

What about cosmologies? Cosmologies have no Killing vectors. We can think of this according to the ADM Hamiltonian H = 0, which means that in a general spacetime there is no definition of a Gaussian type of surface one can evaluate mass-energy. A cosmology has no natural rule for the conservation of not only energy, but momentum and angular momentum. In fact all the generators of the Lorentz group have no global rule for their conservation in cosmologies. As Charlie Parker put it, “Anything goes.” Because of this a general theory of quantum cosmology is a far more difficult problem to work. In fact to go out on a limb I will even say that a complete theory of quantum cosmology is impossible. We can learn things about it. Just as we know about classical cosmology a theory of quantum cosmology with D through N solutions will inform us in ways of quantum cosmology. But, … there will I think always remain a vast gulf of unknown before us.

LC

[Brent Meeker]

Phillip Gibbs derives a conserved current by extending Noether' theorem to 2nd derivatives in the Lagrangian, see attached.  It looks right to me, although the conserved energy includes gravitational potential so it's probably not well defined except in flat or closed universes.

There's also: "Covariant Conservation Laws in General Relativity" by Arthur Komar, PhysRev V113 No. 3, Feb 1959.

Brent

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[Lawrence Crowell]

I have read this some years ago. The question is whether out to second derivative we have conservation laws. From a classical mechanics perspective if we map to a Hamiltonian theory by Legendre transformation it means instead of a doublet of conjugate varaibles, say (q, p). we have a triplet (q, p, dp/dt). This then changes foundations of physics in some big ways, such as with quantum mechanics. One could ponder a whole jet-bundle of such terms up to n-derivatives. There is considerable hesitancy to go there.

LC

[John Clark]

On Mon, Apr 4, 2022 at 8:32 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

When Conservation of Energy FAILS! (Noether's Theorem)

 >> I disagree with his statement about the CMB, for the most part. The universe early on when it was radiation dominate was gravitation dominated by radiation.The radiation was red shifted and that energy in a sense went into gravitational potential energy.

I agree you can redefine what energy means so that it includes the energy in radiation and in matter and in the gravitational field, but I don't see the point of doing so, because you'd end up with something different than what we intuitively think of as energy and you end up with something that is pretty useless because, although you could say how much of this redefined "energy" there was in the entire universe, you still couldn't say how much of this "energy" there was in the curvature of spacetime at every point, so the density of gravitational energy would remain undefined. So this new redefined "energy" would be of no help in trying to figure out how things work. I think it would be better to just say energy is *usually* conserved, but like every law of physics (except maybe the second law of thermodynamics) there are regions of applicability where it doesn't apply, such as when things get very small for General Relativity, or when gravity becomes important for Quantum Mechanics.  

>  If the spatial surface of the universe is an ℝ^3 then the expansion of the space makes no difference. There is an infinite vacuum energy and it just keeps expanding into itself.

That's possible but it seems to me it would violate Occam's razor, the curvature of space in our visible universe can be uniquely defined internally without hypothesizing it's the surface of a higher dimensional object,  Gauss prove that with his "Theorema Egregium" nearly 200 years ago.

John K Clark    See what's on my new list at  Extropolis

gte

[Lawrence Crowell]

On Tuesday, April 5, 2022 at 6:24:27 AM UTC-5 johnk...@gmail.com wrote:

On Mon, Apr 4, 2022 at 8:32 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

When Conservation of Energy FAILS! (Noether's Theorem)

 >> I disagree with his statement about the CMB, for the most part. The universe early on when it was radiation dominate was gravitation dominated by radiation.The radiation was red shifted and that energy in a sense went into gravitational potential energy.

I agree you can redefine what energy means so that it includes the energy in radiation and in matter and in the gravitational field, but I don't see the point of doing so, because you'd end up with something different than what we intuitively think of as energy and you end up with something that is pretty useless because, although you could say how much of this redefined "energy" there was in the entire universe, you still couldn't say how much of this "energy" there was in the curvature of spacetime at every point, so the density of gravitational energy would remain undefined. So this new redefined "energy" would be of no help in trying to figure out how things work. I think it would be better to just say energy is *usually* conserved, but like every law of physics (except maybe the second law of thermodynamics) there are regions of applicability where it doesn't apply, such as when things get very small for General Relativity, or when gravity becomes important for Quantum Mechanics.  

The redshift of photons and the CMB is no different than gravitational redshift of photons sent from the surface of a gravitating body to a region out in space.

 

>  If the spatial surface of the universe is an ℝ^3 then the expansion of the space makes no difference. There is an infinite vacuum energy and it just keeps expanding into itself.

That's possible but it seems to me it would violate Occam's razor, the curvature of space in our visible universe can be uniquely defined internally without hypothesizing it's the surface of a higher dimensional object,  Gauss prove that with his "Theorema Egregium" nearly 200 years ago.

John K Clark    See what's on my new list at  Extropolis

g

That is a different thing. The Theorema Egregium is that the Riemann curvature tensor in 2-dimensions R_{1212} = R is equal to the Ricci scalar.

LC t