Accoring to Noether theorem, energy is a conserved physical quantity related to time translation invariance (symmetry) of a theory. Since our universe is not invariant under time translation - it expands - there is no conservation of energy (according to general relativity).

I don't think it is correct to say that Noether's theorem admits only conservative fields. What the theorem says is that symmetries in the Lagrangian correspond to conservation laws (but not necessarily energy conservation). What we call energy conservation arises as a result of time translation symmetry. In the absence of absolute time, however, two observers won't agree on what "time translation" means. However, they do agree on the meaning of infinitesimal spacetime translations, which leads to the conservation of the energy-momentum tensor. In the non-relativistic limit, this conservation law can be separated into the laws of energy and momentum conservation. General relativity is also invariant under spacetime translations. But one could legitimately propose a theory that is not invariant under spacetime translations, and thus violate energy-momentum conservation.

It's not absolute time that's the obstacle, it's the absence of global time translation invariance that makes energy conservation delicate in the presence of gravity. This, in fact, means that two observers that aren't on the same point in spacetime cannot agree about energy or momentum. These can only be defined at infinity.

On the other hand, it is true that if you have boundaries, you do break translations, thus eenrgy or momentum aren't conserved.

Hi Everyone,

thank you very much for your responses. Two things come to my mind.

(1) If energy conservation is not globally valid for the whole universe, why then do we find it so scrupulously conserved in all observed processes?

(2) Whence comes the excess energy and whither does it go when there is global non-conservation? Outside the universe? But by definition, our universe is all-inclusive and has no outside! Is there some problem here?

Regards.

(1) Because the processes that we do look at, do have this property, are not sensitive to the curvature of space-time.

(2) The notion of excess or deficit doesn't make sense so the question of ``where it goes'' doesn't have a well-defined answer.

Local transformation laws (written in the form of differential equations) do not necessarily translate into global transformation laws (written in the form of integral equations), but they may still remain strictly valid in a small enough neighborhood. As to where a non-conserved quantity goes... well, if it isn't conserved, it doesn't have to go anywhere or come from anywhere! It's just not conserved, meaning that no law exists that the amount you measure today will be the same as the amount you measure tomorrow. On the contrary, if it had to come from somewhere or go somewhere, it'd mean that the quantity _is_ conserved after all, so if it goes down in our neighborhood/universe/whatever, it has to go up somewhere else.

Maybe it is worth to consider the additional (and in parts much older) work on conserved and non-conserved quantities before Noether's Theorem and differential geometry arising in (projective) geometry (Plücker, Clebsch, Lindemann, Klein, Pasch, ...)? And even with respect to physical applications. Of course, nowadays' formulation of gravity may not be the best formalism to discuss such issues... ;-)

Dear Toth,

Then for the globally non-conserved quantity , it means that there will be an INEXPLICABLE surplus or deficit. Something coming out of nothing or going into nothing! I don't know if this will satisfy human curiosity!

Rajat

Charles,

Let's take the case of asymptotically flat space-times where a global energy can be defined.

Rajat: Yes, "something coming out of nothing or going into nothing" is exactly what non-conserved means. Human curiosity is one thing, human expectations are another: we may expect a quantity to be conserved and thus have some sort of an explanation as to where it is coming from or where it is going to, but Nature does not care about our expectations. If it turns out that something is not conserved in Nature, no amount of human curiosity (or human insistence) can force Nature to change her ways.

Charles,

I think what Robert is saying makes sense and it is not necessary that the global value integrated over all space would be infinite just because of asymptotic flatness. It surely would depend on the space-time dependence of the components of the metric tensor and of the stress-energy tensor. So a finite value for energy of the whole universe as has been proposed by Tyron and supported by hawking and Krauss and others is fairly possible in an asymptotically flat space-time. However I'm not sure if their arguments would hold for dark varieties of matter and energy that dominate the universe's energy density.

Rajat.

Dear Toth,

Your observation about the mutual independence of human curiosity and nature's ways would be perfectly valid, if we were ourselves not part and parcel of the very same nature which unfolds its secrets to us through through human curiosity alone. That the question can be posed is enough to send us searching for a satisfactory answer. Whatever we have achieved so far in science is precisely because of this fact only, why should we leave it when we are faced with some seeming conundrum like like a something-nothing interconversion. There must be away out. Let's be positive about our ability to have a logical and scientific explanation for everything.

Rajat: What I was trying to say is that if a quantity in Nature happens to be non-conserved, no amount of human curiosity will change the fact that the questions, "where does it go?" or "where did it come from?" have no meaning. (That is not to say that we cannot discover and understand the exact laws that govern the non-conserved quantity. We just need to make sure that the questions we are asking are in line with what Nature has to offer, and not necessarily with our own expectations and prejudices.)

Thank you very much Toth, for the clarification and for all your very sincere posts to make things clearer. It's nice to have people like explain things.

Regards

Charles,

Tyron published in Nature:

Edward P. Tryon, "Is the Universe a Vacuum Fluctuation?", Nature, vol. 246, p.396–397, 1973.

To me, logically the only contender to the idea of a zero-total-energy universe would be an infinite-total-energy universe. So we have to settle for one of the two. The infinite-total-energy option does not require energy conservation even locally, since any amount of energy can be easily added or subtracted without affecting the infinite total energy. The observed local energy conservation supports the zero-total-energy universe conjecture.

I thank you and Robert for the ongoing very stimulating discussion, which should continue.

Regards.

Dear Rajat,

I do not know if there are any contradictions to have a zero total-energy universe in combination with non-conservative forces. Favouring the idea of zero-total energy, I think it is possible to proceed by demonstrating that gravitational interactions originates from conservative or at least quasi-conservative forces (general relativity). Could we have local energy non-conservation in such a universe? Here we would need a quantum formulation that combines QFT (at least being compatible) with the cosmological scale and providing a better description of black holes etc. Let us summarize what we know and forgive me if this is trivial:

First there exist extensions of Noether’s (first) theorem that provide generalization to non-conservative cases, although I do not know exactly what that means. The interesting case is of course gravity in the theory of general relativity, TGR. For classical Newton gravity (conservative case) the force law, the momentum law and the energy law are all compatible. They may essentially be derived from each other. However, in the case of special relativity, we already note the complication that the mass is dependent on the velocity. It is interesting to note that the relation dEkin =vdp is valid in both classical mechanics and the theory of special of relativity.

However, introducing the gravitational force in the framework of special relativity immediately, the incompatibility between the force- and the energy law shows up. Depending on which law one prefers, the situation looks different. For instance Einstein started from the force law and succeeded to formulate a covariant theory, TGR. The Einstein laws appear to be the correct ones, explaining for instance the presession of Mercury’s perihelion, and being essential to the workings of GPS etc. However, if one starts from the energy law and proceed to use special relativity one may also obtain a perihelion movement, but only half the TGR-value, the error being associated with the neglect of the proper background independence. Obviously the force appears non-conservative in this model, unless the correct metric is incorporated.

One way to model a zero-energy universe would be to start with the simple Klein-Gordon, K-R, Equation (or the Dirac) and write it as matrix problem with the K-R equation being its secular equation. With a non-positive metric this is quite trivial. The extension to the gravitational case is in fact quite straight forward if one pays attention to two analogous matrix problems, i.e. to consistently consider simultaneously the conjugate problem, one for the energy and momentum leading up to a K-R like equation and the one for time and space leading up to the eigen-time expression. In this simple fashion the force might still be conservative (or at least quasi-conservative) and one should be able to establish the, sofar, known physical laws, as well as possibly solutions to present paradoxical cosmological behaviour, e.g. related to dark matter and dark energy, the cosmological constant etc. The question is whether, under the assumption of a zero-energy universe, we would also discover something new, like the teleonomic laws of fundamental importance in biology?

Existence of nonconservative force mean that system is not closed: one can assume that there are undetectable particles (like neutrino), other can say that there are other spatial dimensions, or maybe our universe is not alone and interact with others universes... or you caught God at work!

Dear Yurij,

You are right in stating that non-conservative fores would imply a system that is not closed, but this is true only if there is a surrounding with which there can be energy - exchange, assuming energy conservation for the whole Universe i.e. (system+surrounding). If we assume that the whole physical universe of whatever matter-energy content and whatever number of space-time dimensions, is all that we have, then (non-conservative forces + energy non-conservation) would surely be a logical possibility. The implications would be that there can be energy production or destruction by the non-conservative forces concerned without any contradiction. Could it be related to the kind of violation of energy conservation envisioned by NEILS BOHR in the context of defending the uncertainty principle, for which he drew lots of flak from all and sundry?

Really as you remarked in the end we may catch god at work then !

Regards,

Rajat.

the only conservative force its the deformation of space-time

theory

https://www.researchgate.net/publication/315825530_Theory_of_Empty_Universe_Second_Draft?_iepl%5BviewId%5D=jLKk1vTDMDYJaTUtv0JoUFKW&_iepl%5BprofilePublicationItemVariant%5D=default&_iepl%5Bcontexts%5D%5B0%5D=prfpi&_iepl%5BinteractionType%5D=publicationTitle

Article Theory of Empty Universe (Second Draft)

My feeling is that what we see as conservative values are more like the rule that heads and tails happen equally. I suspect they are probabilistic rules or equilibriums.

that is a cool exercise, what happens if you define some kind of fundamental non conservative force in a close system?