1) Are oscillations the proven fact?

Neutrino oscillations explain the missing Solar neutrinos problem and seems quite a established fact nowadays. In any case mass-energy is conserved and so the oscillations would not affect the geometry of space time. Nevertheless, the change would be so tiny that I cannot imagine any measurable effect. 

No, because the metric doesn't change by this process: spacetime remains flat throughout. 

The metric doesn't change for any particle interaction because we don't have any theory that deals with both. At the very least, the metric should change when a gamma ray pair produces. Well on second thought, maybe not: a graviton is quadrupolar so maybe the basic symmetry of pair production doesn't require one to be emitted. Perhaps loop quantum gravity may say something about this. In any case the lack of theory that includes both particle scattering and gravitation makes even energy balance arguments suspect.

Of course there is such a theory, that describes the dynamics of the metric and its manifestation as gravitational radiation,which is classical, namely general relatvity. The source is the energy-momentum tensor of matter. This coupling is so small for neutrinos and any other elementary particles, that it leads to completely unobservable effects. 

However the statement that the neutrinos ``change mass in flavor oscillations'' is incorrect: the mass eigenstates are superpositions of flavor eigenstates and vice versa. It's not possible to define the mass of a flavor eigenstate, nor the flavor of a mass eigenstate. 

Neutrinos, obviously, have an energy-momentum tensor, thus couple to the metric, therefore, give rise to gravitational radiation-which for neutrinos that are relevant for observing flavor oscillations, thus are not very energetic, is unobservable. 

However the statement that such radiation has anything to do with the oscillations themselves is not correct. That is the issue here.

One could try a strictly gravitational mock up of the pair production scattering. A large dumb-bell shaped comet approaches a large planet and breaks up into 2 pieces. In some cases the 2 pieces both leave the vicinity of the planet. The 2 pieces are the analogs of the photons. The planets gravity field is the analog of the nuclear Coulomb field that the gamma ray interacts with to cause pair production. I  believe this interaction requires a change in the metric and gravitons to be emitted. After this analysis you can decide whether quantum effects may cut off the weak effects expected in particle scattering. 

 Мomentum is conserved only in one frame of reference! Sums of momentums are equal before interacting and after it.

Мomentums are different in different frame of reference, Indeed, Sum of momentums of particle flow is equal zero in own frame of reference, but is non-zero in any different frame of reference.

I do not think so.

As far as I know, in the standard description of neutrino oscillation there is no energy change: do you have any explicit reference where it says otherwise?

It should go like this: a neutrino is produced in some interaction with a well-defined flavour (that is, in a flavour eigenstate), and with given momentum distribution (as a wave packet in momentum space). The flavour eigenstate can then be described as a quantum superposition of mass eigenstates. Each wave component with a given energy will propagate with a slightly different velocity, according to its mass. There is no mass change: the components of different mass are all there from the beginning, it is just their relative phase which changes during the propagation (according to Schroedinger's equation), influencing the probability of measuring a given flavour when the neutrino is finally detected.

At no point is the conservation of energy or momentum violated in the process.

Conversely, if the neutrino emitted a gravitational wave, either total energy or total momentum conservation would necessarily be violated, just as it would be in the case of a free electron emitting an electromagnetic wave (one or more photons).

But we lack any theory connecting neutrino flavor shift with gravitation so of course present theory says four momentum is conserved. And no experiment would be sensitive to small four-momentum imbalances.

I think it may be worth clarifying my point:

flavour oscillation of neutrinos can be quite well understood using standard quantum mechanics and field theory, and if we look at it in this standard framework, I see no reason to expect that gravitational radiation should accompany it.

Then, of course, we have no completely satisfactory theory of quantum gravitation, and it may well be that phenomena we do not expect, and are not explained by currently established theories, actually happen in nature... It would not be the first time; still, this is by no means a necessity.

 Francisco says: "Each wave component with a given energy will propagate with a slightly different velocity, according to its mass." Suppose that we do a thought experiment and imagine that we are in the same frame of reference as a neutrino that is initially 100% an electron neutrino. Therefore, the electron neutrino appears stationary to us and its de Broglie wavelength is infinite. What do we see at a later time when the electron neutrino has become a muon neutrino? Is the muon neutrino still stationary? Is there any difference in the gravitational acceleration produced by the two different states? These questions are open to all.

My answer to John: the situation you describe is tricky.

In fact, if we initially have an electron neutrino at rest, we have a state with well-defined total 3-momentum (in the given reference frame, of course, where it is 0), but ill-defined total energy: the state is a superposition of states which different mass, and since we have no 3-momentum, the mass is equal to the energy. I do not think that this is what is actually produced when a neutrino is emitted.

More generally, and taking a particle-physics viewpoint, it seems to me that if we insist on producing a flavour eigenstate with completely defined 4-momentum (and so fully delocalized, with infinite de Broglie wave length; it does not matter if it is at rest or not), then by necessity we have that at most one of the mass components can be exactly on-shell (satisfy the relation E^2 = m^2 + p^2), and so propagate indefinitely. So, it appears that in this case, we are selecting which mass eigenstate will actually propagate a priori, so to speak. That's why I specifically talked about a "wave packet" in my original post (which is also closer to what will be produced in a realistic process: there will always be some spread in momentum).

I admit that it is the first time that I think about this from this point of view, so forgive me if I made a stupid error!

Francesco:  I appreciate your attempt at an answer. You have confirmed my suspicion that neutrino flavor oscillation is not completely understood. I will now propose a possible explanation which makes sense to me because it permits all these hypothetical questions to be answered using conceptually understandable physics. First, we know that the rate of neutrino flavor oscillation changes when neutrinos pass through a star and experience the MSW effect which is a type of resonance with electrons. We also know that there is a big gap in our knowledge when 80% of the mass in the universe is dark matter and we have no idea what it is. Something that we think that we understand must be wrong. The reason that neutrinos are not considered to be candidates for dark matter is that even at a temperature of 1° K they would have velocity exceeding the escape velocity of a galaxy. However, if there was a mechanism which permits neutrinos to lose kinetic energy (cool) relative to the local CMB rest frame of reference and reach a temperature in the range of 0.01° K, then neutrinos would be a great candidate for dark matter. I cannot go into details, but they would achieve the proper mass distribution around a galaxy and they are the lowest energy particle which makes them most likely to be formed.

Now I will pose a question. Suppose that it was possible to measure the rate of neutrino flavor oscillation in inter-galactic space which I define as outside of the dark matter volume of any galaxy. Would the rate of neutrino flavor oscillation be the same in inter-galactic space as it is at our location in the Milky Way galaxy? I am proposing that it would be different because I think that dark matter is a cloud of roughly 0.01° K neutrinos which are interacting with the MeV neutrinos that we are using in our experiments. When you postulate such an interaction with other neutrinos, then things that are hard to explain in a total vacuum become easy to explain. This short post does not permit more detail but the emission of gravitational waves when neutrinos interact is the mechanism which provides the necessary removal of kinetic energy (cooling) relative to the CMB rest frame.

Aleksei,   Thank you for your answer. I am really trying to find out if there is a conceptually understandable explanation of neutrino flavor oscillation if we assume observing this from a rest frame of reference (infinite de Broglie wavelength). The problem is that all the explanations I am seeing imply neutrinos moving in a highly relativistic frame of reference and a beat between two different de Broglie wavelengths. However, when the effect is quantified such as Δm213 = 2.3 x 10-3 eV2 this seems to be referencing the rest mass (rest energy). Is there conservation of energy and conservation of momentum in the rest frame of reference?

John, you suggested gravitational radiation as a means of cooling neutrinos for consistency with dark matter models but extra cooling relative to the CMB occurs anyway because neutrinos are not massless like the CMB photons. The temperature of the cosmic neutrino background is presently well below that of the CMB, though not quite as low as 0.01K. See Theo M. Nieuwenhuizen, "Do non-relativistic neutrinos constitute the dark matter?" Europhysics Letters 86, 59001 (2009). It includes a discussion of the accelerated cooling process. It is often (incorrectly) assumed that the neutrino temperature is much higher because of work prior to the late 90s where neutrinos were simply taken to be massless. I think Nieuwenhuizen is correct to infer the presence of 1.5eV fermions in the galaxy cluster he studied, but there are strong reasons to believe that active neutrinos cannot be responsible. I personally suspect that active neutrinos are down in the 0~0.05eV range and that the lensing effects in the studied galaxy cluster are due to sterile neutrinos (which due to their larger mass would cool faster and would have been gravitationally trapped by clusters for some time now).

 Robin, In your answer you said, " I personally suspect that active neutrinos are down in the 0~0.05eV range." This small average value of the neutrino energy increases the implied paradox if the neutrino mass/energy change is viewed in the rest frame of reference. While the energies are not known, the difference in the square of the masses (expressed as eV2) is known. If I take your proposed upper limit of 0.05 eV for the average neutrino energy and apply the known difference in the energy squared between electron and tau neutrinos (Δm213 = 2.3 x 10-3 eV2 ), it works out that the transition would require about a 50% change in mass/energy. Clearly this is a violation of both the conservation of energy and the conservation of momentum if this transition is viewed from an arbitrary frame of reference. I am proposing that this paradox is eliminated if the flavor oscillation involves an interaction with the dark matter that permeates the galaxy. Going one step further, if there is some mass change, there should be the emission of a small gravitational wave. It does not matter if the gravitational wave is experimentally measurable, the important part is that there would be a loss of kinetic energy which would tend to cool the neutrino.    

Any moving mass gives off gravity waves.