A test for virtual particles by measuring gravity fluctuations possible? Ok to begin I will begin by talking briefly about my discussions with my Quantum Mechanics (specializes in Particle physics) professor and my Cosmology Professor (who studies particle physics with respect to cosmology).
The question I asked my professors is since virtual article in the electromagnetism interactions are electron-positron pairs and they have a mass. Provided we have a sensitive enough experiment in principle. 
Taking into account the expected mass of something like the sun and even calculate the mass that escapes in the form of energy we should expect it to either be constant or a small decay. This would cause the gravity measurements to either be constant of the force to weaken overtime. 
What we should not expect is these to be a small sinusoidal oscillations in the gravity that correspond to the frequency at which the electron-positron pairs pop in and out. And if we get these corresponding oscillations then what we a measuring is the the virtual particles adding small mass then disappearing and repeating this cycle.
My Quantum Mechanics professor says since at the end of the Feynman diagrams energy is conserved we should not see any perturbed mass sinusoidal oscillations. 
My Cosmology Professor says it may be possible. 
My questions is who is more correct in this regard is my experiment possible Provided it is sensitive enough?
Thanks by the way I have only learned Quantum Mechanics so far including perturbation theory, spin, position and momentum space transforms. Please when you reply try explain while taking into account my understanding. 
update note- I am talking about the Quantum fluctuations caused by the Lamb Shift in which the electron-positron pairs appear and disappear. This would cause gravity fluctuations with the same frequency as the Quantum fluctuations caused by the Lamb Shift
 A: The Lamb shift 

According to the hydrogen Shrodinger equation solution, the energy levels of the hydrogen electron should depend only on the principal quantum number n. In 1951, Willis Lamb discovered that this was not so - that the 2p(1/2) state is slightly lower than the 2s(1/2) state resulting in a slight shift of the corresponding spectral line (the Lamb shift).

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There can be a self interaction of the electron by exchange of a photon as sketched in the Feynman diagram . 


These virtual electrons never pop in and out, let alone in any sinusoidal fashion, they are a mathematical analogue useful for calculations and a help to the intuition, and they are just part of the exchange diagrams necessary to calculate crossections of interactions. The only hypothesis for the generation of real electrons and other particles from virtual pairs  in the diagrams comes next to the gravitational field of black holes, called Hawking radiation, and the energy needed for the particle to be on mass shell is taken from the strong gravitational field of the black hole.
Back to  the Lamb shift :

This "smears out" the electron position over a range of about 0.1 fermi (Bohr radius = 52,900 fermis). This causes the electron spin g-factor to be slightly different from 2. There is also a slight weakening of the force on the electron when it is very close to the nucleus, causing the 2s electron (which has penetration all the way to the nucleus) to be slightly higher in energy than the 2p(1/2) electron. 

Note  that the Lamb shift happens because of the smearing out of the virtual electron's position and the effect appears on the g factor, related to the magnetic moment of the electron, not the mass. This is because off mass shell particles have indeterminate mass, as was stated in one of the comments to your question, whereas quantum numbers that define the particle are there.
So your analogue should be the Zeeman effect . You are asking whether the gravitational field of the sun could split spectral lines the way the magnetic and electric fields do. In astrophysics this effect of the magnetic fields is studied  and that is the way one know of the existence of the magnetic fields. A gravitational field effect might be there  but it would be flooded out by the sun's magnetic field : 
Feynman diagrams at the vertices have coupling constants . The effect of magnetic fields gives a 1/137 at each vertex. The gravitational field given that gravitons would have to be exchanged in the appropriate diagrams , gives a 10^-39 at each vertex. An impossible to measure difference in the Zeeman effect.
