suppose there is a scale able to measure weight with an uncertainty of $10^{-9}kg$ . On the scale, an airtight plastic chamber is placed. Initially, a fly of mass $10^{-5}kg$ is sitting at the bottom of the chamber, which sits on the scale. At a later point in time the fly is flying around the chamber. Will there be a difference in the observed weight as measured by the scale when the fly is sitting at the bottom of the chamber compared to when it is flying around the chamber at some point in time? If so, what does the value of this difference depend on (I am most concerned with the case where the fly has not touched any surface of the container in enough time for the scale to reach some equilibrium value (or do the pressure variations induced from the flies wings cause constant fluctuations in the scale)?
Newtonian Mechanics – Effects of a Flying Fly Inside a Sealed Box on a Scale
airweight
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You ask, "How do we describe mass to the aliens, who don't know about our (g)?" This is an example of a class of questions referred to by Martin Gardner as "Ozma problems." The classic Ozma problem is how we describe to aliens the distinction between right and left; the answer is that we do it by describing the weak nuclear force.
Your statement of your Ozma problem seems a little ambiguous to me. Essentially you're asking how we describe to the aliens a unit of gravitational mass. (You don't say so explicitly, but it seems clear from context that you don't mean inertial mass.) Futhermore, there is a distinction bewteen active gravitational mass (the ability to create spacetime curvature) and passive gravitational mass (what we measure with a balance). Not only that, but your question could be interpreted as asking whether we can compare with the aliens and see whether the value of the gravitational constant $G$ is the same in their region of spacetime as it is in ours.
We can easily establish 1 g as a unit of inertial mass. For example, we can say that it's the inertia of a certain number of carbon-12 atoms.
The equivalence principle holds for us, so presumably it holds in experiments done by the aliens as well. This establishes that our 1 g unit of inertial mass can also be used as a unit for the passive gravitational mass of test particles.
You didn't ask about active gravitational mass, but the equivalence of active and passive gravitational mass is required by conservation of momentum, and has also been verified empirically in Kreuzer 1968. Cf. Will 1976 and Bartlett 1986.
The other issue is whether $G$ is the same for the aliens as for us. Duff 2002 has an explanation of the fact that it is impossible to test whether unitful constants vary between one region of spacetime and another. However, there are various unitless constants that involve $G$, such as the ratio of the mass of the electron to the Planck mass.
A more fundamental difficulty in the fundamental definition of mass is that general relativity doesn't seem to offer any way of defining a conserved, global, scalar measure of mass-energy. See, e.g., MTW, p. 457
Bartlett, Phys. Rev. Lett. 57 (1986) 21.
Duff, 2002, "Comment on time-variation of fundamental constants," http://arxiv.org/abs/hep-th/0208093
Kreuzer, Phys. Rev. 169 (1968) 1007
MTW: Misner, Thorne, and Wheeler, Gravitation, 1973.
Will, “Active mass in relativistic gravity: Theoretical interpretation of the Kreuzer experiment,” Ap. J. 204 (1976) 234, available online at adsabs. harvard.edu.
For any every day life purpose, they weight the same. But there is a difference:
Weight vs mass
Weight and mass are different things. The mass is a measurement of the amount of matter (alternatively, the opposition of a body to change trajectory), whereas the weight is the strength with the Earth pulls objects towards it. The difference is subtle: an astronaut in orbit has mass, but no weight; a scale measures weight, but it transforms it to mass.
Now think of a helium balloon as it goes up. Why does it go up, if it has some mass, and thus the Earth pulls it downwards? Because its weight is less than the air around, so the Earth is pulling from it more, and so it has preference.
Or, in other words, Archimedes principle holds. If you measure it, the balloon has negative weight.
Feathers and lead
If you weight a ton (mass) of feathers and a ton of lead, you will get that the lead is slightly heavier: the air around the feathers is pushing it upwards more than the air around the lead, because feathers take much more space, and thus, more air helps with buoyancy.
A bag full of air
Now we get to your question. We have two rigid identical boxes, one empty, and other full of air. The second one will weight a bit more, and also have more mass. The effect of the air is the same for both, as they have the same volume.
Now, take two soft plastic bag and fill one with air. Again, the mass of the one with air will be greater, but the weight will be the same. Why? Because the one that is empty has also a smaller volume.
Do a thought experiment: take the rigid box full with air and put it on a scale. Carefully open the lid. Nothing has changed, the weight is the same. In the soft bags, the extra mass added by the bag gets exactly cancelled out by the buoyancy (by Archimedes' principle) provided by the extra volume.
Perception
All this efects are small. For volumens of a liter, we are talking about differences of 1g. As you righfully note, the density increases, and we tend to feel it as heavier.
Best Answer
If you had a perfect scale, the reading would fluctuate based on
$$\delta w = m\ddot{x}_{cm}$$
$\delta w$ is the size of the fluctuation in the reading, $m$ the total mass on the scale (including fly and air), and $\ddot{x}_{cm}$ the acceleration of the center of mass.
Integrating over time,
$$\int_{time} \delta w(t) = m\Delta(\dot{x}_{cm})$$
Here, $\Delta(\dot{x}_{cm})$ is the change in velocity of the center of mass over the period you observe the readings. Because the velocity of the center of mass cannot change very much, if you integrate the fluctuations over time, you wind find that their average tends towards zero. If the fly begins and ends in the same place and the air is still, the fluctuations integrate out to exactly zero.
Whenever the fly is accelerating up, we expect the reading to be a little higher than normal. When the fly accelerates down, we expect the reading to be a little lower than normal. If the fly hovers in a steady state, the reading will be the same as if the fly were still sitting on the bottom.
A real scale cannot adjust itself perfectly and instantaneously, so we would need to know more details of the scale to say more about the real reading.