I wonder how a massless object would behave on Earth. To make things concrete, consider the object as a sphere of radius R, and zero mass. It also interacts with matter like a wood pencil would.
My guess is that if R is of the order of an angstrom, such an object would behave a bit like the fastest molecules of hydrogen in the atmosphere, though its speed would be that of light. In other words, it would suffer a random walk motion, bouncing on other molecules until it would leave the Earth in a finite and non zero time. If the object is macroscopic, when left untouched, it would appear to stand still. But what would happen if someone were to push it? Would it keep moving with a constant speed, and if so, which speed? My guess is that it would eventually stop because of air viscosity. So essentially, it wouldn't behave that much differently to an ordinary object, except that it would take almost no effort to push it. It would still take an effort because of air viscosity. And its speed would be the one our arm moves it with.
Am I correct? Or did I miss some/many things?
Edit: I now think a macroscopic such object would not be kickable in the sense that its motion would stop instantly after the force of one's arm or foot is exerted on it. This is due to the drag force of the air.
To make things crystal clear, let's discuss about the details of the thought experiment. The object has to satisfy relativity. It has zero mass, but interacts with ordinary matter in such a way that it is undeformable and interacts elastically with anything that collides with it. It is put at sea level under atmospheric pressure, in air with -initially at least- no wind. It is therefore under the gravitational field of the Earth and any possible drag/buoyant force that may exist because of the surrounding air.
I now think the object would rise like a helium balloon (though quicker than a He balloon of the same volume), but when kicked sideways, would not move at all once the foot or arm stops to touch it. That is so because of the drag force due to the viscous air surrouding the object, which would cancel any net motion with respect to the local wind speed. It would still rise because the buoyant force is continuously applied. But there is also drag. It's not clear to me whether the object would rise with constant speed or not (I guess not, but I have trouble imagining it would continuously accelerate while having a zero mass!).