Confused about volume, density and mass, help!

I got into an argument with my friend, which cast confusion on my understanding of density and its relationship to volume. I'm hoping to get some clarity. The argument involved describing density in terms of volume. Let's say you define a sphere in empty space. You choose a point, apply the formula for a sphere, and now you have a sphere. Not a sphere OF anything other than space, just a spacial, theoretical sphere. No particles, massless or otherwise (this is a thought experiment). What is the density of that sphere? Is it zero, or is it undefined? I know density can be defined as p = m/v. But in a theoretical sphere, which HAS volume, should we call mass zero, because there is none? Or is it undefined because a theoretical sphere really isn't related to mass at all? If it IS undefined, does that mean it makes no sense to relate density to volume because density is only a property of mass?

The answer, to me, seems to be that in fact it makes no sense to talk about the density of a massless object. Sorry if I answered my own question, but I would still like clarity. If someone could help guide me through the assumptions I'm making about reality and math and how they relate, I'd really appreciate it.

Theoretically speaking, in order for something (usually particles) to be massless, it has to be travelling in the speed of light. In quantum theory, uncertainty principle states that the position and momentum of such particle cannot be accurately determined, therefore it is not possible to measure the volume of the particle. So it is generally assumed that the volume is too small and/or insignificant.

Regarding your question, I think that the simple answer would be; a massless object would have no volume nor density, or immeasurable. Take your pick.

• That's interesting! So, if we simply image a sphere, it's not valid to say that sphere is massless, because it's not really even a THING.. just an idea in our brains imposed by our brains on the space we inhabit? – chris c Feb 27 '16 at 15:15

It is simple, the mass density is zero because m is zero (as there is no mass). Using the definition m/v, v is non-zero, 0/v = 0. If v was also zero, then it would be meaningless or undefined.

However energy density of the sphere of empty space is non-zero. I think it is known to be negative energy density. The negative energy of empty space is suspected to be the dark energy.

Using mass energy equivalence, In one sense, you can say that mass density of that imaginary sphere is negative.

• So let me ask you this. Let's say you define a sphere in empty space, then you put an object in there sphere, say a rock. Can you now meaningfully say the sphere is more dense? It seems to me you still can't say that, because the object in the sphere is not a part of the sphere. The only relation the sphere has to that rock is that the rock exists within the sphere. The rock doesn't give the sphere mass, and therefore it seems that it's not meaningful to say that the sphere IS dense is any sense. – chris c Feb 27 '16 at 15:27