How does mass density affect gravity?

If I have two equal mass of objects, and one is less dense but spread over a larger volume, how does their gravity differ from two masses of equal mass and density? (assuming the given volumes do not intersect.)

• Per equivalence principle it doesn't. – CuriousOne Jul 7 '15 at 19:24
• Do you mean gravity as it would feel far away from the obects on a third body, or just their gravitational pull on each other? – user81619 Jul 7 '15 at 19:41
• on each other, I think. – punkerplunk Jul 8 '15 at 1:36

2 Answers

If we assume that the objects are spherically symmetric, then as long as you are outside the object the gravitational field is not dependant on the radius (and therefore density) of the object.

Classically (i.e. non-relativitically) this result is due to Gauss' law. When we move to general relativity the result is due to Birkhoff's theorem.

• Assuming they are not spheres and their shapes are not necessarily symmetric. I'm curious about the SHAPE of the mass affecting it's gravity, so I don't see the classical model as being specific enough. Would space follow such an object's topography, allowing me to consider object's gravitation as being be akin to something like the Ricci Flow? I'm looking into Birkhoff and related Schwarzchild but I don't see them making cases for amorphic masses – punkerplunk Jul 8 '15 at 23:06

If one has two spherical objects with same mass $m$ and densities $\rho_1$ and $\rho_2$, the gravitational field is the same in the two cases $\rho_1=\rho_2$ and $\rho_1\neq\rho_2$. This is a consequence of Gauss' law (see answer of John Rennie).

However, if the mass distribution is not spherical, the gravitational field does depend not only on the densities of the two mass distributions, but also on the geometrical shapes. The situation gets more complicated.

Example

I consider here a situation where the dependence of the gravitational field on the density and shapes of the charge distributions is very pronounced. Consider a comparison between A) two spherical objects with mass $m$ and density $\rho$; and B) one spherical object with density $\rho_1$, and the a thin foil with the same mass $m$ but with a very small density $\rho_2\ll \rho_1$. At large distances from these two objects, the gravitational field is the same in the two cases A) and B). Indeed, it is the same as of a single sphere of mass $2m$. At small distances however, the gravitational field is completely different. Consider for example the point located at the same distance $d$ between the two spheres in A) and between the sphere and the thin foil in case B). Suppose also that in the case B) the distance $d$ is smaller than the dimensions of the thin foil.

Case A) the gravitational field in the middle point is simply zero, since the contributions of the two spheres are equal and opposite and cancel each other.

Case B) the gravitational field of the thin foil is largely negligible (it is proportional to the mass density in this case) if the distance $d$ is much smaller than the linear dimension of the foil, and the only contribution to the field in this point is that of the sphere. In this limit (linear dimension much greater than $d$) the foil can be thought as an infinite plane with finite mass $m$ but with density $\rho_2=0$ (see also the related problem of the electric field of an infinite charged plane).