Aspherical gravitational monopoles

We show how to construct non-spherically-symmetric extended bodies of uniform density behaving exactly as pointlike masses. These ``gravitational monopoles’’ have the following equivalent properties: (i) they generate, outside them, a spherically-symmetric gravitational potential $M\mathrm{/}\mathrm{\mid }x-x_O\mathrm{\mid }$; (ii) their interaction energy with an external gravitational potential $U(x)$ is $- M U(x_O)$;and(iii)alltheirmultipolemoments(oforder$l \geq 1$) with respect to their center of mass $O$ vanish identically. The method applies for any number of space dimensions. The free parameters entering the construction are: (1) an arbitrary surface $\mathrm{\Sigma }$ bounding a connected open subset $\mathrm{\Omega }$ of $R^3$; (2) the arbitrary choice of the center of mass $O$ within $\mathrm{\Omega }$; and (3) the total volume of the body. An extension of the method allows one to construct homogeneous bodies which are gravitationally equivalent (in the sense of having exactly the same multipole moments) to any given body.