Aspherical gravitational monopoles

Abstract

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/b x − x ob ; (ii) their interaction energy with an external gravitational potential U( x) is −MU( x o) ; and (iii) all their multipole moments (of order ℓ ⩾ 1) with respect to their centre 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 Σ bounding a connected open subset Ω of R 3; (2) the arbitrary choice of the centre of mass O within Ω; 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. Though our method generally assumes that the domain Ω is bounded (which leads to bounded monopoles with closed internal cavities), it can also generate unbounded monopoles, with exponentially decreasing thicknesses at infinity, and cylindriclike internal cavities reachable from infinity. This may be useful for optimizing the shape of test masses in high-precision Equivalence Principle experiments, such as the planned Satellite Test of the Equivalence Principle (STEP). By suppressing the couplings to gravity gradients, one can design, with great flexibility in the choice of shapes, differential accelerometers made of nested bodies, which are (exactly or exponentially) insensitive to all external gravitational disturbances. Alternatively, one can construct nested bodies of arbitrary densities having identical (or proportional) sequences of multipole moments, thereby also suppressing any differential acceleration caused by external gravity gradients.