Aspherical gravitational monopoles
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文摘
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 potentialM /
x − xo; (ii) their interaction energy with an external gravitational potentialU (x) is −MU(xo); 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 R3; (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.
x − xo; (ii) their interaction energy with an external gravitational potentialU (x) is −MU(xo); 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 R3; (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.