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The problem of describing isolated rotating bodies in equilibrium in GeneralRelativity has so far been treated under the assumption of the circularitycondition in the interior of the body. For a fluid without energy flux, thiscondition implies that the fluid flow moves only along the angular direction,i.e. there is no convection. Using this simplification, some recent studieshave provided us with uniqueness and existence results for asymptotically flatvacuum exterior fields given the interior sources. Here, the generalisation ofthe problem to include general sources is studied. It is proven that theconvective motions have no direct influence on the exterior field, and hence,that the aforementioned results on uniqueness and existence of exterior fieldsapply equally in the general case.Comment: 8 pages, LaTex, uses iopart style files. To appear in Class. Quatum  Gra

Influence of general convective motions on the exterior of isolated rotating bodies in equilibrium