Towards a characterization of fields leading to black hole hair

In the present work, it is shown that an asymptotically flat spherical black hole can have a nontrivial signature of any field, for an exterior observer, if the energy-momentum tensor of the corresponding field is either trace-free or if the trace falls off at least as rapidly as the inverse cube of the radial distance. In the absence of a general 'No Hair Theorem', this result can provide a characterization of the fields leading to a black hole hair.