Symmetry inheritance of scalar fields

Abstract

Abstract. Matter fields don't necessarily have to share the symmetries with the spacetime they live in. When this happens, we speak of the symmetry inheritance of fields. In this paper we classify the obstructions of symmetry inheritance by the scalar fields, both real and complex, and look more closely at the special cases of stationary and axially symmetric spacetimes. Since the symmetry noninheritance is present in the scalar fields of boson stars and may enable the existence of the black hole scalar hair, our results narrow the possible classes of such solutions. Finally, we define and analyse the symmetry noninheritance contributions to Komar mass and angular momentum of the black hole scalar hair. Keywords: scalar fields, symmetry inheritance, no-hair theorems, boson stars Imperfect accordance of spacetime and fieldsThe simplest version of a relativistic theory is modeled by matter fields on a fixed, possibly curved spacetime. From the perspective of general theory of relativity, such fields can be considered as mere test fields since they don't participate in gravitational field equations and thus do not affect the spacetime itself. Hence, it doesn't come as a surprise that it is not necessary for such fields to have the same symmetries as the background spacetime. For example, in a typical relativistic course we shall encounter scalar fields (solutions to Klein-Gordon equation) and electromagnetic fields (solutions to Maxwell's equations) which come in various shapes and forms, possessing more or less symmetries, in a sheer contrast with the underlying Minkowski spacetime, which is maximally symmetric.On the other hand, if we don't neglect the backreaction of the matter fields on the spacetime and look at the exact solutions to the gravitational field equations,

Static black holes with axial symmetry in asymptoticallyAdS4spacetime

“…In both cases the mechanism which allowed for the existence of such BHs was the symmetry noninheritance of the matter fields [7]. That is, the matter fields possess a dependence on the azimuthal angle φ, which, however, does not manifest at the level of the energy-momentum tensor.…”

“…As argued in this work, when allowing for a sufficiently general matter content, one finds BHs which are static and axially symmetric only. This is achieved by allowing for matter fields that do not share the symmetries with the spacetime they live in [7].…”

Quasinormal modes and greybody factors of a four-dimensional Lifshitz black hole with z = 0 $z=0$

“…Therefore, it is not necessary for such fields to have the same symmetries as the background spacetime. On the other hand, if one considers the backreaction of the matter fields over the spacetime, in order to look for exact solutions to the field equations, the relation between symmetries of the spacetime and the matter fields is not trivial; for a recent study about symmetry inheritance of scalar fields, see Smolić (2015) and references therein. In the case considered here, the gravitational field equations imply that the trace of the stressenergy tensor must vanish, due to the fact that the Bach tensor is traceless; therefore, if one goes beyond the probefield approximation, this implies that the stress-energy tensor of the matter fields must be traceless.…”

Breaking of scaling symmetry by massless scalar on de Sitter

“…One might try to employ the observation that the massless solution (5) is defined up to a constant in order to remove the divergent term above. This though does not work, as (12) with the divergent constant removed simply does not satisfy the massless equation of motion (4). The proper conclusion is that the scaling solution of our problem is singular in the massless limit, and (5) does not represent a valid physical solution.…”

“…Remarkably, the energymomentum tensor in (22) respects dilatation symmetry, even though the field profile in (20) does not. This is a cosmological example of the phenomenon of (perturbative) symmetry non-inheritance, which has attracted significant attention in recent literature [11,12,13,14,15].…”