Non-gravitating scalars and spacetime compactification

Abstract

We discuss role of partially gravitating scalar fields, scalar fields whose energy-momentum tensors vanish for a subset of dimensions, in dynamical compactification of a given set of dimensions. We show that the resulting spacetime exhibits a factorizable geometry consisting of usual four-dimensional spacetime with full Poincaré invariance times a manifold of extra dimensions whose size and shape are determined by the scalar field dynamics. Depending on the strength of its coupling to the curvature scalar, the vacuum expectation value (VEV) of the scalar field may or may not vanish. When its VEV is zero the higher-dimensional spacetime is completely flat and there is no compactification effect at all. On the other hand, when its VEV is nonzero the extra dimensions get spontaneously compactified. The compactification process is such that a bulk cosmological constant is utilized for curving the extra dimensions.

Non-gravitating scalar field in the FRW background

“…This question was addressed by [10] in the Minkowski spacetime and in the next section we will obtain similar results for FRW and anti-de Sitter backgrounds. As in [10,11], we find that the classical nongravitating solution allows space dependent scalar field profile in all these cases, including the Minkowski spacetime. However, in every case, this solution is singular.…”

Higher-dimensional AdS waves andppwaves with conformally related sources

“…Here, we have been concerned with the pp-wave and the AdS wave backgrounds in arbitrary dimension. These spacetimes share in common that their coupling to a matter source The independent Einstein equations (18) for the energy-momentum (20) on the background of a D-dimensional pp-wave (3) are given by the following combination…”

“…APPENDIX B: FIELD EQUATIONS FOR HIGHER DIMENSIONAL ADS WAVES All the information following from Einstein equations (18) with energy-momentum (20) on the background of a D-dimensional AdS wave ( 16) is encoded in the following combination with △ = δ ij ∂ i ∂ j . As before the above equations becomes the 2 + 1 dimensional ones studied in Ref.…”

“…In fact, at the vanishing cosmological constant limit (l → ∞), we recover the potential permitted by the pp-wave background (13). Finally, we also mention that this potential is exactly the one arising in the context of scalar fields nonminimally coupled to special geometries without inducing backreaction (the static BTZ black hole [15,16], flat space [17,18], and the generalized (A)dS spacetimes [19]). All these examples share a common feature, namely the existence of nontrivial solutions with a vanishing energy-momentum tensor called stealth configurations.…”

Are stealth scalar fields stable?

“…In all these examples, the scalar field is non-trivial but recently, even more interesting solutions were found in which an inhomogeneous, wave-like field does not gravitate [7][8][9][10][11][12][13].…”