The present paper is an extension of a recent work (Bhattacharya et al. 2010) to the Einstein-Strauss vacuole model with a cosmological constant, where we work out the light deflection by considering perturbations up to order M3 and confirm the light bending obtained previously in their vacuole model by Ishak et al. (2008). We also obtain another local coupling term −5πM2Λ/8 related to Λ, in addition to the one obtained by Sereno (2008, 2009). We argue that the vacuole method for light deflection is exclusively suited to cases where the cosmological constant Λ disappears from the path equation. However, the original Rindler-Ishak method (2007) still applies even if a certain parameter γ of Weyl gravity does not disappear. Here, using an alternative prescription, we obtain the known term −γR/2, as well as another new local term 3πγM/2 between M and γ. Physical implications are compared, where we argue that the repulsive term −γR/2 can be masked by the Schwarzschild term 2M/R in the halo regime supporting attractive property of the dark matter.

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We revisit certain features of an assumed spherically symmetric perfect fluid dark matter halo in the light of the observed data of our galaxy, the Milky Way (MW). The idea is to apply the Faber-Visser approach of combined observations of rotation curves and lensing to a first post-Newtonian approximation to "measure" the equation of state ω(r) of the perfect fluid galactic halo. However, for the model considered here, no constraints from lensing are used as it will be sufficient to consider only the rotation curve observations. The lensing mass together with other masses will be just computed using recent data. Since the halo has attractive gravity, we shall impose the constraint that ω(r) ≥ 0 for r ≤ R MW , where R MW ∼ 200 kpc is the adopted halo radius of our galaxy. The observed circular velocity ℓ = 2v 2 c /c 2 0 from the flat rotation curve and a crucial adjustable parameter D appearing in the perfect fluid solution then yield different numerical ranges of ω(r). It is demonstrated that the computed observables such as the rotation curve mass, the lens mass, the post-Newtonian mass of our galaxy compare well with the recent mass data. We also calculate the Faber-Visser χ− factor, which is a measure of pressure content in the dark matter. Our analysis indicates that a range 0 ≤ ω(r) ≤ 2.8 × 10 −7 for the perfect fluid dark matter can reasonably describe the attractive galactic halo. This is a strong constraint indicating a dust-like CDM halo (ω ∼ 0) supported also by CMB constraints.1 The reason for the appearance of static FLRW background around the imbedded perfect fluid dark matter is already explained in Ref. [1]. The Einstein field equations are solved with perfect fluid stress tensor in both the cases but we sought a static solution from the start. While working on a local problem (flat rotation curve), the scale factor is usually fixed to R 0 = 1 today.

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The famous hoop conjecture by Thorne has been claimed to be violated in curved spacetimes coupled to linear electrodynamics. Hod [10] has recently refuted this claim by clarifying the status and validity of the conjecture appropriately interpreting the gravitational mass parameter M . However, it turns out that partial violations of the conjecture might seemingly occur also in the well known regular curved spacetimes of gravity coupled to nonlinear electrodynamics. Using the interpretation of M in a generic form accommodating nonlinear electrodynamic coupling, we illustrate a novel extension that the hoop conjecture is not violated even in such curved spacetimes. We introduce a Hod function summarizing the hoop conjecture and find that it surprisingly encapsulates the transition regimes between "horizon and no horizon" across the critical values determined essentially by the concerned curved geometries.