Can accretion properties distinguish between a naked singularity, wormhole and black hole?

We first advance a mathematical novelty that the three geometrically and topologically distinct objects mentioned in the title can be exactly obtained from the Jordan frame vacuum Brans I solution by a combination of coordinate transformations, trigonometric identities and complex Wick rotation. Next, we study their respective accretion properties using the Page–Thorne model which studies accretion properties exclusively for $$r\ge r_{\text {ms}}$$ r ≥ r ms (the minimally stable radius of particle orbits), while the radii of singularity/throat/horizon $$r<r_{\text {ms}}$$ r < r ms . Also, its Page–Thorne efficiency $$\epsilon $$ ϵ is found to increase with decreasing $$r_{\text {ms}}$$ r ms and also yields $$\epsilon =0.0572$$ ϵ = 0.0572 for Schwarzschild black hole (SBH). But in the singular limit $$r\rightarrow r_{s}$$ r → r s (radius of singularity), we have $$\epsilon \rightarrow 1$$ ϵ → 1 giving rise to $$100 \%$$ 100 % efficiency in agreement with the efficiency of the naked singularity constructed in [10]. We show that the differential accretion luminosity $$\frac{d{\mathcal {L}}_{\infty }}{d\ln {r}}$$ d L ∞ d ln r of Buchdahl naked singularity (BNS) is always substantially larger than that of SBH, while Eddington luminosity at infinity $$L_{\text {Edd}}^{\infty }$$ L Edd ∞ for BNS could be arbitrarily large at $$r\rightarrow r_{s}$$ r → r s due to the scalar field $$\phi $$ ϕ that is defined in $$(r_{s}, \infty )$$ ( r s , ∞ ) . It is concluded that BNS accretion profiles can still be higher than those of regular objects in the universe.