Gravitational Collapse and Singularity Formation in Brans-Dicke Gravity

TL;DR

This work investigates gravitational collapse in Brans–Dicke gravity driven solely by the BD scalar field with a self‑interaction potential $V(Φ)$, excluding ordinary matter. The authors derive the BD field equations, implement a spatially flat FRW interior, and solve numerically for the scalar field $Φ(a)$ near singularity under two energy‑density scalings $ρ\sim 1/a$ and $ρ\sim -\ln a$. They show that a central curvature singularity forms with diverging Kretschmann invariant and analyze the causal structure by matching to a generalized Vaidya exterior; under certain boundary radii the singularity is globally naked. Using an extended Tipler criterion, they find that the $ρ\sim 1/a$ collapse yields a Tipler‑strong singularity for all $ω>0$, while the $ρ\sim -\ln a$ case is Tipler‑weak, highlighting rich phenomenology of scalar‑tensor gravity and potential challenges to Cosmic Censorship.

Abstract

We investigate gravitational collapse driven solely by a self-interacting Brans--Dicke (BD) scalar field in the absence of ordinary matter. In this framework, the spacetime dynamics are governed solely by the scalar field $Φ$, endowed with a self-interaction potential $V(Φ)$ and non-minimally coupled to the Ricci scalar through the Brans--Dicke action. We numerically solve for the evolution of $Φ(t)$ and the corresponding potential $V(Φ)$ in order to track the collapse dynamics leading to singularity formation. Our analysis demonstrates that, for the energy densities $ρ\approx 1/a$ and $ρ\approx -\ln a$, the collapse inevitably leads to the formation of a central curvature singularity while consistently satisfying the weak energy condition. We further examine the causal structure of the resulting singularity and find that future-directed null geodesics originating from the singularity can propagate to future null infinity, making the singularity globally visible. The strength of the singularity is also examined by extending Tipler's strong curvature condition to the Brans--Dicke field equations. Overall, our findings indicate that gravitational collapse in scalar--tensor gravity can give rise to scenarios that challenge the Cosmic Censorship Conjecture, while underscoring the potential observational relevance of singularities formed through BD scalar-field-driven collapse.

Figures (6)

• Figure 1: Variation of $\Phi(a)$ with $a(t)$ for $\rho=1/a$ when $\omega=6$ (red) and $\omega=10^{6}$ (blue).
• Figure 2: Plot for the potential $V(\Phi)$ against the BD scalar field $\Phi$ for $\rho=1/a$ when $\omega=6$.
• Figure 3: Variation of $\Phi(a)$ with $a(t)$ for $\rho=-\ln{a}$ when $\omega=6$ (red) and $\omega=10^{6}$ (blue).
• Figure 4: Plot for $V(\Phi)$ against the BD scalar field $\Phi$ for $\rho=-\ln{a}$ when $\omega=6$.
• Figure 5: A finite interior region ($g^-_{\mu\nu}$), sourced only by the Brans–Dicke scalar field ($T^{(\mathrm m)}_{ab}=0$), is smoothly matched at $r=r_b$ to an exterior generalized Vaidya spacetime ($g^+_{\mu\nu}$). The collapse culminates in a central naked singularity, with outgoing null geodesics escaping to the exterior.