Tipler Naked Singularities in $N$ Dimensions

TL;DR

This work extends the analysis of strong curvature naked singularities from four-dimensional gravitational collapse to arbitrary finite dimensions $N$, focusing on spherically symmetric, type-I matter fields with regular initial data under mild energy conditions. By introducing the positive root condition (PRC) and simple positive root condition (SPRC) and employing Clarke and Krolak's curvature growth condition (CGC), the authors derive necessary and sufficient criteria for naked singularities and establish when such singularities are of strong curvature type. They demonstrate that in $N=4$ and $N=5$, strong curvature naked singularities can occur for central, massless (Misner-Sharp mass vanishing at the center) singularities given a specific scaling parameter $\alpha=(N-1)/(N-3)$, while for $N\ge6$ the CGC is not satisfied by past-incomplete geodesics associated with naked singularities, hinting at possible restoration of cosmic censorship in higher dimensions. The results illuminate the dimension-dependent nature of phase transitions between black holes and naked singularities and connect these outcomes to the behavior of spacetime curvature near singularities, with implications for the validity of CCC in higher dimensions and potential links to quantum gravity scenarios.

Abstract

A spacetime singularity, identified by the existence of incomplete causal geodesics in the spacetime, is called a (Tipler) strong curvature singularity if the volume form acting on independent Jacobi fields along causal geodesics vanishes in the approach of the singularity. It is called naked if at least one of these causal geodesics is past incomplete. Here, we study the formation of strong curvature naked singularities arising from spherically symmetric gravitational collapse of general type-I matter fields in an arbitrarily finite number of dimensions. In the spirit of Joshi and Dwivedi [26], and Goswami and Joshi [31], we first construct regular initial data in terms of matter variables and geometric quantities, subject to the dominant and null energy conditions. Using this initial data, we derive two distinct (but not mutually exclusive) conditions, which we call the positive root condition (PRC) and the simple positive root condition (SPRC), that serve as necessary and sufficient conditions, respectively, for the existence of naked singularities. In doing so, we generalize the results of [26] and [31]. We further restrict the PRC and the SPRC by imposing the curvature growth condition (CGC) of Clarke and Krolak [24] on all causal curves that satisfy the causal convergence condition. The CGC gives a sufficient condition for the naked singularities implying the PRC and implied by the SPRC, to be of strong curvature type; thereby also implying the $C^2$ inextendibility of the spacetime. Using the CGC, we extend the results of [28] (that hold for dimension $N=4$) to the case $N=5$, showing that strong curvature naked singularities can occur in this case. However, for the case $N\geq6$, we show that past-incomplete causal curves that identify naked singularities do not satisfy the CGC. These results shed light on the validity of the cosmic censorship conjectures in arbitrary dimensions.

Figures (2)

• Figure 1: On the LHS, we have the conformal diagram for the well known Oppenheimer-Synder-Datt homogeneous dust collapse, which results in the formation of a black hole. Moreover, in the center we have the conformal diagram for the LTB inhomogeneous dust collapse scenario with a locally naked singularity, whereas on the RHS we have the conformal diagram for the LTB inhomogeneous dust collapse scenario with a globally naked singularity PhysRevD.19.2239PhysRevD.109.064019. The shaded regions represent the MGHDs of some initial data hypersurface $\Sigma$. Further, $H^{+}(\Sigma)$ denotes the future Cauchy horizon of $\Sigma$, $\mathscr{E}$ denotes the event horizon, and $\mathscr{N}$ denotes the naked singularity. Note that for the locally naked scenario, $\mathscr{I}^{+}$ is complete and $H^{+}(\Sigma) \cap \mathscr{I}^{+}=\emptyset$, whereas for the globally naked scenario, $\mathscr{I}^{+}$ is incomplete and $H^{+}(\Sigma) \cap \mathscr{I}^{+} \neq \emptyset$. Additionally for both the scenarios, the MGHDs can be extended (to the unshaded region) which then does not admit $\Sigma$ as a Cauchy surface, and hence is not globally hyperbolic. Hence, globally naked singularities violate both WCC and SCC, whereas locally naked singularities only violate the SCC.
• Figure 2: On the LHS, we have cartoon depicting the concepts of ingoing and outgoing geodesic congruences. At any point $p\in M$, we may define two vectors $\mathcal{N}^{-} (p)$ and $\mathcal{N}^+ (p)$, that form the tangents to ingoing and outgoing curves $\gamma_i$ and $\gamma_o$ through that point. In the theory of gravitational collapse, the terminology "ingoing" and "outgoing" is often used for geodesics running into or emanating from a singularity. In the latter case, we refer to the singularity as being naked. On the RHS, we have a cartoon depicting a trapped surface. For such surfaces, ingoing and outgoing geodesics converge in the future (or past).

Theorems & Definitions (40)

• Conjecture 1
• Conjecture 2
• Definition 1
• Remark 1
• Definition 2
• Remark 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6