[Paper Review] Gravitational collapse with equation of state

Sanjay B. Sarwe, R. V. Saraykar+1 more|arXiv (Cornell University)|Jul 13, 2012

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper investigates spherical gravitational collapse of a perfect fluid with a linear isentropic equation of state $p = k\rho$ using exact solutions to Einstein's equations. It demonstrates that both black holes and naked singularities emerge as end states depending on the parameter $k$, with even small pressure ($k > 0$) leading to locally naked singularities in inhomogeneous dust collapse, challenging the cosmic censorship hypothesis.

ABSTRACT

We investigate here gravitational collapse of a perfect fluid with a linear isentropic equation of state $p = k ρ$. A class of collapse models is given which is a family of solutions to Einstein equations and the final fate of collapse is analyzed in terms of the formation of black holes and naked singularities. The collapse evolves from a regular initial data and the positivity of energy conditions and other physical regularity conditions are satisfied. As we provided here an explicit class, this gives useful insights into the endstates of collapse with a physically reasonable and relevant equation of state and for the cosmic censorship hypothesis.

Motivation & Objective

• To investigate the role of a linear isentropic equation of state $p = k\rho$ in determining the final fate of spherical gravitational collapse.
• To analyze whether naked singularities persist as collapse end states when pressure is included, particularly in inhomogeneous models.
• To construct exact solutions to Einstein's equations for perfect fluid collapse with $p = k\rho$, satisfying energy conditions and regularity.
• To test the cosmic censorship hypothesis under physically realistic conditions by varying $k$ and initial data.

Proposed method

• Constructs a family of exact solutions to Einstein's equations for spherically symmetric, inhomogeneous perfect fluid collapse with $p = k\rho$, $0 \leq k \leq 1$.
• Imposes marginally bound condition ($b(r) = 0$) and uses a specific mass function $M(r,v)$ that satisfies regularity and energy conditions.
• Analyzes the apparent horizon curve $t_{ah}(r)$ via integral expressions involving $\chi_2(v_{ah})$, derived from the metric and fluid dynamics equations.
• Evaluates the sign of $\chi_2(v_{ah})$ to determine whether the apparent horizon is increasing or decreasing near the singularity, indicating visibility of the central singularity.
• Uses the Joshi-Dwivedi root equation framework to classify the nature of the central singularity as black hole or naked singularity.
• Performs numerical and analytical analysis of the integrand in $\chi_2(v_{ah})$ for $k = 1/10$, $k = 1/100$, and $k = 0$ (dust) to compare horizon behavior.

Experimental results

Research questions

• RQ1 Does the inclusion of a linear equation of state $p = k\rho$ with $k > 0$ still allow for the formation of naked singularities in inhomogeneous gravitational collapse?
• RQ2 How does the value of $k$ influence the formation of black holes versus naked singularities in perfect fluid collapse?
• RQ3 Can the apparent horizon curve be increasing near the central singularity, indicating a locally naked singularity, when pressure is non-zero?
• RQ4 What is the role of initial density $m_0$ and higher-order derivatives $f^{\prime\prime\prime}(0)$, $f^{\prime\prime\prime\prime}(0)$ in determining the visibility of the singularity?
• RQ5 Does the cosmic censorship hypothesis remain valid when pressure is introduced into inhomogeneous dust collapse models?

Key findings

• For $k = 1/10$, the integrand in $\chi_2(v_{ah})$ is negative, leading to $\chi_2(v_{ah}) > 0$, which implies the apparent horizon curve is increasing near the singularity, indicating a locally naked singularity.
• Even for small values of $k = 1/100$, the apparent horizon curve increases, showing that minimal pressure can lead to naked singularity formation.
• In the dust limit ($k = 0$), the apparent horizon curve decreases ($t_{ah}(r) < t_0$), consistent with black hole formation and event horizon formation.
• The visibility of the central singularity depends critically on the sign of $\chi_2(v_{ah})$, which is determined by $m_0$, $k$, $f^{\prime\prime\prime}(0)$, and $f^{\prime\prime\prime\prime}(0)$.
• Non-central singularities cannot be visible because $M/v$ becomes unbounded as $v \to 0$ for $r > 0$, making the region timelike and thus not causally accessible.
• The model confirms that both black holes and naked singularities arise naturally as end states for perfect fluid collapse with $p = k\rho$, depending on $k$ and initial data, challenging the cosmic censorship hypothesis.