A physically interesting singularity in general relativity

TL;DR

The paper investigates a timelike naked singularity arising in the Weyl curvature within an inhomogeneous generalization of the Einstein static universe. It shows that near the singularity the metric behaves as $R(x)\to \alpha x^{1/2}$ and $B(x)\to \beta x^{-1/4}$, with the Weyl invariant diverging as $C^2\to \frac{3}{4\beta^4}\,x^{-3}$ while matter density vanishes for $w>0$, and that scalar waves and geodesics remain well-posed and complete, respectively. The authors compute a positive Komar mass contribution $M_K=\frac{1}{4}\alpha^2$ from each singularity, which reduces the necessity for a large positive cosmological constant and show that gradient terms in the static equations act like an effective $\Lambda$. Numerical solutions demonstrate one- and two-singularity configurations, with the static solutions yielding $4\pi\langle \rho+3p\rangle/\Lambda$ values near unity for single poles and up to $\sim 10^3$ for asymmetric two-pole cases, indicating significant matter content relative to $\Lambda$ in certain setups. Stability analysis reveals a spectrum of real eigenfrequencies, often with a single unstable mode for $w=1/3$ (as in ESU) and sometimes multiple unstable modes depending on parameters, implying inhomogeneous expansion/contraction tendencies. Extending to time dependence, the paper derives a Buchert-like acceleration equation that includes a gradient energy term $\langle (\partial^i B \partial_i B)/B^2 \rangle$ which can drive positive expansion, and identifies a scaling law that maps static solutions to families of solutions with invariant ratios, suggesting that singularities can meaningfully influence cosmic expansion and potentially lessen the need for a separate cosmological constant.

Abstract

We explore what appears to be a very benign timelike and naked singularity of the Weyl curvature scalar. This singularity arises in an inhomogeneous generalization of the Einstein static universe. The matter energy density is everywhere finite and it vanishes at the singularity. The spacetime still enjoys causal geodesic completeness and unambiguous wave dynamics. Each singularity contributes to the Komar mass of the spacetime, and this reduces the required size of the positive cosmological constant. In an expanding universe, the effect of the singularities is to push the acceleration towards more positive values.

Figures (3)

• Figure 1: The one-singularity solutions with various input values of $w$ and $\rho_0+3p_0$ shown on each plot. The resulting values of ($\alpha$, $\beta$) at the antipodal singularity and the value of $4\pi\langle \rho+3p\rangle/\Lambda$ are: a) (116,6.56) and 1.06, b) (18.05,0.411) and 1.07, c) (9.8,1.37) and 1.01, d) (43.1,4.83) and 1.01. The $\rho(x)$ profile (red curve) is arbitrarily scaled to fit the plot.
• Figure 2: The two-singularity solutions with $4\pi(\rho_1+3p_1)=\Lambda$ and with various input values of $w$ and ($\alpha$, $\beta$) at the $x=0$ singularity shown on each plot. The resulting values of $4\pi\langle \rho+3p\rangle/\Lambda$ are: a) 1.064, b) 1.10 , c) 1.87, d) 1000.
• Figure 3: The first four $\delta B(x)$ and $\delta R(x)$ eigenfunctions and the velocity profiles, labeled by their eigenfrequencies, corresponding to Fig. \ref{['f2']}a. A common overall scale of the velocities (green curves) is adjusted to fit the plots. The velocity of the unstable mode (first plot) is nonzero but small.