An invariant characterization of the quasi-spherical Szekeres dust models

Abstract

The quasi-spherical Szekeres dust solutions are a generalization of the spherically symmetric Lemaitre–Tolman–Bondi dust models where the spherical shells of constant mass are non-concentric. The quasi-spherical Szekeres dust solutions can be considered as cosmological models and are potentially models for the formation of primordial black holes in the early universe. Any collapsing quasi-spherical Szekeres dust solution where an apparent horizon covers all shell-crossings that will occur can be considered as a model for the formation of a black hole. In this paper we will show that the apparent horizon can be detected by a Cartan invariant. We will show that particular Cartan invariants characterize properties of these solutions which have a physical interpretation such as: the expansion or contraction of spacetime itself, the relative movement of matter shells, shell-crossings and the appearance of necks and bellies.

Appendix: Frame independent curvature invariants

As in the case of the spherically symmetric metrics, the components in (38) vanish on the apparent horizon $R = 2M$, while the components in (39) do not. This relationship is reflected in the vanishing of the Cartan invariant $\rho$ relative to the invariant coframe chosen by the Cartan–Karlhede algorithm. Taking the zeroth order and first order SPIs:

$$\begin{aligned} I_1 = C_{abcd} C^{abcd} = \Psi _2,~~R = R^a_{~a} = 8 \Phi _{11}, \end{aligned}$$

(59)

along with the quadratic first order SPIs:

$$\begin{aligned} \begin{aligned}&I_3 = C_{abcd;e} C^{abcd;e},~~I_{3a} = C_{abcd;e} C^{ebcd;a}, I_5 = I_{1;a}I_1^{~;a}, \\&J_1 = R_{ab;c} R^{ab;c},~~ J_2 = R_{ab;c} R^{ac;b},~~J_3 = R_{;a}R^{;a}, \end{aligned} \end{aligned}$$

(60)

we can produce the following algebraically independent SPIs:

$$\begin{aligned}&(\mu - \rho )(\mu - \rho + 8 \epsilon ), \\&\epsilon ( \mu - \rho - \epsilon ), \\&\rho \mu - 2 |\tau |^2, \\&\mu \Delta \ln (\Phi _{11}) + 4 \rho \Delta \ln (\Phi _{11}) + 8 \rho \mu + 16 \rho \epsilon - 8 \rho ^2 - 9 \rho \mu \frac{\Psi _2 }{ \Phi _{11}}, \\&2^2 5 \rho \Delta \ln (\Phi _{11}) - 2^5 \epsilon \Delta \ln (\Phi _{11}) + 2^5 \rho \mu + 2^6 \rho \epsilon + 2^5 \rho ^2 - 6^2 \rho \mu \frac{\Psi _2}{\Phi _{11}} + 3^2 \rho \mu \frac{\Psi _2^2}{\Phi _{11}^2}, \\&2^7 (\Delta \ln (\Phi _{11})^2 - 2^8 \rho \Delta \ln (\Phi _{11}) + 2^8 \mu \Delta \ln (\Phi _{11}) + 2^9 \epsilon \Delta \ln (\Phi _{11}) + 6^2 2^3 |\tau |^2 \frac{\Psi _2^2}{\Phi _{11}^2}. \end{aligned}$$

The six SPIs in (59) and (60) are polynomials in terms of six Cartan invariants:

$$\begin{aligned}\Delta \ln (\Phi _{11}), \rho , \mu , \epsilon , |\tau |^2,~~and~~ \frac{\Psi _2}{\Phi _{11}}.\end{aligned}$$

Locally, it is possible to express $\rho$ (or $\mu$) as a function of these SPIs in order to detect the horizon when the Jacobian of these polynomials in terms of the six Cartan invariants is non-zero. However, this will introduce additional regions where the SPIs will vanish, giving rise to the possibility of the incorrect detection of the apparent horizon.