An invariant characterization of the quasi-spherical Szekeres dust models

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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.

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  1. 1.

    If a Szekeres dust-model admits a symmetry, there will only be three functionally independent invariants [31], since \(\epsilon \ne 0\) in these solutions.

  2. 2.

    If there is a symmetry, then \(dim(H_2) = dim(H_1)=0\) and \(t_2 = t_1 = 3\), and so the algorithm still stops.

  3. 3.

    Since \(u_a = dt\), the projection operator \(h_{ab} = g_{ab} + u_a u_b = g_{ab} + 2(\ell _a + n_a) (\ell _b + n_b)\) was used to compute the Ricci scalar, \({^3} \mathcal {R}\), of the hypersurfaces \(t=const\). To recover the form in [17] we notice that \(\dot{Y} = 0\) and so \(\tilde{M} = \frac{1}{2} \tilde{K} Y\).

  4. 4.

    Since the discriminant SPIs built from the Weyl and Ricci tensors, along with the covariant derivatives of these tensors do not vanish anywhere, these tensors cannot be of alignment type II or more special.


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We would like to thank Ismael Delgado Gaspar and Daniele Gregoris for useful discussions at the beginning of this project. The work was supported by NSERC of Canada (A.C.), and through the Research Council of Norway, Toppforsk grant no. 250367: Pseudo-Riemannian Geometry and Polynomial Curvature Invariants: Classification, Characterisation and Applications (D.M.).

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Appendix: Frame independent curvature invariants

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:

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

along with the quadratic first order SPIs:

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

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.

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Coley, A.A., Layden, N. & McNutt, D.D. An invariant characterization of the quasi-spherical Szekeres dust models. Gen Relativ Gravit 51, 164 (2019) doi:10.1007/s10714-019-2647-6

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