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.

We investigate the dependence of the complete system of 22 Lick indices on overall metallicity scaled from solar abundances, [ M H ], from the solar value, 0.0, down to the extremely-metal-poor (XMP) value of -6.0, for late-type giant stars (MK luminosity class III, log g = 2.0) of MK spectral class late-K to late-F (3750 < T eff < 6500 K) of the type that are detected as "fossils" of early galaxy formation in the Galactic halo and in extra-galactic structures. Our investigation is based on synthetic index values, I, derived from atmospheric models and synthetic spectra computed with PHOENIX in LTE and Non-LTE (NLTE), where the synthetic spectra have been convolved to the spectral resolution, R, of both IDS and SDSS (and LAMOST) spectroscopy. We identify nine indices, that we designate "Lick-XMP", that remain both detectable and significantly [ M H ]dependent down to [ M H ] values of at least ∼ −5.0, and down to [ M H ] ∼ −6.0 in five cases, while also remaining well-behaved (single-valued as a function of [ M H ] and positive in linear units). For these nine, we study the dependence of I on NLTE effects, and on spectral resolution. For our LTE I values for spectra of SDSS resolution, we present the fitted polynomial coefficients, C n , from multi-variate linear regression for I with terms up to third order in the independent variable pairs (T eff , [ M H ]), and (V − K, [ M H ]), and compare them to the fitted C n values of Worthey et al. (1994) at IDS spectral resolution. For this fitted I data-set we present tables of LTE partial derivatives, ∂I ∂T eff | [M/H] , ∂I ∂[M/H] | T eff , ∂I ∂(V −K) | [M/H] , and ∂I ∂[M/H] | (V −K) , that can be used to infer the relation between a given difference, ∆I, and a difference ∆T eff or ∆(V − K), or a difference ∆[ M H ], while the other parameters are held fixed. For Fe-dominated Lick indices, the effect of NLTE is to generally weaken the value of I at any give T eff and [ M H ] values.As an example of the impact on stellar parameter estimation, for late-type giants of inferred T eff 4200 K, an Fe-dominated I value computed in LTE that is too strong might be compensated for by inferring a T eff value that is too large.

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