General solutions of Einstein's spherically symmetric gravitational equations with junction conditions

Einstein's spherically symmetric interior gravitational equations are investigated. Following Synge's procedure, the most general solution of the equations is furnished in case $T^{1}_{1}$ and $T^{4}_{4}$ are prescribed. The existence of a total mass function, $M(r,t)$, is rigorously proved. Under suitable restrictions on the total mass function, the Schwarzschild mass $M(r,t)=m$, implicitly defines the boundary of the spherical body as $r=B(t)$. Both Synge's junction conditions as well as the continuity of the second fundamental form are examined and solved in a general manner. The weak energy conditions for an \emph{arbitrary boost} are also considered. The most general solution of the spherically symmetric anisotropic fluid model satisfying both junction conditions is furnished. In the final section, various exotic solutions are explored using the developed scheme including gravitational instantons, interior $T$-domains and $D$-dimensional generalizations.Comment: 23 pages, 1 figure, uses AMS packages. Updated version has corrected typos as well as added comments and extension regarding ISLD junction conditions. Accepted for publication in Journal of Mathematical Physic

Repeated and Folded DNA Sequences and Their Modular Ag₁₀⁶⁺ Cluster

Molecular silver clusters are optical chromophores, and species with distinct spectra form with DNA strands. One such hybrid chromophore is a violet cluster bound to repeated C₂X sequences where X ≠ C. We varied the number of C₂X repeats and the X nucleobase and consider three observations. First, different lengths of (C₂A)_<i>y</i> and (C₂T)_<i>y</i> strands with <i>y</i> = 3–12 identify a minimal (C₂X)₆ scaffold that forms a specific Ag₁₀ adduct. This cluster has a +6 oxidation state, absorbs between 400–450 nm, and folds its DNA host. Second, different X nucleobases alter the (C₂X)₆ binding site. The natural nucleobases preferentially form the Ag₁₀⁶⁺ cluster and yield strong circular dichroism. These ligands coordinate via their heteroatoms, and the N3 of thymine was identified via cluster fluorescence that varies with pH. In contrast, abasic sites and imidazole substitutions suppress circular dichroism and diminish the number of silver adducts. These observations suggest that a (C₂X)₆ coordinates Ag₁₀⁶⁺ via multiple nucleobases. Third, beyond the minimal (C₂X)₆ binding site, longer strands still form Ag₁₀⁶⁺ but can also coordinate additional Ag⁺ adducts. The added Ag⁺ do not perturb the Ag⁰ and their spectra and thus may partition to open C₂X subunits outside the core (C₂X)₆–Ag₁₀⁶⁺ complex. Thus, these modular complexes distinguish oxidized and reduced silvers. Collectively, these three observations suggest that the DNA and silver cluster play complementary roles: a repeated C₂X sequence stabilizes the Ag₁₀⁶⁺ cluster, while the cluster folds its host. We specifically suggest that the Ag⁺ within Ag₁₀⁶⁺ cross-links remote C₂X subunits and the Ag⁰ coordinate with mismatch sites in a hairpin-like secondary structure. Distinct roles for Ag⁺ and Ag⁰ within a cluster are considered in light of the X-ray spectra of related complexes