More varieties of spherically symmetric and axially symmetric solutions are found, such as the Reissner–Nordström black hole as well as de Sitter and anti-de Sitter variations thereof. Rotating black holes are also given a healthy dose of attention. An old but relatively less-known fact invoked in this chapter is how the rotating Kerr solution can be extracted from an analytic continuation of the spherically symmetric Schwarzschild solution. The same relation is known between the Kerr–Newman and the Reissner–Nordström. Maximally symmetric solutions with the cosmological constant, de Sitter, and anti-de Sitter, are also explored under various coordinate choices.

In preparation for the ADM formulation of General Relativity, we quickly scan Dirac's theory of constrained systems. How to deal with dynamics when the number of variables is larger than the true degrees of freedom is at issue. Starting from a familiar classical mechanics with Lagrange multipliers, we classify constraints into the first class and the second class. The former is particularly relevant for field theories with gauge redundancies, as is the case with General Relativity. Again, the Maxwell theory is invoked as a prototype, with the Gauss constraint given a unique meaning as the generator of the gauge redundancy.

Although the metric is clearly one of the minimal necessities for physics in curved spacetimes, the orthonormal frame is often more sensible as the bearer of the Riemannian geometry. A hallmark of the covariantly constant metric is how the Riemann curvature can at best rotate tensors, whose characteristics are lost in the Christoffel symbol. The Maurer–Cartan alternative addresses this cleanly by introducing a bigger set of variables, the vielbein, which defines an SO-valued connection 1-form, also known as the spin connection, and leads us to the curvature 2-form on par with the ordinary Yang–Mills field strength. Related issues, such as how the Riemann tensor in a general basis differs from the common commutator definition, are also addressed. Several highly symmetric geometries are offered as examples.

We model the Einstein equation, which eventually determines the spacetime metric, after the Maxwell equations. The Bianchi identity of the electromagnetic field strength is required by the charge–current conservation, which inspires the conserved energy-momentum and a symmetric rank-2 tensor that should be divergence-free as a mathematical identity. The universal Bianchi identity of the curvature 2-form is shown to build the divergence-free Einstein tensor as the requisite symmetric tensor, leading us to the Einstein equation. The Newtonian limit fixes the relative coefficient, via the weak field approximation that also leads to gravitational waves. Some rudimentary explorations of the latter are offered.

Once the proper time is recognized as the only viable notion of time, relativistic gravity as an external force arises naturally via the analogy of how one introduces the metric in Newtonian dynamics in curvilinear coordinates. The resulting action principle comes with a key property that the time parameter choice should be entirely irrelevant to the dynamics, which is, in turn, used to simplify the action by choosing the parameter to be the proper time of the particle in question. With the metric supplied later by the gravitational field equation, we discover that the Kepler problem elevates to a fully relativistic one straightforwardly. This chapter closes with the application of all these to the light-bending phenomena.

Olivine and low-Ca pyroxene compositional distributions show a hiatus between H and L, but not between L and LL. Because H, L, and LL chondrites show systematic changes in many characteristics, they must have formed in close proximity. H/L and L/LL chondrites may be anomalous members of one of the major OC groups or representatives of OC bodies of intermediate composition. A few highly reduced OC are either H chondrites that underwent whole-rock reduction or are members of otherwise-unsampled reduced OC bodies. IIE irons likely represent a fourth, reduced OC group. R chondrites resemble OCs but have more matrix material, higher 17O, and are much more oxidized. H, L, and LL chondrites show increasing degrees of oxidation with petrologic type. The bulk chemical and bulk isotopic compositions of OCs show systematic variations among the four principal groups. Metal-silicate fractionation was a nebular process that may have been caused in part by loss of metal from chondrules. OC oxidation state is heterogeneous on global and kilometer-size scales, and homogeneous on meter and smaller size scales. OC bulk O-isotopic composition is heterogeneous on global size scales and homogeneous on km and smaller size scales.

CAI formation began 4.567 Ga ago and ferromagnesian chondrules formed 2-2.7 Ma later. The order of OC parent-body accretion may have been (from earliest to latest) IIE, H, L, LL. Ordinary chondrites formed in the Inner Solar System along with other noncarbonaceous materials. 26Al decay was the primary asteroidal heat source. Ordinary chondrites have been modeled as being a significant component of Earth. Each OC asteroid was subject to major collisions. These are marked by peaks in cosmic-ray exposure (CRE) age distributions: for example, 45% of H chondrites have a CRE age of ~7.5 Ma. The U/Th-He ages of L chondrites are lower than those of H or LL chondrites due to the collisional breakup of the L parent body ~470 Ma ago. The lower maturity of OC asteroidal regoliths compared to lunar regolith is due to OC asteroids’ experiencing a lower micrometeorite flux, lower average projectile velocities, more-significant spallation processes, and having an ultramafic composition. Some OC are associated with abundant non-OC material; these include OC clasts in Cumberland Falls (aubrite), Almahata Sitta (anomalous ureilite), Bencubbin (CBa chondrite), Galim (EH/LL breccia), and Kaidun (carbonaceous-chondrite breccia).

Carbonaceous (CC) and noncarbonaceous (NC) materials have nonoverlapping isotopic compositional ranges. The CC groups include all carbonaceous chondrites, Eagle Station pallasites, and several groups of iron meteorites (IIC, IID, IIF, IIIF, IVB); they likely formed in the Outer Solar System. The NC groups include ordinary, enstatite and R chondrites, Howardites-Eucrites-Diogenites (HEDs), ureilites, angrites, lunar meteorites, martian meteorites, main-group pallasites, the Earth, and the remaining iron groups (IAB, IC, IIAB, IIE, IIIAB, IIIE, IVA); they probably formed in the Inner Solar System. Proto-Jupiter may have accreted rapidly and functioned as a barrier, hindering the radial drift of carbonaceous-chondrite-related materials toward the Inner Solar System, preserving the isotopic dichotomy.

Tiny presolar grains include C polymorphs, carbides, nitrides, oxides, silicates, metallic Fe-Ni, and organic compounds. Rare CAIs and AOAs contain refractory oxides and silicates. Major phases in type-3 OC include olivine and low-Ca pyroxene with variable FeO/(FeO+MgO), metallic Fe-Ni, troilite, and nearly exclusively within chondrules, crystallites of Ca-pyroxene, rare pigeonite, and tiny grains of merrillite. Whole-rock thermal metamorphism produced secondary phases: orthopyroxene, diopside, chromite, ilmenite, rutile, phosphate, and plagioclase. Diffusion facilitated by metamorphism causes increasing compositional homogeneity in olivine and pyroxene. Some minerals and mineraloids are formed at high shock pressure. These include lingunite and maskelynite from plagioclase; ahrensite, asimowite, poirierite, ringwoodite, and wadsleyite from olivine; akimotoite, bridgmanite, hemleyite, hiroseite, and majorite from orthopyroxene; chenmingite and xieite from chromite; tuite from merrillite; wangdaodeite from ilmenite; and TiO2-II from rutile. Parent-body aqueous alteration produced phyllosilicates, Ni-rich sulfides, Ni-rich metal phases, carbides, oxides, and small calcite crystals.

The first asteroid, Ceres, was discovered on the first night of the nineteenth century. There are now more than 800,000 numbered asteroids. Numerous properties link most meteorite groups to asteroids. These include cooling rates, the presence in some specimens of solar-wind gas, formation ages and CRE ages, orbital parameters, and the retrieval of chondritic material from asteroids visited by spacecraft. In addition, the spectral reflectance properties of meteorites match those of particular asteroids. Space-weathering can account for differences between OC spectra and those of S-complex and Q-complex asteroids. Ordinary-chondrite parent asteroids probably initially had an onion-shell-like structure due to internal heating by 26Al. These bodies were likely collisionally disrupted and gravitationally reassembled while still hot.

Chondrule types include porphyritic (FeO-poor and FeO-rich), barred olivine, radial pyroxene, granular, and cryptocrystalline. Chondrules in unequilibrated OC tend to have unfractionated refractory lithophile abundances; metallic components include one enriched in refractory siderophiles and one in common and volatile siderophiles. Although most chondrules are a few hundred µm in diameter, microchondrules (0.8-40 µm) and macrochondrules (0.5–5 cm) also occur. Compound chondrules include enveloping, sibling, and adhering varieties. Some chondrules have fine-grained rims, others igneous rims. Relict grains survived the most recent chondrule melting. Calcium–aluminum-rich inclusions (CAIs) and amoeboid olivine aggregates (AOAs) are very rare. Matrix material occurs mainly as chondrule rims and isolated lumps. Carbon-rich aggregates and clasts contain poorly graphitized C, amorphous C, metallic Fe-Ni, and minor chromite. A few H chondrites contain halite. Opaque assemblages include metallic Fe-Ni, sulfide, and oxides. Some shocked OC contain metallic Cu. Large metal nodules formed by impact-induced vaporization and fractional condensation. Many shocked OC exhibit silicate darkening, and many are breccias with a variety of clasts. Some OC are regolith breccias enriched in solar-wind-implanted noble gases.

The principal methods used in experimental and observational science typically involve hypothesis testing, follow-ups on serendipitous discoveries, the use of new analytical tools (instrumental, numerical, or statistical) to examine extant samples or data sets, the acquisition of new samples to analyze, and the formulation of theoretical models. Many studies, including those in meteoritics and cosmochemistry, employ several of these methods.

Ordinary-chondrite petrologic types range from unheated type-3.00 to highly recrystallized type-6 samples. During metamorphism, olivine grains in ferroan chondrules become depleted in Cr2O3 as Cr2O3 variability decreases. Matrix olivine grains become more ferroan and bulk matrix loses C, metal, and sulfide. Metamorphism induces whole-rock loss of some H2O, C, noble gases, and volatile mobile elements. Type-6 OC contain homogeneous mafic silicates. The maximum metamorphic temperatures in OC ranged from ~200-260ºC for type 3.00 to ~820-930ºC for type 6. Asteroids were heated mainly by 26Al early in Solar System history and by collisions afterwards. Some OC were annealed after shock. Shock also caused microstructural mineral dislocations. Brecciation and melting can occur during collisions. Indigenous water may be mobilized during heating, leading to aqueous alteration of matrix material and chondrule glass. Carbide-magnetite assemblages and fayalite-silica associations are produced during alteration. Some asteroids of OC composition were melted – IVA iron meteorites were derived from the metal core of one such differentiated asteroid.

Orbits have been calculated for dozens of meteorite falls; they match those of many Near-Earth Asteroids. The Kirkwood Gaps are hiatuses in histograms of asteroid semi-major axes; few asteroids occur in these gaps. The gaps are caused by repeated gravitational tugs by Jupiter on asteroids with orbital periods that are simple fractions of Jupiter’s orbital period. Among the most efficient mechanisms for delivering asteroids to the Earth is the υ6 resonance associated with Saturn. The resonance occurs when there is a simple numerical ratio between the precession frequency of an asteroid’s longitude of perihelion and that of the mean precession frequency of Saturn’s longitude of perihelion.

OC parent bodies accreted from a mix of chondrules, chondrule fragments, grains of metallic Fe-Ni and sulfide, porous aggregates of fine-grained dust, and rare CAIs, AOAs, and tiny presolar grains. After accretion, the OC asteroids underwent thermal metamorphism, mainly due to the decay of 26Al. They initially developed onion-shell structures but suffered disruption and reassembly while still hot. Subsequent collisions produced a variety of breccias on each body. The L parent asteroid was destroyed by a catastrophic collision ~470 Ma ago.

Paleomagnetic measurements of relict dusty olivine-bearing chondrules in LL3.01 Semarkona reveal that >1.22 Ma after CAI formation, the region of the solar nebula between ~1 and 3 AU from the Sun had a magnetic field strength of ~54 µT. This is comparable to the current geomagnetic field at the Earth’s surface.