A differentiable manifold, defined with the help of collections of charts, comes with basic notions of calculus before the introduction of the metric. We start with the definitions of vectors as directional derivatives and 1-forms via the natural dual pairing and build up general tensors from these two. Partial derivatives for functions extend to the Lie derivative, while a special subclass of tensors known as differential forms admits the so-called exterior derivative. We develop calculus based on these most basic structures, ending with the Stokes theorem. This sets the stage for the Riemannian geometry, given in two alternate forms in Chapters 4 and 5.

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.

This chapter introduces quantum computation by comparing classical and quantum computers. Core concepts including qubits, superposition, and entanglement are introduced, setting the stage for deeper exploration. Various quantum computing models are summarized, with a focus on the circuit and topological models. The chapter explains why quantum computing is necessary, especially for tasks beyond classical computing’s limits. It discusses existing quantum platforms and provides an overview of their capabilities and limitations. The chapter also offers a brief historical perspective, touches on computational energy efficiency, and forecasts a quantum future where quantum and classical computing work in tandem. This groundwork provides essential insights into quantum computation’s potential and upcoming chapters’ explorations of algorithmic and theoretical principles.

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.