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This Chapter describes the geometry of twistor space of a 4-dimensional manifold. We motivated the twistor space as a geometrical construction that realises the action of the conformal group in 4D as the direct analog of that in 2D. This explains why the coordinates of a 4D space can be naturally put into a 2x2 matrix. We describe both the complexified version of the twistor space, as well as treat all the 3 possible signatures in detail. We then specialise to the case of Euclidean twistors, and describe how the twistor space can be interpreted as the total space of the bundle of almost complex structures of a 4D Riemannian manifold. Quaternionic Hopf fibration and its relation to the Euclidean twistor space is desccribed. We then describe the geometry of 3-forms in seven dimensions, and describe two different G2 structures on the 7-sphere. We end with a description of a lift of the usual twistor construction of integrable almost complex structures into seven dimensions. This is based on the notion of nearly parallel G2 structures.
This is the first central Chapter of the book that describes Riemannian geometry using Cartan's notion of soldering. Gravity first appears in this Chapter as a dynamical theory of a collection of differential forms rather than a metric. We describe thegeneral notion of geometric structures and then specialise to the case of a geometric structure corresponding to a metric. We describe the notion of a spin connection, its torsion, and then present examples of caclulations of Riemann curvature in the tetrad formalism. We then describe the Einstein-Cartan formulation of GR in terms of differential forms, and present its teleparallel version. We introduce the idea of the pure connection formulation, and compute the corresponding actino perturbatively. We then describe the so-called MacDowell-Mansouri formulation. We briefly describe the computations necessary to carry out the dimensional reduction from 5D to 4D. We then describe the so-called BF formulation of 4D GR, which in particular allows to determine the pure connection action in a closed form. We then describe the field redefinitions that are available when one works in BF formalism, and the associated formulation of BF-type plus potential for the B field.
are considered: chiral Einstein-Cartan, and chiral pure connection one. It is explained why chiral 4D perturbative formalisms are particularly powerful - they work with the minimal possible number of auxiliary fields to achieve polynomiality of the action. Spinors and differential operators that are motivated by spinors play a particularly important role in this Chapter, and so Lorentzian signature spinors are reviewed here in some detail. We also treat Yang-Mills theory and show how its chiral first order formalism gives the most powerful perturbative description. We end by describing how to gauge-fix the pure connection action on an arbitrary Einstein background. This produces a very simple perturbative description, remarkably more economic than the usual metric one.
We define gravity as a gauge theory with soldering. We discuss some analogies between gravity and YM theory that become visible via the chiral description of GR. We end with a provocative remark that assigns significance to the demonstrated by this book fact that there are so many non-obviously equivalent reformulations of General Relativity.
We review, in concise manner, the standard Einstein-Hilbert metric formulation of GR, together with related to it desccriptions such as the first order Palatini formalism, as well as the Einstein-Schroedinger pure affine formulation. We also describe the linerisation of the Einstein-Hilbert action around an arbitrary Einstein background, with associated geometric notions such as the Lichnerowicz operator. Some less standard aspects are also reviewed, for example we show how to explicitly rewrite the Einstein-Hilbert action in terms of the metric.
This short Chapter describes a particular modification of 4D General Relativity that is "geometrically natural" in the sense explained. A Bianchi I setup solution of the modified theory is worked out, in particular to illustrate that the modified theory appears to be simpler than GR in the types of functions that get produced.
This Chapter describes the chiral pure connection formulation of 4D GR, which is singled out from all other reformulations because of the economy of the description that arises. We first obtain the chiral pure connection Lagrangian, and explain how the metric arises from a connection. We also discuss the reality conditions. We then introduce notions of definite and semi-definite connections, and discuss the question of whether the pure connection action can be defined non-perturbatively. The question is that of selecting an appropriate branch of the square root of a matrix that appears in the action. Many examples are looked at to get a better feeling for how this connection formalism works. Thus, we describe the Page metric, Bianchi I as well as Bianchi IX setups, and the spherically symmetric problem. All these are treated by the chiral pure connection formalism, to illustrate its power. We also give here the connection description of the gravitational instantons, and in particular describe the Fubini-Study metric. We also show how to use the connection formalism to describe some Ricci-flat metrics, and illustrate this on the examples of Schwarzschild and Eguchi-Hanson metrics. We finish with the description of the chiral pure connection perturbative description of GR.
This is a short Chapter introducing an index-free notation for 2+1 gravity, which encodes all objects into 2x2 matrix-valued one-forms. We then describe the Chern-Simons formulation of 2+1 gravity, as well as the pure spin connection formulation.
Historical introduction into the topic of formulations of General Relativity with their associated mathematical formalisms. We review the history of the idea that gravity is geometry, as well as the history of development of main geometical ideas of modern differential geometry. We start with metric geometry of Riemann, Levi and Civita, and describe the main contributions of Cartan. We touch upon the fact that spinors and differential forms are closely related, first observed by Chevalley. We give arguments for why formalisms based on differential forms may be superrior to the metric one.
This is the longest Chapter of the book introducing the "chiral" formulations of 4D GR. The most important concept here is that of self-duality. We describe the associated decomposition of the Riemann tensor, and then the chiral version of Einstein-Cartan theory, together with its Yang-Mills analog. The geometry that is necessary to understand the fact that the knowledge of the Hodge star is equivalent to the knowledge of the conformal metric is explained in some detail. We also describe the different signature pseudo-orthogonal groups in 4 dimensions, and in particular explain that it is natural to put coordinates of a 4D space into a 2x2 matrix, the fact that is going to play central role in the later description of twistors. The notions of the chiral part of the spin connection, as well as the chiral soldering form are introduced. We give an example of a computation of Riemann curvature using the chiral formalism, to illustrate its power. We then describe the Plebanski formulation, as well as its linearisation. This allows to derive the linearisation of the chiral pure connection action, to be studied in full in the following Chapter. We describe coupling to matter in Plebanski formalism, and then some alternative descriptions related to Plebanski formalism.
This Chapter describes, in concise manner, aspects of differential geometry that are necessary to follow the developments of this book. We give several definitions of the concept of the manifold, illustrated by a number of examples. We then define differential forms, which are viewed as the most primitive objects one can put on a manifold. We define their wedge product and the operation of exterior differentiation. We then define the notions necessary to define the integration of differential forms. After this we define vector fields, their Lie bracket, interior product, then tensors. We then describe the Lie derivative. We briefly talk about distributions and their integrability conditions. Define metrics and isometries. Then define Lie groups, discuss their action on manifolds, then define Lie algebras. Describe main Cartan's isomoprhisms. Define fibre bundles and the Ehresmann connections. Define principal bundles and connections in them. Describe the Hopf fibration. Define vector bundles and give some canonical examples of the latter. Describe covariant differentiation. Briefly reivew Riemannian geometry and the affine connection. We end this Chapter with a description of spinors and their relation to differential forms.
This monograph describes the different formulations of Einstein's General Theory of Relativity. Unlike traditional treatments, Cartan's geometry of fibre bundles and differential forms is placed at the forefront, and a detailed review of the relevant differential geometry is presented. Particular emphasis is given to general relativity in 4D space-time, in which the concepts of chirality and self-duality begin to play a key role. Associated chiral formulations are catalogued, and shown to lead to many practical simplifications. The book develops the chiral gravitational perturbation theory, in which the spinor formalism plays a central role. The book also presents in detail the twistor description of gravity, as well as its generalisation based on geometry of 3-forms in seven dimensions. Giving valuable insight into the very nature of gravity, this book joins our highly prestigious Cambridge Monographs in Mathematical Physics series. It will interest graduate students and researchers in the fields of theoretical physics and differential geometry.
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