Differential forms and the geometry of general relativity (Q2841477)

In this book, the author outlines an interesting path to relativity and shows its various stages on the way, referring for example to his previous works: [the author and \textit{C. A. Manogue},`` The Geometry of vector calculus'', {\url{http://www.math.oregonstate.edu/BridgeBook}}, 2009--2013 and the author, The Geometry of special relativity. Boca Raton, FL: CRC Press (2012; Zbl 1251.83001)], to \textit{R. D'Inverno}'s book [Einführung in die Relativitätstheorie. Weinheim: VCH (1995; Zbl 0855.53045)], and to [\textit{D. Bachman}, A geometric approach to differential forms. Boston, MA: Birkhäuser (2006; Zbl 1108.58001)].NEWLINENEWLINEThe first two parts of the book provide an introduction to general relativity (the geometry of black holes, Einstein's equation and cosmological models), and the third part describes the formalism behind the key geometric idea in general relativity, namely the curvature. The author uses differential forms instead of the standard tensor calculus. In this original approach, there are only three tensors of rank-2: the metric tensor, the energy-momentum tensor, and the Einstein tensor, which are seen here as vector-valued 1-forms.NEWLINENEWLINEThe first part, ``Spacetime geometry'', requires elementary notions such as the line element, circle and hyperbola trigonometry, geodesics, and a symmetry principle, prior to a detailed study of the Schwarzschild geometry, based on geodesics and symmetry. By applying similar techniques to those used for extending the much simpler Rindler geometry, one obtains the maximally extended Schwarzschild geometry, called the Kruskal geometry. The four regions in which the horizons divide the Kruskal geometry are: the original Schwarzschild region (the righthand asymptotic region), the black hole region (the upper quadrant), the white hole region (the lower quadrant), and the second asymptotic region on the left, which cannot communicate with the Schwarzschild region. The author emphasizes that the Kruskal geometry does not correspond to any objects existing in nature and white holes cannot be exemplified, but he presents a realistic model of a black hole, namely that obtained by a collapsing star, which shrinks inside its Schwarzschild radius. The last sections of this part are devoted to Reissner-Nordström and Kerr geometry, which deal respectively with charged and rotating black holes. A rotating black hole is a solution of Einstein's vacuum equation. When the angular velocity of the black hole vanishes, the Kerr line element reduces to the Schwarzschild line element. The most general black hole solution is given by the Kerr-Newman geometry, which generalizes the Kerr geometry, by including charge. It can be also obtained by a construction involving a complex rotation applied to the Reissner-Nordström geometry. The author mentions that a black hole is completely characterized by its mass, charge, and angular velocity.NEWLINENEWLINEThe second part of the book, ``General relativity'', recalls first the basic properties of differential forms, the three key principles: principle of relativity, equivalence principle and Mach's principle, and then the main problems related to vectors in Minkowski space, Earth distance, 2-forms in Minkowski 4-space, orthonormal frames and vector potentials. By using rain coordinates (a certain coordinate system adapted to the radial geodesics), the author relates curvature to geodesic deviation. He proves that this relationship is geometric invariant, by computing (in Section A.3) the curvature 2-forms in the original Schwarzschild coordinates, and showing that these expressions are formally the same as those in rain coordinates. The chapter concerning Einstein's equation deals with the relationship between curvature and ``matter'' (described by a vector-valued 1-form \(\vec{T}\), called the energy momentum tensor). The Einstein tensor is in fact \(G_{ij}=R_{ij}-\frac{1}{2}g_{ij}R,\) where \(R_{ij}\) is the Ricci tensor and \(R\) is the scalar curvature. Einstein's equation has the form \(G_{ij}=\frac{8\pi G}{c^2}T_{ij}\), and its generalization is \(G_{ij}+\Lambda g_{ij}=\frac{8\pi G}{c^2}T_{ij}\), where \(\Lambda\) is the cosmological constant. Other important chapters of general relativity cover cosmological models and Solar System applications. The cosmological principle asserts that the universe is homogeneous (i.e. it can be foliated by spacelike hypersurfaces, such that all points in a given surface are equivalent) and isotropic at each point (i.e. there exists a family of observers \(\vec{u}_s\), such that all directions orthogonal to \(\vec{u}_s\) are equivalent). One can assume that the surfaces are labeled by using ``cosmic time,'' that is \(t\) is proper time according to these observers. In this context, the author presents some simple models of the universe. He recalls the Robertson-Walker metric, which involves two parameters: \(k\), which determines the shape and the topology of the universe (the universe is finite if \(k = 1\), and infinite otherwise) and \(a(t)\) (a function of cosmic time), which determines the scale of the universe, seen as the surface of a balloon of radius a(t). Writing Einstein's equation for the Robertson-Walker metric, it follows that a not empty physically realistic model has strictly positive energy density (\(\rho>0\)) and nonnegative pressure density (\(p \geq0\)). If \(\dot{a}>0\) now, and \(''a<0\) always, it follows that \(a\) was zero at some time in the past. Thus, Robertson-Walker models with \(\Lambda\leq 0\) (and with expansion now) must have a singularity in the past, called the ``Big Bang''. This principle is a special case of the much more general principle in relativity, called ``gravity attracts''. More exactly, reasonable matter (positive energy density) has a singularity, as for example the Big Bang (in the past), the ``Big Crunch'' for cosmological models, or a black hole formed by a collapsing star (in the future). Since in present universe is ``matter dominated'', some appropriate models are those with vanishing pressure density, i.e. with \(p =0\), called Friedmann-Robertson-Walker models. There are three possible vacuum Friedmann solutions, the resulting spacetimes being called de Sitter spaces (when \(\Lambda>0\)), anti de Sitter spaces (for \(\Lambda>0\) and \(k=-1\)) and Minkowski spaces (when \(\Lambda=0\) and \(k=0\) or \(k=-1\)).NEWLINENEWLINEThe third part of the book, ``Differential forms'', begins with some basic notions such as the differential, integrands, vector calculus, and continues with the algebra of differential forms, Hodge duality, formulas for the Laplacian in polar coordinates, Maxwell's equations, integration of differential forms, connections (the Cartan structure equations and the existence of the unique Levi-Civita connection), geodesics, curvature, Gauss's Theorema Egregium about intrinsic curvature and the Gauss-Bonnet theorem relating geometry to topology. The applications are related to the equivalence problem (a coordinate transformation taking one line element to another, e.g. the transformation between rectangular and polar coordinates), Lagrangians, spinors, topology, and integration on the sphere. To simplify computations, the author works in an orthonormal basis, which avoids mistaking coordinate singularities for physical ones, and recovering many standard results in \(\mathbb{R}^3\).NEWLINENEWLINEThe author inserts suggestive pictures and images which make the book more attractive and easier to read. The paper addresses not only specialists and graduate students, but even advanced undergraduates, due to its interactive structure containing questions and answers.NEWLINENEWLINEFurther information about the book, including a wiki version, is available at:NEWLINENEWLINE{\url{http://relativity.geometryof.org/DFGGR/bookinfo}}.