Abstract
Probably the central chapter of the book, deriving the Einstein equations and discussing its main physical and geometric aspects. Includes a detailed illustration of the properties of the dynamical energy-momentum tensor (with several useful examples), and a derivation of the equations of motion for “extended” test bodies with non-trivial internal structure (intrinsic spin, quadrupole moment, etc.). Of special interest (not easily available in the textbook literature): explicit computation of all boundary contributions arising from the variation of the Einstein–Hilbert action, and of their exact cancellation through the variation of the York–Gibbons–Hawking boundary term.
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Notes
- 1.
In particular, with an appropriate choice of the boundary action, it is always possible to obtain a total action which is simply quadratic in the connection (and thus contains the square of the first derivatives of the metric, without second derivatives), and which reproduces the same equations of motion as any other choice of S EH+S YGH (see e.g. [32]).
- 2.
See for instance [52]. The vacuum energy density is exactly vanishing only in the case of supersymmetric field-theory models. At low energy, however, supersymmetry is expected to be broken, and the vacuum of the broken phase has a non-zero energy density.
- 3.
See for instance Particle Data Group, at http://pdg.lbl.gov, for the last updated results.
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Gasperini, M. (2013). The Einstein Equations for the Gravitational Field. In: Theory of Gravitational Interactions. Undergraduate Lecture Notes in Physics. Springer, Milano. https://doi.org/10.1007/978-88-470-2691-9_7
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