Abstract
This chapter introduces the Cartan approach to differential geometry, the vielbein and the spin connection, discusses Bianchi identities and their relation with gauge invariances and eventually introduces Einstein field equations. The dynamical equations of gravity and their derivation from an action principle are developed in a parallel way to their analogues for electrodynamics and non-Abelian gauge theories whose structure and features are constantly compared to those of gravity. The linearization of Einstein field equations and the spin of the graviton are then discussed. After that the bottom-up approach to gravity is discussed, namely, following Feynman’s ideas, it is shown how a special relativistic linear theory of the graviton field could be uniquely inferred from the conservation of the stress-energy tensor and its non-linear upgrading follows, once the stress-energy tensor of the gravitational field itself is taken into account. The last section of this chapter contains the derivation of the Schwarzschild metric from Einstein equations.
Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there…
Richard Feynman
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- 1.
See Chap. 2 of Volume 2 for more information on Laplace and its contributions to gravitational theories.
- 2.
From now one we choose p=1 which is the physically most interesting case.
- 3.
All leptons and quarks fall into this category.
- 4.
For the conventions on spinors and gamma-matrix algebra see Appendix A.
- 5.
Note that from now on we omit the factor \({\frac {1}{2}}\) introduced in the electromagnetic analogue of (5.3.29), namely in (5.3.10). There such a factor was used in order for F μν to have the traditional normalization
. Omitting this factor the normalization of F
μν
becomes the less traditional one: 
. Paying this moderate price all formulae become much neater and deprived of annoying prefactors. - 6.
An important comment is obligatory at this point. A fundamental theorem of Lie Algebra Theory, states that the Killing metric of a compact semi-simple Lie algebra is always negative definite. So, as long as, the gauge group G is chosen compact, as it is the case in all standard gauge theories of non-gravitational interactions, the kinetic term of the gauge fields defined by the action (5.3.31) turns out to have correct positive definiteness properties and the corresponding quanta have physical propagators. In the case of non-compact gauge algebras the action (5.3.31) introduces negative norm states in the spectrum. This does not mean that non-compact gauge groups are altogether forbidden. Actually they appear in supergravity theories yet the corresponding kinetic terms have a more sophisticated structure which takes care of unitarity.
- 7.
Théophile Ernest de Donder (1872–1957) was a Belgian mathematician and physicist. His most famous work dating 1923 concerns a correlation between the Newtonian concept of chemical affinity and the Gibbsian concept of free energy. Prof. de Donder was among the very first scientists who studied Einstein General Relativity and was one of its sustainers since its very beginning. He was a personal close friend of Einstein. His main field of activity was the thermodynamics of irreversible processes and the his work can be considered the early basis of the full-fledged development of this subject performed by Ilya Prigogine, the famous Russian born Belgian chemical-physicist who received the 1977 Nobel Prize.
. Omitting this factor the normalization of F
μν
becomes the less traditional one: 
. Paying this moderate price all formulae become much neater and deprived of annoying prefactors.References
Lord, E.A.: A theorem on stress-energy tensors. J. Math. Phys. 17, 37 (1976)
Einstein, A.: Die Feldgleichungen der Gravitation. in: Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, pp. 844–847 (1915)
Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie. Ann. Phys. 49 (1916)
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Frè, P.G. (2013). Einstein Versus Yang-Mills Field Equations: The Spin Two Graviton and the Spin One Gauge Bosons. In: Gravity, a Geometrical Course. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5361-7_5
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