Abstract
This chapter treats Einstein’s theory of gravitation, namely general relativity, as well as modifications and extensions thereof. The chapter starts with the foundational notion of “equivalence principle(s)” which more or less lead to describe the gravitational interaction in terms of a metric geometry with a linear connection. Therefore in a further section, these differential-geometric notions are introduced in order to discriminate different geometries. These are specialized to metric-compatible linear connections, leading to a Riemann-Cartan geometry. In case of vanishing torsion this becomes a Riemannian geometry in terms of which Einstein’s theory is formulated. Since needed in coupling spinors to a gravitational field, these geometries are also described in terms of tetrads and spin connections. Given any of these geometric entities, the equivalence principle allows to formulate the “minimal coupling” recipe in order to couple any matter distribution to gravitation. The dynamics of the gravitational field is deduced from action functionals both for metric and for tetrad general relativity. The relation among the canonical, the Belinfante and the Hilbert energy-momentum tensor is derived. Optional forms of the action for general relativity are considered and the role of boundary terms is discussed. Another section deals with the notions of covariance and invariance in the context of symmetries. As for the stilldisputed positioning of energy-momentum conservation in general relativity, the Klein-Noether identities are analyzed, and expressions for total and quasi-local energy are discussed. As regards to extensions and modifications of general relativity, amongst others Einstein-Cartan gravity is outlined and more general “Poincaré gauge theories” and “de Sitter gauge theories” are derived following the recipe for building Yang-Mills theories with respect to a given symmetry group. Further more speculative ideas about gravity in higher dimensions (with the remarkable class of Lanczos-Lovelock actions) are sketched, since again these are determined in their structure completely by the quest for diffeomorphism symmetry.
Mit der Einsteinschen Relativitätstheorie hat das menschliche Denken über den Kosmos eine neue Stufe erklommen. Es ist, als wäre plötzlich eine Wand zusammengebrochen, die uns von der Wahrheit trennte: Nun liegen Weiten und Tiefen vor unserem Erkenntnisblick entriegelt dar, deren Möglichkeiten wir vorher nicht einmal ahnten. Der Erfassung der Vernunft, welche dem physikalischen Weltgeschehen innewohnt, sind wir einen gewaltigen Schritt nähergekommen. [549]
From the first English translation: “Einstein’s theory of relativity has advanced our ideas of the structure of the cosmos a step further. It is as if a wall which separated us from Truth has collapsed. Wider expanses and greater depths are now exposed to the searching eye of knowledge, regions of which we had not even a presentiment. It has brought us much nearer to grasping the plan that underlies all physical happening.”
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- 1.
Indeed there is a seminal journal titled “General Relativity and Gravitation”.
- 2.
“Nun zum Namen Relativitätstheorie. Ich gebe zu, dass dieser nicht glücklich ist und zu philosophischen Mißverständnissen Anlaß gegeben hat. Der Name Invarianz-Theorie würde die Forschungs-Methode der Theorie bezeichnen, leider aber nicht den materiellen Gehalt der Theorie.” Letter to E. Zschimmer 30.9. 1921
- 3.
M. Planck coined the name Relativtheorie, which was changed to Relativitätstheorie by A. Bucherer in 1906.
- 4.
As mentioned in Sect. 3.1.2 this only holds true for point transformations.
- 5.
These are comprehensively treated in [282], [439].
- 6.
Together with the influence of the motion on clocks according to special relativity, one can calculate how both relativity theories affect the relative time measurements in GPS satellites and on earth. If these effects were not taken into consideration, GPS systems would show deviations of more than 10 km already within a day.
- 7.
According to Y. Ne’eman and Dj. Šijači infinite-dimensional representations do exist; see [380].
- 8.
It is a scalar only if \(\,\det \mathcal {K}=1=\det \mathcal {J}\). Indeed, since Einstein originally was not fully aware of the concept of a tensor density, he was inclined to assume that the general coordinate transformations are to be restricted to those for which \(\,\det \mathcal {K}=1=\det \mathcal {J}\)–even two weeks before he presented the ultimate theory to the Prussian Academy of Science on Nov. 25th, 1915.
- 9.
The identically-named covariant derivatives in general relativity and in Yang-Mills theory have common differential-geometric roots. In both cases, the connection is defined as a section in a specific fibre bundle; for more of this advanced geometry see Appendix E.6 and [192].
- 10.
Observe: contortion but not “contorsion” as you can find it in many papers dealing with the tetrad description of spacetime geometry.
- 11.
Not to be confused with the curvature scalar which is introduced below.
- 12.
In \(D>1\) dimensions, combinatorics yields \(\,\frac{D^2(D^2-1)}{12}\,\) independent components.
- 13.
Named after the Italian mathematician Luigi Bianchi (1856–1928).
- 14.
At other places I will denote the inverse tetrad by \(E^\mu _I\); this is advantages in short-hand expressions in terms of matrix relations.
- 15.
Here I anticipate that the antisymmetry is closely related to the fact that gravity, if formulated in a Riemann-Cartan geometry, can be interpreted as a gauge theory of the Poincaré group with its antisymmetric Lorentz generators; see Sect. 7.6.3.
- 16.
Here I use the wording of [78].
- 17.
This gives rise to the question why nature likes it mini: As simple as it can be, but not too simple.
- 18.
I have chosen to call this ’geometrodynamics’, a term originally coined by J.A. Wheeler; however for an ambitious program geared to quantize GR together with the other fundamental interactions.
- 19.
This is merely an assumption, maybe put forward by Einstein (see pages 48 and 49 in [162]) in observing that the energy-momentum tensors for electrodynamics and for hydrodynamis are symmetric.
- 20.
“With this, we have finally completed the general theory of relativity as a logical structure.”
- 21.
Written for instance in the introduction of [373]
- 22.
More about Constrained Dynamics in Appendix C.
- 23.
As W. Pauli states in his classical text [406], that this formalism was first investigated by Einstein [158].
- 24.
There is a photograph of Einstein taken on his visit at CalTech in 1931 where he is seen writing it on a blackboard. This became a widespread postcard.
- 25.
- 26.
In the context of GR, this is the group of diffeomorphism, but the definition applies to any symmetry group.
- 27.
The field equations derived from the Hilbert-Einstein action exhibit an additional scale invariance \(\,g_{\mu \nu } \rightarrow \lambda ^2 g_{\mu \nu }\), under which \(R_{\mu \nu } \rightarrow R_{\mu \nu }\,\) and \(\,R \rightarrow \lambda ^{-2} R.\) This is another example of a Lie symmetry not being a Noether symmetry.
- 28.
A nice example and illustration of the distinction between these two notions of diffeomorphism invariance can be found in [451].
- 29.
Strictly speaking, we need to distinguish the case of no gravitation at all–which amounts to \(\,g_{\mu \nu } =\eta _{\mu \nu }{,} \,D_\mu =\partial _\mu \)–from the case of a fixed gravitational background field for which there is no (dynamical) part for the gravitational field in the total Lagrangian, but where covariant derivatives–now defined with respect to the background metric–still remain in the expressions.
- 30.
see the appendix of [174] for misunderstandings about these notions.
- 31.
In the 1980’s there was some quarrel about a factor \(\tfrac{1}{2}\) compared to the original Komar superpotential and the related Moeller pseudotensor. This is settled in the meantime; see e.g [312].
- 32.
To my knowledge, these by now very common techniques originate from [448] and were rediscovered and refined in [1].
- 33.
Here I refrain from a mathematically proper defintion which you find in any advanced textbook on GR (e.g. [526]). Informally, an asymptotically flat spacetime is a manifold for which at large distances from some region the geometry becomes indistinguishable from that of Minkowski spacetime or–referring to solutions of the field equations–the gravitational field, as well matter fields become negligible in magnitude.
- 34.
The existence of a timelike Killing vector field for a metric means that coordinates can be found such that the metric components are independent of the time variable. These are also called stationary spacetimes, the most well-known representatives being the Schwarzschild black holes, and the rotating and charged ones.
- 35.
Strictly speaking, their proof needs the assumption that the “dominant energy condition” is valid.
- 36.
In Witten’s energy expression occurs technically a spinor quantity. In hindsight it was observed that this comes about because the generator of diffeomorphism is the square of the generator of supersymmetry transformations [281].
- 37.
This touches the question of whether there will ever exist a final theory of everything, a point to be discussed in the conclusion, Sect. 9.2.
- 38.
Astoundingly it is highly non-trivial to find the explicit expression, various incorrect ones have appeared in the literature. For a thorough analysis see [569]; see also Appendix F.3.2 for the appropriate expressions in terms of differential forms.
- 39.
Observe that neither the Einstein(-Cartan) tensor \(\tilde{G}\,\) nor the energy-momentum tensor \(\Sigma \,\) are presumed to be symmetric.
- 40.
This differs from previous expressions, since I adopt the conventions of M. Blagojević [49]. His definitions of \(\Sigma , L, P\) differ from those used elsewhere in this text by factors (-i).
- 41.
I thank M. Blagojević for pointing this out to me.
- 42.
Einstein himself was not only aware of teleparallel gravitation but actively followed this approach in what he called “Fernparallelismus” [464].
- 43.
Geometrically, this is a fibre bundle with base space \(\,M_4\,\) and fibre \(\mathbf {dS_4}\). The de Sitter group acts on the matter fields in each fibre.
- 44.
Be reminded of conventions for the group generators and other related objects different from those elsewhere in this text.
- 45.
Interestingly there are various ways to characterize these models, as also will be seen at other places in this book.
- 46.
A typical representative of this approach is the brane world.
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Sundermeyer, K. (2014). General Relativity and Gravitation. In: Symmetries in Fundamental Physics. Fundamental Theories of Physics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-06581-6_7
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