Abstract
The hard core of the theory consists of Einstein’s field equation, which relates the metric field to matter. After a discussion of the physical meaning of the curvature tensor, we shall first give a simple physical motivation for the field equation and will then show that it is determined by only a few natural requirements (Lovelock theorem), with two coupling constants. One is just Newtons gravitational constant, and the other is the much discussed cosmological constant, whose observational magnitude is a complete mystery for present day fundamental physics. This long chapter of about ninety pages, is devoted to various qualitative aspects of Einstein’s field equation. We treat the Lagrangian formalism, the role of diffeomorphism invariance, the tetrad formalism, energy, momentum, and angular momentum for isolated systems, the initial value problem, the 3+1 formalism, causality of matter propagation, and conclude with a careful treatment of the general relativistic Boltzmann equation.
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Notes
- 1.
For an excellent historical account of Einstein’s struggle which culminated in the final form of his gravitational field equations, presented on November 25 (1915), we refer to A. Pais, [71]. Since the publication of this master piece new documents have been discovered, which clarified what happened during the crucial weeks in November 1915 (see, e.g., [72]).
- 2.
The reader is invited to generalize the proof of Theorem 13.11 (given in DG, Sect. 16.1) to vector fields along maps.
- 3.
To simplify notation, we use for the induced covariant derivative of fields along the map H the same letter ∇ (instead of \(\tilde{\nabla}\) in the quoted sections of DG).
- 4.
Among historians, there have recently been controversial discussions after the claim in [78] that D. Hilbert had not found the definite equations a few days earlier. Archival material reveals that Hilbert revised the proofs of his paper [79] in a major way, after he had seen Einstein’s publication from November 25. We may never know exactly what happened in November 1915. I prefer to quote from a letter of Einstein to Hilbert on December 20, 1915: “On this occasion I feel compelled to say something else to you that is of much more importance to me. There has been a certain ill-feeling between us, the cause of which I do not wish to analyze. I have struggled against the feeling of bitterness attached to it, and this with success. I think of you again with unmixed congeniality and I ask that you try to do the same with me. Objectively it is a shame when two real fellows who have managed to extricate themselves somewhat from this shabby world do not give one another pleasure.”
- 5.
In higher dimensions additional terms are possible. We shall state Lovelock’s general result at the end of Sect. 3.6.
- 6.
In these early years, Einstein believed that GR should satisfy what he named Mach’s principle, in the sense that the metric field should be determined uniquely by the energy-momentum tensor (see Einstein’s paper [80]). His intention was to eliminate all vestiges of absolute space. For a detailed discussion we refer to the last part of his Princeton lectures [83]. Only later Einstein came to realize that the metric field is not an epiphenomenon of matter, but has an independent existence, and his enthusiasm for Mach’s principle decreased. In a letter to F. Pirani he wrote in 1954: “As a matter of fact, one should no longer speak of Mach’s principle at all.” For more on this, see [71].
- 7.
Actually, Einstein considered this term earlier in a footnote in his first review paper on GR early in 1916, but discarded it without justification.
- 8.
An interesting functional of g, which in four dimensions is locally a total divergence, is the Gauss–Bonnet scalar discussed in Exercise 3.18.
- 9.
In the light of recent historical studies it is no more justified to attribute the action principle entirely to Hilbert. For a detailed discussion and references we refer to [104]. Beside Einstein and Hilbert, also H.A. Lorentz published a series of papers on general relativity based on a variational principle.
- 10.
For a proof, as part of a global theorem, see [105].
- 11.
This has been emphasized, for instance, by D. Giulini in [35], p. 105-.
- 12.
Using dg αβ =ω αβ +ω βα one easily derives from \(\varOmega_{\;\beta}^{\alpha}=d\omega_{\;\;\beta}^{\alpha}+\omega_{\;\; \lambda}^{\alpha}\wedge\omega_{\;\;\beta}^{\lambda}\) that \(\varOmega_{\beta\gamma}=d\omega_{\beta\gamma}-\omega_{\sigma\beta }\wedge\omega_{\;\;\gamma}^{\sigma}\). In a similar manner, one obtains from the first structure equation \(d\theta_{\beta}-\omega_{\;\;\beta}^{\sigma}\wedge\theta_{\sigma}=0 \).
- 13.
Remark (for readers familiar with connections in principal fiber bundles). The identity (3.192) and its derivation can be interpreted globally on the frame bundle, when ω is regarded as an so(1,3)-valued connection form and θ as the canonical ℝn-valued 1-form (soldering form) [106]. In standard terminology (see [41, 42]), ∗t α is a pseudo-tensorial 3-form on the principal frame bundle.
- 14.
The use of pseudotensors appeared to many researchers in GR to violate the whole spirit of this generally invariant theory, and criticism of Einstein’s conservation law was widespread (see, e.g., the discussions of Einstein with F. Klein, [77]). Einstein defended his point of view in detail in [75]. Note that the three forms (3.193) exists globally if the spacetime manifold is parallelizable. In (3.195) the sum ∗T α +∗t α is then a three form which is globally closed.
- 15.
This discussion follows partly [36], Sect. 4.2.11.
- 16.
A pseudo-Riemannian manifold is geodesically complete if every maximal geodesic is defined on the entire real line. For Riemannian manifolds there is another notion of completeness. Introduce the Riemannian distance d(p,q) between two points as the infimum of L(γ) for all piecewise smooth curve segments from p to q. This makes the manifold into a metric space whose topology coincides with the original one. According to an important theorem of Hopf and Rinow, a Riemannian manifold is complete as a metric space if and only if it is geodesically complete. (Another equivalent statement is that any closed bounded subset is compact.) For such a space each pair of points can be connected by a geodesic. For proofs see, e.g., [48].
- 17.
Consider a 3-dimensional Riemannian manifold (N,h) with volume form \(\varOmega=\sqrt{h} dx^{1}\wedge dx^{2}\wedge dx^{3}\). If one writes the equation DG, (14.53),
$$i_X\varOmega=\langle X,N\rangle\, dS, $$in terms of coordinates, one easily finds dx j∧dx k=N i η ijk dS.
- 18.
Consider the following situation in linear algebra. Let (V,g) be a Minkowski vector space, T a linear map of V which is symmetric, 〈v 1,Tv 2〉=〈Tv 1,v 2〉, and assume that there is a timelike eigenvector of T:Tu=ρu. Let V ⊥ be the orthogonal complement of u in V. Then V splits into the direct orthogonal sum V=ℝu⊕V ⊥. The symmetry of T implies that V ⊥ is invariant under T. Restricted to V ⊥, with the induced Euclidean metric, T is—since it is symmetric—diagonizable.
- 19.
Quote from the textbook [57] by L.C. Evans on partial differential equations. We highly recommend this well written clear graduate text to all those who are eager to read a detailed modern introduction to this vast field of mathematics.
- 20.
For a compact \(\mathcal{S}\) this is a special case of the Yamabe problem: Show that on a compact Riemannian manifold of dimension ≥ 3 there always exists a metric with constant scalar curvature. This problem is solved, but for complete non-compact manifolds there are only a few results.
- 21.
For example, there exist functions a(t,x) and u(t,x) in C ∞(ℝ2,ℂ) with supports in {(x,t)∈ℝ2:t>0}, such that
$$\partial_t u+a \partial_xu=0, $$but do not vanish everywhere. Such an u and u≡0 are both solutions with the initial condition u=0.
- 22.
Physicists usually do not care about such theorems, because they take the result for granted. As a warning, we mention that there are even linear PDE without singular points that have no solution anywhere. A famous example was constructed by H. Lewy that also highlights the importance of analyticity in the Cauchy–Kovalevskaya theorem (see Chap. 8 of [58]).
- 23.
Gaussian normal coordinates are introduced in Sect. 3.9.3.
- 24.
For a simple proof of (3.265) chose the function S as one of the coordinates, say x 1. Along Σ the left-hand side of (3.263) has the form \(g^{11}\partial_{1}^{2}\psi\) + terms which are all determined by the Cauchy data on Σ (since these involve tangential derivatives of ψ and ∇ψ on Σ). Hence, \(\partial_{1}^{2}\psi\) is only determined by the Cauchy data and the differential equation (3.263) if g 11≠0, that is if (3.265) holds.
- 25.
Note that \((\partial_{t}\bar{K})_{\;i}^{i}\not=\partial_{t}K_{\;i}^{i}\); we find from (3.291) (since \(\partial_{t}, \bar{L}_{\bar{\beta}}\) are derivations commuting with contractions)
$$(\partial_t-\bar{L}_{\bar{\beta}}\bar{K})_{\;i}^i=( \partial_t-L_{\bar{\beta}})H-2\alpha(\bar{K}\cdot\bar{K})_{\;i}^i. $$ - 26.
We adapt the proof in Sect. 4.3 of [10].
- 27.
For this we refer to the treatise [126], Sect. XII.2.
- 28.
If x μ are coordinates of M then the dx μ form in each point p∈M a basis of the cotangent space \(T^{\ast}_{p}M\). The bundle coordinates of \(\beta\in T^{\ast}_{p} M \) are then (x μ,β ν ) if β=β ν dx ν and x μ are the coordinates of p. With such bundle coordinates one can define an atlas, by which T ∗ M becomes a differentiable manifold.
- 29.
Note that in Minkowski spacetime we get for a constant time section \(\operatorname{vol}_{\varSigma}=dx^{123}\wedge dp^{123}\).
- 30.
Since this book contains only a modest introduction to cosmology, we give references to some useful recent textbooks.
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Straumann, N. (2013). Einstein’s Field Equations. In: General Relativity. Graduate Texts in Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5410-2_3
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