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General Relativity: Subtle Is the Lord

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Evolution of Black Holes in Anti-de Sitter Spacetime and the Firewall Controversy

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

In this chapter, we introduce the main ideas of Einstein’s theory of General Relativity. We make precise some important terms that have been mentioned in the previous chapter, such as black hole and event horizon. For later use, we also introduce Penrose diagrams and anti-de Sitter spacetime. Finally, we will briefly discuss the AdS/CFT correspondence, and the role of general relativity in that context. Some parts of this chapter consist of the author’s own, perhaps biased and rather philosophical, opinion, on some aspects of general relativity.

A certain king had a beautiful garden, and in the garden stood a tree which bore golden apples. These apples were always counted, and about the time when they began to grow ripe it was found that every night one of them was gone. [...] As the clock struck twelve he heard a rustling noise in the air, and a bird came flying that was of pure gold; and as it was snapping at one of the apples with its beak, the gardener’s son jumped up and shot an arrow at it [...] it dropped a golden feather from its tail, and then flew away. [...] it was worth more than all the wealth of the kingdom: but the king said, ‘One feather is of no use to me, I must have the whole bird.

The Golden Bird, The Brothers Grimm

The original version of this book was revised: New contents have been replaced. The erratum to this book is available at 10.1007/978-3-662-48270-4_7

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Notes

  1. 1.

    This is, however, not necessarily true if the manifold is non-orientable; see Chap. 1, Sect. 7 of [1].

  2. 2.

    Though not positive definite in the Lorentzian case.

  3. 3.

    I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice... – Albert Einstein [2].

  4. 4.

    Of course, since \(\nabla _X: Y \mapsto \nabla _XY \in TM\), it is not a tensor in the strict sense of the word; but any tensor \(T: T_pM \times T_p^*M \rightarrow \mathbb {R}\) can also be viewed naturally as a map \(T: T_pM \rightarrow T_pM\). It is in this sense that \(\nabla _X\) is not a tensor. If one wishes to be more accurate, one could say that \(\nabla _X\) is not an endomorphism of the \(\mathcal {F}(M)\)-module TM.

  5. 5.

    There is unfortunately no accepted convention of the sign of the curvature tensor, or even which index is the one to be put “upstairs.” Exercise extreme caution when reading the literature.

  6. 6.

    Spivak’s Volume 2 [3] has a nice explanation of the Riemann curvature tensor. Essentially, it comes about from an integrability condition for the existence of solution, when trying to solve for \(g(\partial /\partial x^a, \partial /\partial x^b)=\delta _{ab}\).

  7. 7.

    Sometimes, we carelessly refer to \(R_{ab}\) as the Ricci tensor, or that \(g_{ab}\) is the metric tensor, instead of the components of these tensors in a particular basis. In the “abstract-index notation,” they are actually referring to the tensors themselves. However, such practice can be confusing to beginners. For example, it may cause people to ask whether “coordinate \(x^a\) is a vector” (c.f. \(V^a\), the components of a vector \(V=V^a\frac{\partial }{\partial x^a}\)). My humble opinion is: learn the geometric objects properly, and then go ahead and abuse the notations all you want—but not before you know what you are doing!

  8. 8.

    I chose to explain what an action is, albeit very briefly and not very rigorously, because a mathematician reader may not be familiar with this concept.

  9. 9.

    A vector field X is called a Killing vector field with respect to the metric g if the Lie derivative \(\pounds _Xg=0\). Essentially, this is saying that the geometry on the manifold M determined by the metric g does not change if we move along the flow of the Killing vector field.

  10. 10.

    In linearized gravity, the metric tensor has components \(g_{ab}=\eta _{ab} + h_{ab}\), where \(\eta \) is the background (here, flat) metric, and h is the perturbation (in the solar system, \(|h_{ab}|\sim GM_\odot /(R_\odot c^2) \sim 10^{-6}\)). Roughly speaking, quantization of \(h_{\mu \nu }\) gives the graviton—it is an excitation of the background metric field.

  11. 11.

    There was one thing that really riled many of the general relativists about string theory: in string theory [...] the geometry of spacetime, the be-all and end-all of general relativity, seemed to disappear. It was all about describing a force [...] – Pedro G. Ferreira. [10].

  12. 12.

    Indeed, even the “one-metric” theory of massive gravity requires two metrics, but one of which serves as a fixed—nonzero—background for the dynamical metric.

  13. 13.

    In GR, torsion vanishes identically by construction. The role of torsion in other theories of gravity is theory-dependent: in some theories such as the Einstein–Cartan theory [34,35,36], torsion couples to spin; however, it is also possible to use only torsion (without curvature), to construct a theory that is, surprisingly, equivalent to GR, which is often called “TEGR (teleparallel equivalent of general relativity)” [37]. TEGR reminds us that theories should not be confused with reality—the latter involves observed phenomena, e.g., a falling apple, while the former are attempts to understand said observations. It is possible that there can be more than one theory, which are different in terms of their mathematical structures, yet provide equivalent physical predictions.

  14. 14.

    Or for that matter, the “principle of general covariance”—(almost?) any theory can be made general covariant. See the debate about this issue in [39].

  15. 15.

    Readers interested in thought experiments would also enjoy [40].

  16. 16.

    The study of causal structure is an important aspect of Lorentzian geometry, and we refer the readers to [8, 47] for more details.

  17. 17.

    Note that both U, V and u, v are dimensionless.

  18. 18.

    Readers with a complex analysis background are encouraged to read [49].

  19. 19.

    For even more details, see [50].

  20. 20.

    A maximally symmetric spacetime has, in d-dimensions, a total of d(d+1)/2 Killing vectors. See Lemma (9.28) of [8].

  21. 21.

    In d-dimensions, \(\Lambda =-\frac{(d-1)(d-2)}{2L^2}\).

  22. 22.

    In d-dimensions, the Ricci tensor satisfies \(R_{ab}=\frac{2\Lambda }{d-2}g_{ab}\). Thus, the scalar curvature satisfies \(R=-\frac{d(d-1)}{L^2}\).

  23. 23.

    Note that in writing \(U=\arctan (u)\), it is really \(\tan U = u/1\), where 1 is \(u=1\) in the corresponding unit of length.

  24. 24.

    This “center” is arbitrary in the same sense that all points in de Sitter spacetime are a “center” from which everything else moves away from—there is no real center in such an expanding universe.

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  1. Nordic Institute for Theoretical Physics, Stockholm, Sweden

    Yen Chin Ong

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Ong, Y.C. (2016). General Relativity: Subtle Is the Lord. In: Evolution of Black Holes in Anti-de Sitter Spacetime and the Firewall Controversy. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48270-4_2

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