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A Century of Black Hole Physics: From Classical Geometry to Hawking Radiation and the Firewall Controversy

  • Chapter
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

This introductory chapter aims to provide a history of the field, from the early days when Einstein first formulated his general theory of relativity, and discoveries of black hole solutions in the theory, to the later debates about the as yet unresolved information loss paradox and the firewall controversy today. Most of this chapter is written in the style of a semipopular science article. The aim is to convey, at least partially, the results of this thesis to a wider audience, who are not necessarily trained in physics beyond that of their high school education. The use of equations will be kept to a minimum (some equations are included since they represent major milestones in the history of black hole physics; these will be further elaborated on in Chap. 2). Some technical statements are provided in footnotes and boxes.

“All right,” said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone.

“Well! I’ve often seen a cat without a grin,” thought Alice; “but a grin without a cat! It’s the most curious thing I ever saw in my life!”

Alice in Wonderland, Lewis Carroll

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Notes

  1. 1.

    Strictly speaking, this is not a correct statement. For a sufficiently large black hole, one does not feel much tidal force at the event horizon. It is the curvature of spacetime around a black hole that is warped in such a way as to prevent anything from escaping.

  2. 2.

    General relativity has been tested again and again throughout the century, and passed with flying colors. Together with its underlying intricate geometric foundation and healthy causal structure, general relativity deserves the title “the most beautiful theory in physics” [3, 4].

  3. 3.

    In John Wheeler’s now famous words [5], “spacetime tells matter how to move; matter tells spacetime how to curve.”

  4. 4.

    Wheeler did not actually invent the term. It was suggested to him a few weeks earlier during another lecture. A member of the audience suggested it after he presumably got tired of hearing Wheeler repeatedly saying “gravitationally completely collapsed object.” Also, the term goes a couple of years back at least—Science News Letter used the term “black hole” already in January 1964 [7]. Half a century later, it is no longer clear who actually first coined the phrase. Wheeler of course deserves the credit for he popularized its usage.

  5. 5.

    The peculiar property of the event horizon is not obvious in the original coordinates employed by Schwarzschild [6]. He had followed Einstein’s idea that coordinate system that satisfies the “unimodular condition” \(\det {(g)}=-1\) is somehow better.

  6. 6.

    It was only in 1939 that Robert Oppenheimer and Hartland Snyder showed the collapse of a pressureless homogeneous fluid sphere does lead to the formation of a black hole [9]. The suspicion that even in a complete vacuum, a sufficiently strong gravitational field (in the form of a gravitational wave) can form black holes was only proved in 2008 by Christodoulou [10].

  7. 7.

    Technically, we can check that the scalar quantity called the Kretschmann scalar, \(K=R_{abcd}R^{abcd}\), is finite at the horizon, but is infinite at \(r=0\).

  8. 8.

    As pointed out in [14], Georges Lemaître, among others, already pointed out that a coordinate change can remove the \(r=2M\) singularity back in 1933. Somehow the results were either ignored or forgotten.

  9. 9.

    In realistic stellar collapse, some amounts of angular momentum can be lost due to shredding of mass prior to a complete gravitational collapse.

  10. 10.

    This is however not entirely a fair statement. It is true that astrophysical black holes tend to discharge fairly quickly, but surely it is not difficult to find black holes with sufficiently low charge.

  11. 11.

    In precise mathematical language, a spacetime is stationary if it admits a timelike Killing vector field, and is static if it admits a hypersurface-orthogonal timelike Killing vector field.

  12. 12.

    Note that the usual statement one finds in many textbooks and literature reads “stationary spherically symmetric vacuum spacetime must be static.” This is at best misleading. The correct mathematical statement of Birkhoff’s theorem is: any spherically symmetric spacetime satisfying the Einstein vacuum field equations must have an extra Killing vector field V, in addition to the three Killing vector fields we already have from spherical symmetry. There is no requirement that V has to be timelike. In the interior of a Schwarzschild black hole, V is in fact spacelike, and therefore the interior spacetime is not static.

  13. 13.

    Technically, the theorem says that all multipole moments of the asymmetric body are radiated away in the form of gravitational (and/or electromagnetic) waves. Mass, electric charge, and angular momentum are protected due to the fact that these are (geometrically) conserved quantities.

  14. 14.

    Even in the case without an electric field, the proof (Hawking–Carter–Robinson’s theorem [34–36]) that the Kerr solution is unique requires additional assumptions that do not seem to be physical, e.g., that the spacetime is real-analytic, a stronger condition than just being smooth. Recent advancements in the field include proving uniqueness without the analyticity assumption, provided that a scalar identity is assumed to be satisfied on the bifurcation 2-sphere (Ionescu–Klainerman [37]), and proving uniqueness without the analyticity condition assuming that the spacetime is, in some technical sense, “close” to being Kerr (Alexakis–Ionescu–Klainerman [38]).

  15. 15.

    In general relativity, the theory becomes trivial if we do not specify some conditions for the matter field, since given any \(T_{ab}\), one can always in principle find the corresponding metric \(g_{ab}\) that solves the Einstein field equations. Various energy conditions are thus devised to specify how “physically reasonable” matter fields should behave. The weak energy condition stipulates that for every timelike vector field V, the matter density observed by a local observer is always nonnegative, i.e., \(\rho :=T_{ab}V^aV^b \geqslant 0\). For a recent review on energy conditions, see [47].

  16. 16.

    The first law does depend on the Dominant Energy Condition (DEC) in the sense that the proof requires the 0th law, which requires the DEC. However a weaker form of the 0th law exists without energy conditions; it is just the geometric statement that a “sufficiently regular” (a condition which may not hold for all black holes) Killing horizon must be a bifurcation surface, and the surface gravity will be constant. If one assumes this weaker 0th law, then the 1st law can be derived without assuming energy conditions. See [47] for further discussions.

  17. 17.

    Note that there is still a crucial difference here that is not always mentioned—the Second Law of Thermodynamics turns out to be a statistical law (a priori thermodynamics makes sense without statistical mechanics at its foundation); entropy can and does occasionally go down due to fluctuations. On the other hand, the area law of a black hole horizon is strictly geometrical; the area cannot (classically) “fluctuate” downward. This only becomes consistent if one takes into account the fact that quantum mechanically the energy conditions may not hold, and so the area may indeed fluctuate downward.

  18. 18.

    For a more detailed history, see [51].

  19. 19.

    A curiosity: \(\hbar \) cancels out in the expression TdS, so we could not know from thermodynamical laws alone, where the \(\hbar \) is hiding (a related question was recently explored in [54]). Formally, one only needs a cutoff \(\ell \) that has the right dimension to make T nonzero and S finite. In fact, recently it was emphasized by Erik Curiel that even classical black holes are “hot,” i.e., one does not need quantum mechanics to justify black hole thermodynamics [55]—“Does the use of quantum field theory in curved spacetime offer the only hope for taking the analogy seriously? I think the answer is ‘no.’ [...] the analogy between classical black hole mechanics and classical thermodynamics should be taken more seriously, without the need to rely on or invoke quantum mechanics.”

  20. 20.

    The Casimir effect is an example of “indirect detection” of virtual particles.

  21. 21.

    This has basis in the time–energy uncertainty principle: \(\Delta E \Delta t \gtrsim \hbar \). One could borrow large energy \(\Delta E\) from the vacuum as long as one returns it in a short time interval \(\Delta t\).

  22. 22.

    The cartoon picture is of course just a cartoon picture and should not be taken too seriously (see Box 1.2). However, even at this level, it should be clarified that the negative energy particle does not have negative energy with respect to a comoving observer. Due to the Killing vector field switching from timelike to spacelike beyond the horizon, what is seen as negative energy outside becomes positive energy inside. This is consistent with the fact that local observers should not expect to see a real particle with negative energy, either inside or outside the black hole.

  23. 23.

    It is a common misunderstanding that all accelerating observers see a thermal radiation. The spectrum of the radiation depends on the motions of the observer (more technically, on the curvature and torsions of the Frenet–Serret frame that defines the worldline of the observer. See, e.g., [59, 60]. This is a nice example of elementary differential geometry being applicable to spacetime physics).

  24. 24.

    This is not necessarily true for black holes with other asymptotic geometries, for example, it is not true for large black holes in anti-de Sitter spacetime; there an infalling observer sees no Hawking radiation [61].

  25. 25.

    In the original derivation of Hawking, these modes are actually transplanckian when they are first created near the horizon. However, subsequent works have shown that Hawking radiation is a much more generic phenomenon. In particular it is independent on the cutoff scale imposed on the wavelengths in the theory. See, e.g., the Refs. [62] and [63].

  26. 26.

    The locally measured temperature of the Hawking radiation measured by an observer who is following an orbit of a Killing vector field \(\xi ^a\) normal to the horizon, is given by \(T_{\text {BH}}/\sqrt{\xi ^a\xi _a}\), where \(T_{\text {BH}}\) is the (asymptotic) Hawking temperature.

  27. 27.

    At the quantum level, reversing time is not sufficient since physics is not time-reversal invariant, but CPT invariant. However, the same idea holds, mutatis mutandis.

  28. 28.

    The fact that this does not happen in real life is merely because a book has a vastly lower entropy than burned ashes. Low entropy states give rise to an arrow of time in the universe; see, e.g., [66–71].

  29. 29.

    We will give a more detailed introduction to quantum information in Appendix B.

  30. 30.

    More generally, there is the Rényi entropy [73], \(S_\alpha = \frac{1}{1-\alpha }\log \left( \sum _{i=1}^n p_i^\alpha \right) \), \(\alpha \in [0, \infty )\). The limiting case \(\alpha \rightarrow 1\) yields the Shannon entropy \(S_1 = -\sum _{i=1}^n p_i \log p_i\).

  31. 31.

    Spaghettification refers to the process in which an infalling object is stretched vertically and compressed horizontally by the tidal force of a black hole. Note that spaghettification can happen way before one reaches the horizon if the black hole is sufficiently small: that is why we chose a large black hole for our poor elephant.

  32. 32.

    Locality means roughly that quantum fields at different points of space do not interact with one another. This should not be confused with “non-locality” of quantum entanglement.

  33. 33.

    More technically, the absence of entanglement means that the field configuration across the event horizon is generically not continuous, which leads to a divergent local energy density. We recall that the quantum field Hamiltonian contains terms like \((\partial _x \varphi )^2\). The derivative is divergent at some \(x=R\), if the field configuration is not continuous across R.

  34. 34.

    This is a generic statement. Complexity theory tells us that given a configuration on a \(n \times n\) chess board, we could determine the winning strategy in \({{2^n}}^c\) steps for some constant c, i.e., it is what a computer scientist would call an “EXPTIME complete” problem. However for specific small n this is a manageable task. For black holes we could imagine that they are covered by configuration of 0’s and 1’s on each Planck sized square tiling their horizon. This yields \(n \sim 10^{77}\) for a solar-mass black hole (This is the ratio of black hole area over Planck area—in the units we use in this thesis, \(\hbar \) is an area: \(\hbar \approx 3 \times 10^{-66}\) \(\text {cm}^{2}\). So, \({4\pi (2 M_\odot )^2}/{\hbar } \approx 3.77 \times 10^{77}\)). Nevertheless we cannot rule out the possibility that quantum gravity has novel features that may make computation easy.

  35. 35.

    Scott Aaronson proposed a rather appropriate acronym HARD for “HAwking Radiation Decoding” during his talk at the “Rapid Response Workshop: Black Holes: Complementarity, Fuzz, or Fire?”, held at the KITP in Santa Barbara on August 19–30, 2013.

  36. 36.

    “The loser will reward the winner with an encyclopedia of the winner’s choice, from which information can be recovered at will.”

  37. 37.

    Perhaps gravity is more similar to a condensed matter system than being a fundamental interaction—gravity could be “emergent” from some as yet unknown degrees of freedom. Such ideas of “emergent gravity” can be dated back to Sakharov [97].

  38. 38.

    “Pace particle physicists, general relativity simply cannot be comprehended as a theory describing a dynamical ‘force’ at all.” [47].

  39. 39.

    Technically, it is a closed—and thus finite—Friedmann–Lemaître–Robertson–Walker (FLRW) universe, but it can be arbitrarily huge.

  40. 40.

    Here we are referring to the Arnowitt–Deser–Misner (ADM) mass, which will be reviewed in Chap. 3.

  41. 41.

    We will provide a short introduction to the Seiberg–Witten instability in Appendix D.

  42. 42.

    I thank Stephen Hsu for emphasizing this point to me during our conversations when he visited Taipei in 2011.

  43. 43.

    Since they are defined locally based on the behavior of light rays, while the very notion of the event horizon requires a full knowledge of the entire spacetime.

  44. 44.

    “ER” refers to the Einstein–Rosen [123] bridge, a non-traversable wormhole in a maximally extended Schwarzschild manifold, while “EPR” refers to the Einstein–Podolsky–Rosen paradox [124], a thought experiment that involved quantum entanglement.

  45. 45.

    John Baez and Jamie Vicary examined 3-dimensional topological field theory, and found that the process of particle pair-creation is identical to the process of wormhole formation. The entanglement between the particles is thus “fake entanglement,” which is indeed not subjected to the monogamy theorem [126].

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    Yen Chin Ong

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Ong, Y.C. (2016). A Century of Black Hole Physics: From Classical Geometry to Hawking Radiation and the Firewall Controversy. 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_1

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