Black holes are quantum complete Marc Michael Schneider Munchen¨ 2018 Black holes are quantum complete Marc Michael Schneider Dissertation an der Fakult¨at fur¨ Physik der Ludwig{Maximilians{Universit¨at Munchen¨ vorgelegt von Marc Michael Schneider aus Trier Munchen,¨ den 6. August 2018 Erstgutachter: Prof. Dr. Stefan Hofmann Zweitgutachter: Prof. Dr. Peter Mayr Tag der mundlichen¨ Prufung:¨ 14. September 2018 Only those who will risk going too far can possibly find out how far one can go. { Thomas Stearns Eliot { (Preface to Transit of Venus: Poems) Contents Zusammenfassung xi Abstract xiii 1. Concerning singularities 1 2. The fellowship of completeness 7 2.1. Classical completeness . .7 2.2. Quantum mechanical completeness . .9 2.3. Quantum-mechanical versus classical completeness . 14 2.4. Geodesic completeness . 17 2.5. Quantum-mechanical probes of space-time singularities . 21 3. The two singularities 27 3.1. The gravitational singularity . 28 3.2. The generalised Kasner singularity . 36 4. The return of regularity 45 4.1. The Schr¨odinger representation of quantum field theory . 46 4.1.1. Functional calculus . 47 4.1.2. Flat space-time formulation . 50 4.1.3. Curved space-time formulation . 59 4.2. The quantum completeness criterion . 66 viii Contents 4.3. Quantum probing of Schwarzschild . 72 4.3.1. Ground state analysis . 73 4.3.2. Gaussian deviations: Excited states . 79 4.3.3. Influence of polynomial self-interactions . 84 4.3.4. Stress-energy tensor of quantum probes . 91 4.4. Charge conservation inside the black hole . 96 4.5. Quantum probing of the Kasner space-time . 103 4.5.1. Ground state analysis . 104 4.5.2. Mode functions . 106 5. Conclusion 109 A. Fourier transformation 115 B. Riccati differential equation 117 C. Publication: Classical versus quantum completeness 119 D. Publication: Non-Gaussian ground-state deformations near a black-hole singu- larity 127 E. Publication: Information carriers closing in on the black-hole singularity 135 F. Kasner analysis and connexion to the Schwarzschild case 141 G. Heisenberg analysis of charge conservation inside a black hole 143 Acknowledgements 159 List of figures 2.1. Potential V(x) with steps. 15 2.2. Potential V(x) with spikes. 17 2.3. Penrose diagram of the negative mass Schwarzschild solution . 25 3.1. Penrose diagram of the Schwarzschild solution . 29 3.2. Penrose diagram of the fully extended Schwarzschild solution. 32 4.1. Relation between functions f(x) and functionals F[f]............. 47 4.2. Schematical illustration of the scattering operator from two to n particles. 60 4.3. Plot of the normalisation N(0)(t) for t (0; 0:25M) ............. 78 2 4.4. Plot of the stress-energy tensor T00 for N(Λ) f1; 5g and t [0; 0:25] .. 93 h i 2 2 4.5. Plot of the stress-energy tensor T00 for t [0; 0:5].............. 95 h i 2 4.6. Plot of the probabilistic current. 102 Zusammenfassung Vollst¨andigkeit ist ein ¨außerst wichtiges Konzept in der theoretischen Physik. Die Haupt- idee besagt, dass eine Bewegung oder ein Freiheitsgrad eindeutig und fur¨ alle Zeiten de- finiert ist. Die Kriterien, die Vollst¨andigkeit in klassischen und quantenmechanischen Theo- rien beschreiben, sind je nach Theorie unterschiedlich. Sie sind eng mit dem spezifischen Messprozess der jeweiligen Theorie verknupft.¨ Unvollst¨andigkeit geht oft mit der Pr¨asenz einer Singularit¨at einher. Die Existenz einer Singularit¨at ist untrennbar gegeben durch die Freiheitsgrade der Theorie. In Allgemeiner Relativit¨atstheorie sind Raumzeiten charakterisiert durch die L¨ange geo- d¨atischer Kurven. Dieses Kriterium fußt auf der klassischen Punktteilchenbeschreibung mittels Differentialgeometrie und fuhrte¨ zur Entwicklung der Singularit¨atentheoreme nach Hawking und Penrose. Ein Quantendetektor in einer dynamischen Raumzeit kann nicht durch Quantenmecha- nik im engeren Sinne beschrieben werden, denn es ist nicht m¨oglich eine konsistente re- lativistische Einteilcheninterpretation einer Quantentheorie zu formulieren. Infolgedessen ubertragen¨ wir den Terminus der Vollst¨andigkeit auf Situationen deren einzig ad¨aquate Be- schreibung durch Quantenfeldtheorie auf gekrummten¨ Raumzeiten erfolgen kann. Um Uni- tarit¨atsverletzungen, welche sich in endlicher Zeit ereignen, aufl¨osen zu k¨onnen, nutzen wir die Schr¨odingerdarstellung der Quantenfeldtheorie, da diese eine Zeitaufl¨osung erm¨oglicht. Sinnhaftigkeit der Persistenzamplitude des Wellenfunktionales, d.h. Wahrscheinlichkeits- verlust oder Stabilit¨at werden mit Vollst¨andigkeit in Verbindung gebracht. Anhand eines schwarzen Loches der Schwarzschildgattung wenden wir unser Kriteri- um an und testen Vereinbarkeit mit freien, massiven Skalarfeldern. Abweichungen vom xii Zusammenfassung Gauß’schen Ansatz fur¨ das Wellenfunktional, angeregte Zust¨ande und Selbstwechselwir- kung der Testfelder, werden betrachtet und deren gutartige Entwicklung wird gezeigt. Die Analyse wird auf eine weitere Klasse von Raumzeiten, den Kasner-Raumzeiten an- gewandt, die hohe Relevanz durch die Vermutung von Belinskii, Khalatnikov und Lifshitz haben. Abstract Completeness is a very important concept in theoretical physics. The main idea is that the motion or the degree of freedom is uniquely defined for all times. The criterion for completeness is different for classical and quantum theories. This corresponds to a specific measurement process in the corresponding theories. Incompleteness is often related to the occurrence of a singularity. The notion of a singularity is closely related to the corresponding degree of freedom. In general relativity space-times are characterised by the extendibility of geodesic curves. This criterion founded on the point particle description through differential geometry has given rise to the singularity theorems of Hawking and Penrose. A quantum probing of dynamical space-times can not be described with quantum me- chanics because there is no consistent relativistic one particle interpretation of a quantum theory. Hence, we extend the notion for completeness to situations where the only adequate description is in terms of quantum field theory on curved space-times. In order to analyse unitarity violations occurring during a finite time, we use the Schr¨odinger representation of quantum field theory which allow for time resolution. Consistency of the wave-functional's persistency amplitude, i.e. probability loss or stability, will be connected to completeness. For a Schwarzschild type back hole we apply the criterion and probe with free massive scalar fields for consistency. Furthermore, deviations from Gaussianity, i.e. excited states and self-interaction of the probing fields are derived and consistency is showed for those deformations. The analysis is furthermore applied to Kasner space-times, which have high relevance due to the conjecture of Belinskii, Khalatnikov, and Lifshitz. 1 Concerning singularities Considering the case that an random person on the street asks you about black holes and the big bang, most people which are educated in science would inevitably come to the point where they have to talk about singularities, but this will trigger the next question: What is a singularity?\. The concept is known to every physicist, although the notion is " not very familiar to non-physicists or non-mathematicians because it has no equivalence in their daily life, at least, however, it seems to be important in order to understand what a black hole or the big bang is. During a day without thoughts about physics or mathematics the concept of singularities is not needed. From a heuristic point of view one would deny that a system in nature becomes singular, e.g. reaches an infinite value of energy. Amongst physicists, it is clear that the occurrence of a singularity is equivalent of having a severe problem our theory. The black hole and the big bang singularities are a direct outcome of Einstein's equations, hence, they are predicted to occur by general relativity. It is, however, not clear that these solutions can be reached dynamically. Hence, they could also be a mathematical artefact. Usually the term singularity\ is connected to the situation where at least one observable " could be measured to grow unbounded. Experimentalists have never reported an infinite value of a measurable quantity. This fact motivates to question the physical justification for the concept. Let us recall the definition of singularities in mathematics. The definition of a singular point is [Simon, 2015a]: Definition 1. If Ω is a region, f A(Ω) (all analytic functions on Ω) and z0 ∂Ω, we 2 2 say that f is regular at z0 if and only if there is δ > 0, g analytic in Dδ(z0) (disk of radius δ about z0) so that f(z) = g(z) for all z Ω Dδ(z0). If f is not regular at z0, we say that 2 \ 2 1. Concerning singularities f is singular at z0. The above definition says that there exists a point z0 for which the function acquires an infinite value: f(z0) > M; M R. 8 2 The language of physics is mathematics, therefore, it is very natural that mathematical concepts are also present in physics. Singularities appear in different theories; in some they signal a breakdown of the description. Before we specify the notion in physical examples we want to elucidate what is generally meant by breakdown of the description\. Physical " theories are devoted to specific energy (or length) scales, below this energy scales the theory is effectively describing the system while beyond the scale the theory loses its predictability. In general, the pure presence of a singularity starts to become a problem only in conjunc- tion with the possibility to dynamically reach the singularity in a finite amount of time. Considering the mathematical definition, the first singularity, ever deduced in physics, oc- curred in the theory of gravitation. Newton's law of the attraction between two massive mM bodies admits a singular value at the origin. The formula F = -GN r2 suggests that the force at r = 0 will be infinitely strong. This is clearly a pathology of the theory signalling its breakdown close to this point.