Physics Area { PhD course in Astroparticle Physics Doctoral Thesis Beyond General Relativity: Modified Theories and Non-Singular Black Holes Candidate: Advisor: Francesco Di Filippo Prof. Stefano Liberati Academic Year 2019 { 2020 ii iii Abstract Einstein's general relativity is an extremely elegant and successful theory. Recent obser- vations coming from the LIGO/VIRGO collaboration as well as from the Event Horizon Telescope give us the possibility to perform precision test of general relativity in regimes never tested before. Even though all the current observations are in perfect agreement with the predictions of general relativity, there are several reasons to study extensions of the theory. From the experimental point of view, we are forced to include a dark sector for the matter and energy content of the universe to explain the cosmological data. Whereas from a conceptual point of view, the theory is not perturbatively renormalizable, and it predicts the formation of spacetime singularities. This thesis studies possible modifications of general relativity both considering spe- cific theories of modified gravity and implementing a model independent approach. In the first part of the thesis, we study a specific class of modified theory of gravity which has the peculiarity of propagating the same number of degrees of freedom of general relativity. The existence of these theories apparently challenges the distinctive role of general relativity as the unique non-linear theory of massless spin-2 particles. However, we provide strong evidence that these theory are actually equivalent to general relativity in vacuum. In the second part of the thesis, we focus on the problem of black hole singularities which are unavoidably present in general relativity. However, it is reasonable to assume that there will be a mechanism preventing their formation in a full theory of quantum gravity. Without specifying any theory of quantum gravity or the nature of such mech- anism, simply assuming a minimal set of kinematical constraints, we classify and study the properties of non-singular spacetime with a trapping horizon. Contrary to what one might expect, the set of regular geometries that arises is remarkably limited. Further- more we show that it is very difficult to construct a self consistent geometry without any long range effect. This give us further motivation to study the phenomenology of non-singular black holes. To this end, we provide a set of parameters describing the deviations from classical black holes, and we review the possible observational channels that can measure or constrain them. iv Contents Abstract iii Publication list vii Introduction1 1 Minimally modified theories of gravity5 1.1 A class of Hamiltonian theories . .6 1.1.1 Poincar´ealgebra . .8 1.2 A class of Lagrangian theories . .9 1.2.1 Linearized field equations . 10 1.2.2 Propagator . 12 1.2.3 On-shell 3-point amplitudes . 13 1.2.4 Interlude, the most general theories compatible with these results 14 1.3 Higher orders in the perturbative expansion . 15 1.4 Genuinely different theories . 20 1.4.1 Theories with no gravitons . 20 1.4.2 Theories with a modified dispersion relation for gravitons . 21 1.5 Coupling to matter . 23 1.5.1 Algebra of constraints for minimal coupling . 23 1.6 Conclusions . 26 2 Causal hierarchy beyond general relativity. 29 2.1 The standard general relativity causal hierarchy . 29 2.2 Multi-metric frameworks . 31 2.3 Infinite signal velocities . 33 2.3.1 Hypersurface orthogonality of the aether . 34 2.3.2 Uniqueness of the aether field . 35 2.3.3 Universal horizons . 36 2.4 Conclusions . 38 v vi CONTENTS 3 Geodesically complete black holes 39 3.1 Avoiding singularities: Beyond Penrose's theorem . 40 3.2 Geometric setting . 42 3.3 Taxonomy of non-singular geometries . 44 3.4 Conclusions . 58 4 On the viability of non-singular black holes 61 4.1 Evanescent horizons and everlasting horizons . 61 4.1.1 Time independent geometry . 62 4.1.2 Instability of the inner horizon . 64 4.1.3 Time dependent scenario . 68 4.1.4 (Ir)relevance of the cosmological constant . 70 4.2 One way and asymptotic hidden wormholes . 71 4.2.1 One way hidden wormhole . 71 4.2.2 Asymptotic hidden wormhole . 72 4.3 Conclusions . 73 5 Phenomenology of non-singular black holes 75 5.1 A taxonomy of proposals . 75 5.1.1 Regular black holes . 75 5.1.2 Bouncing geometries . 76 5.1.3 Quasi-black holes . 78 5.1.4 Wormholes . 80 5.1.5 From local to global classification on geometries. 81 5.2 Phenomenological parametrization . 82 5.3 Electromagnetic waves . 84 5.3.1 Orbiting stars . 84 5.3.2 Infalling matter close to the gravitational radius . 86 5.3.3 Accretion disks around supermassive black holes . 90 5.3.4 UCOs shadows . 98 5.3.5 Bursts . 99 5.4 Gravitational waves . 100 5.4.1 Coalescence of compact objects . 100 5.4.2 Echoes in the late-time ringdown . 102 5.5 Conclusions . 106 General conclusions and outlook 109 A Extracting the Poincar´ealgebra 113 B Brief review of on-shell techniques 117 C Regularity conditions for vanishing radius 123 D Evaporation time 127 Bibliography 135 Publication list This thesis is based on the following list of publications. P1 R. Carballo Rubio, F. Di Filippo, S. Liberati. Minimally modified theories of gravity: a playground for testing the uniqueness of general relativity. JCAP 06 (2018), 026, (2018), arXiv:1802.02537. P2 R. Carballo Rubio, F. Di Filippo, S. Liberati, C. Pacilio, M. Visser. On the viability of regular black holes. JHEP 07 (2018), 023, arXiv:1805.02675. P3 R. Carballo Rubio, F. Di Filippo, S. Liberati, M. Visser. Phenomenological aspects of black holes beyond general relativity. Phys. Rev. D 98 (2018) no.12, 124009, arXiv:1809.08238. P4 R. Carballo Rubio, F. Di Filippo, N. Moynihan. Taming higher-derivative interactions and bootstrapping gravity with soft theorems. JCAP 10 (2019), 030, arXiv:1811.08192. P5 R. Carballo Rubio, F. Di Filippo, S. Liberati, M. Visser. Opening the Pandora's box at the core of black holes. Class. Quant. Grav. 37 (2020) no.14, 145005, arXiv:1908.03261. P6 R. Carballo Rubio, F. Di Filippo, S. Liberati, M. Visser. Geodesically complete black holes. Phys. Rev. D 101 (2020), 084047, arXiv:1911.11200. P7 R. Carballo Rubio, F. Di Filippo, S. Liberati, M. Visser. Causal hierarchy in modified gravity. Preprint, arXiv:2005.08533. vii viii Introduction General relativity represents a very elegant theory from a conceptual point of view and a very successful one from the experimental side. Thanks to the recent observations coming from the LIGO/VIRGO collaboration [1,2,3,4,5,6] as well as from the Event Horizon Telescope [7,8,9, 10, 11, 12], we have the possibility to perform precision test of general relativity in regimes never tested before. Remarkably, the outcomes of all these observations are in perfect agreement with the prediction of general relativity. Regardless of the phenomenological success, there are good reasons to study modi- fications of the theory. From an experimental point of view, cosmological data cannot be explained within general relativity only with the matter and the energy that we observe. The most simple explanation is given by the ΛCDM model where one adds two addiional \dark" cosmological fluids: a pressurless dust and a cosmological con- stant, avoiding any modification of the gravitational interaction. Nonetheless, it is also possible to explain the cosmological data without introducing dark matter and dark energy, but by modifying general relativity. This possibility is reinforced by the fact the ΛCDM is not free from problems. For instance, there is a naturalness problem for the value of the cosmological constant which is very different from the one which can be naively estimated [13] and different measurements of the Hubble constant at different redshifts lead to values in tension with each other [14]. Even stronger reasons to modify general relativity come from theoretical arguments. General relativity predicts the formations of spacetime singularities. In 1939 Oppen- heimer and Snyder studied the gravitational collapse of a spherically symmetric ball of pressureless dust [15], obtaining that the endpoint of the collapse is a singular black hole. However, it was widely believed that the formation of the singularity was simply due to the highly idealized scenario and that a realistic collapse would not end up with the formation of a black hole. In fact, something similar happens in Newtonian gravity, where a spherically symmetric collapse produces a region of infinite density , but an ar- bitrary small amount of angular momentum will prevent the formation of such a region. While preliminary results appeared to confirm this possible resolution of singularities [16], Penrose's 1965 singularity theorem [17] proved them as the natural end points of general relativity. This theorem states that if a trapped region is formed, the spacetime cannot be geodesically complete, implying that there must exist at least one geodesic ending in a Cauchy horizon or in a singularity. If we assume that the spacetime is globally hyperbolic, there cannot be any Cauchy horizon, and thus a singularity has 1 2 to form. Subsequent results extended the theorem to cosmological scenarios [18] and got rid of the global hyperbolicity assumption [19] showing that in general relativity singularities are going to be unavoidably present both as an endpoint in gravitational collapse and in cosmology. A possible approach to accept the presence of singularities in general relativity, is to assume that while the theory will unavoidably produce them, it will also always hide them as well. The precise mathematical formulation of this concept is given by the cosmic censorship conjecture [20].