General Relativity and Quantum Cosmology (gr-qc)

Abstract: Recently there is progress to resolve the issue regarding the non-existence of the Cauchy horizon inside the static, charged, and spherically symmetric black holes. However, when we generically extend the black holes' spacetime, they are not just static but can be dynamical, thus the interior of black holes does not remain the same as the static case when we take into account the dynamical evolution of black holes. Then, our aim in this paper is to provide a few constructive insights and guidelines regarding this issue by revisiting a few examples of the gravitational collapse of spherically symmetric charged black holes using the double-null formalism. Our numerical results demonstrate that the inside of the outer horizon is no longer static even in late time, and the inner apparent horizon exists but is not regular. The inner apparent horizon can be distinguished clearly from the Cauchy horizon. The spherical symmetric property of black holes allows the inner horizon to be defined in two directions, i.e., the differentiation of the areal radius vanishes along either the out-going or the in-going null direction. Moreover, the Cauchy horizon can be generated from a singularity. Still, the notion of the singularity can be subtle where it can have a vanishing or non-vanishing areal radius; the corresponding curvature quantities could be finite or diverge, although the curvatures can be greater than the Planck scale. Finally, we show some examples that the "hair" which is associated with the matter field on the inner horizon is not important to determine the existence of the Cauchy horizon; rather, the hair on the outer horizon might play an important role on the Cauchy horizon. Therefore, the dynamic properties of the interior of charged black holes could shed light for us to understand deeply about the Cauchy horizon for the extensions of no-Cauchy-horizon theorems.

Abstract: Complexifying space time has many interesting applications, from the construction of higher dimensional unification, to provide a useful framework for quantum gravity and to better define some local symmetries that suffer singularities in real space time. In this context here spacetime is extended to complex spacetime and standard general coordinate invariance is also extended to complex holomorphic general coordinate transformations. This is possible by introducing a non Riemannian Measure of integration, which transforms avoiding non holomorphic behavior . Instead the measure transforms according to the inverse of the jacobian of the coordinate transformation and avoids the traditional square root of the determinant of the metric $\sqrt{-g}$. which is not globally holomorphic , or the determinant of the vierbein which is sensitive to the vierbein orientations and not invariant under local lorentz transformations with negative determinants. It is impossible to introduce a cosmological constant term in the complex holomorphic invariant theory action, but a cosmological term appears as an integration constant in the equations of motion. The ideas can be generalized to theories of extended objects.

Abstract: The decrease of both the rolling speed of the inflaton and the sound speed of the curvature perturbations can amplify the curvature perturbations during inflation so as to generate a sizable amount of primordial black holes. In the ultraslow-roll inflation scenario, it has been found that the power spectrum of curvature perturbations has a $k^4$ growth. In this paper, we find that when the speed of sound decreases suddenly, the curvature perturbations becomes scale dependent in the infrared limit and the power spectrum of the curvature perturbation only has a $k^2$ growth. Furthermore, by studying the evolution of the power spectrum in the inflation model, in which both the sound speed of the curvature perturbations and the rolling speed of the inflaton are reduced, we find that the power spectrum is nearly scale invariant at the large scales to satisfy the constraint from the cosmic microwave background radiation observations, and at the same time can be enhanced at the small scales to result in an abundant formation of primordial black holes. In the cases of the simultaneous changes of the sound speed and the slow-roll parameter $\eta$ and the change of the sound speed preceding that of the slow-roll parameter $\eta$, the power spectrum can possess a $k^6$ growth under certain conditions, which is the steepest growth of the power spectrum reported so far.

Abstract: We demonstrate that Schwarzschild spacetime has a conformal extension and that, beyond null infinity, there is a black hole with a timelike singularity. In conformal extended spacetime, every null infinity is a killing horizon with vanishing surface gravity. When a quantized massless scalar field is taken into this spacetime and different vacuums for the field are defined, thermal radiation coming from the extended black hole could be observed. This makes sense, much like the thermal radiation coming from the white hole. The Unruh effect is therefore plausible in conformal extended Schwarzschild spacetime. It is shown that the thermal radiation coming from past null infinity in Schwarzschild spacetime, which is difficult to imagine as the result of any physical process, is the result of the reduction of the thermal radiation passing through past null infinity in conformal extended spacetime. We also present a conformal extension of Kerr spacetime for the first time. Then, by examining a quantized massless scalar field on this spacetime, we get the meaningful conclusion that there is thermal radiation coming from a different rotating black hole passing through past null infinity. Similarly, the result of the Kerr black hole is consistent with that of the Schwarzschild black hole.

Abstract: We study the inflationary model constructed from a Spatially Covariant Gravity (SCG). The Lagrangian for the SCG in our consideration is expressed as the polynomial of irreducible SCG monomials where the total number of derivatives of each monomial is two, and the theory propagates two tensorial degrees of freedom of gravity up to the first order in cosmological perturbations. The condition for having two tensorial degrees of freedom studied earlier in literature for such theories is derived in vacuum. We extend the condition for having two tensorial degrees of freedom to the case where a scalar field is included by imposing a gauge-fixing. We apply the resulting SCG to describe inflationary universe. The observational predictions such as the scalar spectral index and tensor-to-scalar ratio from this model are investigated. We find that the tensor-to-scalar ratio in this model can either be in the order of unity or be small depending on the parameter of the model.

Abstract: The Weyl geometry promises potential applications in gravity and quantum mechanics. We study the relationships between the Weyl geometry, quantum entropy and quantum entanglement based on the Weyl geometry endowing the Euclidean metric. We give the formulation of the Weyl Ricci curvature and Weyl scalar curvature in the $n$-dimensional system. The Weyl scalar field plays a bridge role to connect the Weyl scalar curvature, quantum potential and quantum entanglement. We also give the Einstein-Weyl tensor and the generalized field equation in 3D vacuum case, which reveals the relationship between Weyl geometry and quantum potential. Particularly, we find that the correspondence between the Weyl scalar curvature and quantum potential is dimension-dependent and works only for the 3D space, which reveals a clue to quantize gravity and a understanding why our space must be 3D if quantum gravity is compatible with quantum mechanics. We analyze numerically a typical example of two orthogonal oscillators to reveal the relationships between the Weyl scalar curvature, quantum potential and quantum entanglement based on this formulation. We find that the Weyl scalar curvature shows a negative dip peak for separate state but becomes a positive peak for the entangled state near original point region, which can be regarded as a geometric signal to detect quantum entanglement.

Abstract: As shown by Marunovic and Murkovic, non-minimal d-stars, composite structures consisting of a boson star and a global monopole non-minimally coupled to the general relativistic field, can have extremely high gravitational compactness. In a previous paper we demonstrated that these ground-state stationary solutions are sometimes additionally characterized by shells of bosonic matter located far from the center of symmetry. In order to investigate the question of stability posed by Marunovic and Murkovic, we investigate the stability of several families of d-stars using both numerical simulations and linear perturbation theory. For all families investigated, we find that the most highly compact solutions, along with those solutions exhibiting shells of bosonic matter, are unstable to radial perturbations and are therefore poor candidates for astrophysically-relevant black hole mimickers or other highly compact stable objects.

Abstract: These notes are self-contained, with the first six chapters used for a one-semester course with recommended texts by Wald, Misner, Thorne, and Wheeler (MTW), and, particularly for gravitational waves, by Schutz and by Thorne and Blandford. In its treatment of topics covered in these standard texts, the presentation here typically includes steps between equations that are skipped in Wald or MTW. Treatments of gravitational waves, particle orbits in black-hole backgrounds, the Teukolsky equation, and the initial value equations are motivated in part by the dramatic discoveries of gravitational waves from the inspiral and coalescence of binary black holes and neutron stars, advances in numerical relativity, and the expected launch of the LISA space-based observatory. Students are assumed to have encountered special relativity, but these notes give a detailed presentation with a geometrical orientation, starting with with time dilation and length contraction and including relativistic particles, fluids, electromagnetism, and curvilinear coordinates. Chaps. 2-5 cover curvature, the Einstein equation, relativistic stars, and black holes. Chap. 6, on gravitational waves, includes a discussion of detection and of noise in interferometric detectors. Chap. 7, on the initial value problem, has a section on the form of the equations used in numerical relativity. Its notation is that used, for example, in Baumgarte and Shapiro and Shibata; the presentation here is taken in part from the text by Friedman and Stergioulas. The notes also have a chapter on the Newman-Penrose formalism and the Teukolsky equation. Following that is a chapter on black-hole thermodynamics and a final chapter on the gravitational action and on conserved quantities for asymptotically flat spacetimes, using Noether's theorem.