DeWitt boundary condition is consistent in Hořava-Lifshitz quantum gravity

The concept of spacetime loses its usual interpretation at the essential singularity of a black hole. In consequence, all laws of physics must fail at this classical singularity. This unphysical behavior of spacetime at the singularity originates from general relativity. In order to have a consistent description of spacetime, this singularity must disappear in a quantum mechanical description of spacetime which is expected to be given by a quantum theory of gravity. In this paper, we therefore attempt to describe the quantum nature of spacetime in the vicinity of the (classical) singularity of a black hole. We take the Kantowsi–Sachs representation for the interior spacetime of a black hole and include inevitable vacuum fluctuations of matter field in the Klein–Gordon representation. Hence we obtain the Wheeler–DeWitt equation for the black hole interior and solve this equation exactly yielding a general expression for the interior wave function of the black hole. Admissible wave functions consistent with the DeWitt boundary condition implies that the Hilbert space has three nonoverlapping sectors distinguished by the relative character of the eigenvalues. Regular quantum black holes with admissible and well-behaved wave function having no singularity can exist only in two of those sectors. However, the remaining sector does not contain any regular quantum black hole.

We investigate the quantum evolution of the metric operators for Bianchi-Type I model universes in the Heisenberg picture in order to remove the need to consider the wave function of the universe and interpret its “spin” variables. The calculation is analogous to that of the Zitterbewegung of the Dirac electron. We consider the behavior of the metric near the classical singularity, and consider the curvature there. Although factor ordering questions preclude the presentation of an unambiguous result for the curvature invariants, it does seem that the classical t −4 divergence of the Kretschmann scalar is not removed by quantization.

By using the classically solvable minimal massless scalar minisuperspace model we study two examples of wave functions of the universe in the semiclassical limit. The first one consists of a Lorentzian component and a Euclidean component and admits a clear semiclassical interpretation as a superposition of universes. The second one consists of a Lorentzian (or Euclidean) component and another oscillatory component which corresponds to (neither Lorentzian nor Euclidean) complex classical solutions. This example has some resemblance to Hawking's minimal massive scalar minisuperspace model. We suggest a possible way of recovering the classical interpretation in such a case.

We present a well-tempered DeWitt wave function, which vanishes at the classical big-bang singularity, in Ho\v{r}ava-Lifshitz (HL) cosmology with tensor perturbation, both analytically and numerically. In general relativity, the DeWitt wave function is ill-behaved once the tensor perturbation is taken into account. This is essentially because the amplitude of the perturbation diverges at the singularity and the perturbative expansion completely breaks down. On the other hand, in HL gravity it is known that the higher dimensional operators required by the perturbative renormalizability render the tensor perturbation scale-invariant and regular all the way up to the singularity. In this paper we analytically show that in $d+1$ dimensional HL gravity the DeWitt wave function for tensor perturbation is indeed well-defined around the classical big-bang singularity. Also, we numerically demonstrate the well-behaved DeWitt wave function for tensor perturbation from the big-bang to a finite size of the Universe.

Quantum gravity is expected to be necessary in order to understand situations in which classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e., the fact that the backward evolution of a classical spacetime inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding spacetime is then modified. One particular theory is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. The main effects are introduced into effective classical equations, which allow one to avoid the interpretational problems of quantum theory. They give rise to new kinds of early-universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function, which allows an extension of quantum spacetime beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of spacetime arising in loop quantum gravity and its application to cosmology sheds light on more general issues, such as the nature of time.Electronic Supplementary MaterialSupplementary material is available for this article at 10.12942/lrr-2008-4.

The Everett-type interpretations of quantum mechanics and quantum cosmology proposed independently by Hartle, Geroch, and Wada are discussed. They essentially involve regarding a strong peak in the wave function as a definite prediction. Wave functions in quantum cosmology are usually peaked about correlations between coordinates and momenta, and methods for identifying such correlations are introduced. The first method involves Wigner's function, a quantum-mechanical analogue of the classical phase-space distribution. The properties of this distribution are discussed and it is shown how it can be of use in describing the emergence of classical behavior from quantum systems. The second method involves a suitably chosen canonical transformation. These methods are applied to harmonic-oscillator examples, which are of relevance to scalar field fluctuations in inflationary universe models. These methods are also applied to WKB wave functions in quantum mechanics and quantum cosmology. The manner in which the wave function becomes peaked about sets of classical solutions is elucidated. This is extended to include inhomogeneous perturbations about minisuperspace in quantum cosmology, and the derivation of the semiclassical Einstein equations, ${G}_{\ensuremath{\mu}\ensuremath{\nu}}$=8\ensuremath{\pi}G〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$〉, from the Wheeler-DeWitt equation is considered. A condition under which they are valid is derived. It is essentially the requirement that the distribution of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$, as a function of the matter modes, is strongly peaked about its average value. Some situations in which this condition is satisfied are discussed.

This paper exploits an analogy with a time-independent Schrodinger equation to propose a concrete interpretation of a 'wavefunction of the universe' psi which solves the Wheeler-DeWitt equation for a minisuperspace quantum cosmology, and then uses this interpretation to define information theoretic notions of entropy. Given a wavefunction psi depending, in a configuration space representation, on some set of variables q0, ., qN, one is instructed to pick one of the qA, say q0 identical to a, as a 'time' and then view psi (Q; a) as an unnormalised conditional probability amplitude for finding the values Q identical to (q1, ., qN) at 'time' a. By identifying the 'natural' measure dQ on an a=constant hypersurface in minisuperspace, rho (Q, Q'; a)dQ varies as psi *(Q; a) psi (Q'; a)dQ is interpreted as representing a density matrix in the Q-representation; and, by viewing the Q as abstract operators for fixed a, rho is interpreted as an abstract density matrix. This interpretation admits no a priori notion of unitary evolution with respect to a 'translation' da, unitarity being viewed instead as a derived concept arising only in a semiclassical limit. The object then is to use rho to define reasonable measures of 'coarse-grained entropy' in accord with physical intuition which could serve as a diagnostic for an 'arrow of time'. Thus, e.g., by interpreting the diagonal components rho ( lambda i, lambda i; a) in some Oa-representation as conditional probabilities pi for measuring the eigenvalues lambda i of some operator Oa, one is led to consider an observable dependent entropy S0(a)=- Sigma ipi log pi. And similarly, at least when the a=constant hypersurfaces are flat, one can map psi into a 'Wigner function' fw and then use that fw to define a 'Wigner function entropy' Sw.

We consider a proposal to define the wave function of the Universe as a sum over spacetimes that eventually inflate. In the minisuperspace model, we explicitly show that a simple family of initial conditions, parametrized by a positive real number $a_0$, can be imposed to realise this prescription. The resulting wave function is found to be proportional to the Hartle-Hawking wave function and its dependence on $a_0$ is only through an overall phase factor. Motivated by this observation, we ask whether it is possible to analytically extend $a_0$ to an extended region $\bar{\mathcal{D}}$ in complex $a_0-$plane, while retaining the Hartle-Hawking form of the wave function. We use the condition for convergence of path integral and a recent theorem due to Kontsevich and Segal, further extended by Witten, to explicitly find $\bar{\mathcal{D}}$. Interestingly, a special point on the boundary of $\bar{\mathcal{D}}$ recovers the exact no-boundary wave function. Following that, we show that our prescription leads to a family of quantum states for the perturbations, which give rise to scale-invariant power spectra. If we demand, as an extra ingredient to our prescription, a matching condition at the "no-boundary point" in $\bar{\mathcal{D}}$, we converge on a unique quantum state for the perturbations.

We implement the no-boundary proposal for the wave function of the universe in an exactly solvable Bianchi IX minisuperspace model with two scale factors. We extend our earlier work (Phys. Rev. Lett. 121, 081302, 2018 / arXiv:1804.01102) to include the contribution from the $\mathbb{C}\text{P}^2 \setminus B^4$ topology. The resulting wave function yields normalizable probabilities and thus fits into a predictive framework for semiclassical quantum cosmology. We find that the amplitude is low for large anisotropies. In the isotropic limit the usual Hartle-Hawking wave function for the de Sitter minisuperspace model is recovered. Inhomogeneous perturbations in an extended minisuperspace are shown to be initially in their ground state. We also demonstrate that the precise mathematical implementation of the no-boundary proposal as a functional integral in minisuperspace depends on detailed aspects of the model, including the choice of gauge-fixing. This shows in particular that the choice of contour cannot be fundamental, adding weight to the recent proposal that the semiclassical no-boundary wave function should be defined solely in terms of a collection of saddle points. We adopt this approach in most of this paper. Finally we show that the semiclassical tunneling wave function of the universe is essentially equal to the no-boundary state in this particular minisuperspace model, at least in the subset of the classical domain where the former is known.

The method of path integration is used to study the effects of quantum fluctuations in the space-time geometry near the classical singularity of general relativity. It is shown that in certain special cases explicit Feynman propagators can be constructed which enable us to evaluate these fluctuationsquantitatively. The cases discussed are (i) the gravitational collapse of a uniform dust ball, (ii) the Friedmann cosmologies, (iii) the axisymmetric Bianchi type I cosmological model, and (iv) the general anisotropic Bianchi type I cosmological model. In all cases discussed here the quantum uncertainty grows to infinity as the classical space-time singularity is approached. In this wider regime of quantum gravitation nonsingular solutions can occur with finite probabilities.

We calculate the four point correlation function for scalar perturbations in the canonical model of slow-roll inflation. We work in the leading slow-roll approximation where the calculation can be done in de Sitter space. Our calculation uses techniques drawn from the AdS/CFT correspondence to find the wave function at late times and then calculate the four point function from it. The answer we get agrees with an earlier result in the literature, obtained using different methods. Our analysis reveals a subtlety with regard to the Ward identities for conformal invariance, which arises in de Sitter space and has no analogue in AdS space. This subtlety arises because in de Sitter space the metric at late times is a genuine degree of freedom, and hence to calculate correlation functions from the wave function of the Universe at late times, one must fix gauge completely. The resulting correlators are then invariant under a conformal transformation accompanied by a compensating coordinate transformation which restores the gauge.

Anisotropies in Kaluza-Klein quantum cosmology

A process of idea filtration in two distinct streams of physics i.e., 1) The dimensionality perspective of spacetime, and 2) The quantum perspective leads us to an understanding of what might be a true reality of all that we perceive. The conclusions arrived at in this paper are a bit perplexing in the sense that our perceived reality could be a manifestation of a combination of 4D + n (n > 0) flat space-time, universal wave function, and cognizance. The work is based on a review and analysis of the main concepts in quantum theory, relativistic physics, and cosmology. Key ideas and conclusions are filtered and logically connected to arrive at what might be a view of the true reality. A significant part of the paper is dedicated to the concept of the “observer” and the “ability of cognizance” that should accompany the “observer”. Though the “observer” is central to modern physics, it is not known what constitutes observation, and the term observer, often open to interpretations, does not have a standard definition and hence, is lacking in clarity. In our analysis, we have argued that the environment, in which the observer-observed system is embedded, emerges as an all-knowing, cognizant, and ideal observer that has the knowledge of the observer-observed system. At a philosophical level, we link to the fundamentals of physics, “consciousness” or “ability of cognizance” as an unavoidable and key element in not only carrying out the observation but perhaps, as believed by many, having a role in shaping the reality perceived. In the review and analysis of another stream of physics, that of General Theory of Relativity (GTR) and cosmology, we examine the question of reality from a cosmological and dimensionality perspective. Research on the 4D and 5D constructs of the universe indicates that the “reality” perceived in 4D spacetime as matter, distance, time, etc., is a manifestation of a higher dimensional 4D + n (n > 0) reality. Theoretical research on this front points towards a 4D-spacetime embedded in a 5D or higher dimensional flat space with matter and energy being a manifestation of the higher dimensions. The flow of logic in this paper leans towards a view of an ultimate true reality that is flat 4D + n (n > 0) space combined with cognizance, universal wave function, and the environment.

We study the time evolution of a wave function for the spatially flat Friedmann–Lemaître–Robertson–Walker Universe governed by the Wheeler–DeWitt equation in both analytical and numerical methods. We consider a Brown–Kuchař dust as a matter field in order to introduce a ‘clock’ in quantum cosmology and adopt the Laplace–Beltrami operator-ordering. The Hamiltonian operator admits an infinite number of self-adjoint extensions corresponding to a one-parameter family of boundary conditions at the origin in the minisuperspace. For any value of the extension parameter in the boundary condition, the evolution of a wave function is unitary and the classical initial singularity is avoided and replaced by the big bounce in the quantum system. Exact wave functions show that the expectation value of the spatial volume of the Universe obeys the classical-time evolution in the late time but its variance diverges.

We investigate the black hole information paradox in the setting of pseudo-complex gravity, a covariant geometric extension of general relativity that introduces a minimal length scale by deforming the spacetime manifold. In this framework, curvature invariants stay finite, and the classical singularity is geometrically regularized via a smooth core. We show that the correction term B/(6r 4) alters the Schwarzschild metric, generating the regularized geometry above, yielding a finite Hawking temperature, and inducing subleading corrections to the Bekenstein–Hawking entropy. Crucially, we demonstrate that the pseudo-complex geometric structure obstructs a clean factorization of the Hilbert space into interior and exterior regions, thereby removing the key assumption behind the standard derivation of the paradox. This structural reinterpretation of entanglement flow offers a new geometric route to unitarity preservation and information recovery. We examine the resulting effects on evaporation dynamics, entropy flow, and thermodynamic behavior. Our predictions are compared with those of generalized uncertainty principles (GUP), loop quantum gravity (LQG), and island-based models, and are summarized in a comparative table. Observable signatures—such as shifts in quasi-normal mode frequencies and the appearance of gravitational wave echoes from the regularized core—suggest that pseudo-complex gravity (pc-Gravity) is a testable, covariant approach to resolving the paradox without invoking firewalls, holography, or exotic quantum states.