Lapse Function Research Articles

A pilot study on canonical gravity with mass dimension one fermions

The mass dimension one (MDO) fermionic field is built on a complete set of dual-helicity eigenspinors of the charge conjugation operator, which obeys the statistic of Fermi-Dirac. These spinors are a potential candidates for the description of dark matter. The redefinition of the dual structure of this object holds a local adjacent theory and completely satisfies the Lorentz invariance. In this work, we investigate important aspects of the interaction of this fermion with gravity in the light of a canonical formulation in ADM formalism. We construct an action via tetrad fields using a manifold on a family of spacelike surfaces Σt that carries the MDO matter field, and additionally we propose a condition for the action to have a term associated with dark energy. We found the Hamiltonian and diffeomorphism constraints at the classical level for gravitational dynamics with the immersion of this material content in space-time, which leads us to the interpretation of the contribution of dark matter energy density, parallel to the lapse function of foliation and its directional flux of energy density in the hypersurface of manifold.

DC conductance and memory in 3D gravity

Transport properties are investigated in the two-dimensional dual dynamics of AdS3 gravity. By providing boundary conditions that deform the ADM lapse and shift functions, we construct a lower dimensional model comprising two copies of chiral boson excitations with anisotropic scaling symmetry. Using bosonization, an electric current is identified. By means of the Kubo formula, we find a DC conductance depending on the level of the theory and the dynamical exponents. The bulk realization of the linear response is related to a type of gravitational memory emerging in the context of near-horizon boundary conditions. The process is adiabatic and represents a permanent spacetime deformation parametrized by anisotropic chiral bosons through a large gauge transformation.

Cosmological perturbations in the spatially covariant gravity with a dynamical lapse function

We investigate the scalar perturbations in a class of spatially covariant gravity theory with a dynamical lapse function. Generally, there are two scalar degrees of freedom due to the presence of the velocity of the lapse function. We treat the scalar perturbations as analogues of those in a two-field inflationary mode, in which one is light mode and the other is the heavy mode. This is justified by the fact that the scalar mode due to the dynamical lapse function becomes infinitely heavy in the limit when the lapse function reduces to be an auxiliary variable. The standard approaches of multiple filed perturbations can be applied to deal with our model. By integrating out the heavy mode and derive the effective theory for the single light field, we find the solution to the single mode in the form of plane waves. Then we calculate the corrections to the power spectrum of the light mode from the heavy mode, by making use of the standard perturbative method of field theory. At last, when the two fields are not weakly coupled, we find a power law mode for the coupled system in large scales.

New first-order formulation of the Einstein equations exploiting analogies with electrodynamics

The Einstein and Maxwell equations are both systems of hyperbolic equations which need to satisfy a set of elliptic constraints throughout evolution. However, while electrodynamics (EM) and magnetohydrodynamics (MHD) have benefited from a large number of evolution schemes that are able to enforce these constraints and are easily applicable to curvilinear coordinates, unstructured meshes, or N-body simulations, many of these techniques cannot be straightforwardly applied to existing formulations of the Einstein equations. We develop a 3+1 a formulation of the Einstein equations which shows a striking resemblance to the equations of relativistic MHD and to EM in material media. The fundamental variables of this formulation are the frame fields, their exterior derivatives, and the Nester-Witten and Sparling forms. These mirror the roles of the electromagnetic 4-potential, the electromagnetic field strengths, the field excitations and the electric current. The role of the lapse function and shift vector, corresponds exactly to that of the scalar electric potential. The formulation, that we name dGREM (for differential forms, General Relativity and Electro-Magnetism), is manifestly first order and flux-conservative, which makes it suitable for high-resolution shock capturing schemes and finite-element methods. Being derived as a system of equations in exterior derivatives, it is directly applicable to any coordinate system and to unstructured meshes, and leads to a natural discretization potentially suitable for the use of machine-precision constraint propagation techniques such as the Yee algorithm and constrained transport. Due to these properties, we expect this new formulation to be beneficial in simulations of many astrophysical systems, such as binary compact objects and core-collapse supernovae as well as cosmological simulations of the early universe.

On the construction of Riemannian three-spaces with smooth inverse mean curvature foliation

Consider a one-parameter family of smooth Riemannian metrics on a two-sphere, {mathscr {S}}. By choosing a one-parameter family of smooth lapse and shift, these Riemannian two-spheres can always be assembled into smooth Riemannian three-space, with metric h_{ij} on a three-manifold Sigma foliated by a one-parameter family of two-spheres {mathscr {S}}_rho . It is shown first that we can always choose the shift such that the {mathscr {S}}_rho surfaces form a smooth inverse mean curvature foliation of Sigma . An integrodifferential expression, referring only to the area of the level sets and the lapse function, is also derived that can be used to quantify the Geroch mass. If the constructed Riemannian three-space happens to be asymptotically flat and the rho -integral of the integrodifferential expression is non-negative, then not only the positive mass theorem but, if one of the {mathscr {S}}_{rho } level sets is a minimal surface, the Penrose inequality also holds. Notably, neither of the above results requires the scalar curvature of the constructed three-metric to be non-negative.

Time orientability and particle production from universal horizons

We discuss particle production in spacetimes endowed with a universal horizon in Einstein--Aether and Horava gravity. We argue that continuity and differentiability of the lapse function require the orientation of the foliation in the interior of the horizon to be reversed with respect to the exterior one. Unless this is allowed, interaction of gravitating scalar fields with the universal horizon leads to unitarity violations in the quantum theory. This property is responsible for particle production by the universal horizon, as we show by computing explicitly its Hawking temperature for all stationary and spherically symmetric spacetimes. We particularize our result to known analytic solutions, including those compatible with observational constraints.

Lorentzian path integral for quantum tunneling and WKB approximation for wave-function

Recently, the Lorentzian path integral formulation using the Picard–Lefschetz theory has attracted much attention in quantum cosmology. In this paper, we analyze the tunneling amplitude in quantum mechanics by using the Lorentzian Picard–Lefschetz formulation and compare it with the WKB analysis of the conventional Schrödinger equation. We show that the Picard–Lefschetz Lorentzian formulation is consistent with the WKB approximation for wave-function and the Euclidean path integral formulation utilizing the solutions of the Euclidean constraint equation. We also consider some problems of this Lorentzian Picard–Lefschetz formulation and discuss a simpler semiclassical approximation of the Lorentzian path integral without integrating the lapse function.

Consistent gauge-fixing conditions in polymerized gravitational systems

For classical gravitational systems the lapse function and the shift vector are usually determined by imposing appropriate gauge fixing conditions and then demanding their preservation with respect to the dynamics generated by a canonical Hamiltonian. Effective descriptions encoding quantum geometric effects motivated by loop quantum gravity for symmetry reduced models are often captured by polymerization of connection (or related) variables in gauge fixing conditions as well as constraints. Usually, one chooses the same form of polymerization in both cases. A pertinent question is if the dynamical stability of the effective gauge fixing conditions under the effective dynamics generated by the polymerized canonical Hamiltonian is provided by the lapse function and the shift vector obtained from the polymerization of their classical counterparts. If this is the case, then we say that gauge fixing and polymerization commute. In this manuscript we investigate these issues and obtain consistency conditions for the commutativity of gauge fixing and polymerization. Our analysis shows that such a commutativity occurs in rather special situations and reveals pitfalls in making seemingly well motivated choices which turn out to be inconsistent with the effective dynamics. We illustrate these findings via examples of symmetry reduced models in the loop quantization of the Schwarzschild interior and Lema\^{\i}tre-Tolman-Bondi (LTB) spacetimes and report the non-commutativity of gauge fixing and polymerization, and inherent limitations of some choices made in the literature with a consistent effective dynamics.

Bona-Masso slicing conditions and the lapse close to black-hole punctures

We consider several families of functions $f(\ensuremath{\alpha})$ that appear in the Bona-Masso slicing condition for the lapse function $\ensuremath{\alpha}$. Focusing on spherically symmetric and time-independent slices we apply these conditions to the Schwarzschild spacetime in order to construct analytical expressions for the lapse $\ensuremath{\alpha}$ in terms of the areal radius $R$. We then transform to isotropic coordinates and determine the dependence of $\ensuremath{\alpha}$ on the isotropic radius $r$ in the vicinity of the black-hole puncture. We propose generalizations of previously considered functions $f(\ensuremath{\alpha})$ for which, to leading order, the lapse is proportional to $r$ rather than a noninteger power of $r$. We also perform dynamical simulations in spherical symmetry and demonstrate advantages of the above choices in numerical simulations employing spectral methods.

Scalar-tensor cosmologies in a minisuperspace formulation: A case study

We revisit a general minisuperspace (MSS) formalism for scalar tensor Friedmann-Lema\^{\i}tre-Robertson-Walker type cosmological models in an arbitrary frame with a perfect fluid source. We discuss how to impose Cauchy data on the corresponding dynamical system in order to reconstruct a standard ($\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$) cosmological model. So far, the integrability of such models has been extensively studied in the Einstein or Jordan frames mainly. We extend these studies to an arbitrary frame taking into account nonminimal coupling between matter and gravity. To this aim we explore a gauge freedom associated with a choice of lapse function. We show that particular isothermal MSS coordinates are related with Einstein frames representing solution equivalent classes and have some invariant meaning. This provides a new universal framework for investigating cosmic evolution in an arbitrary frame: analytical solutions obtained in the Einstein frame can be transformed back to the physical frame by making use of a conformal transformation and field redefinition. We also show how such a technique can be useful when applied to Wheeler-DeWitt quantum cosmology.

The Friedmann–Lemaître–Robertson–Walker metric

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric used to describe the cosmic spacetime is based on the cosmological principle, which assumes homogeneity and isotropy throughout the Universe. It also adopts free-fall conditions via the selection of a constant lapse function, [Formula: see text], regardless of whether or not the chosen energy–momentum tensor [Formula: see text] produces an accelerated expansion. This is sometimes justified by arguing that one may shift the gauge, if necessary, transforming the time [Formula: see text] to a new coordinate [Formula: see text], thereby re-establishing a unitary value for [Formula: see text]. Previously, we have demonstrated that this approach is inconsistent with the Friedmann equations derived using comoving coordinates. In this paper, we advance this discussion significantly by using the Local Flatness Theorem in general relativity to prove that [Formula: see text] in FLRW is inextricably dependent on the expansion dynamics via the expansion factor [Formula: see text], which itself depends on the equation-of-state in [Formula: see text]. One is therefore not free to choose [Formula: see text] arbitrarily without ensuring its consistency with the energy–momentum tensor. We prove that the use of FLRW in cosmology is valid only for zero active mass, i.e. [Formula: see text], where [Formula: see text] and [Formula: see text] are, respectively, the total energy density and pressure in the cosmic fluid.

Relativistic extended uncertainty principle from spacetime curvature

The investigations presented in this study are directed at relativistic modifications of the uncertainty relation derived from the curvature of the background spacetime. These findings generalize previous work that is recovered in the nonrelativistic limit. Applying the $3+1$ splitting in accordance with the ADM formalism, we find the relativistic physical momentum operator and compute its standard deviation for wave functions confined to a geodesic ball on a spacelike hypersurface. Its radius can then be understood as a measure of position uncertainty. Under the assumption of small position uncertainties in comparison to background curvature length scales, we obtain the corresponding corrections to the uncertainty relation in flat space. Those depend on the Ricci scalar of the effective spatial metric, the particle is moving on, and, if there are nonvanishing time-space components of the spacetime metric, there are gradients of the shift vector and the lapse function. Interestingly, this result is applicable not only to massive but also to massless particles. Over all, this is not a covariant, yet a consistently general relativistic approach. We further speculate on a possible covariant extension.

Asymptotic symmetries in Carrollian theories of gravity

Asymptotic symmetries in Carrollian gravitational theories in 3+1 space and time dimensions obtained from “magnetic” and “electric” ultrarelativistic contractions of General Relativity are analyzed. In both cases, parity conditions are needed to guarantee a finite symplectic term, in analogy with Einstein gravity. For the magnetic contraction, when Regge-Teitelboim parity conditions are imposed, the asymptotic symmetries are described by the Carroll group. With Henneaux-Troessaert parity conditions, the asymptotic symmetry algebra corresponds to a BMS-like extension of the Carroll algebra. For the electric contraction, because the lapse function does not appear in the boundary term needed to ensure a well-defined action principle, the asymptotic symmetry algebra is truncated, for Regge-Teitelboim parity conditions, to the semidirect sum of spatial rotations and spatial translations. Similarly, with Henneaux-Troessaert parity conditions, the asymptotic symmetries are given by the semidirect sum of spatial rotations and an infinite number of parity odd supertranslations. Thus, from the point of view of the asymptotic symmetries, the magnetic contraction can be seen as a smooth limit of General Relativity, in contrast to its electric counterpart.

Wave Functional of the Universe and Time

A version of the quantum theory of gravity based on the concept of the wave functional of the universe is proposed. To determine the physical wave functional, the quantum principle of least action is formulated as a secular equation for the corresponding action operator. Its solution, the wave functional, is an invariant of general covariant transformations of spacetime. In the new formulation, the history of the evolution of the universe is described in terms of coordinate time together with arbitrary lapse and shift functions, which makes this description close to the formulation of the principle of general covariance in the classical theory of Einstein’s gravity. In the new formulation of quantum theory, an invariant parameter of the evolutionary time of the universe is defined, which is a generalization of the classical geodesic time measured by a standard clock along time-like geodesics.

Wavefunction of the universe: Reparametrization invariance and field redefinitions of the minisuperspace path integral

We consider the Hartle–Hawking wavefunction of the universe defined as a Euclidean path integral that satisfies the “no-boundary proposal.” We focus on the simplest minisuperspace model that comprises a single scale factor degree of freedom and a positive cosmological constant. The model can be seen as a non-linear σ-model with a line-segment base. We reduce the path integral over the lapse function to an integral over the proper length of the base and use diffeomorphism-invariant measures for the ghosts and the scale factor. As a result, the gauge-fixed path integral is independent of the gauge. However, we point out that all field redefinitions of the scale factor degree of freedom yield different choices of gauge-invariant path-integral measures. For each prescription, we compute the wavefunction at the semi-classical level and find a different result. We resolve in each case the ambiguity in the form of the Wheeler–DeWitt equation at this level of approximation. By imposing that the Hamiltonians associated with these possibly distinct quantum theories are Hermitian, we determine the inner products of the corresponding Hilbert spaces and find that they lead to a universal norm, at least semi-classically. Quantum predictions are thus independent of the prescription at this level of approximation. Finally, all wavefunctions of the Hilbert spaces of the minisuperspace model we consider turn out to be non-normalizable, including the no-boundary states.

Quasinormal modes of scalar field coupled to Einstein's tensor in the non-commutative geometry inspired black hole

We investigate the quasinormal modes (QNMs) of the scalar field coupled to the Einstein's tensor in the non-commutative geometry inspired black hole spacetime. It is found that the lapse function of the non-commutative black hole metric can be represented by a Kummer's confluent hypergeometric function, which can effectively solve the problem that the numerical results of the QNMs are sensitive to the model parameters and make the QNMs values more reliable. We make a careful analysis of the scalar QNM frequencies by using several numerical methods, and find that the numerical results obtained by the new WKB method (the Padé approximants) and the Mashhoon method (Pöschl-Teller potential method) are quite different from those obtained by the asymptotic iterative method (AIM) and time-domain integration method when the non-commutative parameter θ and coupling parameter η are large. The most obvious difference is that the numerical results obtained by the AIM and the time-domain integration method appear a critical value ηc with an increase of η, which leads to the dynamical instability. After carefully analyzing the numeral results, we conclude that the numerical results obtained by the AIM and the time-domain integration method are closer to the theoretical values than those obtained by the WKB method and the Mashhoon method, when the θ and η are large. Moreover, through a numerical fitting, we obtain that the functional relationship between the threshold ηc and the non-commutative parameter θ satisfies ηc=aθb+c for a fixed l approximately. We find that the stability of dynamics can be ensured in the η<ηc(θ,l) region.

On the thermodynamics of the Hayward black hole

In light of the growing interest in the Hayward black hole solution, a detailed study on the corresponding lapse function and its roots is presented. The lapse function is expressed in terms of the classical Schwarzschild radius r s and the Hayward’s parameter l. Both of these quantities are used as thermodynamic variables to find related thermodynamic quantities. In this context, the variable l is associated with a canonical conjugate variable , and a free energy Ξ. Moreover, a second order phase transition is found to appears at l ≈ 0.333r s.

Spatially covariant gravity with a dynamic lapse function

In the framework of spatially covariant gravity, it is natural to extend a gravitational theory by putting the lapse function $N$ and the spatial metric $h_{ij}$ on an equal footing. We find two sufficient and necessary conditions for ensuring two physical degrees of freedom (DoF) for the theory with the lapse function being dynamical by Hamiltonian analysis. A class of quadratic actions with only two DoF is constructed. In the case that the coupling functions depend on $N$ only, we find that the spatial curvature term cannot enter the Lagrangian and thus this theory possesses no wave solution and cannot recover general relativity (GR). In the case that the coupling functions depend on the spatial derivatives of $N$, we perform a spatially conformal transformation on a class of quadratic actions with nondynamical lapse function to obtain a class of quadratic actions with $\dot{N}$. We confirm this theory has two DoF by checking the two sufficient and necessary conditions. Besides, we find that a class of quadratic actions with two DoF can be transformed from GR by disformal transformation.

Local Flexibility for Open Partial Differential Relations

AbstractWe show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K ∈ ℝ every manifold of dimension at least 2 carries a complete C1, 1‐metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C2‐metrics in general. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Higher derivative scalar-tensor theory from the spatially covariant gravity: a linear algebraic analysis

We investigate the ghostfree scalar-tensor theory with a timelike scalar field, with the derivatives of the scalar field up to the third order and with the Riemann curvature tensor up to the quadratic order. There are two kinds of linear space that are isomorphic to each other in the sense of gauge fixing/recovering procedures. One is the set of linearly independent generally covariant scalar-tensor monomials, the other is the set of linearly independent spatially covariant gravity monomials. We concentrate on the subspaces of the spatially covariant gravity, which are spanned by the linearly independent monomials built of the extrinsic curvature, the intrinsic curvature, the lapse function as well as their spatial derivatives. The vectors in these subspaces, i.e., spatially covariant gravity polynomials, automatically propagate at most three degrees of freedom. As a result, their images under the gauge recovering mappings are automatically the subspaces of the generally covariant scalar-tensor theory that propagate up to three degrees of freedom as long as the scalar field is timelike, although the higher derivatives of the scalar field are present generally. We also derive the explicit expressions for the projection matrices, which encode the mappings from the subspaces of the spatially covariant gravity to the subspaces of the generally covariant scalar-tensor theory, up to the fourth order in the total number of derivatives in the spatially covariant gravity. Our formalism and results will be useful in deriving the generally covariant higher derivative scalar-tensor theory without ghost(s).