Ernst Equation Research Articles

Plebanski-Demianski goes NUTs (to remove the Misner string)

We present a general procedure, based on the Ehlers transformation of the Ernst equations, to add the gravitomagnetic mass to the whole Plebanski-Demianski family of solutions. We can efficiently generate a large class of accelerating black holes, such as Reissner-Nordstrom or Kerr-Newman, endowed with the NUT parameter. The full rotating version carries a couple of independent NUT charges, one associated to the black hole and the other to the accelerating Rindler background.The two NUT parameters can be coupled to remove the axial irregularity which causes the Misner string, still remaining with a Lorentzian spacetime, without the need to impose periodic time.All the metrics we build are not of D-type according to the Petrov classification, but type-I, so they belong to a more general category with respect to C-metrics and the Plebanski-Demianski seed.A convenient form of the most general type D black hole solution in general relativity, coupled with Maxwell electromagnetism, is obtained when switching off one of the two NUT parameters.

Non-BPS bubbling geometries in AdS3

We construct large classes of non-BPS smooth horizonless geometries that are asymptotic to AdS3 × S3 × T4 in type IIB supergravity. These geometries are supported by electromagnetic flux corresponding to D1-D5 charges. We show that Einstein equations for systems with eight commuting Killing vectors decompose into a set of Ernst equations, thereby admitting an integrable structure. This feature, which can a priori be applied to other {textrm{AdS}}_Dtimes mathcal{C} settings in supergravity, allows us to use solution-generating techniques associated with the Ernst formalism. We explicitly derive solutions by applying the charged Weyl formalism that we have previously developed. These are sourced internally by a chain of bolts that correspond to regions where the orbits of the commuting Killing vectors collapse smoothly. We show that these geometries can be interpreted as non-BPS T4 and S3 deformations on global AdS3 × S3 × T4 that are located at the center of AdS3. These non-BPS deformations can be made arbitrarily small and should therefore correspond to non-supersymmetric operators in the D1-D5 CFT. Finally, we also construct interesting bound states of non-extremal BTZ black holes connected by regular bolts.

Black holes in a swirling universe

We present a new solution in Einstein's General Relativity representing a Schwarzschild black hole immersed in a rotating universe. Such a solution is constructed analytically by means of the last unexplored Lie point symmetry of the Ernst equations for stationary and axisymmetric spacetimes. This kind of the Ehlers transformation is able to embed any given solution into a rotating background, which is not of NUT type. We analyse the physical properties, ergoregions and geodesics of the new metric, which is regular outside the event horizon and has a well defined thermodynamics. We finally consider the Kerr generalisation.

Stationary generalizations for the Bronnikov-Ellis wormhole and for the vacuum ring wormhole

We analyze possibilities to obtain a globally regular stationary generalization for the ultrastatic wormhole with a repulsive scalar field found by Bronnikov and by Ellis in 1973. The extreme simplicity of this static solution suggests that its stationary version could be obtainable analytically and should be globally regular, but no such generalization has been found. We analyze the problem and find that the difficulty originates in the vacuum theory, since the scalar field can be eliminated within the Eris-Gurses procedure. The problem then reduces to constructing the spinning generalization for the vacuum wormhole sourced by a thin ring of negative tension. Solving the vacuum Ernst equations determines the $g_{00}$, $g_{0\varphi}$ metric components and hence the AMD mass ${\rm M}$ and angular momentum ${\rm J}$, all of these being specified by the ring source. The scalar field can be included into consideration afterwords, but this only affects $g_{rr}$ and $g_{\vartheta\vartheta}$ without changing the rest. Within this approach, we analyze a number of exact stationary generalizations for the wormhole, but none of them are satisfactory. However, the perturbative expansion around the static vacuum background contains only bounded functions and presumably converges to an exact solution. Including the scalar field screens the singularity at the ring source and renders the geometry regular. This solution describes a globally regular spinning wormhole with two asymptotically flat regions. Even though the source itself is screened and not visible, the memory of it remains in $g_{00}$ and $g_{0\varphi}$ and accounts for the ${\rm M}\propto {\rm J}^2$ relation typical for a rotating extended source. Describing stationary spacetimes with an extended source is a complicated problem, which presumably explains the difficulty in finding the solution.

Calculation of multipole moments of axistationary electrovacuum spacetimes

The multipole moments of stationary axially symmetric vacuum or electrovacuum spacetimes can be expressed in terms of the power series expansion coefficients of the Ernst potential on the axis. In this paper we present a simpler, more efficient calculation of the multipole moments, applying methods introduced by B\"ackdahl and Herberthson. For the nonvacuum electromagnetic case, our results for the octupole and higher moments differ from the results already published in the literature. The reason for this difference is that we correct an earlier unnoticed mistake in the power series solution of the Ernst equations. We also apply the presented method to directly calculate the multipole moments of a five-parameter charged magnetized generalization of the Kerr and Tomimatsu-Sato exact solutions.

The hyperbolic Ernst equation in a triangular domain

The collision of two plane gravitational waves in Einstein’s theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann–Hilbert problem. The formulation of the Riemann–Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle’s corners—this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann–Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary.

The hyperbolic Ernst–Maxwell equations in a triangular domain

In a recent paper, we applied Riemann–Hilbert techniques to analyze the Goursat problem for the hyperbolic Ernst equation, which describes the interaction of two colliding gravitational plane waves. Here we generalize this approach to colliding electromagnetic plane waves in Einstein–Maxwell theory.

Ernst Potential of Near-Horizon Extremal Kerr Black Holes

One way to find the solution of black holes is through the Ernst equations that is quite simple instead of solving the Einstein equation. Solution of Ernst equations for Kerr and Kerr-Newman black holes have been achieved in the last century. The magnetized case for those black holes and their Ernst potentials can be found using Harrison transformation. Herein the Ernst potential for extremal rotating Kerr and its magnetized solution is shown. In the end, we also extend this fashion for extremal Kerr-Newman black hole.

Scale-dependent parameterization of groundwater–surface water interactions in a regional hydrogeological model

In regional hydrogeological models groundwater–surface water interaction is generally represented with a Cauchy boundary condition, in which a conductance parameter governs the exchange flux rate. In some models, the conductance is controlled by the streambed properties, since it has generally a lower hydraulic conductivity than the aquifer. However, depending on the specific system and the spatial discretization of the hydrogeological model, aquifer conductance can be a limiting factor for groundwater–surface water interactions. The present study introduces a new expression to represent the aquifer conductance as a function of aquifer properties, surface water network density and model discretization. This expression is based on the Dupuit-Forcheimer theory, the Ernst equation and vertical 2D numerical experiments at the field scale. The main assumptions used to derive our formulation are the presence of a no-flow boundary at the bottom of the hydrogeological model and the homogeneity of the aquifer. The expression is evaluated using simulations with 3D hydrogeological models at different spatial resolutions and compared against previously published parameterization approaches. The results show that the new expression outperforms the other approaches by capturing accurately both the grid-size and the surface water network density dependency of the conductance, which is caused by pressure head losses due to flow within the aquifer grid cell to the surface water, without any additional numerical calculation. Moreover, the proposed expression can be implemented directly in hydrogeological models thereby improving current approaches to represent groundwater–surface water interactions in regional hydrogeological models.

Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations

For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a Dirichlet boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen–Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a Neumann boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann–Hilbert problem on a genus one Riemann surface. By solving this Riemann–Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric.

T1 mapping with the variable flip angle technique: A simple correction for insufficient spoiling of transverse magnetization.

The variable flip angle method derives T1 maps from radiofrequency-spoiled gradient-echo data sets, acquired with different flip angles α. Because the method assumes validity of the Ernst equation, insufficient spoiling of transverse magnetization yields errors in T1 estimation, depending on the chosen radiofrequency-spoiling phase increment (Δϕ). This paper presents a versatile correction method that uses modified flip angles α' to restore the validity of the Ernst equation. Spoiled gradient-echo signals were simulated for three commonly used phase increments Δϕ (50°/117°/150°), different values of α, repetition time (TR), T1 , and a T2 of 85 ms. For each parameter combination, α' (for which the Ernst equation yielded the same signal) and a correction factor CΔϕ (α, TR, T1 ) = α'/α were determined. CΔϕ was found to be independent of T1 and fitted as polynomial CΔϕ (α, TR), allowing to calculate α' for any protocol using this Δϕ. The accuracy of the correction method for T2 values deviating from 85 ms was also determined. The method was tested in vitro and in vivo for variable flip angle scans with different acquisition parameters. The technique considerably improved the accuracy of variable flip angle-based T1 maps in vitro and in vivo. The proposed method allows for a simple correction of insufficient spoiling in gradient-echo data. The required polynomial parameters are supplied for three common Δϕ. Magn Reson Med 79:3082-3092, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Integrable and non-integrable structures in Einstein-Maxwell equations with Abelian isometry group G 2

The classes of electrovacuum Einstein - Maxwell fields (with a cosmological constant), which metrics admit an Abelian two-dimensional isometry group $\mathcal{G}_2$ with non-null orbits and electromagnetic fields possess the same symmetry, are considered. For these fields we describe the structures of so called "nondynamical degrees of freedom" which presence as well as the presence of a cosmological constant change (in a strikingly similar ways) the dynamical equations and destroy their known integrable structures. Modifications of the known reduced forms of Einstein - Maxwell equations -- the Ernst equations and self-dual Kinnersley equations in the presence of non-dynamical degrees of freedom are found and subclasses of fields with non-dynamical degrees of freedom are considered for : (I) vacuum metrics with cosmological constant, (II) vacuum space-times with isometry groups $\mathcal{G}_2$ which orbits do not admit the orthogonal 2-surfaces (none-orthogonally-transitive isometry groups) and (III) electrovacuum fields with more general structures of electromagnetic fields than in the known integrable cases. For these classes, in the case of diagonal metrics, the field equations can be reduced to the only nonlinear equation of the fourth order for one real function $\alpha(x^1,x^2)$ which characterises the element of area on the orbits of the isometry group $\mathcal{G}_2$ . Simple examples of solutions for each of these classes are presented. It is pointed out that if for some two-dimensional reduction of Einstein's field equations in four or higher dimensions, the function $\alpha(x^1,x^2)$ possess a "harmonic" structure, instead of being (together with other field variables) a solution of some nonlinear equations, this can be an indication of possible complete integrability of these reduced dynamical equations for the fields with vanishing of all non-dynamical degrees of freedom.

Ernst formulation of axisymmetric fields inf(R)gravity: Applications to neutron stars and gravitational waves

The Ernst formulation of the Einstein equations is generalised to accommodate $f(R)$ theories of gravity. It is shown that, as in general relativity, the axisymmetric $f(R)$ field equations for a vacuum spacetime that is either stationary or cylindrically symmetric reduce to a single, non-linear differential equation for a complex-valued scalar function. As a worked example, we apply the generalised Ernst equations to derive a $f(R)$ generalisation of the Zipoy-Voorhees metric, which may be used to describe the gravitational field outside of an ellipsoidal neutron star. We also apply the theory to investigate the phase speed of large-amplitude gravitational waves in $f(R)$ gravity in the context of soliton-like solutions that display shock-wave behaviour across the causal boundary.

Numerical study of the Kerr solution in rotating coordinates

The Kerr solution in coordinates corotating with the horizon is studied as a testbed for a spacetime with a helical Killing vector in the Ernst picture. The solution is numerically constructed by solving the Ernst equation with a spectral method and a Newton iteration. We discuss convergence of the iteration for several initial iterates and different values of the Kerr parameters.

Residual symmetries of the gravitational field

We develop a geometric criterion that unambiguously characterizes and also provides a systematic and effective computational procedure for finding all the residual symmetries of any gravitational Ansatz . We apply the criterion to several examples starting with the Collinson Ansatz for circular stationary axisymmetric spacetimes. We reproduce the residual symmetries already known for this Ansatz including their conformal symmetry, for which we identify the corresponding infinite generators spanning the two related copies of the Witt algebra. We also consider the noncircular generalization of this Ansatz and show how the noncircular contributions on the one hand break the conformal invariance and on the other hand enhance the standard translations symmetries of the circular Killing vectors to supertranslations depending on the direction along which the circularity is lost. As another application of the method, the well-known relation defining conjugate gravitational potentials introduced by Chandrasekhar, that makes possible the derivation of the Kerr black hole from a trivial solution of the Ernst equations, is deduced as a special point of the general residual symmetry of the Papapetrou Ansatz. We emphasize how the election of Weyl coordinates, which determines the Papapetrou Ansatz, breaks also the conformal freedom of the stationary axisymmetric spacetimes. Additionally, we study AdS waves for any dimension generalizing the residual symmetries already known for lower dimensions and exhibiting a very complex infinite-dimensional Lie algebra containing three families: two of them span the semidirect sum of the Witt algebra and scalar supertranslations and the third generates vector supertranslations. We manage to comprehend that the infinite connected group is the precise diffeomorphisms subgroup allowing to locally deform the AdS background into AdS waves.

Stationary axially symmetric solutions in Brans-Dicke theory

Stationary, axially symmetric Brans-Dicke-Maxwell solutions are reexamined in the framework of the Brans-Dicke (BD) theory. We see that, employing a particular parametrization of the standard axially symmetric metric simplifies the procedure of obtaining the Ernst equations for axially symmetric electrovacuum space-times for this theory. This analysis also permits us to construct a two parameter extension in both Jordan and Einstein frames of an old solution generating technique frequently used to construct axially symmetric solutions for BD theory from a seed solution of general relativity. As applications of this technique, several known and new solutions are constructed including a general axially symmetric BD-Maxwell solution of Plebanski-Demianski with vanishing cosmological constant, i.e. the Kinnersley solution and general magnetized Kerr-Newman--type solutions. Some physical properties and the circular motion of test particles for a particular subclass of Kinnersley solution, i.e., a Kerr-Newman-NUT--type solution for BD theory, are also investigated in some detail.

Icosahedral Symmetry and Quantum Gravity

In this note we will show that Theta functions are a solution of the icosahedron equation and also a solution of the Ernst equation for the stationary axisymmetric case of Einstein’s gravitational equation.

New solutions of initial conditions in general relativity

We find new classes of exact solutions of the initial momentum constraint for vacuum Einstein’s equations. Considered data are either invariant under a continuous symmetry or they are assumed to have the exterior curvature tensor of a simple form. In general the mean curvature H is non-constant and g is not conformally flat. In the generic case with the symmetry we obtain general solution in an explicit form. In other cases solutions are given up to quadrature. We also find a class of explicit solutions without symmetries which generalizes data induced by the Kerr metric or other metrics related to the Ernst equation. The conformal method of Lichnerowicz, Choquet-Bruhat and York is used to prove solvability of the Hamiltonian constraint if H vanishes. Existence of marginally outer trapped surfaces in initial manifold is discussed.

Magnetic resonance imaging of solid dental restoration materials using 3D UTE sequences: visualization and relaxometry of various compounds

Due to an increasing scientific interest in MR-imaging of carious lesions and teeth, an accurate signal characterization of dental restoration materials is necessary for optimization of MR sequence protocols and evaluation of material degradation. Therefore, signal yield and relaxation behavior of common dental restoration materials in comparison to those of dentine of extracted human teeth were assessed in vitro by ultrashort echo time (UTE) sequences. Eighteen material samples and dentine of two freshly extracted human teeth were investigated on a 3T whole-body clinical MR-scanner. Transverse (T2*) and longitudinal relaxation times (T1) were quantified using a recently published modified Ernst equation that takes relevant in-pulse relaxation effects into account. All investigated samples could be successfully visualized but maximum signal yield was highly variable between samples. T1-values of the investigated dental restoration materials ranged between 28 and 365 ms, whereas T2*-values ranged between 96 and 917 μs. In contrast, T1-values of dentine (T1=545 ms±299 ms) were higher, while T2*-values (T2*=478 μs±271 μs) showed similar values. Dental restoration materials and dentine of extracted human teeth can be visualized by UTE sequences and show a broad range of signal yield and relaxation times.

Comparison of steady- and unsteady-state drainage equations for determination of subsurface drain spacing in paddy fields: a case study in Northern Iran

In Northern Iran paddy fields, waterlogging is a serious problem. Provision of subsurface drainage seems to be a prerequisite for optimal crop production. Considering specific characteristics of the paddies, application of a suitable equation for planning of subsurface drainage systems is of great importance. In this study, five steady-state equations including Hooghoudt, Kirkham, Dagan, Ernst, and Ernst–Hooghoudt; and three unsteady-state equations including Glover–Dumm, Van-Shilfgard, and De-zeeuw–Hellinga were implemented to compute drain spacing. Required field data were obtained from 52 ha of paddy fields of “Haraz Human-Agriculture Resources Development Center,” located in Mazandaran province of Iran. The computed drain spacings were evaluated using dynamic equilibrium concept and DRAINMOD model. The minimum and maximum drain spacings were found to be 26 and 63 m corresponding to Glover–Dumm and Ernst equations, respectively. Based on the dynamic equilibrium concept, the averages of the computed water table depths for the drain spacings obtained by the unsteady-state equations were 0.5–0.63 m as compared with those obtained by the steady-state methods which were in the range of 0–0.43 m. According to the DRAINMOD model simulations, the maximum relative yield was 80.2 % which was obtained when the drain spacing was 26 m and it reduced to 73.5 % when the drain spacing was increased to 63 m. The computed drain spacings suggested by the unsteady-state equations were found to be more suitable than those obtained from the steady-state equations for the study area.