Einstein Case Research Articles

Black hole solutions to Einstein-Bel-Robinson gravity

In this paper, we study the physical properties of black holes in the framework of the recently proposed Einstien-Bel-Robinson gravity. We show that interestingly the theory propagates a transverse and massless graviton on a maximally symmetric background with positive energy. There is also a single ghost-free branch that returns to the Einstein case when β → 0. We find new black hole solutions to the equations, both approximate and exact, the latter being a constant curvature black hole solution, and discuss inconsistencies with metrics that were previously claimed to be approximate solutions to the equations. We obtain the conserved charges of the theory and briefly study the thermodynamics of the black hole solutions.

On noncompact warped product Ricci solitons

AbstractThe goal of this paper is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented as a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian.

Stability and moduli space of generalized Ricci solitons

The generalized Einstein–Hilbert action is an extension of the classic scalar curvature energy and Perelman’s F-functional which incorporates a closed three-form. The critical points are known as generalized Ricci solitons, which arise naturally in mathematical physics, complex geometry, and generalized geometry. Through a delicate analysis of the group of generalized gauge transformations, and implementing a novel connection, we give a simple formula for the second variation of this energy which generalizes the Lichnerowicz operator in the Einstein case. As an application we show that all Bismut-flat manifolds are linearly stable critical points, and admit nontrivial deformations arising from Lie theory. Furthermore, this leads to extensions of classic results of Koiso [1–4] and Podesta, Spiro, Kröncke [5–8] to the moduli space of generalized Ricci solitons. To finish we classify deformations of the Bismut-flat structure on S3 and show that some are integrable while others are not.

Seeded vacuum decay with Gauss-Bonnet

We investigate false vacuum decay catalysed by black holes under the influence of the higher order Gauss-Bonnet term. We study both bubble nucleation and Hawking-Moss types of phase transition in arbitrary dimension. The equations of motion of “bounce” solutions in which bubbles nucleate around arbitrary dimensional black holes are found in the thin wall approximation, and the instanton action is computed. The headline result that the tunnelling action for static instantons is the difference in entropy of the seed and remnant black holes is shown to hold for arbitrary dimension. We also study the Hawking-Moss transition and find a picture similar to the Einstein case, with one curious five-dimensional exception (due to a mass gap). In four dimensions, we find as expected that the Gauss-Bonnet term only impacts topology changing transitions, i.e. when vacuum decay removes the seed black hole altogether, or in a (Hawking-Moss) transition where a black hole is created. In the former case, topology changing transitions are suppressed (for positive GB coupling α), whereas the latter case results in an enhanced transition.

Quasinormal ringing of general spherically symmetric parametrized black holes

The general parametrization of spherically symmetric and asymptotically flat black-hole spacetimes in arbitrary metric theories of gravity was suggested in [3]. The parametrization is based on the continued fraction expansion in terms of the compact radial coordinate and has superior convergence and strict hierarchy of parameters. It is known that some observable quantities, related to particle motion around the black hole, such as the eikonal quasinormal modes, radius of the shadow, frequency at the innermost stable circular orbit, and others, depend mostly on only a few of the lowest coefficients of the parametrization. Here we continue this approach by studying the dominant (low-lying) quasinormal modes for such generally parametrized black holes. We show that, due to the hierarchy of parameters, the dominant quasinormal frequencies are also well determined by only the first few coefficients of the expansion for the so-called moderate black-hole geometries. The latter are characterized by a relatively slow change of the metric functions in the radiation zone near the black hole. The nonmoderate metrics, which change strongly between the event horizon and the innermost stable circular orbit are usually characterized by echoes or by the distinctive (from the Einstein case) quasinormal ringing which does not match the current observational data. Therefore, the compact description of a black-hole spacetime in terms of the truncated general parametrization is an effective formalism for testing strong gravity and imposing constraints on allowed black-hole geometries.

Physical Implications of Pure Lovelock Geometry on Stellar Structure

AbstractAn exact anisotropic star model with a linear barotropic equation of state and with Finch‐Skea potential is constructed within the framework of pure Lovelock gravity. A comparison with the corresponding Einstein model in a suitable limit is easily deduced. Evidently higher curvature effects induced by the Lovelock contributions generate lower densities, pressures, surface tensions, and anisotropy factors when compared to its Einstein counterpart. The maximum moment of inertia is attained for the Einstein case and hence it may be inferred that Lovelock effects soften the equation of state. The model satisfies various stability tests.

Uniformization theorems: Between Yamabe and Paneitz

This paper is devoted to several existence results for a generalized version of the Yamabe problem. First, we prove the remaining global cases for the range of powers $\gamma\in (0,1)$ for the generalized Yamabe problem introduced by Gonzalez and Qing. Second, building on a new approach by Case and Chang for this problem, we prove that this Yamabe problem is solvable in the Poincar\'{e}-Einstein case for $\gamma\in (1,\min\{2,n/2\})$ provided the associated fractional GJMS operator satisfies the strong maximum principle.

On the Einstein condition for Lorentzian 3-manifolds

It is well known that in Lorentzian geometry there are no compact spherical space forms; in dimension 3, this means there are no closed Einstein 3-manifolds with positive Einstein constant. We generalize this fact here, by proving that there are also no closed Lorentzian 3-manifolds (M,g) whose Ricci tensor satisfiesRic=fg+(f−λ)T♭⊗T♭, for any unit timelike vector field T, any positive constant λ, and any nontrivial function f that never takes the value λ. (Observe that this reduces to the positive Einstein case when f=λ.) We show that there is no such obstruction if λ is negative. Finally, the “borderline” case λ=0 is also examined: we show that if λ=0, then (M,g) must be isometric to (S1×N,−dt2⊕h) with (N,h) a Riemannian manifold.

Formal power series for asymptotically hyperbolic Bach-flat metrics

It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein 4-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein equation, we find formal power series for conformally compactifiable, asymptotically hyperbolic Bach-flat 4-metrics expanded about conformal infinity. We also consider Bach-flat metrics in the special case of constant scalar curvature and in the special case of constant Q-curvature. This allows us to determine the free data at conformal infinity and to select those choices that lead to Einstein metrics. The asymptotically hyperbolic mass is part of that free data, in contrast to the pure Einstein case. Higher-dimensional generalizations of the Bach tensor lack some of the geometrical meaning of the 4-dimensional case, but for a generalized Bach equation suited to the Fefferman–Graham technique, we are able to obtain a relatively complete result illustrating an interesting splitting of the free data into low-order “Dirichlet” and high-order “Neumann” pairs.

Dust collapse in 4D Einstein–Gauss–Bonnet gravity

We consider gravitational collapse in the recently proposed 4D limit of Einstein-Gauss-Bonnet gravity. We show that for collapse of a sphere made of homogeneous dust the process is qualitatively similar to the case of pure Einstein's gravity. The singularity forms as the endstate of collapse and it is trapped behind the horizon at all times. However, and differently from Einstein's theory, as a consequence of the Gauss-Bonnet term, the collapsing cloud reaches the singularity with zero velocity, and the time of formation of the singularity is delayed with respect to the pure Einstein case.

The Calabi metric and desingularization of Einstein orbifolds

Consider an Einstein orbifold (M_0, g_0) of real dimension 2n having a singularity with orbifold group the cyclic group of order n in SU (n) which is generated by an nth root of unity times the identity. Existence of a Ricci-flat Kähler ALE metric with this group at infinity was shown by Calabi. There is a natural “approximate” Einstein metric on the desingularization of M0 obtained by replacing a small neighborhood of the singular point of the orbifold with a scaled and truncated Calabi metric. In this paper, we identify the first obstruction to perturbing this approximate Einstein metric to an actual Einstein metric. If M_0 is compact, we can use this to produce examples of Einstein orbifolds which do not admit any Einstein metric in a neighborhood of the natural approximate Einstein metric on the desingularization. In the case that (M_0, g_0) is asymptotically hyperbolic Einstein and non-degenerate, we show that if the first obstruction vanishes, then there does in fact exist an asymptotically hyperbolic Einstein metric on the desingularization. We also obtain a non-existence result in the asymptotically hyperbolic Einstein case, provided that the obstruction does not vanish. This work extends a construction of Biquard in the case n = 2 , in which case the Calabi metric is also known as the Eguchi–Hanson metric, but there are some key points for which the higher-dimensional case differs.

Time dependence of complexity for Lovelock black holes

We study the general time dependence of complexity for holographic states dual to Lovelock black holes using the "Complexity=Action" (CA) proposal. We observe that at early times, the critical time at which the complexity begins to increase is a decreasing function of the higher order coupling constants, which implies that the complexity evolves faster than that of Schwarzschild black holes. At late times, the rate of change of complexity is essentially determined by the generalised Gibbons-Hawking-York boundary term evaluated at the future singularity. In particular, its ratio to black hole mass is a characteristic constant, independent of the higher order couplings. Thus, in the vanishing coupling limit, the result in general does not reduce to that of Schwarzschild black holes, in spite of that the metric reduces to the latter as well as the gravitational action. In fact, the two differ by a constant during the whole time evolution. Including the next-to-leading order term around late times, we find that as the Einstein case, the late time limit is always approached from above, thus violating any conjectured upper bound given by the late time result. For charged Lovelock black holes, we find that with sufficient charge, the complexity roughly behaves the same as the Einstein case. However, for smaller charges, the two have some significant differences. In particular, unlike the Einstein case, in the uncharged limit the complexity growth rate does not match with the neutral case, differing by a constant in the whole time evolution.

Einstein and $$\eta $$-Einstein Sasakian submanifolds in spheres

The aim of this paper is to study Sasakian immersions of compact Sasakian manifolds into the odd-dimensional sphere equipped with the standard Sasakian structure. We obtain a complete classification of such manifolds in the Einstein and $$\eta $$-Einstein cases when the codimension of the immersion is 4. Moreover, we exhibit infinite families of compact Sasakian $$\eta $$-Einstein manifolds which cannot admit a Sasakian immersion into any odd-dimensional sphere. Finally, we show that, after possibly performing a $${{\mathcal {D}}}$$-homothetic deformation, a homogeneous Sasakian manifold can be Sasakian immersed into some odd-dimensional sphere if and only if S is regular and either S is simply connected or its fundamental group is finite cyclic.

4-Index theory of gravity and its relation with the violation of the energy–momentum conservation law

Recently, a 4-index generalization of the Einstein theory has been proposed by Moulin [F. Moulin, Eur. Phys. J. C 77, 878 (2017)]. Using this method, we find the most general 2-index field equations derivable from the Einstein–Hilbert action. The application of Newtonian limit, the role of gravitational coupling constant and the effects of the properties of ordinary energy–momentum tensor in obtaining a 4-index gravity theory have been studied. We also address the results of building Weyl free 4-index gravity theory. Our study displays that both the Einstein and Rastall theories can be obtained as the subclasses of a 4-index gravity theory which shows the power of 4-index method in unifying various gravitational theories. It is also obtained that the violation of the energy–momentum conservation law may be allowed in 4-index gravity theory, and moreover, the contraction of 4-index theory generally admits a non-minimal coupling between geometry and matter field in the Rastall way. This study also shows that, unlike the Einstein case, the gravitational coupling constant of 4-index Rastall theory generally differs from that of the ordinary 2-index Rastall theory.

Impact of the Rastall parameter on perfect fluid spheres

We examine the effects of the Rastall parameter on the behaviour of spherically symmetric static distributions of perfect fluid matter. It was claimed by Visser [Physics Letters B, 782, 83, (2018)] that the Rastall proposition is completely equivalent to the Einstein theory. While many authors have raised contrary arguments, our intention is to analyse the properties of Rastall gravity through variation of the Rastall parameter in the context of perfect fluids spheres that may be used to model neutron stars or cold fluid planets. This analysis also serves to counter the claim that Rastall gravity is equivalent to the standard Einstein theory. It turns out that the condition of pressure isotropy is exactly the same as for Einstein gravity and hence that any known solution of the Einstein equations may be used to study the effects of the Rastall dynamical quantities. Moreover, by choosing the well studied Tolman metrics, we discover that in the majority of cases there is substantial deviation from the Einstein case when the Rastall parameter vanishes and in cases where the Einstein model displays defective behaviour, certain Rastall models obey the well known elementary requirements for physical plausibility. These empirical findings do not support the idea that Rastall theory is equivalent to Einstein theory as several deviations in physical behaviour are displayed as counter-examples.

Projective manifolds modeled after hyperquadrics

We classify all projective manifolds with flat holomorphic conformal structure. The Kähler–Einstein case was treated by Kobayashi and Ochiai, there exists a very short list of possible manifolds. In the non-Kähler–Einstein case a classification was only known in small dimensions: by Kobayashi and Ochiai for complex surfaces and by the authors for projective threefolds. This paper completes the general case.

Volume minimization and conformally Kähler, Einstein–Maxwell geometry

Let $M$ be a compact complex manifold admitting a Kahler structure. A conformally Kahler, Einstein–Maxwell metric (cKEM metric for short) is a Hermitian metric $\tilde g$ on $M$ with constant scalar curvature such that there is a positive smooth function $f$ with $g = f^{2} \tilde g$ being a Kahler metric and $f$ being a Killing Hamiltonian potential with respect to $g$. Fixing a Kahler class, we characterize such Killing vector fields whose Hamiltonian function $f$ with respect to some Kahler metric $g$ in the fixed Kahler class gives a cKEM metric $\tilde g = f^{-2}g$. The characterization is described in terms of critical points of certain volume functional. The conceptual idea is similar to the cases of Kahler–Ricci solitons and Sasaki–Einstein metrics in that the derivative of the volume functional gives rise to a natural obstruction to the existence of cKEM metrics. However, unlike the Kahler–Ricci soliton case and Sasaki–Einstein case, the functional is neither convex nor proper in general, and often has more than one critical points. The last observation matches well with the ambitoric examples studied earlier by LeBrun and Apostolov–Maschler.

Analysis of s-wave, p-wave and d-wave holographic superconductors in Hořava–Lifshitz gravity

In this work, the s-wave, p-wave and d-wave holographic superconductors in the Hořava–Lifshitz gravity are investigated in the probe limit. For this approach, it is shown that the equations of motion for different wave states in Einstein gravity can be written as a unified form, and condensates take place in all three cases. This scheme is then generalized to Hořava–Lifshitz gravity, and a unified equation for multiple holographic states is obtained. Furthermore, the properties of the condensation and the optical conductivity are studied numerically. It is found that, in the case of Hořava–Lifshitz gravity, it is always possible to find some particular parameters in the corresponding Einstein case where the condensation curves are identical. For fixed scalar field mass m, a nonvanishing [Formula: see text] makes the condensation easier than in Einstein gravity for s-wave superconductor. However, the p-wave and d-wave superconductors have T[Formula: see text] greater than the s-wave.

Generalized spheroidal spacetimes in 5-D Einstein\u2013Maxwell\u2013Gauss\u2013Bonnet gravity

The field equations for static EGBM gravity are obtained and transformed to an equivalent form through a coordinate redefinition. A form for one of the metric potentials that generalizes the spheroidal ansatz of Vaidya–Tikekar superdense stars and additionally prescribing the electric field intensity yields viable solutions. Some special cases of the general solution are considered and analogous classes in the Einstein framework are studied. In particular the Finch–Skea ansatz is examined in detail and found to satisfy the elementary physical requirements. These include positivity of pressure and density, the existence of a pressure free hypersurface marking the boundary, continuity with the exterior metric, a subluminal sound speed as well as the energy conditions. Moreover, the solution possesses no coordinate singularities. It is found that the impact of the Gauss–Bonnet term is to correct undesirable features in the pressure profile and sound speed index when compared to the equivalent Einstein gravity model. Furthermore graphical analyses suggest that higher densities are achievable for the same radial values when compared to the 5-dimensional Einstein case. The case of a constant gravitational potential, isothermal distribution as well as an incompressible fluid are studied. All exact solutions derived exhibit an equation of state explicitly.

Exact EGB models for spherical static perfect fluids

We obtain a new exact solution to the field equations for a 5-dimensional spherically symmetric static distribution in the Einstein–Gauss–Bonnet modified theory of gravity. By using a transformation, the study is reduced to the analysis of a single second order nonlinear differential equation. In general the condition of pressure isotropy produces a first order differential equation which is an Abel equation of the second kind. An exact solution is found. The solution is examined for physical admissibility. In particular a set of constants is found which ensures that a pressure-free hypersurface exists which defines the boundary of the distribution. Additionally the isotropic pressure and the energy density are shown to be positive within the radius of the sphere. The adiabatic sound-speed criterion is also satisfied within the fluid ensuring a subluminal sound speed. Furthermore, the weak, strong and dominant conditions hold throughout the distribution. On setting the Gauss–Bonnet coupling to zero, an exact solution for 5-dimensional perfect fluids in the standard Einstein theory is obtained. Plots of the dynamical quantities for the Gauss–Bonnet and the Einstein case reveal that the pressure is unaffected, while the energy density increases under the influence of the Gauss–Bonnet term.