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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.

It is known that Einstein metrics are Bach flat. So, the question may arises whether there exists any Riemannian metric which is Bach flat but not Einstein. In this paper, we answer this by constructing several example on certain class of warped product manifold. Indeed the warped product spaces $$\mathbb {R}\times _f \mathbb {CP}^n$$ and $$\mathbb {R}\times _f \mathbb {CH}^n$$ with the warping function $$f(t) = ke^t$$ (where k is a non zero constant) are non Einstein Bach flat manifolds. These spaces are commonly known as Kenmotsu manifold. Moreover, we prove that a Kenmotsu manifold with parallel Cotton tensor is $$\eta $$ -Einstein and Bach flat. Next, we establish that any $$\eta $$ -Einstein Kenmotsu manifold of dimension $$>3$$ is Bach flat.

In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Weyl tensor with the non-negativity of the complete divergence of the Bach tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. Second, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness of the metric. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.

Homogeneous conformally Einstein manifolds are classified in dimension four. As a consequence we show that any homogeneous strictly Bach-flat four-dimensional manifold is homothetic to one of the examples constructed by Abbena, Garbiero and Salamon in [1] .

We study static spherically symmetric solutions to the vacuum field equations of quadratic gravity in the presence of a cosmological constant $\Lambda$. Motivated by the trace no-hair theorem, we assume the Ricci scalar to be constant throughout a spacetime. Furthermore, we employ the conformal-to-Kundt metric ansatz that is valid for all static spherically symmetric spacetimes and leads to a considerable simplification of the field equations. We arrive at a set of two ordinary differential equations and study its solutions using the Frobenius-like approach of (infinite) power series expansions. While the indicial equations considerably restrict the set of possible leading powers, careful analysis of higher-order terms is necessary to establish the existence of the corresponding classes of solutions. We thus obtain various non-Einstein generalizations of the Schwarzschild, (anti-)de Sitter [or (A)dS for short], Nariai, and Pleba\'{n}ski-Hacyan spacetimes. Interestingly, some classes of solutions allow for an arbitrary value of $\Lambda$, while other classes admit only discrete values of $\Lambda$. For most of these classes, we give recurrent formulas for all series coefficients. We determine which classes contain the Schwarzschild-(A)dS black hole as a special case and briefly discuss the physical interpretation of the spacetimes. In the discussion of physical properties, we naturally focus on the generalization of the Schwarzschild-(A)dS black hole, namely the Schwarzschild-Bach-(A)dS black hole, which possesses one additional Bach parameter. We also study its basic thermodynamical properties and observable effects on test particles caused by the presence of the Bach tensor. This work is a considerable extension of our letter [Phys. Rev. Lett., 121, 231104, 2018]., Comment: 68 pages, matches the published version + contains the table of contents

A generalized Bach tensor B i j t with parameter t ∈ R is introduced. A Riemannian manifold ( M n , g ) is called B t -flat if its generalised Bach tensor B i j t ≡ 0 for some parameter t. In this paper, we first study the rigidity of closed B t -flat Riemannian manifolds with positive constant scalar curvature. When the dimension n = 4 , we prove that all B t -flat manifolds that satisfy a point-wise inequality must be of positive constant sectional curvature. Similar rigidity results are also obtained in terms of the Yamabe invariant. Moreover, using the curvature estimates for the general dimension n ≥ 4 , we obtain an integral inequality for B t -flat manifolds, and prove that the equality occurs if and only if these manifolds are of positive constant sectional curvature. In addition, we also obtain similar rigidity results for complete B t -flat manifolds.

We study static, spherically symmetric vacuum solutions to Quadratic Gravity, extending considerably our previous Rapid Communication [Phys. Rev. D 98, 021502(R) (2018)] on this topic. Using a conformal-to-Kundt metric ansatz, we arrive at a much simpler form of the field equations in comparison with their expression in the standard spherically symmetric coordinates. We present details of the derivation of this compact form of two ordinary differential field equations for two metric functions. Next, we apply analytical methods and express their solutions as infinite power series expansions. We systematically derive all possible cases admitted by such an ansatz, arriving at six main classes of solutions, and provide recurrent formulas for all the series coefficients. These results allow us to identify the classes containing the Schwarzschild black hole as a special case. It turns out that one class contains only the Schwarzschild black hole, three classes admit the Schwarzschild solution as a special subcase, and two classes are not compatible with the Schwarzschild solution at all since they have strictly nonzero Bach tensor. In our analysis, we naturally focus on the classes containing the Schwarzschild spacetime, in particular on a new family of the Schwarzschild-Bach black holes which possesses one additional non-Schwarzschild parameter corresponding to the value of the Bach tensor invariant on the horizon. We study its geometrical and physical properties, such as basic thermodynamical quantities and tidal effects on free test particles induced by the presence of the Bach tensor. We also compare our results with previous findings in the literature obtained using the standard spherically symmetric coordinates., 50 pages, 5 figures; various typos were corrected and formulations improved

In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold $\varrg$, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen.

Einstein manifolds, being critical for the Hilbert-Einstein functional, are central in Riemannian Geometry and Mathematical Physics. A strategy to construct Einstein metrics consists on deforming a given metric by a conformal factor so that the resulting metric is Einstein. In the present Thesis we follow this approach with special emphasis in dimension four. This is the first non-trivial case where the conformally Einstein condition is not tensorial and there are topological obstructions to the existence of Einstein metrics. The conformally Einstein condition is given by a overdetermined PDE-system. Hence the consideration of necessary conditions to be conformally Einstein are of special relevance: the Bach-flat condition is central. In this Thesis we classify four-dimensional homogeneous conformally Einstein manifolds and provide a large family of strictly Bach-flat gradient Ricci solitons. We show the existence of Bach-flat structures given as deformations of Riemannian extensions by means of the Cauchy-Kovalevskaya theorem.

We present, in an explicit form, the metric for all spherically symmetric Schwarzschild-Bach black holes in Einstein-Weyl theory. In addition to the black hole mass, this complete family of spacetimes involves a parameter that encodes the value of the Bach tensor on the horizon. When this additional "non-Schwarzschild parameter" is set to zero the Bach tensor vanishes everywhere and the "Schwa-Bach" solution reduces to the standard Schwarzschild metric of general relativity. Compared with previous studies, which were mainly based on numerical integration of a complicated form of field equations, the new form of the metric enables us to easily investigate geometrical and physical properties of these black holes, such as specific tidal effects on test particles, caused by the presence of the Bach tensor, as well as fundamental thermodynamical quantities., Comment: 5 pages, 3 figures: accepted for publication as a Rapid Communication in Physical Review D

In three dimensions, there exist modifications of Einstein's gravity akin to the topologically massive gravity that describe massive gravitons about maximally symmetric backgrounds. These theories are built on the three-dimensional version of the Bach tensor (a curl of the Cotton-York tensor) and its higher derivative generalizations; and they are on-shell consistent without a Lagrangian description based on the metric tensor alone. We give a generic construction of these models, find the spectra and compute the conserved quantities for the Banados-Teitelboim-Zanelli black hole., Comment: 17 pages, a note added on MMG

Since all Einstein spacetimes are vacuum solutions to quadratic gravity in four dimensions, in this paper we study various aspects of non-Einstein vacuum solutions to this theory. Most such known solutions are of traceless Ricci and Petrov type N with a constant Ricci scalar. Thus we assume the Ricci scalar to be constant which leads to a substantial simplification of the field equations. We prove that a vacuum solution to quadratic gravity with traceless Ricci tensor of type N and aligned Weyl tensor of any Petrov type is necessarily a Kundt spacetime. This will considerably simplify the search for new non-Einstein solutions. Similarly, a vacuum solution to quadratic gravity with traceless Ricci type III and aligned Weyl tensor of Petrov type II or more special is again necessarily a Kundt spacetime. Then we study the general role of conformal transformations in constructing vacuum solutions to quadratic gravity. We find that such solutions can be obtained by solving one non-linear partial differential equation for a conformal factor on any Einstein spacetime or, more generally, on any background with vanishing Bach tensor. In particular, we show that all geometries conformal to Kundt are either Kundt or Robinson-Trautman, and we provide some explicit Kundt and Robinson-Trautman solutions to quadratic gravity by solving the above mentioned equation on certain Kundt backgrounds., Comment: 13 pages, matches the published version

Inspired by the Bach tensor on Riemannian manifolds, we introduce the B -tensor ( B i j ≔ n + 2 n ∇ k ∇ l W i j k l − W i j k l R k l ) on Kahler manifolds. We prove that a compact gradientKahler–Ricci soliton with vanishing B -tensor is Kahler–Einstein. Moreover, we show that a complete non-compact extremal gradient shrinking Kahler–Ricci soliton with vanishing B -tensor is Kahler–Einstein.

In this note, we study a gradient generalized m-quasi-Einstein manifold. That is, there exist two smooth functions f, \({\lambda}\) such that $$R_{ij}+f_{ij}-\frac{1}{m}f_if_j=\lambda g_{ij},$$ where m is a constant. We first obtain some rigidity results on compact gradient generalized m-quasi-Einstein manifolds. Then, we obtain some classifications for the special gradient generalized m-quasi-Einstein manifolds under the assumption that the Bach tensor is flat. In particular, we obtain some even stronger results in dimension three.

The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold \(M\) with boundary \(\partial M\). Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form \(\mathbb {R}^{4}, \mathbb {H}^{4}\) or \(\mathbb {S}^{4}\). Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that \(M\) has divergence-free Bach tensor.

In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach–Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein–Maxwell equations. Inspired by the work of C. LeBrun on Einstein–Maxwell equations on compact Kahler surfaces, we give a variational characterization of solutions to Bach–Merkulov equations as critical points of the Weyl functional. We also show that extremal Kahler metrics are solutions to these equations, although, contrary to the Einstein–Maxwell analogue, they are not necessarily minimizers of the Weyl functional. We illustrate this phenomenon by studying the Calabi action on Hirzebruch surfaces.

We formulate off-shell N=1 superconformal higher spin multiplets in four spacetime dimensions and briefly discuss their coupling to conformal supergravity. As an example, we explicitly work out the coupling of the superconformal gravitino multiplet to conformal supergravity. The corresponding action is super-Weyl invariant for arbitrary supergravity backgrounds. However, it is gauge invariant only if the supersymmetric Bach tensor vanishes. This is similar to linearised conformal supergravity in curved background., 24 pages; V2: published version; V3: sign factor in (5.12) corrected

If there exist a smooth function f on \((M^n, g)\) and three real constants \(m,\rho ,\lambda \) (\(0

We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any $n$-dimensional ($n\geq 4$) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite quotient of $N^{n-k}\times \mathbb{R}^k$, $(k > 0)$, the product of a Einstein manifold $N^{n-k}$ with the Gaussian shrinking soliton $\mathbb{R}^k$. The technique applies also to the steady and expanding cases in all dimensions. In particular, we prove that a three dimensional gradient steady soliton with third order divergence-free Cotton tensor, i.e. with vanishing double divergence of the Bach tensor, is either flat or isometric to the Bryant soliton., Minor typos

We provide a simple derivation of the equivalence between Einstein and Conformal Gravity (CG) with Neumann boundary conditions given by Maldacena. As Einstein spacetimes are Bach flat, a generic solution to CG would contain both Einstein and non-Einstein part. Using this decomposition of the spacetime curvature in the Weyl tensor, makes manifest the equivalence between the two theories, both at the level of the action and the variation of it. As a consequence, we show that the on-shell action for Critical Gravity in four dimensions is given uniquely in terms of the Bach tensor., Comment: Improved discussion on Einstein spaces in CG. Reference list updated. Final version for PRD

We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite L2 norm in dimension 4.

In four dimensions, we prove that the Bach tensor is the only symmetric divergence-free 2-tensor which is also quadratic in Riemann and has good conformal behavior. In n > 4 dimensions, we prove that there are no symmetric divergence-free 2-tensors which are also quadratic in Riemann and have good conformal behavior, nor are there any symmetric divergence-free 2-tensors which are concomitants of the metric tensor gab together with its first two derivatives, and have good conformal behavior.

The goal of this paper is to express the Bach tensor of a four dimensional conformal geometry of an arbitrary signature by the Cartan normal conformal (CNC) connection. We show that the Bach tensor can be identified with the Yang-Mills current of the connection. It follows from that result that a conformal geometry whose CNC connection is reducible in an appropriate way has a degenerate Bach tensor. As an example we study the case of a CNC connection which admits a twisting covariantly constant twistor field. This class of conformal geometries of this property is known as given by the Fefferman metric tensors. We use our result to calculate the Bach tensor of an arbitrary Fefferman metric and show it is proportional to the tensorial square of the four-fold eigenvector of the Weyl tensor. Finally, we solve the Yang-Mills equations imposed on the CNC connection for all the homogeneous Fefferman metrics. The only solution is the Nurowski-Plebanski metric., Comment: 30 pages, no figures, LaTeX, to be published in Class. Quant. Grav

We show that a (2n + 1)-dimensional Sasakian manifold (M, g) with a purely transversal Bach tensor has constant scalar curvature ≥2n(2n+1), equality holding if and only if (M, g) is Einstein. For dimension 3, M is locally isometric to the unit sphere S3. For dimension 5, if in addition (M, g) is complete, then it has positive Ricci curvature and is compact with finite fundamental group π1(M).

If the scalar wave equation satisfies Huygens' principle, then it may always be written in the form and a necessary condition on the metric is that the Bach tensor vanishes.

In a previous paper we showed that the electromagnetic superenergy tensor, the Chevreton tensor, gives rise to a conserved current when there is a hypersurface orthogonal Killing vector present. In addition, the current is proportional to the Killing vector. The aim of this paper is to extend this result to the case when we have a two-parameter Abelian isometry group that acts orthogonally transitive on non-null surfaces. It is shown that for four-dimensional Einstein-Maxwell theory with a source-free electromagnetic field, the corresponding superenergy currents lie in the orbits of the group and are conserved. A similar result is also shown to hold for the trace of the Chevreton tensor and for the Bach tensor, and also in Einstein-Klein-Gordon theory for the superenergy of the scalar field. This links up well with the fact that the Bel tensor has these properties and the possibility of constructing conserved mixed currents between the gravitational field and the matter fields., Comment: 15 pages

We investigate various aspects of a geometric flow defined using the Bach tensor. Firstly, using a well-known split of the Bach tensor components for $(2,2)$ unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like $S^2\times S^2$, $R^2\times S^2$. In addition, we obtain the fixed point condition for general $(2,2)$ manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped four manifolds, we reduce the flow equations to a first order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work., to appear in IJGMMP (2012)

For stationary, asymptotically flat solutions of Einstein's equations, covariant functionals of the metric variables are defined which characterize the Kerr metric uniquely. For instance, we obtain a generalization of the “Bach tensor” to stationary metrics, which vanishes if and only if the solution is Kerr. We also give a new interpretation of the “Schwarzschild-to-Kerr-transformation.” Our results might be applicable to simplify the proof of the uniqueness theorem for stationary black holes.

Explicit sets of spinor nth derivatives of the Riemann curvature spinor for a general spacetime are specified for each n so that they contain the minimal number of components enabling all derivatives of order m to be expressed algebraically in terms of these sets for n

Conformally invariant gravitational field equations on the hand and fourth order field equations on the other were discussed in the early history of general relativity (Weyl Einstein, Bach et al.) and have recently gained some new interest (Deser, P. Gunther, Treder, et al.). The equations Bαβ=0 or Bαβ=ϰTαβ, where Bαβ denotes the Bach tensor and Tαβ a suitable energy-momentum tensor, possess both the mentioned properties. We construct exact solutions ds2=gαβdxαdxβ of the Bach equations: (2, 2)-decomposable, centrally symmetric and pp-wave solutions. The gravitational field gαβ is coupled by Bαβ=ϰTαβ to an electromagnetic field Fαβ=−Fαβ obeying the Maxwell equations or to a neutrino field ϕA obeying the Weyl equations respectively. Among interesting new metrics ds2 there appear some physically well-known ones, such as the De Sitter universe, the Weyl-Trefftz metric. and the plane-fronted gravitational waves with parallel rays (pp-waves) known from Einstein's theory. The solutions are built up by means of special techniques: A separation method for (2, 2)-decomposable solutions, simplification of centrally symmetric metrics by a suitable conformal transformation, and complex function methods for pp-wave solutions.