The Einstein–Maxwell Equations and Conformally Kähler Geometry

Page’s Einstein metric on \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\) is conformally related to an extremal Kähler metric. Here we construct a family of conformally Kähler solutions of the Einstein–Maxwell equations that deforms the Page metric, while sweeping out the entire Kähler cone of \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\). The same method also yields analogous solutions on every Hirzebruch surface. This allows us to display infinitely many geometrically distinct families of solutions of the Einstein–Maxwell equations on the smooth 4-manifolds \({S^2 \times S^2}\) and \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\).     

The extended conformal Einstein field equations with matter: The Einstein–Maxwell field

A discussion is given of the conformal Einstein field equations coupled with matter whose energy–momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to know a priori the location of the conformal boundary. In vacuum this gauge reduces to the so-called conformal Gaussian gauge. These ideas are applied to obtain (i) a new proof of the stability of Einstein–Maxwell de Sitter-like spacetimes; (ii) a proof of the semi-global stability of purely radiative Einstein–Maxwell spacetimes.     

Particle-like solutions of the Einstein–Dirac–Maxwell equations

We consider the coupled Einstein–Dirac–Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Soliton-like solutions are constructed numerically. The stability and the properties of the ground state solutions are discussed for different values of the electromagnetic coupling constant. We find solutions even when the electromagnetic coupling is so strong that the total interaction is repulsive in the Newtonian limit. Our solutions are regular and well-behaved; this shows that the combined electromagnetic and gravitational self-interaction of the Dirac particles is finite.     

Generating Techniques and Analytically Extended Solutions of the Einstein–Maxwell Equations

The correspondence of arbitrary parameters inexact axisymmetric solutions of the Einstein-Maxwellequations constructed with the aid of differentgenerating methods to the analytically extendedparameter sets is discussed and examples of the extendedsolutions are given.     

Nernst Theorem and Entropy of the Axisymmetric Einstein–Maxwell–Dilaton-Axion Black Hole

The free energy and the entropy of scalar field are calculated by brick-wall in the axisymmetric Einstein–Maxwell–Dilaton-axion black hole. It is shown that when the black hole has inner and outer horizons, the entropy is not only related to the area of an outer horizon but also is the function of the area of an inner horizon. When the area of an inner horizon approaches zero, we can obtain the known conclusion. The entropy expressed by location parameter of outer horizon and inner horizons approaches absolute zero. It obeys Nernst theorem. It can be taken as Planck absolute entropy of a black hole.     

All static and electrically charged solutions with Einstein base manifold in the arbitrary-dimensional Einstein–Maxwell system with a massless scalar field

We present a simple and complete classification of static solutions in the Einstein–Maxwell system with a massless scalar field in arbitrary \(n(\ge 3)\) dimensions. We consider spacetimes which correspond to a warped product \(M^2 \times K^{n-2}\), where \(K^{n-2}\) is a \((n-2)\)-dimensional Einstein space. The scalar field is assumed to depend only on the radial coordinate and the electromagnetic field is purely electric. Suitable Ansätze enable us to integrate the field equations in a general form and express the solutions in terms of elementary functions. The classification with a non-constant real scalar field consists of nine solutions for \(n\ge 4\) and three solutions for \(n=3\). A complete geometric analysis of the solutions is presented and the global mass and electric charge are determined for asymptotically flat configurations. There are two remarkable features for the solutions with \(n\ge 4\): (i) Unlike the case with a vanishing electromagnetic field or constant scalar field, asymptotically flat solution is not unique, and (ii) The solutions can asymptotically approach the Bertotti–Robinson spacetime depending on the integrations constants. In accordance with the no-hair theorem, none of the solutions are endowed of a Killing horizon.     

Spectroscopy of the Einstein–Maxwell–Dilaton–Axion black hole

The entropy spectrum of a spherically symmetric black hole was derived via the Bohr–Sommerfeld quantization rule in Majhi and Vagenas’s work. Extending this work to charged and rotating black holes, we quantize the horizon area and the entropy of an Einstein–Maxwell–Dilaton–Axion black hole via the Bohr–Sommerfeld quantization rule and the adiabatic invariance. The result shows the area spectrum and the entropy spectrum are respectively equally spaced and independent on the parameters of the black hole.     

Einstein–Maxwell gravity with additional corrections

Motivated by the E ₈×E ₈ heterotic string theory, we obtain topological black hole solutions of Einstein–Maxwell gravity with additional corrections. We consider the Gauss–Bonnet (GB) and (F _μν F ^μν )² terms as an effective quartic order Lagrangian of gauge–gravity coupling and investigate geometric and thermodynamic properties of the black hole solutions. We also analyze the effects of the GB term as well as the correction of Maxwell field on the properties of the solutions.     

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein–Dirac–Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.     

Cohomological uniqueness of the Cauchy problem solutions for the Einstein–Maxwell equation

In this paper we analyze the Cauchy problem for the Einstein–Maxwell equation in the case of non-characteristic initial hypersurface. To find the correct notions of characteristic and Cauchy data we introduce a complex, which we call the Einstein–Maxwell complex. Then the Cauchy problem acquires correctness in terms of an associated spectral sequence. We define a Cauchy data in such way that they allow us to reconstruct a cohomologously unique formal solution.     

Thermodynamics,phase transitions and Ruppeiner geometry for Einstein–dilaton–Lifshitz black holes in the presence of Maxwell and Born–Infeld electrodynamics

In this paper, we first obtain the higher-dimen-sional dilaton–Lifshitz black hole solutions in the presence of Born–Infeld (BI) electrodynamics. We find that there are two different solutions for the cases of \(z=n+1\) and \(z\ne n+1\) where z is the dynamical critical exponent and n is the number of spatial dimensions. Calculating the conserved and thermodynamical quantities, we show that the first law of thermodynamics is satisfied for both cases. Then we turn to the study of different phase transitions for our Lifshitz black holes. We start with the Hawking–Page phase transition and explore the effects of different parameters of our model on it for both linearly and BI charged cases. After that, we discuss the phase transitions inside the black holes. We present the improved Davies quantities and prove that the phase transition points shown by them are coincident with the Ruppeiner ones. We show that the zero temperature phase transitions are transitions in the radiance properties of black holes by using the Landau–Lifshitz theory of thermodynamic fluctuations. Next, we turn to the study of the Ruppeiner geometry (thermodynamic geometry) for our solutions. We investigate thermal stability, interaction type of possible black hole molecules and phase transitions of our solutions for linearly and BI charged cases separately. For the linearly charged case, we show that there are no phase transitions at finite temperature for the case \( z\ge 2\). For \(z<2\), it is found that the number of finite temperature phase transition points depends on the value of the black hole charge and there are not more than two. When we have two finite temperature phase transition points, there is no thermally stable black hole between these two points and we have discontinuous small/large black hole phase transitions. As expected, for small black holes, we observe finite magnitude for the Ruppeiner invariant, which shows the finite correlation between possible black hole molecules, while for large black holes, the correlation is very small. Finally, we study the Ruppeiner geometry and thermal stability of BI charged Lifshtiz black holes for different values of z. We observe that small black holes are thermally unstable in some situations. Also, the behavior of the correlation between possible black hole molecules for large black holes is the same as for the linearly charged case. In both the linearly and the BI charged cases, for some choices of the parameters, the black hole system behaves like a Van der Waals gas near the transition point.     

Current algebra Ward identities in the renormalized σ model

The many-current Ward identities corresponding to the Gell-Mann current algebra are discussed in the renormalized model. The Ward identities are verified in the case of the SU(2)×SU(2) chiral symmetry. In the SU(3)×SU(3) case the uniqueness of the Adler-Bardeen anomaly is proved using the Wess-Zumino consistency conditions.On leave of absence from the University of Genova (Italy).Chercheur Associé au C.N.R.S. (C.P.T./Marseille).     

The Carter constant and Petrov classification of the Vaidya–Einstein–Kerr spacetime

The existence of the Carter constant in the Vaidya–Einstein–Kerr (VEK) spacetime and its relation to the Petrov type is investigated. This spacetime is an example of a black hole in an asymptotically non-flat background. We construct the Carter constant and obtain the Killing tensor in the VEK spacetime. The Newman–Penrose formalism is employed to obtain the spin coefficients. We present a complete (Petrov) classification of the VEK spacetime and the special case of the non-rotating Vaidya–Einstein–Schwarzschild spacetime. We demonstrate explicitly that both spacetimes are of type-D.     

Uniqueness of static,isotropic low-pressure solutions of the Einstein–Vlasov system

Letters in Mathematical Physics - In Beig and Simon (Commun Math Phys 144:373–390, 1992) the authors prove a uniqueness theorem for static solutions of the Einstein–Euler system which...