Joan Josep Ferrando, Salvador Mengual
We analyze the field equations for the perfect fluid solutions admitting a group G$_3$ of isometries acting on orbits S$_2$ whose curvature has a gradient that is tangent to the fluid flow (T-models). We propose several methods to integrate the field equations and we present the general solution without the need to calculate any integral.
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Applications of the monodromy transform approach to construction of exact solutions of electrovacuum Einstein - Maxwell field equations are considered. Examples of new solutions are given.
Sachs equations and plane waves II: Isometries and conformal isometries
本文研究四维及以上平面波时空的对称性，分析了等距同构与共形等距同构的结构，给出含Bernoulli移位的不可分类真空时空实例，并完成两类形式平面波的对称分类。
The generalized Einstein - Maxwell field equations which arise from a truncated bosonic part of the low - energy string gravity effective action in four dimensions (the so called Einstein-Maxwell - axion - dilaton theory) are considered. The integrable structure of these field equations for D=4 space-times with two commuting isometries is elucidated. We express the dynamical part of the reduced equations as integrability conditions of some overdetermined $4\times 4$-matrix linear system with a spectral parameter. The remaining part of the field equations are expressed as the conditions of exis…
In this work we find the Killing vector fields of the riemannian submanifolds of the thermodynamic phase space of an ideal gas and show that the isometry group corresponding to them is homomorphic to the euclidean group $E(2)$. We also give the embedding of these submanifolds in the euclidean space $(R^{3},\ delta)$.
Isometries of N=1 4D supergravity
本文利用超场方法将广义相对论中的Killing方程推广到N=1四维超引力，使其涵盖包含Rarita-Schwinger场的完整超引力多重态，并通过对旋量-矢量方程进行松弛以容纳非零各向同性Rarita-Schwinger场。
By extending the notion of Lie derivative to distribution-valued tensor fields of order $m$, Lie derivatives with respect to $C^k$ vector fields, $k\geqslant m+1$, can be shown to be well defined. Geometric symmetries, definable in terms of these Lie derivatives, can then be considered. In particular, for spacetimes of low regularity, geometric symmetries generated by vector fields of regularity as low as $C^1$ are definable. We find that the conformal algebra of the impulsive plane wave spacetime (with $+$ polarization) described by the continuous Baldwin-Jeffery-Rosen form of the metric has …
In four dimensions one can use the chiral part of the spin connection as the main object that encodes geometry. The metric is then recovered algebraically from the curvature of this connection. We address the question of how isometries can be identified in this "pure connection" formalism. We show that isometries are recovered from gauge transformation parameters satisfying the requirement that the Lie derivative of the connection along a vector field generating an isometry is a gauge transformation. This requirement can be rewritten as a first order differential equation involving the gauge t…
Two examples of the use of differential geometry in plasma physics are given: The first is the computation and solution of the constraint equations obtained from the Riemann metric isometry of the twisted flux tube. In this case a constraint between the Frenet torsion and curvature is obtained for inhomogeneous helical magnetic flux tube axis. In the second one, geometrical and topological constraints on the current-carrying solar loops are obtained by assuming that the plasma filament is stable. This is analogous to early computations by Liley [(Plasma Physics (1964)] in the case of hydromagn…
In this paper the isometries of the dual space were investigated. The dual structural equations of a Killing tensor of order two were found . The flat space case was analyzed in details.
Generalizations and challenges for the spacetime block-diagonalization
本文重新审视了具有等距的块对角度规的几何背景，分析了多种物质源满足等距相容块对角形式的充分条件，并将静态时空中不存在零电磁场的定理推广至任意维度及非线性电磁场。
In this paper, it is shown that the covariant representation (CR) transforming the Dirac field under de Sitter isometries is equivalent to a direct sum of two unitary irreducible representations (UIRs) of the $Sp(2,2)$ group transforming alike the particle and antiparticle field operators in momentum representation. Their basis generators and Casimir operators are written down for the first time finding that these representations are equivalent to a UIR from the principal series whose canonical labels are determined by the fermion mass and spin. The properties of the conserved observables (i. …
An effective chiral model of a plane-symmetric gravitational field is considered. Isometries of the target space of the model are described and exact solutions corresponding to the isometric ansatz method are obtained. New exact solutions are found using the method of functional parameters. The solutions obtained are Bäcklund transforms of solutions of the d'Alembert equation to those of the Einstein equations.
For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein and Einstein--Maxwell equations as represented by their Ernst potentials. This hierarchy contains three arbitrary rational functions of an auxiliary complex parameter. They are constructed using the so called `monodromy transform' approach and our new method for the solution of the linear singular integral equation form of the reduced Einstein equations. The solutions presented, which describe inhomogeneous cosmological models or gravitational and electromagnetic waves and their in…
Dynamical Horizon Segments and Spacetime Isometries
本文证明动力学视界段邻域内允许的Killing场受严格限制：一般为旋转场且保持截面不变；若超曲面正交，则所有自旋多极矩为零，与球对称情形不可区分。
Birkhoffs Theorem and Lie Symmetry Analysis
本文通过李对称分析和诺特点对称性方法，重新表述比尔霍夫定理，探讨爱因斯坦真空场方程的对称性及守恒量。
We construct perfect fluid metrics corresponding to spacelike surfaces invariant under a 1-dimensional group of isometries in 3-dimensional Minkowski space. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI_0 and VII_0. The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state.
The perfect fluid solutions admitting a group G$_3$ of isometries acting on orbits S$_2$ whose curvature has a gradient which is tangent to the fluid flow (T-models) are studied from a thermodynamic approach. All the admissible thermodynamic schemes are obtained, and the solutions compatible with the generic ideal gas equation of state are studied in detail. The possible physical interpretation of some previously known T-models is also analyzed.
A brief summary of results on kinematic self-similarities in general relativity is given. Attention is focussed on locally rotationally symmetric models admitting kinematic self-similar vectors. Coordinate expressions for the metric and the kinematic self-similar vector are provided. Einstein's field equations for perfect fluid models are investigated and all the homothetic perfect fluid solutions admitting a maximal four-parameter group of isometries are given.
The acoustic spacetime corresponding to perturbed Friedman-Lemaitre-Robertson-Walker universe inherit the space isometries from the original FLRW model, but essentially differs in dynamics. The scale factor manifestly depends on the equation of state of the matter content. Despite the higher complexity of the background evolution the perturbation equation in this space is substantially simpler: the density perturbations obey d'Alembert equation. Canonical formalism reconstructed in the acoustic spacetime enables one to employ the Klein-Gordon scalar product. Consequently, the Fourier decomposi…
The fluid-gravity correspondence provides us with explicit spacetime metrics that are holographically dual to (non-)relativistic nonlinear hydrodynamics. The vacuum Einstein equations, in the presence of a Killing vector, possess solution-generating symmetries known as spacetime Ehlers transformations. These form a subgroup of the larger generalized Ehlers group acting on spacetimes with arbitrary matter content. We apply this generalized Ehlers group, in the presence of Killing isometries, to vacuum metrics with hydrodynamic duals to develop a formalism for solution-generating transformations…
We consider a static self-gravitating system consisting of perfect fluid with isometries of an $(n-2)$-dimensional maximally symmetric space in Lovelock gravity theory. A straightforward analysis of the time-time component of the equations of motion suggests a generalized mass function. Tolman-Oppenheimer-Volkoff like equation is obtained by using this mass function and gravitational equations. We investigate the maximum entropy principle in Lovelock gravity, and find that this Tolman-Oppenheimer-Volkoff equation can also be deduced from the so called "maximum entropy principle" which is origi…
We comprehensively study Galilean and Carrollian hydrodynamics on arbitrary backgrounds, in the presence of a matter/charge conserved current. For this purpose, we follow two distinct and complementary paths. The first is based on local invariance, be it Galilean or Carrollian diffeomorphism invariance, possibly accompanied by Weyl invariance. The second consists in analyzing the relativistic fluid equations at large or small speed of light, after choosing an adapted gauge, ADM-Zermelo for the former and Papapetrou-Randers for the latter. Unsurprisingly, the results agree, but the second appro…
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