The Algebra of the Riemann Curvature Tensor in General Relativity: Preliminaries

In a four-dimensional curved space-time it is well-known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley-Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.     

M-Eigenvalues of the Riemann Curvature Tensor of Conformally Flat Manifolds

We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold. The expressions of Meigenvalues and M-eigenvectors are presented in this paper. As a special case,M-eigenvalues of conformal flat Einstein manifold have also been discussed,and the conformal the invariance of M-eigentriple has been found. We also reveal the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We prove that the M-eigenvalue can determine the Riemann curvature tensor uniquely. We also give an example to compute the Meigentriple of de Sitter spacetime which is well-known in general relativity.     

The Mathematicians’ Happy Hunting Ground: Einstein’s General Theory of Relativity

There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry. —Hermann Weyl, “Relativity as a Stimulus to Mathematical Research,” pp. 536–537.     

The Complex Algebra of Physical Space: A Framework for Relativity

A complex and, equivalently, hyperbolic extension of the algebra of physical space (APS) is discussed that allows one to distinguish space-time vectors from paravectors of APS, while preserving the natural origin of the Minkowski space-time metric. The CAPS formalism is Lorentz covariant and gives expression to persistent vectors in physical space as time-like planes in space-time. Commuting projectors ${P_{\pm} = \frac{1}{2} (1 \pm h)}$ project CAPS onto two-sided ideals, one of which is APS. CAPS has the same dimension as the space-time algebra (STA) if both are considered real algebras, and it distinguishes covariant roles of elements, as does STA. Its structure, however, is closer to APS, with a volume element that belongs to the center of the algebra and a simple relation between space-times of opposite signature. Furthermore, CAPS, unlike STA, distinguishes point-like space-time inversion of a Dirac spinor from a physical rotation. To illustrate its use, CAPS is applied to the Dirac equation and to the fundamental symmetry transformations of the equation and Dirac spinors. The physical interpretations of both the equation and the spinor are clarified, and it is seen that the space-time frame ${\{\gamma_{\mu}\}}$ arises fully from relative vectors and does not imply the existence of an absolute space-time frame.     

The C*-envelope of the Tensor Algebra of a Directed Graph

Given an arbitrary countable directed graph G we prove the C*- envelope of the tensor algebra coincides with the universal Cuntz- Krieger algebra associated with G. Our approach is concrete in nature and does not rely on Hilbert module machinery. We show how our results extend to the case of higher rank graphs, where an analogous result is obtained for the tensor algebra of a row-finite k-graph with no sources.     

Judith R. Goodstein: Einstein’s Italian Mathematicians – Ricci,Levi-Civita,and the Birth of General Relativity

Mathematische Semesterberichte -     

一般扭算子李代数

由扭算子构成的扭算子李代数在李代数理论中占有重要的位置,首先构造了一般形式的扭顶点算子Z~σ(E_(ij),α,β,z),然后给出了一般扭算子李代数g(G,l)[σ],研究了一般扭顶点算子所具有的性质.     

On Preservation of the Riemann Tensor With Respect to Some Mappings of Affinely Connected Space

This paper is devoted to geodesic and almost geodesic mappings of affinely connected spaces. We find conditions which ensure that the Riemann tensor is an invariant geometric object with respect to the studied mappings. In this work we present an example of a non-trivial geodesic mapping between the flat spaces.     

广义相对论之守恒定律与重力波辐射

本文证明爱因斯坦之广义相对论运动方程能演变成为物质守恒定律。虽然获得的重力场能量动量张量相等于Ｌａｎｄａｕ－Ｌｉｆｓｈｉｔｚ所得的．但演变过程之物理概念是不同的。除此之外，本文分别从物体运动方程和测地线方程着手，提供了两个寻找重力波辐射的方法。     

Applications of Fixed Point Theorems to the Vacuum Einstein Constraint Equations with Non-Constant Mean Curvature

In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer’s fixed point theorem (known methods use Schauder’s fixed point theorem), while the second one uses the concept of half-continuity coupled with the introduction of local supersolutions. These methods allow to: unify some recent existence results, simplify many proofs (for instance, the one of the main theorems in Dahl et al., Duke Math J 161(14):2669–2697, 2012) and weaken the assumptions of many recent results.     

良序套张量积的代数中的强不可约算子

设Ｈ为复的可分无限维Ｈｉｌｂｅｒｔ空间，称有界线性算子Ｔ为强不可约的，如果与Ｔ可交换的幂等算子只有０和Ｉ．王宗尧、蒋春澜、纪有清等人证明了在任何一个套的套代数中都存在大量的强不可约算子，并且找到了它们的酉轨道闭包．本文考虑有限个套的张量积的代数中强不可约算子的存在性问题。证明了：对复平面上任何一个连通完备集σ、总存在一个对角算子Ｎ和它的一个范数可以任意小的紧摄动Ｔ＝Ｘ＋Ｋ，使得Ｔ是一个强不可约算子、Ｔ在有限个良序套的张量积的代数中，并且σ（Ｔ）＝σｌｒｅ（Ｔ）＝σ（Ｎ）＝σｌｒｅ（Ｎ）＝σ进一步，文章还对具有单点谱的算子和良序套与正交补为良序套的张量积的代数进行了讨论，得到了一些结果．     

On the Validity of the Definition of Angular Momentum in General Relativity

We exam the validity of the definition of the ADM angular momentum without the parity assumption. Explicit examples of asymptotically flat hypersurfaces in the Minkowski spacetime with zero ADM energy–momentum vector and finite non-zero angular momentum vector are presented. We also discuss the Beig–Ó Murchadha–Regge–Teitelboim center of mass and study analogous examples in the Schwarzschild spacetime.     

The limits of Riemann solutions to the isentropic magnetogasdynamics

The objective of this note is to prove that the Riemann solutions of the isentropic magnetogasdynamics equations converge to the corresponding Riemann solutions of the transport equations by letting both the pressure and the magnetic field vanish. The delta shock wave can be obtained as the limit of two shock waves and the vacuum state can be obtained as the limit of two rarefaction waves. Moreover the relation between the speed of formation of singular density and those of the vanishing pressure and the vanishing magnetic field is discussed in detail.     

An Expression for the Riemann Tensor of a Submanifold Given by a System of Equations in a Riemannian Space

We derive an expression for the Riemann tensor of a submanifold given implicitly by a system of independent equations in a Riemannian space. In particular, we prove a formula for the internal curvature of a two-surface in a three-dimensional Riemannian space. Some applications of the formula are given.     

Einstein Equations on a 5-Manifold with a Causal Structure in the Absence of Matter Fields

A 5-manifold with a restricted smooth structure and an appropriate group of coordinate transformations including general relativity and gauge transformations is considered. An explicit expression for the Riemannian curvature of a 4-dimensional vector distribution is obtained, which implies the classical Einstein and Maxwell equations. Bibliography: 14 titles.     

The Quiver of an Algebra Associated to the Mantaci-Reutenauer Descent Algebra and the Homology of Regular Semigroups

We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola’s results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao’s algebra, which is related to the Mantaci-Reutenauer descent algebra.     

The Inner Cauchy Horizon of Axisymmetric and Stationary Black Holes with Surrounding Matter in Einstein–Maxwell Theory: Study in Terms of Soliton Methods

We use soliton methods in order to investigate the interior electrovacuum region of axisymmetric and stationary, electrically charged black holes with arbitrary surrounding matter in Einstein–Maxwell theory. These methods can be applied since the Einstein–Maxwell vacuum equations permit the formulation in terms of the integrability condition of an associated linear matrix problem. We find that there always exists a regular inner Cauchy horizon inside the black hole, provided the angular momentum J and charge Q of the black hole do not vanish simultaneously. Moreover, the soliton methods provide us with an explicit relation for the metric on the inner Cauchy horizon in terms of that on the event horizon. In addition, our analysis reveals the remarkable universal relation (8πJ)² + (4πQ ²)² = A ⁺ A ^?, where A ⁺ and A ^? denote the areas of event and inner Cauchy horizon, respectively.