Ricci curvature and metric in causal spacetimes
Javier Lafuente-López
本文研究了因果时空中的里奇曲率与度量关系，证明了在可行时空条件下，保持里奇张量的因果微分同胚必为相似变换。
一个可行的时空是指允许存在完整的类时测地线的时空。证明了如果一个时空是可行的，那么保持两个时空之间里奇张量的因果微分同胚必定是相似变换。
A viable spacetime is one that admits a complete timelike geodesic. It is shown that a causal diffeomorphism preserving the Ricci tensor between two spacetimes is necessarily a homothety, if one of them is viable.
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We find all Ricci semi-symmetric as well as all conformally semi-symmetric spacetimes. Neither of these properties implies the other. We verify that only conformally flat spacetimes can be Ricci semi-symmetric without being conformally semi-symmetric and show that only vacuum spacetimes and spacetimes with just a $Λ$-term can be Ricci semi-symmetric without being conformally semi-symmetric.
We consider the kinematics of specific fluid spacetimes admitting timelike congruences of Ricci Solitons. These fluids includes string cloud, string fluid, perfect fluid, radially symmetric fluid, anisotropic fluid and relativistic magneto-fluid. Results are obtained and important physical aspects are discussed.
Pseudo generalized Ricci-recurrent spacetimes and modified gravity
该研究引入并刻画伪广义Ricci递推时空并验证其存在性，证明其相关共形平坦情形为伪拟Einstein时空、其广义Robertson-Walker形式为理想流体时空，还推导了其在两类修正引力下的能量条件。
Pseudo generalized Ricci-recurrent spacetimes with certain applications to modified gravity
本文引入伪广义Ricci递推时空并证明其存在，分析了其在广义Robertson-Walker时空与完美流体时空中的性质，还研究了其在$f(\mathcal{R})$引力下的能量条件。
In the present paper we discuss about a set of geometric and physical properties of hyper-generalised quasi-Einstein spacetime. At the beginning we discuss about pseudosymmetry over a hyper-generalised quasi-Einstein spacetime. Here we discuss about W2-Ricci pseudosymmetry, Z-Ricci pseudosymmetry, Ricci pseudosymmetry and projective pseudosymmetry over a hyper-generalised quasi-Einstein spacetime. Later on we take over Ricci symmetric hyper-generalised quasi-Einstein spacetime and derive a set of important geometric and physical theorems over it. Moving further we consider some physical applic…
In this paper geometrical aspects of perfect fluid spacetime with torse-forming vector field ξare discribed and Ricci soliton in perfect fluid spacetime with torse-forming vector field ξare determined. Conditions for the Ricci soliton to be expanding, steady or shrinking are also given.
Almost Gradient Ricci Solitons on Static Spacetime
本文研究静态时空上几乎梯度里奇孤子的几何性质，确定其存在条件，证明在真空静态时空下退化为稳态孤子，并探讨膨胀孤子满足薛定谔方程、能量条件及底流形为爱因斯坦流形等结果。
Local regularity of spacetimes under Ricci curvature and Lie derivative conditions
本文研究了4维时空的局部正则性，通过Ricci曲率和Lorentzian度量关于任意类时向量场的Lie导数的先验界，推导出一般的正则性估计。
The present paper is to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field ξ in connection with conformal Ricci-Yamabe metric and conformal η-Ricci-Yamabe metric. Here we have delineated the conditions for conformal Ricci-Yamabe soliton to be expanding, steady, or shrinking. Later, we have acquired Laplace equation from conformal η-Ricci-Yamabe soliton equation when the potential vector field ξ of the soliton is of gradient type. Lastly, we have designated perfect fluid with Robertson-Walker spacetime and some applications of physics and gravity.
Ricci solitons and curvature inheritance on Robinson-Trautman spacetimes
本文研究Robinson-Trautman时空上的Ricci孤立子与曲率继承，引入广义曲率继承和广义Lie继承概念，证明该时空在特定条件下允许多种孤立子并实现广义继承与共线。
We prove the linear stability of Schwarzschild-Tangherlini spacetimes and their Anti-de Sitter counterparts under Ricci flow for a special class of perturbations. This is useful in the choice of suitable initial conditions in numerical Ricci-flow-based algorithms for obtaining new solutions to the Einstein equation when the cosmological constant is zero or negative. The Ricci flow is a first-order renormalization group (RG) flow in string theory, and its solutions are believed to approximate string field theory processes in certain cases. Thus this result offers insights into the off-shell sta…
Conformally flat pseudo-Riemannian manifolds with generalized Ricci recurrent, $(GR)_n$ structure are completely classified in this short report. A conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is shown to be either a de Sitter space or an anti-de Sitter space. In particular, a conformally flat generalized Ricci recurrent spacetime must be either a de Sitter spacetime or an anti-de Sitter spacetime.
Identifying any conformally round metric on the $2$-sphere with a unique cross section on the standard lightcone in the $3+1$-Minkowski spacetime, we gain a new perspective on $2d$-Ricci flow on topological spheres. It turns out that in this setting, Ricci flow is equivalent to a null mean curvature flow first studied by Roesch--Scheuer along null hypersurfaces. Exploiting this equivalence, we can translate well-known results from $2d$-Ricci flow first proven by Hamilton into a full classification of the singularity models for null mean curvature flow in the Minkowski lightcone. Conversely, we…
Ollivier-Ricci curvature for causal sets
利用洛伦兹最优传输引入因果集的Ollivier-Ricci曲率，通过离散化方法恢复时间型Ricci曲率，并证明局部到全局传播结果和类时Bonnet-Myers定理。
Complete $(2+1)$-dimensional Ricci flow spacetimes
本文研究完备的(2+1)维Ricci流时空，证明其世界线的存在性、起始时刻的可数性，非负曲率下的结构性质，并应用于二维Ricci孤立子的向量场完备性问题。
The causal structure of a strongly causal spacetime is particularly well endowed. Not only does it determine the conformal spacetime geometry when the spacetime dimension n >2, as shown by Malament and Hawking-King-McCarthy (MHKM), but also the manifold dimension. The MHKM result, however, applies more generally to spacetimes satisfying the weaker causality condition of future and past distinguishability (FPD), and it is an important question whether the causal structure of such spacetimes can determine the manifold dimension. In this work we show that the answer to this question is in the aff…
We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of [32]. Results of Bamler-Kleiner [8] and Haslhofer [29] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup r…
In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the secondvariation of the {\it time arrival functional} along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of the critical points of the time arrival functional is investigated and a Morse index theorem in the context of Finsler spacetime is presented.
Static Vacuum Spacetimes with $Λ<0$ as Attractors of the Ricci-Harmonic Flow
研究了Λ<0的静态渐近超几何解的动态稳定性，证明了正质量定理与Ricci-谐波流的稳定性之间的关系。
Timelike Ricci bounds for low regularity spacetimes by optimal transport
该研究针对低正则性时空，利用最优输运方法证明了分布意义下类时Ricci有下界的C¹正则整体双曲时空满足类时测度收缩性质及相关几何不等式，C¹,¹正则下还成立更强曲率-维数条件且耦合与测地线唯一。
On the causality of gravitational pp-wave spacetimes: A null version of Anderson's rigidity theorem
本文证明平面波是唯一区分且强因果的引力 pp-波时空，并据此证明安德森刚性定理的空版本：满足完备性与强因果等条件的 4 维 Ricci 平坦 Brinkmann 时空必为平面波时空的商。
In 2002, Isenberg-Mazzeo-Pollack (IMP) constructed a series of vacuum initial data sets via a gluing construction. In this paper, we investigate some local geometry of these initial data sets as well as implications regarding their spacetime developments. In particular, we state conditions for the existence of outer trapped surfaces near the center of the IMP gluing neck and thence use a generalization of the Penrose incompleteness theorem to deduce null incompleteness of the resulting spacetimes.
B List has recently studied a geometric flow whose fixed points correspond to static Ricci flat spacetimes. It is now known that this flow is in fact Ricci flow modulo pullback by a certain diffeomorphism. We use this observation to associate to each static Ricci flat spacetime a local Ricci soliton in one higher dimension. As well, solutions of Euclidean-signature Einstein gravity coupled to a free massless scalar field with nonzero cosmological constant are associated to shrinking or expanding Ricci solitons. We exhibit examples, including an explicit family of complete expanding solitons wh…
Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes
本文利用 q-Lorentz-Wasserstein 最优输运与概率测度 q-测地线，证明 Finsler 时空在加权 Ricci 曲率类时下界条件下满足 TCD_q(K,N)，并导出尖锐类时 Brunn-Minkowski 不等式。
In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off-diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/Ricci flow models with nontrivial torsion to configurations with the Levi-Civita connection is allowed in some specific physical circumstances by constraining the class of integral varieties for the Eins…
On the structure of noncollapsed Ricci flow limit spaces
本文研究了闭里奇流在熵有界下的模空间弱紧性，并揭示了极限空间的结构，证明正则部分具有里奇流时空结构，奇点集的余维至少为四。
Several uniqueness results on compact maximal hypersurfaces in a wide class of sta- bly causal spacetimes are given. They are obtained from the study of a distinguished function on the maximal hypersurface, under suitable natural first order conditions of the spacetime. As a consequence several applications to Geometric Analysis are given.
Globally hyperbolic spacetimes admitting infinitely many causal (and timelike) homotopy classes of curves joining two prescribed points, are exhibited and discussed.
Characterizations of a spacetime of quasi-constant sectional curvature and $\mathcal{F}(\mathcal{R})$-gravity
本文研究拟常截面曲率时空，证明其与宇宙现状一致且为Robertson Walker时空，并分析Ricci对称性、Petrov分类及$\mathcal{F}(\mathcal{R})$-引力中的能量条件。
Characterizations of Perfect fluid spacetimes obeying $f(\mathcal{R})$-gravity equipped with different gradient solitons
本文研究服从$f(\mathcal{R})$-引力下带四类孤立子度量的理想流体时空，证明孤立子存在性与演化条件，推导梯度型势向量场对应的Poisson方程，得出暗物质时代相关定理。
On pp-waves with lightlike parallel spinors
本文研究了具有光状平行旋量的pp波时空，通过参数化方法展示了其与Ricci平坦度量的一一对应关系，并探讨了null Ricci曲率的影响。
In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion components. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forwa…
Solitons and geometrical structures in a perfect fluid spacetime
该研究围绕理想流体时空的几何结构展开，分析了不同曲率张量下的几何特征，确定了η-Ricci、η-Einstein孤子及Ricci孤子的各类演化条件，并在梯度势场情形下推导出Poisson方程，丰富了相关微分几何研究。
In this work we will focus on the causal character of Carter Spacetime (see B. Carter, Causal structure in space-time, Gen. Rel. Grav. 1 4 337-406, 1971). The importance of this spacetime is the following: for the causally best well behaved spacetimes (the globally hyperbolic ones), there are several characterizations or alternative definitions. In some cases, it has been shown that some of the causal properties required in these characterizations can be weakened. But Carter spacetime provides a counterexample for an impossible relaxation in one of them. We studied the possibility of Carter sp…
In the present paper we discuss about a set of geometric properties and physical applications of a mixed quasi-Einstein spacetime$[M(QE)_{4}]$, which is a special type of nearly quasi-Einstein spacetime$[N(QE)_{4}]$. Firstly we consider a nearly quasi-Einstein manifold$[N(QE)_{n}]$ along with a mixed quasi-Einstein manifold$[M(QE)_{n}]$ and study some geometric conditions with respect to different curvature tensors on them. Then we consider a mixed quasi-Einstein spacetime and study $W_{2}$-Ricci pseudosymmetry and projective Ricci pseudosymmetry on it. Then we study the conditions $H \cdot S=…
We consider the usual causal structure $(I^+,J^+)$ on a spacetime, and a number of alternatives based on Minguzzi's $D^+$ and Sorkin and Woolgar's $K^+$, in the case where the spacetime metric is continuous, but not necessarily smooth. We compare the different causal structures based on three key properties, namely the validity of the push-up lemma, the openness of chronological futures, and the existence of limit causal curves. Recall that if the spacetime metric is smooth, $(I^+,J^+)$ satisfies all three properties, but that in the continuous case, the push-up lemma fails. Among the proposed…
In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices, we prove that every complete CMC hypersurface having either bounded hyperbolic angle or bounded height is maximal. Our conclusions follow from general mean curvature estimates for spacelike hypersurfaces. In case where the spacetime is a Lorentzian product with spatial factor of nonnegative Ricci curvature and sectional curvatures bounded below, we also show…
The idea is considered that a quantum wormhole in a spacetime foam can be described as a Ricci flow. In this interpretation the Ricci flow is a statistical system and every metric in the Ricci flow is a microscopical state. The probability density of the microscopical state is connected with a Perelman's functional of a rescaled Ricci flow.
We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery parameter varies. We prove that the flow with surgery subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.
On the causal discontinuity of Morse spacetimes
本研究构造了一个四维Morse时空反例，其临界点指标为2且各向异性足够大，导致因果不连续，从而否证了Borde-Sorkin猜想在一般情形下的成立。
Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some natural topological assumptions (fulfilled, for example, if one of the conjugacy classes of deck transformations containing a closed timelike curve is finite) hold. As a consequence, any compact Lorentzian manifold conformal to a static spacetime is geodesically connected by causa…
In General Relativity the metric can be recovered from the structure of the lightcones and a measure giving the volume element. Since the causal structure seems to be simpler than the Lorentzian manifold structure, this suggests that it is more fundamental. But there are cases when seemingly healthy causal structure and measure determine a singular metric. Here it is shown that this is not a bug, but a feature, because big-bang and black hole singularities are instances of this situation. But while the metric is special at singularities, being singular, the causal structure and the measure are…
We give an example of a spacetime with a continuous metric which is globally hyperbolic and exhibits causal bubbling. The metric moreover splits orthogonally into a timelike and a spacelike part. We discuss our example in the context of energy conditions and the recently introduced synthetic timelike curvature-dimension (TCD) condition. In particular we observe that the TCD-condition does not, by itself, prevent causal bubbling.
We construct an example which shows that two isocausal spacetimes, in the sense introduced by García-Parrado and Senovilla, may have c-boundaries which are not equal (more precisely, not equivalent, as no bijection between the completions can preserve all the binary relations induced by causality). This example also suggests that isocausality can be useful for the understanding and computation of the c-boundary.
The group of conformal diffeomorphisms and the group of causal automorphisms on two-dimensional globally hyperbolic spacetimes are clarified. It is shown that if spacetimes have non-compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Minkowski spacetime, and if spacetimes have compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Einstein's static universe.
The causal structure of the c-completion of warped spacetimes
本文借助 Lorentzian 预长度空间研究整体双曲 GRW 时空的未来因果完备化，证明在年代拓扑 Hausdorff 条件下其因果完备化仍为整体双曲预长度空间。
One of the central difficulties of settling the $L^2$-bounded curvature conjecture for the Einstein -Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of Enstein-Vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface.
This is the third and last in our series of papers concerning rough solutions of the Einstein vacuum equations expressed relative to wave coordinates. In this paper we prove an important result concerning Ricci defects of microlocalized solutions, stated and used in the proof of the crucial Asymptotics Theorem in our second paper "The causal structure of the microlocalized rough Einstein metrics"
Geometrical Properties of a Point-like Global Monopole Spacetime
本文研究点状全球单极子（PGM）时空的几何性质，证明其具备多种伪对称结构、拟爱因斯坦特性与曲率相关对称性，还验证了不同型曲率张量的曲率继承和共线概念不等价。
Causal completion of a globally hyperbolic conformally flat spacetime
本文研究全局双曲时空的因果完备化，证明在单连通且共形平坦条件下其为同胚于S×[0,1]的带边拓扑流形，并刻画三类时空族的完备化。
It is shown that the space of null geodesics of a star-shaped causally simple subset of Minkowski space is contactomorphic to the canonical contact structure in the spherical cotangent bundle of $\mathbb{R}^n$. In the $3$-dimensional case we prove a similar result for a large class of causally simple contractible subsets of an arbitrary globally hyperbolic spacetime applying methods from the theory of contact-convex surfaces. Moreover we prove that under certain assumptions the space of null geodesics of a causally simple spacetime embeds with smooth boundary into the space of null geodesics o…
We establish the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with $ε$-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie-Yeroshkin and Kuwae-L…
New general results of non-existence and rigidity of spacelike submanifolds immersed in a spacetime, whose mean curvature is a time-oriented causal vector field, are given. These results hold for a wide class of spacetimes which includes globally hyperbolic, stationary, conformally stationary and pp-wave spacetimes, among others. Moreover, applications to the Cauchy problem in General Relativity, are presented. Finally, in the case of hypersurfaces, we also obtain significant consequences in Geometrical Analysis, solving new Calabi-Bernstein and Dirichlet problems on a Riemannian manifold.
It is shown that the space of null geodesics of a causally simple Lorentzian manifold is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions. Examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones.
Causal convergence conditions through variable timelike Ricci curvature bounds
该研究提出非光滑整体双曲正则长度度量时空概念，引入带可变Ricci曲率界的熵类时曲率维数条件，统一广义相对论能量条件，推广相关几何估计与奇点定理，给出稳定性结果与类时几何不等式。
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