L. Ake Hau, Armando J. Cabrera Pacheco, Didier A. Solis
In this work we provide the full description of the upper levels of the classical causal ladder for spacetimes in the context of Lorenztian length spaces, thus establishing the hierarchy between them. We also show that global hyperbolicity, causal simplicity, causal continuity, stable causality and strong causality are preserved under distance homothetic maps.
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In this work we revisit the notion of the (future) causal completion of a globally hyperbolic spacetime and endow it with the structure of a Lorentzian pre-length space. We further carry out this construction for a certain class of generalized Robertson-Walker spacetimes.
A classical result in Lorentzian geometry states that a strongly causal spacetime is globally hyperbolic if and only if the Lorentzian distance is finite valued for every metric choice in the conformal class. It is proven here that a non-total imprisoning spacetime is globally hyperbolic if and only if for every metric choice in the conformal class the Lorentzian distance is continuous. Moreover, it is proven that a non-total imprisoning spacetime is causally simple if and only if for every metric choice in the conformal class the Lorentzian distance is continuous wherever it vanishes. Finally…
We consider Lorentzian manifolds as examples of partially ordered measure spaces, sets endowed with compatible partial order relations and measures, in this case given by the causal structure and the volume element defined by each Lorentzian metric. This places the structure normally used to describe spacetime in geometrical theories of gravity in a more general context, which includes the locally finite partially ordered sets of the causal set approach to quantum gravity. We then introduce a function characterizing the closeness between any two partially ordered measure spaces and show that, …
Causal set quantum gravity is a Lorentzian approach to quantum gravity, based on the causal structure of spacetime. It models each spacetime configuration as a discrete causal network of spacetime points. As such, key questions of the approach include whether and how a reconstruction of a sufficiently coarse-grained spacetime geometry is possible from a causal set. As an example for the recovery of spatial geometry from discrete causal structure, the construction of a spatial distance function for causal sets is reviewed. Secondly, it is an open question whether the path sum over all causal se…
Causal properties of Lorentzian symmetric spaces are investigated in the paper. The global hyperbolicity of the Cahen--Wallach Lorentzian symmetric spaces is proved.
A Closeness Function on Coarse Grained Lorentzian Geometries
本研究基于随机因果集的离散区间构建粗粒度洛伦兹几何的贴近度函数，虽弱于洛伦兹Gromov-Hausdorff距离，但可对大型因果集数值计算，为因果集理论类连续行为提供定量度量并定义弱收敛条件。
In this conference published in 1997 some problems on the geodesics of a Lorentzian manifold concerning causality and infinite-dimensional variational methods, are pointed out. Even though a big progress on many of these questions have been carried out since then, some computations in this paper may be useful and have not been published elsewhere. Among them, for example, the following one (Section 3). Consider a spacetime which can be written globally as a product $R x M$, such that the natural vector field associated to the coordinate $t$ in $R$ is timelike. When is this spacetime globally h…
A first step towards implementing a notion of coarse graining in an intrinsically Lorentzian, discrete quantum- gravity approach, namely causal set quantum gravity is taken. It makes use of an abstract notion of scale, based on counting the number of discrete elements. To that end, the space of actions for causal set quantum gravity is written in a matrix-model-like language, and a flow equation for the effective action of the model is derived from the path integral.
It is shown that the warped product spacetime P=M *_f H, where H is a complete Riemannian manifold, and the original spacetime M share necessarily the same causality properties, the only exceptions being the properties of causal continuity and causal simplicity which present some subtleties. For instance, it is shown that if diamH=+\infty, the direct product spacetime P=M*H is causally simple if and only if (M,g) is causally simple, the Lorentzian distance on M is continuous and any two causally related events at finite distance are connected by a maximizing geodesic. Similar conditions are fo…
We give a brief introduction to causal fermion systems with a focus on the geometric structures in space-time.
We define and study a new kind of relation between two diffeomorphic Lorentzian manifolds called {\em causal relation}, which is any diffeomorphism characterized by mapping every causal vector of the first manifold onto a causal vector of the second. We perform a thorough study of the mathematical properties of causal relations and prove in particular that two given Lorentzian manifolds (say $V$ and $W$) may be causally related only in one direction (say from $V$ to $W$, but not from $W$ to $V$). This leads us to the concept of causally equivalent (or {\em isocausal} in short) Lorentzian manif…
We recast the tools of ``global causal analysis'' in accord with an approach to the subject animated by two distinctive features: a thoroughgoing reliance on order-theoretic concepts, and a utilization of the Vietoris topology for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call $K^+$, and in terms of it we formulate extended definitions of concepts like causal curve and global hyperbolicity. In particular we prove that, in a spacetime $\M$ which is free of causal cycles, one may define a causal curve simply as a compact connected subset…
We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that existence of time functions remains true on domains of dependence with continuous metrics, and that $C^{0,1}$ differentiability of the metric suffices for many key results of the smooth causality theory.
We investigate the causal hierarchy in various modified theories of gravity. In general relativity the standard causal hierarchy, (key elements of which are chronology, causality, strong causality, stable causality, and global hyperbolicity), is well-established. In modified theories of gravity there is typically considerable extra structure, (such as: multiple metrics, aether fields, modified dispersion relations, Horava-like gravity, parabolic propagation, etcetera), requiring a reassessment and rephrasing of the usual causal hierarchy. We shall show that in this extended framework suitable …
For the construction of the Lorentzian path integral for gravity one faces two main questions: Firstly, what configurations to include, in particular whether to allow Lorentzian metrics that violate causality conditions. And secondly, how to evaluate a highly oscillatory path integral over unbounded domains. Relying on Picard-Lefschetz theory to address the second question for discrete Regge gravity, we will illustrate that it can also answer the first question. To this end we will define the Regge action for complexified variables and study its analytical continuation. Although there have bee…
En Route to Reduction: Lorentzian Manifolds and Causal Sets
本文从哲学还原视角出发，分析因果集理论通过普朗克标度复现洛伦兹流形、以泊松撒点为核心概率方法的研究路径，论证其内格尔式还原的可行性并探讨功能主义变体前景。
Causal spinfoam vertex for 4d Lorentzian quantum gravity
本文提出一种四维洛伦兹量子引力的因果spinfoam顶点，通过Toller T矩阵和费曼iε表示，探讨因果数据在不同模型中的表现及大自旋极限下的选择机制。
This thesis start by a review of different approaches to classical and quantum gravity. The main theme is Lorentzian Gromov Hausdorff theory which is an active diffeomorphism invariant theory on the space of Lorentz spaces (think about globally hyperbolic spacetimes). It is argued why such theory might be of significant importance for Lorentzian approaches to quantum gravity such as causal set theory and Lorentzian dynamical triangulations
Causal structure and light-conical singularities in Lorentzian simplicial quantum gravity
本文研究洛伦兹Regge引力中的光锥奇点，开发算法采样四维三角剖分构型空间，发现光锥奇点普遍存在，分析其结构及因果不规则性对引力路径积分因果条件选择的启示。
The theory of noncommutative geometry provides an interesting mathematical background for developing new physical models. In particular, it allows one to describe the classical Standard Model coupled to Euclidean gravity. However, noncommutative geometry has mainly been developed using the Euclidean signature, and the typical Lorentzian aspects of space-time, the causal structure in particular, are not taken into account. We present an extension of noncommutative geometry à la Connes suitable the for accommodation of Lorentzian structures. In this context, we show that it is possible to recove…
To solve the path integral for quantum gravity, one needs to regularise the space-times that are summed over. This regularisation usually is a discretisation, which makes it necessary to give up some paradigms or symmetries of continuum physics. Causal dynamical triangulations regularises the path integral through a simplicial discretisation that introduces a preferred time foliation. The first part of this thesis presents three articles on causal dynamical triangulations. The first article shows how to obtain a multicritical 2d model by coupling the theory to hard dimers. The second explores …
We formulate a non-perturbative lattice model of two-dimensional Lorentzian quantum gravity by performing the path integral over geometries with a causal structure. The model can be solved exactly at the discretized level. Its continuum limit coincides with the theory obtained by quantizing 2d continuum gravity in proper-time gauge, but it disagrees with 2d gravity defined via matrix models or Liouville theory. By allowing topology change of the compact spatial slices (i.e. baby universe creation), one obtains agreement with the matrix models and Liouville theory.
We give a covariant definition of closeness between (time oriented) Lorentzian metrics on a manifold M, using a family of functions which measure the difference in volume form on one hand and the difference in causal structure relative to a volume scale on the other hand. These functions will distinguish two geometric properties of the Alexandrov sets $ A(p,q), \tilde{A} (p,q) $ relative to two space time points $q$ and $p$ and metrics $g$ and $ \tilde{g} $. It will be shown that this family generates uniformities and consequently a topology on the space of Lorentzian metrics which is Hausdorf…
Several classes of directed acyclic graphs have been investigated in the last two decades, in the context of the Causal Set Program, in search for good discrete models of spacetime. We introduce some statistical indicators that can be used for comparing these graphs and for assessing their closeness to the ideal Lorentzian causal sets ('causets') -- those obtained by sprinkling points in a Lorentzian manifold. In particular, with the reversed triangular inequality of Special Relativity in mind, we introduce 'longest/shortest path plots', an easily implemented tool to visually detect the extent…
The causal relation $K^+$ was introduced by Sorkin and Woolgar to extend the standard causal analysis of $C^2$ spacetimes to those that are only $C^0$. Most of their results also hold true in the case of spacetimes with degeneracies. In this paper we seek to examine $K^+$ explicitly in the case of Lorentzian topology changing Morse spacetimes containing isolated degeneracies. We first demonstrate some interesting features of this relation in globally Lorentzian spacetimes. In particular, we show that $K^+$ is robust and that it coincides with the Seifert relation when the spacetime is stably c…
We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive the objects of our quantum geometry: the spin space, the tangent space endowed with a Lorentzian metric, connection and curvature. In order to get the correspondence to differential geometry, we construct examples of causal fermion systems by regularizing Dirac sea configurations i…
Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch's splitting. A revision
本文综述时间函数与 Cauchy 超曲面可光滑化问题的最新成果，并围绕因果连续、稳定因果及整体双曲时空给出自足证明。
Studying transition amplitudes in (2+1)-dimensional causal dynamical triangulations, Cooperman and Miller discovered speculative evidence for Lorentzian quantum geometries emerging from its Euclidean path integral. On the basis of this evidence, Cooperman and Miller conjectured that Lorentzian de Sitter spacetime, not Euclidean de Sitter space, dominates the ground state of the quantum geometry of causal dynamical triangulations on large scales, a scenario akin to that of the Hartle-Hawking no-boundary proposal in which Lorentzian spacetimes dominate a Euclidean path integral. We argue against…
We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In particular, for stationary spacetimes, we give a simple characterization of when they are causally continuous or globally hyperbolic (including in the latter case, when $S$ is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. Causality suggests that the role of completeness in many results o…
In this work we define and study the relations between Lorentzian Manifolds given by the diffeomorphisms which map causal future directed vectors onto causal future directed vectors. This class of diffeomorphisms, called proper causal relations, contains as a subset the well-known group of conformal relations and are deeply linked to the so-called causal tensors. If two given Lorentzian manifolds are in mutual proper causal relation then they are said to be causally isomorphic: they are indistinguishable from the causal point of view. Finally, the concept of causal transformations for Lorentzi…
The causality structure of two-dimensional manifolds with degenerate metrics is analysed in terms of global solutions of the massless wave equation. Certain novel features emerge. Despite the absence of a traditional Lorentzian Cauchy surface on manifolds with a Euclidean domain it is possible to uniquely determine a global solution (if it exists), satisfying well defined matching conditions at the degeneracy curve, from Cauchy data on certain spacelike curves in the Lorentzian region. In general, however, no global solution satisfying such matching conditions will be consistent with this data…
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…
Lorentzian metric spaces and their Gromov-Hausdorff convergence
本文提出了一种不依赖辅助度量的抽象洛伦兹 Gromov-Hausdorff 距离与收敛理论，定义了有界洛伦兹度量空间，证明了其等距同构与同胚性质，并验证了因果曲线、预长度空间及截面曲率界限在 GH 极限下的稳定性。
The causal structure is a quintessential element of continuum spacetime physics and needs to be properly encoded in a theory of Lorentzian quantum gravity. Established spin foam (and tensorial group field theory (TGFT)) models mostly work with relatively special classes of Lorentzian triangulations (e.g. built from spacelike tetrahedra only), obscuring the explicit implementation of the local causal structure at the microscopic level. We overcome this limitation and construct a full-fledged model for Lorentzian quantum geometry the building blocks of which include spacelike, lightlike and time…
Lorentzian Spectral Geometry with Causal Sets
研究离散洛伦兹谱几何，通过因果矩阵导出的算子谱等几何不变量识别因果集，提出两种分类方法并在多达9个元素的因果集上测试，其中结合扰动的新方法能完全区分因果集，并探讨了在量子引力路径积分中的应用。
We compute Teitelboim's causal propagator in the context of canonical loop quantum gravity. For the Lorentzian signature, we find that the resultant power series can be expressed as a sum over branched, colored two-surfaces with an intrinsic causal structure. This leads us to define a general structure which we call a ``causal spin foam''. We also demonstrate that the causal evolution models for spin networks fall in the general class of causal spin foams.
Cosmology in Lorentzian Regge calculus: causality violations, massless scalar field and discrete dynamics
本文在洛伦兹Regge微积分中构建了平坦宇宙学模型，发现类空子单元导致因果性违背，引入无质量标量场后证明了其单调演化及在连续极限下恢复Friedmann方程，为有效自旋泡沫路径积分研究奠定基础。
A locally finite, causal and quantal substitute for a locally Minkowskian principal fiber bundle $\cal{P}$ of modules of Cartan differential forms $\omg$ over a bounded region $X$ of a curved $C^{\infty}$-smooth differential manifold spacetime $M$ with structure group ${\bf G}$ that of orthochronous Lorentz transformations $L^{+}:=SO(1,3)^{\uparrow}$, is presented. ${\cal{P}}$ is the structure on which classical Lorentzian gravity, regarded as a Yang-Mills type of gauge theory of a $sl(2,\com)$-valued connection 1-form $\cal{A}$, is usually formulated. The mathematical structure employed to mo…
In this work we introduce the taxicab and uniform products for Lorentzian pre-length spaces. We further use these concepts to endow the space $D(R\times_T X)$ of causal diamonds with a Lorentzian length space structure, closely relating its causal properties with its geometry as a metric space furnished with its associated Hausdorff distance. Among the general results, we show that this space is geodesic and globally hyperbolic for complete $X$.
Null distance and convergence of Lorentzian length spaces
本文将零距离概念从光滑时空推广到洛伦兹长度空间，并基于该距离研究扭曲积洛伦兹长度空间的Gromov-Hausdorff收敛，首次证明了其与合成曲率界的相容性。
We introduce a canonical, compact topology, which we call weakly causal, naturally generated by the causal site of J. D. Christensen and L. Crane, a pointless algebraic structure motivated by certain problems of quantum gravity. We show that for every four-dimensional globally hyperbolic Lorentzian manifold there exists an associated causal site, whose weakly causal topology is co-compact with respect to the manifold topology and vice versa. Thus, the causal site has the full information about the topology of space-time, represented by the Lorentzian manifold. In addition, we show that there e…
Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a $C^*$-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' nonco…
Null geodesics, causal structure, and matter accretion in Lorentzian-Euclidean black holes
本文进一步探索洛伦兹-欧几里得黑洞模型，通过分析零测地线发现光子无法穿过事件视界，加强中心奇点不可达结论，并讨论了时空因果结构与物质吸积过程。
A change of spatial topology in a causal, compact spacetime cannot occur when the metric is globally Lorentzian. One can however construct a causal metric from a Riemannian metric and a Morse function on the background cobordism manifold, which is Lorentzian almost everywhere except that it is degenerate at each critical point of the function. We investigate causal structure in the neighbourhood of such a degeneracy, when the auxiliary Riemannian metric is taken to be Cartesian flat in appropriate coordinates. For these geometries, we verify Borde and Sorkin's conjecture that causal discontinu…
Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems
本文研究形式为 $I\ imes_f X$ 的洛伦兹扭曲积推广，即广义锥作为洛伦兹长度空间，探讨其因果性、综合曲率界与度量曲率的关系，并证明非正下类时曲率界蕴含不完备类时测地线存在的奇点定理。
Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary $\partial V$ for a standard conformally stationary spacetime V = R x M, suggests a natural compactification $M_B$ associated to any Riemannian metric on M or, more generally, to any Finslerian one. The corresponding boundary $\partial_B M$ is constructed in terms of Busemann-type functions. Roughly, $\partial_B M$ represents the set of all the directions in M including both, asymptotic and "finite" (or "incomplete") directions. This Busemann boundary $\partial_…
Conformal transformations of metric spaces and Lorentzian pre-length spaces
本文提出在度量空间和洛伦兹预长度空间中定义共形变换，并分析其性质，包括共形长度、角度不变性及因果结构的等价关系。
Time functions on Lorentzian length spaces
本文在 Lorentzian（预）长度空间框架下建立时间函数的经典存在性理论，用 $K$-因果性、因果连续性及 Cauchy 时间函数刻画相关性质，证明无需流形结构。
Lorentzian Gromov-Hausdorff convergence and pre-compactness
本文引入洛伦兹空间的Gromov-Hausdorff收敛概念，基于因果钻石的ε-网和时间分离函数，证明了洛伦兹版本的Gromov预紧定理，并应用于全局双曲时空和曲率驱动的预紧性。
Further observations on the definition of global hyperbolicity under low regularity
本文澄清了洛伦兹长度空间中全局双曲性定义与闭锥结构及拓扑预序空间定义的一致性，指出该定义仅依赖因果闭序和因果凸包的紧致保持性，与度量、时间序关系及洛伦兹距离无关。
Because no closed timelike curve (CTC) on a Lorentzian manifold can be deformed to a point, any such manifold containing a CTC must have a topological feature, to be called a timelike wormhole, that prevents the CTC from being deformed to a point. If all wormholes have horizons, which typically seems to be the case in space-times without exotic matter, then each CTC must transit some timelike wormhole's horizon. Therefore, a Lorentzian manifold containing a CTC may nevertheless be causally well behaving once its horizon's are deleted. For instance, there may be a Cauchy-like surface through wh…
The causal spacetimes admitting a covariantly constant null vector provide a connection between relativistic and non-relativistic physics. We explore this relationship in several directions. We start proving a formula which relates the Lorentzian distance in the full spacetime with the least action of a mechanical system living in a quotient classical space time. The timelike eikonal equation satisfied by the Lorentzian distance is proved to be equivalent to the Hamilton-Jacobi equation for the least action. We also prove that the Legendre transform on the classical base corresponds to the mus…
Truly Lorentzian quantum cosmology
本文提出将迷你超空间路径积分限制在正尺度因子平方的洛伦兹构型上，发现这能显著改变期望值、实现奇点避免，并表明半经典鞍点近似并非总是有效，需重新审视隧穿、无边界提议、反弹宇宙学及暴胀量子起源等相关工作。
Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry
该研究在洛伦兹几何中建立格罗莫夫重构定理，提出归一化有界洛伦兹度量测度空间的同构概念与三种收敛性定义并证明基本性质，给出其在因果集理论时空重构问题中的应用。
A Lorentzian analog for Hausdorff dimension and measure
本文在洛伦兹长度空间上定义单参数典范体积测度，引入类似豪斯多夫维数的几何维数以区分闵可夫斯基时空中的类空与零子空间，并提出因果加倍条件与因果加倍测度，为合成时空的塌缩定义和类时曲率界研究提供工具。
The full causal ladder of spacetimes is constructed, and their updated main properties are developed. Old concepts and alternative definitions of each level of the ladder are revisited, with emphasis in minimum hypotheses. The implications of the recently solved ``folk questions on smoothability'', and alternative proposals (as recent isocausality), are also summarized.
We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a $\mathcal{C}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an $α$-Hölder continuous Lorentzian metric admit a $\mathcal{C}^{1,\fracα{4}}$-parametrization.
The c-completion of Lorentzian metric spaces
本文重新审视洛伦兹度量空间概念，在极大普遍性下构建了包含低正则性时空的c-完备化，并证明其自身具有洛伦兹度量空间结构且能精确完备原空间并反映其因果性质。
We consider pseudoconvexity properties in Lorentzian and Riemannian manifolds and their relationship in static spacetimes. We provide an example of a causally continuous and maximal null pseudoconvex spacetime that fails to be causally simple. Its Riemannian factor provides an analogous example of a manifold that is minimally pseudoconvex, but fails to be convex.
Causal structure and topology change in (2+1)-dimensional simplicial gravity
研究(2+1)维简形引力中因果结构与拓扑变化，通过分类洛rentzian四面体因果类型，揭示顶点和hinge因果性的独立性及其对离散量子引力的影响。
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