J. A. R. Cembranos, J. Gigante Valcarcel, F. J. Maldonado Torralba
In this work we review the study of singularities in Poincaré gauge theories of gravity. Since one of the most recent studies uses the appearance of black hole regions of arbitrary dimension as an indicator of singular behaviour, we also give some explicit examples of these structures and study how particles behave around them.
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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.
Chruściel, Isenberg, and Pollack constructed a class of vacuum cosmological spacetimes that do not admit Cauchy surfaces with constant mean curvature. We prove that, for sufficiently large values of the gluing parameter, these examples are both future and past null geodesically incomplete. The authors are honored to dedicate this paper to Robert Bartnik on the occasion of his 60th birthday.
We study the issue of length renormalization in the context of fully covariant gravity theories as well as non-relativistic ones such as Hořava-Lifshitz gravity. The difference of their symmetry groups implies a relation between the lengths of paths in spacetime in the two types of theory. Provided that certain asymptotic conditions hold, this relation allows us to transfer analytic criteria for the standard spacetime length to be finite to the Perelman length to be likewise finite, and therefore formulate conditions for geodesic incompleteness in partially covariant theories. We also discuss …
Throughout the study of the geodesics of some popular spherically symmetric regular black holes, we hereby prove that the analytically extended Hayward black hole is geodetically incomplete. The simplest extension of the Culetu-Simpson-Visser's non-analytic smooth black hole is also geodetically incomplete, with the exception of the antipodal continuation of the radial geodesics. However, the huge ambiguity in the extension of non analytic spacetimes is tantamount of geodesic incompleteness and such spacetimes do not solve the singularity issue unless at least all the extensions turn out to be…
Supplementing earlier literature by e.g. Tipler, Clarke, & Ellis (1980), Israel (1987), Thorne, (1994), Earman (1999), Senovilla & Garfinkle (2015), Curiel (2019ab), and Landsman (2021ab), I provide a historical and conceptual analysis of Penrose's path-breaking 1965 singularity (or incompleteness) theorem. The emphasis is on the nature and historical origin of the assumptions and definitions used in-or otherwise relevant to-the theorem, as well as on the discrepancy between the (astro)physical goals of the theorem and its actual content: even if its assumptions are met, the theorem fails to p…
Unboundable Spacetimes with Metric Singularities and Matching Metrics and Geodesics: A Black-White Hole and a Big Crunch-Bang
广义相对论的奇点定理利用因果测地线不完备性的概念作为时空奇点存在的判据。因果曲线的不完整性意味着粒子存在的结束和/或开始，这是一个事件。在普遍接受的方法中，奇点不被纳入时空。因此，时空是事件不完全的。从虚无中创造，奇点是无法无天的根源。绕过这些概念性问题的一个直接方法是在时空中引入度量奇点，然后在奇点处匹配度量和因果测地线。为此，我们假设时空流形是无界的，所以奇点只能是内部的。测地线的匹配是通过弱化其光滑性条件来实现的。这种方法被应用于一个黑白洞和一个大的嘎吱声。
An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified concusions may arise, showing that those conclusions will fail only in special cases, at least some of which may be described. These are the so-called rigidity theorems, and have many important examples in the especialized literature. In this paper, we prove rigidity results for generalized plane waves and certain globally hyperbolic spacetimes in the prese…
Past incompleteness of a bouncing multiverse
根据经典的GR，反德西特（AdS）泡沫在多元宇宙中终止于大紧缩奇点。然而，已经证明，基本理论可以解决这些奇异性，并用非奇异反弹来代替它们。这可能对多元宇宙的开始有重要的影响。已知宇宙时空中的测地线是过去不完全的，只要测地线上的平均膨胀率为正，但不清楚如果测地线反复穿过嘎吱作响的AdS泡，后一个条件是否满足。我们在一个简单的多元宇宙模型中研究这个问题，其中时空由FRW区域拼凑而成。结论是，即使存在AdS反弹，时空仍然是过去不完整的。
Some remarks on marginally trapped surfaces and geodesic incompleteness
在最近的一篇论文中，Eichmair，Galloway和Pollack证明了一个基于全局双曲时空非紧初始数据集上边缘外捕获面（MOTS）存在性的Gannon-Lee型奇异性定理。然而，人们可能想知道，在可能的时空非整体双曲扩张中，相应的不完全测地线是否仍然是完全的。在本文中，给出了该结果的一些变体，其中因果关系假设较弱，从而表明答案一般是否定的，至少如果假定的扩展没有封闭的类时曲线。然而，不像在MOTS的情况下，只有外出家庭的正常测地线的约束，我们发现有必要在我们的证明，也施加一个弱收敛条件的正常测地线的传入家庭。换句话说，我们认为边际捕获面（MTS）在时间时空，介绍了一个通用MTS的自然概念。特别是，霍金-彭罗斯型奇异性定理被证明在时间时空的维度大于2包含一个通用MTS。这样的表面自然出现的横截面的准局部概括的黑洞视界，如动力和捕获视界。最后，我们在初始数据集的MTS的存在的一些意见。
Epicycles and Poincaré Resonances in General Relativity
测地线偏差法提供了任意背景时空中测地线的解析近似。因此，该方法在许多实际情况下是有用的工具。在本说明中，我们指出了一些微妙的应用程序的方法有关的长期运动，在第一次以及在更高的顺序。特别是，我们工作的一般二阶贡献束缚轨道史瓦西时空，并表明，它提供了很好的分析结果，一直到最里面的稳定的圆形轨道。
On the geodesic incompleteness of spacetimes containing marginally outer trapped surfaces
在最近的一篇论文中，Eichmair，Galloway和Pollack证明了一个基于全局双曲时空非紧初始数据集上边缘外捕获面（MOTS）存在性的Gannon-Lee型奇异性定理。这一结果要求MOTS在适当的意义上是通用的。本着同样的精神，作者证明了该结果的一些变体，这些变体适用于时空上较弱的因果条件，但它们涉及（一般）边际捕获面（MTS）而不是MOTS，即，大多数结果还需要一个关于引入的正规零测地线族收敛的条件。然而，最近的许多文献都集中在MOTS，而不是MTS作为描述黑洞的准本地替代品，因为它们可以说是更自然，更容易在许多情况下处理。因此，我们有必要问一问，在多大程度上可以推断出只存在MOTS时奇点的存在。在这篇文章中，我们将讨论这个问题，并证明在较弱的因果条件下（特别是因果简单时空），在一般MOTS存在的情况下，奇点仍然会出现。此外，如果我们假设MOTS是一个紧凑的空间区域的边界，Penrose-Hawking型奇异性定理可以建立包含一般MOTS的时间时空。
Geodesics around gravitational dislocations
本文提出了一种基于齐次函数的球对称重力直接Poincaré规范理论位错解的生成方法，该方法使Cartan挠率为零。给出了静态的空间支撑位错解和时间相关解。光子沿测地线运动，类似于固体物理中位错周围的电子所描述的测地线。还发现了快子扇区。
Geodesics at Sudden Singularities
我们表明，爱因斯坦方程的一般解决方案，描述方法的非均匀和各向异性的突然时空奇点不经历测地线不完整性。这推广了为各向同性和均匀宇宙建立的结果。进一步讨论的弱点的奇异性也包括在内。
Noncommutative spaces of geodesics provide an alternative way of introducing noncommutative relativistic kinematics endowed with quantum group symmetry. In this paper we present explicitly the seven noncommutative spaces of time-, space- and light-like geodesics that can be constructed from the time-, space- and light- versions of the $κ$-Poincaré quantum symmetry in (3+1) dimensions. Remarkably enough, only for the light-like (or null-plane) $κ$-Poincaré deformation the three types of noncommutative spaces of geodesics can be constructed, while for the time-like and space-like deformations bo…
Possible relationship between initial conditions for inflation and past geodesic incompleteness of the inflationary spacetime
本文提出双测度理论模型，揭示暴胀初始条件与时空过去测地不完备性相关，并证明在特定条件下暴胀阶段必然存在。
We point out that past-incompleteness of geodesics in FLRW spacetimes does not necessarily imply that these spacetimes start from a singularity. Namely, if a test particle that follows such a trajectory has a non-vanishing velocity, its energy was super-Planckian at some time in the past if it kept following that geodesic. That indicates a breakdown of the particle's description, which is why we should not consider those trajectories for the definition of an initial singularity. When one only considers test particles that do not have this breakdown of their trajectory, it turns out that the on…
Possible relationship between initial conditions for inflation and past geodesic incompleteness of the inflationary spacetime
本文在双测度理论单标量场模型中证明，当初始动能密度大于梯度能密度且标量场初始值处于特定区间时，暴胀条件得以满足，同时暴胀时空的类时测地线无法在过去方向延拓到边界超曲面之外，从而将暴胀初始条件与时空过去测地不完备性联系起来。
We treat the problem of defining, and characterising in a practical way, an appropriate class of distinguished curves for Poincaré-Einstein manifolds, and other conformally singular geometries. These "generalised geodesics" agree with geodesics away from the conformal singularity set and are shown to satisfy natural "boundary conditions" at points where they meet or cross the metric singularity set. We also characterise when they coincide with conformal circles. In the case of (Poincaré-)Einstein manifolds, we are able to provide a very general theory of first integrals for these distinguished…
Non-Commutativity in Geodesic Congruences
该研究通过弗罗贝尼乌斯定理推广测地汇向量对易关系，提出时空非对易性的几何实现，引入NCEC修正聚焦相关定理，为暗能量演化提供纯几何起源。
In this work we have studied the non-geodesical behaviour of particles with spin 1/2 in Poincaré gauge theories of gravity, via the WKB method and the Mathisson-Papapetrou equation. We have analysed the relation between the two approaches and we have argued the different advantages associated with the WKB approximation. Within this approach, we have calculated the trajectories in a particular Poincaré gauge theory, discussing the viability of measuring such a motion.
The Singularity Problem for Space-Times with Torsion
本文讨论了有挠时空中奇点的严格理论问题。我们将测地线定义为切向量通过平行输运移动的曲线。这与其他作者所做的不同，因为他们对测地线的定义只涉及Christoffel联络，尽管他们研究的是有挠的理论。我们提出了一个初步的定义奇点是基于类时或零测地线的不完全性，即使理论与扭转的粒子的路径不是测地线。对有挠宇宙学模型的测地线方程的研究表明，该定义具有物理相关性。它也可以被激发，正如文献中所做的那样，评论具有挠率的时空的因果结构相对于广义相对论不会改变。然后，我们证明了如何扩展霍金的奇点定理没有因果关系的假设，时空的ECSK理论。这是通过研究ECSK理论中的广义Raychaudhuri方程、共轭点存在的条件和极大类时测地线的性质而实现的。只要挠率张量满足一定的条件，霍金定理就可以推广。因此，如果不满足这些附加条件，我们的结果也可以解释为无奇性定理。换句话说，事实证明，在基于ECSK理论的封闭宇宙学模型中，奇点的出现比广义相对论中更不普遍。我们的工作是在文献中与以前的文件进行比较。有一些相关的差异，因为我们依赖于不同的测地线定义，我们保持ECSK理论的场方程在其原始形式，而不是铸造他们在一个类似于广义相对论的形式与修改的能量动量张量，
Quadrupolar tidal effects destroy the integrability of black hole geodesics: analytic proof and numerical evidence of chaos
该研究通过解析证明与数值诊断，证实具有潮汐诱导四极矩的非旋转天体的潮汐动力学不可积，破坏了克尔测地线的可积性，揭示了混沌现象。
Volume Singularities in General Relativity
本文提出广义相对论中体积奇点的新概念，证明霍金定理的奇点也是体积奇点，并指出黑洞中体积奇点总被事件视界隐藏，关联彭罗斯宇宙监督猜想。
Geodesic completeness, curvature singularities and infinite tidal forces
该研究揭示潮汐力发散的非奇点成因，证明曲率奇点处度规无法定义故时空测地不完备，明确粒子无法到达曲率奇点。
In the present work some generalizations of the Hawking singularity theorems in the context of $f(R)$ theories are presented. The assumptions are of these generalized theorems is that the matter fields satisfy the conditions $\bigg(T_{ij}-\frac{g_{ij}}{2} T\bigg)k^i k^j\geq 0$ for any generic unit time like field, that the scalaron takes bounded positive values during its evolution, and that the resulting space time is globally hyperbolic. Then, if there exist a Cauchy hyper surface $Σ$ for which the expansion parameter $θ$ of the geodesic congruence emanating orthogonally from $Σ$ satisfies s…
Geometric effects of torsion on black hole ringdown and shadows in Poincaré gauge gravity
研究庞加莱规范引力中类似雷斯纳 - 诺德斯特龙黑洞，用六阶WKB近似等方法研究挠率修正背景上的微扰和测试场，发现挠率对黑洞铃宕有影响，通过与观测比较推导挠率参数约束，揭示其在黑洞阴影和铃宕上留下相关印记。
Internal symmetry in Poincare gauge gravity
本文在4维Poincare规范理论中发现大型内部对称性，通过推广Riemann-Cartan几何引入射影不变的挠率与非度量组合S，在四维下获得足以满足标准模型的SO(11,9)对称性，且该对称性与引力解耦符合Coleman-Mandula定理。
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