Singularities In General Relativity Research Articles

Volume singularities in general relativity

We propose a new notion of singularity in general relativity which complements the usual notions of geodesic incompleteness and curvature singularities. Concretely, we say that a spacetime has a volume singularity if there exist points whose future or past has arbitrarily small spacetime volume: in particular, smaller than a Planck volume. From a cosmological perspective, we show that the (geodesic) singularities predicted by Hawking’s theorem are also volume singularities. In the black hole setting, we show that volume singularities are always hidden by an event horizon, prompting a discussion of Penrose’s cosmic censorship conjecture.

On the visibility of singularities in general relativity and modified gravity theories

We investigate the global causal structure of the end state of a spherically symmetric marginally bound Lemaitre–Tolman–Bondi (LTB) (Lemaitre 1933 Ann. Soc. Sci. Brux. A 53 51, Tolman 1934 Proc. Natl Acad. Sci. USA 20 169–76, Bondi 1947 Mon. Not. Astron. Soc. 107 410) collapsing cloud (which is well studied in general relativity) in the framework of modified gravity having the generalized Lagrangian in the action. Here R is the Ricci scalar, and is a constant. By fixing the functional form of the metric components of the LTB spacetime, using up the available degree of freedom, we realize that the matching surface of the interior and the exterior metric are different for different values of α. This change in the matching surface can alter the causal property of the first central singularity. We depict this by showing a numerical example. Additionally, for a globally naked singularity to have physical relevance, a congruence of null geodesics should escape from such singularity to be visible to an asymptotic observer for infinite time. For this to happen, the first central singularity should be a nodal point. We here give a heuristic method to show that this singularity is a nodal point by considering the above class of theory of gravity, of which general relativity is a particular case.

Superradiance of massive scalar particles around rotating regular black holes* *Supported in part by the National Natural Science Foundation of China (11675081, 12175108)

Regular black holes, as part of an important attempt to eliminate the singularities in general relativity, have been of wide concern. Because the superradiance associated with rotating regular black holes plays an indispensable role in black hole physics, we calculate the superradiance related effects, i.e., the superradiance instability and the energy extraction efficiency, for a scalar particle with a small mass around a rotating regular black hole, where the rotating regular black hole is constructed by the modified Newman-Janis algorithm. We analytically give the eigenfrequency associated with instability and the amplification factor associated with energy extraction. For two specific models, i.e., the rotating Hayward and Bardeen black holes, we investigate how their regularization parameters affect the growth of instability and the efficiency of energy extraction from the two rotating regular black holes. We find that the regularization parameters give rise to different modes of the superradiance instability and the energy extraction when the rotation parameters vary. There are two modes for the growth of superradiance instability and four modes for the energy extraction. Our results show the diversity of superradiance in the competition between the regularization parameter and the rotation parameter for rotating regular black holes.

On holonomy singularities in general relativity and the Cloc0,1-inextendibility of space-times

This paper investigates the structure of gravitational singularities at the level of the connection. We show in particular that for FLRW space-times with particle horizons a local holonomy, which is related to a gravitational energy, becomes unbounded near the big-bang singularity. This implies the Cloc0,1-inextendibility of such FLRW space-times. Again using an unbounded local holonomy, we also give a general theorem establishing the Cloc0,1-inextendibility of spherically symmetric weak null singularities which arise at the Cauchy horizon in the interior of black holes. Our theorem does not presuppose the mass-inflation scenario and in particular applies to the Reissner–Nordström-Vaidya space-times, as well as to space-times which arise from small and generic spherically symmetric perturbations of two-ended subextremal Reissner–Nordström initial data for the Einstein–Maxwell scalar field system. In previous work, Luk and Oh proved the C2-formulation of strong cosmic censorship for this latter class of space-times—and based on their work we improve this to a Cloc0,1-formulation of strong cosmic censorship.

Absence of covariant singularities in pure gravity

The assumptions of the Hawking–Penrose singularity theorem are not covariant under field redefinitions. Following the works on the covariant formulation of quantum field theory initiated by Vilkovisky and DeWitt in the 80s, we propose to study singularities in field space, where the spacetime metric is treated as a coordinate along with the other fields in the theory. From this viewpoint, a spacetime singularity might be just a singularity in the field-space coordinates, analogously to the standard coordinate singularities in General Relativity. Objects invariant under field-space coordinate transformations can then reveal whether certain spacetime singularity is indeed singular. We recall that observables in quantum field theory are scalar functionals in field space. Therefore, in principle, spacetime singularities corresponding to regular field-space curvature invariants would not affect physical observables. In this paper, we show that the field-space Kretschmann scalar for a certain choice of the DeWitt field-space metric is everywhere finite. This fact could be interpreted as an indication that no singularities actually exist in pure gravity for any gravitational action. In particular, all vacuum singularities of General Relativity result from an unhappy choice of field variables. The extension to the case in which matter fields are present, as required by singularity theorems, is left for future development.

Analysis for Science Librarians of the 2020 Nobel Prize in Physics: Luminous Bodies, Black Holes, and How They Come Together at the Milky Way’s Heart

ABSTRACT The 2020 Nobel Prize in Physics was awarded to Roger Penrose, Reinhard Genzel, and Andrea Ghez. Penrose was awarded half of the Nobel Prize for creating a groundbreaking black hole theorem that drew on a half-century of theoretical speculations about singularities in general relativity. Ghez and Genzel share the remaining half of the prize for their research teams’ decades of precise observations of the center of the Milky Way, determining from stellar orbits that a supermassive compact object lies at the heart of the galaxy, consistent with predictions for a roughly 4 million solar-mass black hole. This paper describes the science behind the prize, each scientist’s career, and concludes with a citation analysis of each laureate that draws on Web of Science and Scopus data.

Light Speed Invariant Solution and Its Enlightenment of Field Equation of General Relativity

A systematic examination of the basic theory of general relativity is made, the meaning of coordinates again is emphasized, the confusion caused by unclear meaning of coordinates in the past is corrected, and the expression of the theory is made more accurate. Firstly, the equation of Einstein’s gravitational field is solved in the usual coordinate system, the existence of light speed invariant solution in the spherically symmetric gravitational field is proved, and in the same time, the solution is determined. It turns out that black holes are not an inevitable prediction of general relativity. The more exact formulas for calculating the curvature of light on the surface of the Sun and the precession angle of the orbit of Mercury are given, and the convergence of general relativistic gravity and special relativistic mechanics under the weak field approximation is realized. Finally, it is shown that the coupling coefficient of the gravitational field equation is not unique. Modifying this coefficient is an ideal project to eliminate the singularities of general relativity on the condition keeping the field equation concise and elegant, and moreover, it reveals that dark matter and dark energy are the negative energy field in the matter, the expansion of the universe is the appearance of the gradual formation of galaxies in accordance with fractal rules, not only the space between galaxies is expanding but also the galaxies themselves are also expanding, new matter is continuously generated in the celestial bodies, for the first time, the unity of fractal geometry and cosmic dynamics of general relativity is realized, and the formation and evolution of galaxies are brought into the fractal generation mode. This is a living and vivacious universe in which all aspects are gradually strengthening, in sharp contrast to the dying universe under the current cosmological framework.

Nonsingular black holes from charged dust collapse: A concrete mechanism to evade interior singularities in general relativity

In this paper, we examine the gravitational collapse of a nonrelativistic charged perfect fluid interacting with a dark energy component. Given a simple factor for the energy transfer, we obtain a nonsingular interior solution which naturally matches the Reissner–Nordström–de Sitter exterior geometry. We also show that the interacting parameter is proportional to the overall charge of the final black hole thus formed. For the case of quasi-extremal configurations, we propose a statistical model for the entropy of the collapsed matter. This entropy extends Bekenstein’s geometrical entropy by an additive constant proportional to the area of the extremal black hole.

How does a dark compact object ringdown?

A generic feature of nearly out-of-equilibrium dissipative systems is that they resonate through a set of quasinormal modes. Black holes - the absorbing objects par excellence - are no exception. When formed in a merger, black holes vibrate in a process called "ringdown", which leaves the gravitational-wave footprint of the event horizon. In some models of quantum gravity which attempt to solve the information-loss paradox and the singularities of General Relativity, black holes are replaced by regular, horizonless objects with a tiny effective reflectivity. Motivated by these scenarios, here we develop a generic framework to the study of the ringdown of a compact object with various shades of darkness. By extending the black-hole membrane paradigm, we map the interior of any compact object in terms of the bulk and shear viscosities of a fictitious fluid located at the surface, with the black-hole limit being a single point in a three-dimensional parameter space. We unveil some remarkable features of the ringdown and some universal properties of the light ring in this framework. We also identify the region of the parameter space which can be probed by current and future gravitational-wave detectors. A general feature is the appearance of mode doublets which are degenerate only in the black-hole limit. We argue that the merger event GW150914 already imposes a strong lower bound on the compactness of the merger remnant of approximately 99% of the black-hole compactness. This places model-independent constraints on black-hole alternatives such as diffuse "fuzzballs" and nonlocal stars.

The Light Speed Invariant Solution of the Field Equation of General Relativity

The fundamental concepts of general relativity are systematically rearranged. Firstly, the equations of Einstein’s gravitational field are solved in the usual sphere coordinate system, and the existence of the invariant solution of the speed of light in the spherically symmetric gravitational field is proved, also determine the solution. It is revealed that black holes are not the inevitable prediction of general relativity. Black holes do not exist, and the so-called gravitational waves sent by black holes need to be explained in other ways. Correcting the conceptual confusion caused by unclear radial coordinates in the past, two more accurate formulas for calculating the curvature of light on the surface of the sun and the precession angle of the orbit of Mercury are given. Finally, discuss that the coupling constant of the gravitational field equation needs to be modified, which is a shortcut to eliminate various singularities of general relativity. It is proved that the celestial bodies and galaxies expand with the expansion of the universe, and new matter is generated continuously in the celestial bodies, which provides a theoretical basis for the fractal generation model of galaxy formation.

Entropic locking of Action Complexity at cosmological singularities

We study the relation between entropy and Action Complexity (AC) for various examples of cosmological singularities in General Relativity. The complexity is defined with respect to the causal domain of dependence of the singular set, and the entropy is evaluated on the boundary of the same causal domain. We find that, contrary to the situation for black hole singularities, the complexity growth near the singularity is controlled by the dynamics of the entropy S, with a characteristic linear relation. This formula is found to apply to singularities with vanishing entropy as well as those with diverging entropy. In obtaining these results it is crucial to take into account the AC expansion counterterm, whose associated length scale must be chosen sufficiently large in order to ensure the expected monotonicity properties of the complexity.

Through the Big Bang in inflationary cosmology

Singularities in General Relativity are regions where the description of spacetime in terms of a pseudo-Riemannian geometry breaks down. The theory seems unable to predict the evolution of the physical degrees of freedom around and beyond such regions. In a recent paper, the author and collaborators challenged this view by providing an example of a singularity at which Einstein's equations can be rewritten in a form that satisfies an existence and uniqueness theorem, thereby predicting that each solution can be continued uniquely through the singularity. This result was obtained under the assumption of homogeneity (but not isotropy), and requires the presence of a massless free scalar field. This paper extends the result to N scalar fields with a potential, the only requirement being that it does not grow too fast. In particular, the result is compatible with inflationary potentials, e.g. Starobinsky's. This brings us one step closer to the goal of extending the original result to realistic cosmologies.

Space-time Singularities vs. Topologies in the Zeeman—Göbel Class

In this article we first observe that the Path topology of Hawking, King and MacCarthy is an analogue, in curved spacetimes, of a topology that was suggested by Zeeman as an alternative topology to his so-called Fine topology in Minkowski spacetime. We then review a result of a recent paper on spaces of paths and the Path topology, and see that there are at least five more topologies in the class $\mathfrak{Z}-\mathfrak{G}$ of Zeeman-G\"obel topologies which admit a countable basis, incorporate the causal and conformal structures, but the Limit Curve Theorem fails to hold. The "problem" that L.C.T. does not hold can be resolved by "adding back" the light-cones in the basic-open sets of these topologies, and create new basic open sets for new topologies. But, the main question is: do we really need the L.C.T. to hold, and why? Why is the manifold topology, under which the group of homeomorphisms of a spacetime is vast and of no physical significance (Zeeman), more preferable from an appropriate topology in the class $\mathfrak{Z}-\mathfrak{G}$ under which a homeomorphism is an isometry (G\"obel)? Since topological conditions that come as a result of a causality requirement are key in the existence of singularities in general relativity, the global topological conditions that one will supply the spacetime manifold might play an important role in describing the transition from the quantum non-local theory to a classical local theory.

Spherical symmetric dust collapse in vector-tensor gravity

There is a viable vector-tensor gravity (VTG) theory, whose vector field produces repulsive forces leading to important effects. In the background universe, the effect of these forces is an accelerated expansion identical to that produced by vacuum energy (cosmological constant). Here, we prove that another of these effects arises for great enough collapsing masses which lead to Schwarzschild black holes and singularities in general relativity (GR). For these masses, pressure becomes negligible against gravitational attraction and the complete collapse cannot be stopped in the context of GR; however, in VTG, a strong gravitational repulsion could stop the falling of the shells towards the symmetry center. A certain study of a collapsing dust cloud is then developed and, in order to undertake this task, the VTG equations in comoving coordinates are written. In this sense and, as it happens in general relativity for a pressureless dust ball, three different solutions are found. These three situations are analyzed and the problem of the shell crossings is approached. The apparent horizons and trapped surfaces, whose analysis will lead to diverse situations, depending on certain theory characteristic parameter value, are also examined.

General relativity as a hybrid theory: The genesis of Einstein's work on the problem of motion

In this paper I describe the genesis of Einstein's early work on the problem of motion in general relativity (GR): the question of whether the motion of matter subject to gravity can be derived directly from the Einstein field equations. In addressing this question, Einstein himself always preferred the vacuum approach to the problem: the attempt to derive geodesic motion of matter from the vacuum Einstein equations. The paper first investigates why Einstein was so skeptical of the energy-momentum tensor and its role in GR. Drawing on hitherto unknown correspondence between Einstein and George Yuri Rainich, I then show step by step how his work on the vacuum approach came about, and how his quest for a unified field theory informed his interpretation of GR. I show that Einstein saw GR as a hybrid theory from very early on: fundamental and correct as far as gravity was concerned but phenomenological and effective in how it accounted for matter. As a result, Einstein saw energy-momentum tensors and singularities in GR as placeholders for a theory of matter not yet delivered. The reason he preferred singularities was that he hoped that their mathematical treatment would give a hint as to the sought after theory of matter, a theory that would do justice to quantum features of matter.

Weak null singularities in general relativity

We construct a class of spacetimes (without symmetry assumptions) satisfying the vacuum Einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2-sphere. The metric of these spacetimes extends continuously beyond the singularities while the Christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries. The construction shows moreover that the singularities are stable in a suitable sense. These singularities are stronger than the impulsive gravitational spacetimes considered by Luk and Rodnianski, and conjecturally they are present in the interior of generic black holes arising from gravitational collapse.

Stability of Schwarzschild singularity in non-local gravity

In a previous work, it was shown that all Ricci-flat spacetimes are exact solutions for a large class of non-local gravitational theories. Here we prove that, for a subclass of non-local theories, the Schwarzschild singularity is stable under linear perturbations. Thus, non-locality may be not enough to cure all the singularities of general relativity. Finally, we show that the Schwarzschild solution can be generated by the gravitational collapse of a thin shell of radiation.

Black hole solutions in de Rham-Gabadadze-Tolley massive gravity

We present a detailed study of the static spherically symmetric solutions in de Rham-Gabadadze-Tolley (dRGT) theory. Since the diffeomorphism invariance can be restored by introducing the St\"{u}ckelberg fields $\phi^a$, there is new invariant $I^{ab}=g^{\mu\nu}\partial_{\mu}\phi^a\partial_\nu\phi^b$ in the massive gravity, which adds to the ones usually encountered in general relativity (GR). In the unitary gauge $\phi^a=x^\mu\delta_\mu^a$, any inverse metric $g^{\mu\nu}$ that has divergence including the coordinate singularity in GR would exhibit a singularity in the invariant $I^{ab}$. Therefore, there is no conventional Schwarzschild metric if we choose unitary gauge. In this paper, we obtain a self-consistent static spherically symmetric ansatz in the nonunitary gauge. Under this ansatz, we find that there are seven solutions including the Schwarzschild solution, Reissner-Nordstr\"{o}m solution and five other solutions. These solutions may possess an event horizon depending upon the physical parameters (Schwarzschild radius $r_s$, scalar charge $S$ and/or electric charge $Q$). If these solutions possess an event horizon, we show that the singularity of $I^{ab}$ is absent at the horizon. Therefore, these solutions may become candidates for black holes in dRGT.

Black-and-white hole as a space-time with integrable singularity

We discuss the problem of singularities in general relativity and emphasize the distinction that should be made between what is understood to be mathematical and physical singularities. We revise examples of space-times that conventionally contain a singularity which, in a sense, does not manifest itself physically. A special attention is paid to the case of integrable singularities for which we propose a well-defined mathematical procedure used to extend the space-time beyond the singularity. We argue that this type of singularity may connect the interior of a black hole with a newly born universe (a space-time referred to as black-and-white hole) giving a resolution to the problem of initial high density and symmetry of the universe. We exemplify by presenting toy models of eternal and astrophysical black-and-white holes.

Origins and development of the Cauchy problem in general relativity

The seminal work of Yvonne Choquet-Bruhat published in 1952 demonstrates that it is possible to formulate Einstein's equations as an initial value problem. The purpose of this article is to describe the background to and impact of this achievement, as well as the result itself. In some respects, the idea of viewing the field equations of general relativity as a system of evolution equations goes back to Einstein himself; in an argument justifying that gravitational waves propagate at the speed of light, Einstein used a special choice of coordinates to derive a system of wave equations for the linear perturbations on a Minkowski background. Over the following decades, Hilbert, de Donder, Lanczos, Darmois and many others worked to put Einstein's ideas on a more solid footing. In fact, the issue of local uniqueness (giving a rigorous justification for the statement that the speed of propagation of the gravitational field is bounded by that of light) was already settled in the 1930s by the work of Stellmacher. However, the first person to demonstrate both local existence and uniqueness in a setting in which the notion of finite speed of propagation makes sense was Yvonne Choquet-Bruhat. In this sense, her work lays the foundation for the formulation of Einstein's equations as an initial value problem. Following a description of the results of Choquet-Bruhat, we discuss the development of three research topics that have their origin in her work. The first one is local existence. One reason for addressing it is that it is at the heart of the original paper. Moreover, it is still an active and important research field, connected to the problem of characterizing the asymptotic behaviour of solutions that blow up in finite time. As a second topic, we turn to the questions of global uniqueness and strong cosmic censorship. These questions are of fundamental importance to anyone interested in justifying that the Cauchy problem makes sense globally. They are also closely related to the issue of singularities in general relativity. Finally, we discuss the topic of stability of solutions to Einstein's equations. This is not only an important and active area of research, it is also one that only became meaningful thanks to the work of Yvonne Choquet-Bruhat.