Special Spacetimes Research Articles

CR structures and asymptotically flat spacetimes

We discuss the existence, arising by analogy to that in algebraically special spacetimes, of a unique CR structure realized on null infinity for (almost) any asymptotically flat Einstein or Einstein–Maxwell spacetime.

Twisting null geodesic congruences, scri, H-space and spin-angular momentum

The purpose of this work is to return, with a new observation and rather unconventional point of view, to the study of asymptotically flat solutions of Einstein equations. The essential observation is that from a given asymptotically flat spacetime with a given Bondi shear, one can find (by integrating a partial differential equation) a class of asymptotically shear-free (but, in general, twisting) null geodesic congruences. The class is uniquely given up to the arbitrary choice of a complex analytic world line in a four-parameter complex space. Surprisingly, this parameter space turns out to be the H-space that is associated with the real physical spacetime under consideration. The main development in this work is the demonstration of how this complex world line can be made both unique and also given a physical meaning. More specifically, by forcing or requiring a certain term in the asymptotic Weyl tensor to vanish, the world line is uniquely determined and becomes (by several arguments) identified as the ‘complex centre of mass’. Roughly, its imaginary part becomes identified with the intrinsic spin-angular momentum while the real part yields the orbital angular momentum. One should think of this work as developing a generalization of the properties of the algebraically special spacetimes in the sense that the term that is forced here to vanish is automatically vanishing (among many other terms) for all the algebraically special metrics. This is demonstrated in the several given examples. It was, in fact, an understanding of the algebraically special metrics and their associated shear-free null congruence that led us to this construction of the asymptotically shear-free congruences and the unique complex world line. The Robinson–Trautman metrics and the Kerr and charged Kerr metrics with their properties are explicit examples of the construction given here.

Bianchi identities in higher dimensions

A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension n. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in n − 4 spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers–Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.

Spacetime without Reference Frames: An Application to the Velocity Addition Paradox

Much conceptualisation in contemporary physics is bogged down by unnecessary assumptions concerning a specific choice of coordinates which often leads to misunderstandings and paradoxes. Considering an absolute (coordinate-free) formulation of special relativistic spacetime, we show clearly that the velocity addition paradox emerged because the use of coordinates obscures that the space of relativistic observers is ‘more relative’ than the space of non-relativistic observers.

The holographic principle for general backgrounds

We aim to establish the holographic principle as a universal law, rather than a property only of static systems and special spacetimes. Our covariant formalism yields an upper bound on entropy which applies to both open and closed surfaces, independently of shape or location. It reduces to the Bekenstein bound whenever the latter is expected to hold, but complements it with novel bounds when gravity dominates. In particular, it remains valid in closed FRW cosmologies and in the interior of black holes. We give an explicit construction for obtaining holographic screens in arbitrary spacetimes (which need not have a boundary). This may aid the search for non-perturbative definitions of quantum gravity in spacetimes other than AdS. More details, references and examples can be found in papers by Bousso R (1999 J. High Energy Phys. JHEP07(1999)004, JHEP06(1999)028).

Persistence and Space-Time

Although considerations based on contemporary space-time theories, such as special and general relativity, seem highly relevant to the debate about persistence, their significance has not been duly appreciated. My goal in this paper is twofold: 1) to reformulate the rival positions in the debate (i.e. endurantism, or three-dimensionalism, and perdurantism, or four-dimensional, the doctrine of temporal parts) in the framework of special relativistic space-time; and 2) to argue that, when so reformulated, perdurantism exhibits explanatory advantages over endurantism. The argument builds on the fact that four-dimensional entities extended in space as well as time are relativistically invariant in a way three-dimensional entities are not.

On the Papapetrou field in vacuum

In this paper we study the electromagnetic fields generated by a Killing vector field in vacuum spacetimes (Papapetrou fields). The motivation of this work is to provide new tools for the resolution of Maxwell's equations as well as for the search, characterization, and study of exact solutions of Einstein's equations. The first part of this paper is devoted to an algebraic study in which we give an explicit and covariant procedure to construct the principal null directions of a Papapetrou field. In the second part, we focus on the main differential properties of the principal directions, studying when they are geodesic, and in that case we compute their associated optical scalars. With this information we get the conditions that a principal direction of the Papapetrou field must satisfy in order to be aligned with a multiple principal direction of the Weyl tensor in the case of algebraically special vacuum spacetimes. Finally, we illustrate this study using the Kerr, Kasner and pp waves spacetimes.

A Note on Algebraically Special, Asymptotically Simple and Empty Spacetime

It is proved that subject to certain physically reasonable restrictions on the curvature of spacetime, an algebraically special, asymptotically simple and empty spacetime is necessarily a copy of the Minkowski space. This imposes further requirements a nontrivial asymptotically simple and empty spacetime has to satisfy.

On `Diagonal spacetimes admitting inheriting conformal Killing vector fields'

In a recent paper it is stated that if a perfect-fluid diagonal spacetime with barotropic equation of state satisfying the energy positive condition admits a proper inheriting conformal Killing vector field, then the solution is either FLRW, a very special plane-symmetric spacetime, or a solution which is conformal to a Bianchi type I spacetime. Nevertheless, this result is more general than the authors claim because in the process of obtaining the solutions one of the Einstein equations was incidentally forgotten, and therefore, their statement applies not only to perfect fluids but to more general anisotropic sources with two equal pressures. Thanks to the remaining equation, it is possible to establish a more restrictive statement that characterizes the Petrov type-D Allnutt solution.

The asymptotic structure of algebraically special spacetimes

It is shown that the only vacuum algebraically special spacetime that is asymptotically simple is the Minkowski space. The structure of null infinity for algebraically special spacetimes satisfying a subset of the vacuum equations is shown to be particularly simple in terms of the coordinates used in Kerr's original reduction of the field equations. With the assumption of the existence of a global future null infinity, (in the sense of having topology ), past null infinity can be constructed with the same topology and canonically identified with future null infinity. This identifies the corresponding asymptotic data. The Bondi momentum on corresponding cuts of future null infinity and past null infinity sums to zero. In particular, the Bondi energy is negative or zero on one of future or past null infinities, thus implying singularities or negative energy densities in the interior if the spacetime is non-trivial. In appendix A the shear and radiation fields at null infinity are all derived. In appendix B the Cauchy - Riemann structure on the space of geodesics of the congruence underlying these spacetimes is considered and the auxiliary structures required to determine the spacetime are characterized intrinsically with respect to the Cauchy - Riemann structure.

Quantum fractals in boxes

A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all positions , the graph of the evolution of P is a fractal curve with dimension . For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension . When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension , simple arguments suggest that the dimension of the time fractal is , and that of the space fractal is .

Integration in the GHP formalism I: A coordinate approach with applications to twisting type N spaces

The compacted spin coefficient (ghp) formalism is clearly more concise and efficient than the older Newman-Penrose formalism. Yet few people use it when integration of the field equations is involved, Held being the notable exception. However, to most workers in the field, Held's approach seems far removed from the usual Newman-Unti (nu) type integration procedure. This paper and a subsequent one are concerned with integration within theghp formalism. In this first paper we develop aghp coordinate-style integration procedure modelled closely on thenu procedure whereas in the second paper we present aghp operator-style integration procedure along the lines suggested by Held. For simplicity of illustration we restrict the discussion to algebraically special vacuum spacetimes. We show clearly the similarities and differences between the two approaches, and compare their respective efficiencies. To deal with a concrete example, we illustrate the two methods by once more considering the problem of twisting typeN vacuum solutions to Einstein's field equations. TheGhp approach enables us to have a comprehensive overview of this much discussed problem and gain new insight into the relationship between various results derived in a number of different formalisms.

Stationary strings and principal Killing triads in 2 + 1 gravity

A new tool for the investigation of (2 + 1)-dimensional gravity is proposed. It is shown that in a stationary (2 + 1)-dimensional space-time, the eigenvectors of the covariant derivative of the timelike Killing vector form a rigid structure, the principal Killing triad. Two of the triad vectors are null, and in many respects they play the role similar to the principal null directions in the algebraically special 4D space-times. It is demonstrated that the principal Killing triad can be efficiently used for classification and study of stationary 2 + 1 space-times. One of the most interesting applications is a study of minimal surfaces in a stationary space-time. A principal Killing surface is defined as a surface formed by Killing trajectories passing through a null ray, which is tangent to one of the null vectors of the principal Killing triad. We prove that a principal Killing surface is minimal if and only if the corresponding null vector is geodesic. Furthermore, we prove that if the (2 + 1)-dimensional space-time contains a static limit, then the only regular stationary timelike minimal 2-surfaces that cross the static limit, are the minimal principal Killing surfaces. A timelike minimal surface is a solution to the Nambu-Goto equations of motion and hence it describes a cosmic string configuration. A stationary string interacting with a (2 + 1)-dimensional rotating black hole is discussed in detail.

Some D type metric forms for spacetimes with shear, nongeodesic, and nondiverging principal null congruences

In the context of a null tetrad formalism it is assumed that the two real vectors of the tetrad have neither expansion nor rotation and that they are nongeodesic and/or have shear. This assumption is rarely used in search of algebraically special spacetimes. General relations for the connection and for the Ricci and Weyl tensors are presented. An additional assumption that these vectors are principal leads to Petrov type D spacetimes, reducible in some cases to conformally flat ones. Twenty-two explicit metric forms, probably hitherto unknown, are found.

Spacetimes in which the Ricci equations characterize the Riemann tensor

It has recently been asked whether a fourth-order tensorK with all the algebraic symmetries of a Riemann tensor, and which satisfies the Ricci equations (with covariant derivative constructed from the metricg in the usual way), is always equal to the Riemann tensorR of the metricg; and a positive answer has been given for a generic tensorK in any nonflat 4-dimensional spacetime. In this paper it is shown that the result is also true in a generic 4-dimensional spacetime for any nontrivial tensorK. In addition, those special spacetimes where the result fails are given explicitly in terms of the Petrov types of their Weyl and Plebanski tensors.

Vacuum polarization of scalar and spinor fields near closed null geodesics.

Hawking has recently proposed a ``chronology protection conjecture,'' which states that closed timelike curves cannot form in the real Universe. The most likely mechanism for enforcing this conjecture, if it is correct, is a divergent vacuum polarization at the Cauchy horizon (``chronology horizon'') where closed timelike curves first try to form. Hawking has proved that, if the chronology horizon is compactly generated, then it contains one or more smoothly closed null geodesics. Because it seems likely that all the horizon's generators emerge from these closed null geodesics, a sufficiently strongly divergent vacuum polarization at the closed null geodesics is likely to destroy the chronology horizon completely and thereby prevent closed timelike curves from forming. In this paper we compute the details of the divergence near the closed null geodesics, in a generic spacetime with a compactly generated chronology horizon---thereby generalizing earlier computations, in special spacetimes, by Hiscock and Konkowski, Kim and Thorne, and Frolov. We carry out the computation for both a conformal scalar field and a two-component spinor field. We show that, for an observer who will pass through a point on the closed null geodesic after a small interval of proper time \ensuremath{\delta}t, the leading-order divergence is always proportional to (\ensuremath{\delta}t${)}^{\mathrm{\ensuremath{-}}3}$ and has the same tensorial structure as the stress-energy of a null fluid moving along the closed null geodesic. We also show that, by contrast with flat spacetime, there is in general no cancellation between the divergent vacuum energies of a combination of fields that, in flat spacetime, would be related by supersymmetry: two conformal scalar fields and one two-component spinor field. We discuss the implications of these results for Hawking's chronology protection conjecture.

Simply-transitive homothety groups

In this paper the spin-coefficient formalism of Penrose and Rindler is applied to the problem of finding space-times that admit a simply-transitive group of homotheties. The method is illustrated by an application to algebraically special vacuum space-times. Vacuum metrics with a fivedimensional homothety group are also considered.

On Behavior of Weyl's Gauge Field11The project supported in part by National Natural Science Foundation of China.

We consider a system, consisting of a metric tensor , a scalar field , a Weyl's gauge field and a scalar matter field , which is invariant under general coordinate transformation and Weyl's gauge transformation. Two kinds of identities and field equations are given and discussed. A special space-time with is considered in a gauge-independent manner. We point out that in a correct treatment where is not regarded as an independent variable, an auxiliary condition for Weyl's gauge field cannot be obtained. Therefore Weyl's gauge field can be treated as a usual field.

Separation of variables for the Klein-Gordon equation in special Stackel spacetimes

The theorem concerning separation of variables for the Klein-Gordon equation in electrovac spacetimes is proved. The sufficient conditions for separability in special Stackel spacetimes are stated. The useful relationships for arbitrary special Stackel spacetimes are found.

Algebraically special spacetimes

The Newman-Penrose formalism for algebraically special spacetimes, with or without twist, is recast in terms of weighted quantities defined at conformal null infinityJ+. Weighted differential operators, also defined atJ+, are introduced that are special cases of those defined in a recent extension of the Geroch-Held-Penrose formalism. The “solution,” including spin coefficients. Weyl tensor components, and reduced equations, is expressed rather concisely in terms of these weighted variables and operators. Its form invariance under the remaining freedom in the choice of tetrad and coordinate system now becomes evident.