In this work we analyse asymptotically flat, spherically symmetric spacetimes in which an event horizon is present without any trapped surfaces. We identify two types of such spacetimes, each related to the asymptotic behaviour (in time) of one of the two degrees of freedom of the metric. We study the causal structure of both types, showing that one almost always has a Cauchy horizon beyond which it is extendable, while the other is inextendable but has two separate future null infinity regions on either side of the horizon. We also study what energy conditions can be satisfied by the matter around the horizon. Some of these spacetimes were first introduced in an earlier work in which semiclassical effects near black-hole horizons were analysed. Here we generalise this analysis to a larger family of geometries.
We consider how causality and micro-causality are realised in QED in curved spacetime. The photon propagator is found to exhibit novel non-analytic behaviour due to vacuum polarization, which invalidates the Kramers-Kronig dispersion relation and calls into question the validity of micro-causality in curved spacetime. This non-analyticity is ultimately related to the generic focusing nature of congruences of geodesics in curved spacetime, as implied by the null energy condition, and the existence of conjugate points. These results arise from a calculation of the complete non-perturbative frequ…
Recently, an axially symmetric spacetime with causality violation is appeared in Ref. [1]. There, author presented a type III metric with vanishing expansion, shear and twist. The matter-energy represents a pure radial field with a negative cosmological constant. The spacetime is asymptotically anti-de Sitter space in the radial direction. Actually, this work with different for the metric was already published in Ref. [4]. Therefore, the current paper Ref. [1] is a duplicate one of the previous work.
We investigate the motion of electrically charged test particles in spacetimes with closed timelike curves, a subset of the black hole or wormhole Reissner-Nordström-NUT spacetimes without periodic identification of time. We show that, while in the wormhole case there are closed worldlines inside a potential well, the wordlines of initially distant charged observers moving under the action of the Lorentz force can never close or self-intersect. This means that for these observers causality is preserved, which is an instance of our weak chronology protection criterion.
Slice-stretching effects are discussed as they arise at the event horizon when geodesically slicing the extended Schwarzschild black-hole spacetime while using singularity excision. In particular, for Novikov and isotropic spatial coordinates the outward movement of the event horizon (``slice sucking'') and the unbounded growth there of the radial metric component (``slice wrapping'') are analyzed. For the overall slice stretching, very similar late time behavior is found when comparing with maximal slicing. Thus, the intuitive argument that attributes slice stretching to singularity avoidance…
It has been known for a long time that vacuum polarization in QED leads to a superluminal low-frequency phase velocity for light propagating in curved spacetime. Assuming the validity of the Kramers-Kronig dispersion relation, this would imply a superluminal wavefront velocity and the violation of causality. Here, we calculate for the first time the full frequency dependence of the refractive index using world-line sigma model techniques together with the Penrose plane wave limit of spacetime in the neighbourhood of a null geodesic. We find that the high-frequency limit of the phase velocity (…
We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. This provides an abstract mathematical setting in which one can study causality independent of geometry and differentiable structure.
Spacetime must be foliable by spacelike surfaces for the quantum mechanics of matter fields to be formulated in terms of a unitarily evolving state vector defined on spacelike surfaces. When a spacetime cannot be foliated by spacelike surfaces, as in the case of spacetimes with closed timelike curves, a more general formulation of quantum mechanics is required. In such generalizations the transition matrix between alternatives in regions of spacetime where states {\it can} be defined may be non-unitary. This paper describes a generalized quantum mechanics whose probabilities consistently obey …
A mathematical definition of classical causality over discrete spacetime dynamics is formulated. The approach is background free and permits a definition of causality in a precise way whenever the spacetime dynamics permits. It gives a natural meaning to the concepts of cosmic time, spacelike hypersurfaces and timelike or lightlike flows without assuming the notion of a background metric. The concepts of causal propagators and the speed of causality are discussed. In this approach the concepts of spacetime and dynamics are linked in an essential and inseparable whole, with no meaning to either…
We present an axially symmetric spacetime which contains closed timelike curves, and hence violates the causality condition. The metric belongs to type III in the Petrov classification scheme with vanishing expansion, shear and twist. The matter-energy represents a pure radiation field with a negative cosmological constant. The spacetime is asymptotically anti-de Sitter space in the radial direction.
In this paper a systematic study of the causal structure and global causality properties of multiwarped spacetimes is developed. This analysis is used to make a detailed description of the causal boundary of these spacetimes. Some applications of our results in examples of physical interest, for instance, in the context of Maldacena's conjecture, are considered.
We try to constrain the noncommutativity length scale of the theoretical model given in Ref. [1] using the observational data from ACBAR, CBI and five year WMAP. The noncommutativity parameter is not constrained by WMAP data, however ACBAR and CBI data restrict the lower bound of its energy scale to be around 10 TeV. We also derive an expression for the amount of non-causality coming from spacetime noncommutativity for the fields of primordial scalar perturbations that are space-like separated. The amount of causality violation for these field fluctuations are direction dependent.
For n+1 dimensional asymptotically AdS spacetimes which have holographic duals on their n dimensional conformal boundaries, we show that the imposition of causality on the boundary theory is sufficient to prove positivity of mass for the spacetime when n > 2, without the assumption of any local energy condition. We make crucial use of a generalization of the time-delay formula calculated in gr-qc/9404019, which relates the time delay of a bulk null curve with respect to a boundary null geodesic to the Ashtekar-Magnon mass of the spacetime. We also discuss holographic causality for the negative…
In this paper, we address the problem of causality violation in the solutions of Einstein equations and seek possible causality restoration mechanisms in modifed theories of gravity. We choose for the above problem, the causality violation due to the existence of closed time-like curves in the context of Kerr-Newman black hole. We first revisit and quantify the details of the causality violation in the Kerr-Newman spacetime. We then show that the issue is also existent in two of the modified solutions to the Kerr Newman spacetime: The Non-Commutativity inspired solution and the f(R)-Gravity mo…
It is widely held that plane wave spacetimes are causally well behaved which means that they are free from causal pathologies such as closed timelike curves or closed null geodesics. In this paper, we show that one can get closed null geodesics in a plane wave metric by analytically solving the geodesic equations.
By definition a spacetime is stably causal if it is possible to widen the light cones all over the spacetime without spoiling causality. We prove that if the spacetime is at least non-total imprisoning then it is stably causal provided the light cones can be widened outside any compact arbitrarily large set, i.e. in a neighborhood of infinity, without spoiling causality. Furthermore, we prove that the new causality level `compact stable causality' can be obtained as the antisymmetry condition of a new causal relation which we identify, but it cannot be obtained as a causal stability condition …
It is proven that K-causality coincides with stable causality, and that in a K-causal spacetime the relation K^+ coincides with the Seifert's relation. As a consequence the causal relation "the spacetime is strongly causal and the closure of the causal relation is transitive" stays between stable causality and causal continuity.
Suppose a spacetime $M$ is a quotient of a spacetime $V$ by a discrete group of isometries. It is shown how causality conditions in the two spacetimes are related, and how can one learn about the future causal boundary on $M$ by studying structures in $V$. The relations between the two are particularly simple (the boundary of the quotient is the quotient of the boundary) if both $V$ and $M$ have spacelike future boundaries and if it is known that the quotient of the future completion of $V$ is past-distinguishing. (That last assumption is automatic in the case of $M$ being multi-warped.)
We observe that Khovanov homology detects causality in $(2+1)$-dimensional globally hyperbolic spacetimes whose Cauchy surface is homeomorphic to $\mathbb R^2$
In Einstein-Gauss-Bonnet gravity, for a group of warped product spacetimes, we get a generalized master equation for the perturbation of tensor type. We show that the "effective metric" or "acoustic metric" for the tensor perturbation equation can be defined even without a static condition. Since this master equation does not depend on the mode expansion, the hyperbolicity and causality of the tensor perturbation equation can be investigated for every mode of the perturbation. Based on the master equation, we study the hyperbolicity and causality for all relavent vacuum solutions of this theor…
A theory governing the metric and matter fields in spacetime is {\it locally causal} if the probability distribution for the fields in any region is determined solely by physical data in the region's past, i.e. it is independent of events at space-like separated points. General relativity is manifestly locally causal, since the fields in a region are completely determined by physical data in its past. It is natural to ask whether other possible theories in which the fundamental description of space-time is classical and geometric -- for instance, hypothetical theories which stochastically coup…
We construct a static axisymmetric wormhole from the gravitational field of two Schwarzschild particles which are kept in equilibrium by strings (ropes) extending to infinity. The wormhole is obtained by matching two three-dimensional timelike surfaces surrounding each of the particles and thus spacetime becomes non-simply connected. Although the matching will not be exact in general it is possible to make the error arbitrarily small by assuming that the distance between the particles is much larger than the radius of the wormhole mouths. Whenever the masses of the two wormhole mouths are diff…
This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian Wüthrich and under contract with Oxford University Press. (More information at www.beyondspacetime.net.) This chapter introduces causal set theory and identifies and articulates a 'problem of space' in this theory.
We formulate the generalization of the Legendrian Low conjecture of Natario and Tod (proved by Nemirovski and myself before) to the case of causally simple spacetimes. We prove a weakened version of the corresponding statement. In all known examples, a causally simple spacetime $(X, g)$ can be conformally embedded as an open set into some globally hyperbolic $(\widetilde X, \widetilde g)$ and the space of light rays in $(X, g)$ is an open submanifold of the space of light rays in $(\widetilde X, \widetilde g)$. If this is always the case, this provides an approach to solving the conjectures re…
Given a (d+1)-dimensional spacetime (M,g), one can consider the set N of all its null geodesics. If (M,g) is globally hyperbolic then this set is naturally a smooth (2d-1)-manifold. The sky of an event x in M is the set X of all null geodesics through x, and is an embedded submanifold of N diffeomorphic to S^{d-1}. Low conjectured that if d=2 then x,y are causally related in M iff X,Y are linked in N. We prove Low's conjecture for a (large) class of static spacetimes.
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…
Generalized quantum mechanics is used to examine a simple two-particle scattering experiment in which there is a bounded region of closed timelike curves (CTCs) in the experiment's future. The transitional probability is shown to depend on the existence and distribution of the CTCs. The effect is therefore acausal, since the CTCs are in the experiment's causal future. The effect is due to the non-unitary evolution of the pre- and post-scattering particles as they pass through the region of CTCs. We use the time-machine spacetime developed by Politzer [1], in which CTCs are formed due to the id…
We develop a framework that facilitates the study of the causal structure of spacetimes with a causally preferred foliation. Such spacetimes may arise as solutions of Lorentz-violating theories, e.g. Horava gravity. Our framework allows us to rigorously define concepts such as black/white holes and to formalize the notion of a `universal horizon', that has been previously introduced in the simpler setting of static and spherically symmetric geometries. We also touch upon the issue of development and prove that universal horizons are Cauchy horizons when evolution depends on boundary data or as…
Let $X$ be a $(2+1)$-dimensional globally hyperbolic spacetime with a Cauchy surface $Σ$ whose universal cover is homeomorphic to $\mathbb{R}^2$. We provide empirical evidence suggesting that the Jones polynomial detects causality in $X$. We introduce a new invariant of certain tangles related to the Conway polynomial, and prove that the Conway polynomial does not detect the connected sum of two Hopf links among relevant 3-component links, which suggests that the Conway polynomial does not detect causality in the scenario described.
Hawking's stable causality implies Sorkin and Woolgar's K-causality. The work investigates the possible equivalence between the two causality requirements, an issue which was first considered by H. Seifert and then raised again by R. Low after the introduction of K-causality. First, a new proof is given that a spacetime is stably causal iff the Seifert causal relation is a partial order. It is then shown that given a K-causal spacetime and chosen an event, the light cones can be widened in a neighborhood of the event without spoiling K-causality. The idea is that this widening of the light con…
The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact two-dimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group G_q over all compact (finite genus) oriented 2-surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU(2…
The causal properties of curved spacetime, which underpin our sense of time in gravitational theories, are defined by the null cones of the spacetime metric. In classical general relativity, it is assumed that these coincide with the light cones determined by the physical propagation of light rays. However, the quantum vacuum acts as a dispersive medium for the propagation of light, since vacuum polarisation in QED induces interactions which effectively violate the strong equivalence principle (SEP). For low frequencies the phenomenon of gravitational birefringence occurs and indeed, for some …
A brief overview of some open questions in general relativity with important consequences for causality theory is presented, aiming to a better understanding of the causal structure of the spacetime. Special attention is accorded to the problem of fundamental causal stability conditions. Several questions are raised and some of the potential consequences of recent results regarding the causality problem in general relativity are presented. A key question is whether causality violating regions are locally allowed. The new concept of almost stable causality is introduced; meanwhile, related cond…
Suppose we assume that (a) information about a black hole is encoded in its Hawking radiation and (b) causality is not violated to leading order in gently curved spacetime. Then we argue that spacetime cannot just be described as a manifold with a shape; it must be given an additional attribute which we call `thickness'. This thickness characterizes the spread of the quantum gravity wavefunctional in superspace -- the space of all 3-geometries. Low energy particles travel on spacetime without noticing the thickness parameter, so they just see an effective manifold. Objects with energy large en…
In this paper we investigated the causality problem present in the recent work about the Doplicher-Fredenhagen-Roberts-Amorim (DFRA) noncommutative framework which analyzed the complex scalar field. To accomplish this task we provided a brief review of the main ingredients of the problem and we demonstrated precisely that the DFRA algebra obeys the rules of the Canonical Commutation Relations algebra. This fact permitted us to prove the form of the DFRA operators previously constructed in the usual way. After that, we introduced the solution of its Klein-Gordon equation with a source term. Its…
To confront relativity theory with observation, it is necessary to split spacetime into its temporal and spatial components. The (1+3) timelike threading approach involves restrictions on the gravitational potentials $(g_{μν})$, while the (3+1) spacelike slicing approach involves restrictions on $(g^{μν})$. These latter coordinate conditions protect chronology within any such coordinate patch. While the threading coordinate conditions can be naturally integrated into the structure of Lorentzian geometry and constitute the standard coordinate conditions in general relativity, this circumstance …
A new interpretation of the causality implementation in the Lienard-Wiechert solution raises new doubts against the validity of the Lorentz-Dirac equation and the limits of validity of the Minkowski structure of spacetime.
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 present a topologically trivial, non-vacuum solution of the Einstein's field equations in four-dimensions, which is regular everywhere. The metric admits circular closed timelike curves, which appear beyond the null curve, and these timelike curves are linearly stable under linear perturbations. Additionally, the spacetime admits null geodesics curve which are not closed, and the metric is of type D in the Petrov classification scheme. The stress-energy tensor anisotropic fluid satisfy the different energy conditions and a generalization of Equation-of-State parameter of perfect fluid $p=ω\…
A unifying framework for the study of causal relations is presented. The causal relations are regarded as subsets of M x M and the role of the corresponding antisymmetry conditions in the construction of the causal ladder is stressed. The causal hierarchy of spacetime is built from chronology up to K-causality and new characterizations of the distinction and strong causality properties are obtained. The closure of the causal future is not transitive, as a consequence its repeated composition leads to an infinite causal subladder between strong causality and K-causality - the A-causality sublad…
If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segré classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the generalized eigenvector formalism reduce the configuration space to the c…
We introduced few years ago a new notion of causality for noncommutative spacetimes directly related to the Dirac operator and the concept of Lorentzian spectral triple. In this paper, we review in a non-technical way the noncommutative causal structure of many toy models as almost commutative spacetimes and the Moyal-Weyl spacetime. We show that those models present some unexpected physical interpretations as a geometrical explanation of the Zitterbewegung trembling motion of a fermion as well as some geometrical constraints on translations and energy jumps of wave packets on the Moyal spacet…
The causal set approach to quantum gravity embodies the concepts of causality and discreteness. This article explores some foundational and conceptual issues within causal set theory.
In Lovelock theories, gravity can travel faster or slower than light. The causal structure is determined by the characteristic hypersurfaces. We generalise a recent result of Izumi to prove that any Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom of a Lovelock theory. Hence gravitational signals cannot escape from the region inside such a horizon. We investigate the hyperbolicity of Lovelock theories by determining the characteristic hypersurfaces for various backgrounds. First we consider Ricci flat type N spacetimes. We show that characteristic hyper…
The causal properties of the family of Kerr-de Sitter spacetimes are analyzed and compared to those of the Kerr family. At first we show that a Kerr-de Sitter spacetime can be viewed as an assembly of Carter's blocks i.e. four dimensional spacetime regions contained within Killing horizons or a Killing horizon and the asymptotic de Sitter region. From this perspective and leaving aside topological identifications, the causal properties of a Kerr de Sitter spacetime are determined by the causal properties of the individual Carter's blocks viewed as spacetimes in their own right. We show that an…
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…
We show that the causal properties of asymptotically flat spacetimes depend on their dimensionality: while the time-like future of any point in the past conformal infinity $\mathcal{I}^-$ contains the whole of the future conformal infinity $\mathcal{I}^+$ in $(2+1)$ and $(3+1)$ dimensional Schwarzschild spacetimes, this property (which we call the Penrose property) does not hold for $(d+1)$ dimensional Schwarzschild if $d>3$. We also show that the Penrose property holds for the Kerr solution in $(3+1)$ dimensions, and discuss the connection with scattering theory in the presence of positive ma…
We show that that vector field-based models of the ether generically do not have a Hamiltonian that is bounded from below in a flat spacetime. We also demonstrate that these models possess multiple light cones in flat or curved spacetime, and that the non-lightlike characteristic is associated with an ether degree of freedom that will tend to form shocks. Since the field equations (and propagation speed) of this mode is singular when the timelike component of the ether vector field vanishes, we demonstrate that linearized analyses about such configurations cannot be trusted to produce robust a…
A number of techniques in Lorentzian geometry, such as those used in the proofs of singularity theorems, depend on certain smooth coverings retaining interesting global geometric properties, including causal ones. In this note we give explicit examples showing that, unlike some of the more commonly adopted rungs of the causal ladder such as strong causality or global hyperbolicity, less-utilized conditions such as causal continuity or causal simplicity do not in general pass to coverings, as already speculated by one of the authors (EM). As a consequence, any result which relies on these causa…
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…
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 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…
