Future Null Cone Research Articles

Improved decay estimates and C2-asymptotic stability of solutions to the Einstein-scalar field system in spherical symmetry

We investigate the asymptotic stability of solutions to the characteristic initial value problem for the Einstein (massless) scalar field system with a positive cosmological constant. We prescribe spherically symmetric initial data on a future null cone with a wider range of decaying profiles than previously considered. New estimates are then derived in order to prove that, for small data, the system has a unique global classical solution. We also show that the solution decays exponentially in (Bondi) time and that the radial decay is essentially polynomial, although containing logarithmic factors in some special cases. This improved asymptotic analysis allows us to show that, under appropriate and natural decaying conditions on the initial data, the future asymptotic solution is differentiable, up to and including spatial null-infinity, and approaches the de Sitter solution, uniformly, in a neighborhood of infinity. Moreover, we analyze the decay of derivatives of the solution up to second order showing the (uniform) [Formula: see text]-asymptotic stability of the de Sitter attractor in this setting. This corresponds to a surprisingly strong realization of the cosmic no-hair conjecture.

Global solutions to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in Bondi coordinates

We consider a characteristic initial value problem, with initial data given on a future null cone, for the Einstein (massless) scalar field system with a positive cosmological constant, in Bondi coordinates. We prove that, for small data, this system has a unique global classical solution which is causally geodesically complete to the future and decays polynomially in radius and exponentially in Bondi time, approaching the de Sitter solution.

Small sphere limit of the quasi-local energy with anti de-Sitter space reference

In [13], a new quasi-local energy is introduced for spacetimes with a non-zero cosmological constant. In this article, we study the small sphere limit of this newly defined quasi-local energy for spacetimes with a negative cosmological constant. For such spacetimes, the anti de-Sitter space is used as the reference for the quasi-local energy. Given a point $p$ in a spacetime $N$, we consider a canonical family of surfaces approaching $p$ along its future null cone and evaluate the limit of the quasi-local energy. The optimal embedding equation which identifies the critical points of the quasi-local energy is solved in order to evaluate the limit. Using the optimal embedding, we show that the limit recovers the stress-energy tensor of the matter field at $p$. For vacuum spacetimes, the quasi-local energy vanishes to a higher order. In this case, the limit of the quasi-local energy is related to the Bel-Robinson tensor at $p$.

Evaluating Small Sphere Limit of the Wang–Yau Quasi-Local Energy

In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point $p$ in a spacetime $N$, we consider a canonical family of surfaces approaching $p$ along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving an "optimal embedding equation" whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in [7]. Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at $p$. For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding equation is more complicated. Nevertheless, we are able to show that there exists a solution which is a local minimum and that the limit of its quasi-local energy is related to the Bel-Robinson tensor. Together with earlier work [7], this completes the consistency verification of the Wang-Yau quasi-local energy with all classical limits.

Linking and Causality in Globally Hyperbolic Space-times

The classical linking number lk is defined when link components are zero homologous. In [15] we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes. Here we apply alk to the study of causality in Lorentzian manifolds. Let M m be a spacelike Cauchy surface in a globally hyperbolic space-time (X m+1, g). The spherical cotangent bundle ST * M is identified with the space $$\mathcal N$$ of all null geodesics in (X,g). Hence the set of null geodesics passing through a point $$x\in X$$ gives an embedded (m−1)-sphere $$\mathfrak S_x$$ in $$\mathcal N=ST^*M$$ called the sky of x. Low observed that if the link $$(\mathfrak S_x, \mathfrak S_y)$$ is nontrivial, then $$x,y\in X$$ are causally related. This observation yielded a problem (communicated by R. Penrose) on the V. I. Arnold problem list [3,4] which is basically to study the relation between causality and linking. Our paper is motivated by this question. The spheres $$\mathfrak S_x$$ are isotopic to the fibers of $$(ST^*M)^{2m-1}\to M^m.$$ They are nonzero homologous and the classical linking number lk $$(\mathfrak S_x, \mathfrak S_y)$$ is undefined when M is closed, while alk $$(\mathfrak S_x, \mathfrak S_y)$$ is well defined. Moreover, alk $$(\mathfrak S_x, \mathfrak S_y)\in {\mathbb{Z}}$$ if M is not an odd-dimensional rational homology sphere. We give a formula for the increment of alk under passages through Arnold dangerous tangencies. If (X,g) is such that alk takes values in $${\mathbb{Z}}$$ and g is conformal to $$\widehat{g}$$ that has all the timelike sectional curvatures nonnegative, then $$x, y\in X$$ are causally related if and only if alk $$(\mathfrak S_x, \mathfrak S_y)\neq 0$$ . We prove that if alk takes values in $${\mathbb{Z}}$$ and y is in the causal future of x, then alk $$(\mathfrak S_x, \mathfrak S_y)$$ is the intersection number of any future directed past inextendible timelike curve to y and of the future null cone of x. We show that x,y in a nonrefocussing (X, g) are causally unrelated if and only if $$(\mathfrak S_x, \mathfrak S_y)$$ can be deformed to a pair of S m-1-fibers of $$ST^*M\to M$$ by an isotopy through skies. Low showed that if (X, g) is refocussing, then M is compact. We show that the universal cover of M is also compact.

Cylindrically Symmetric Models of Gravitational Collapse

In this thesis we examine two main problems. Firstly, we attempt to match the most general cylindrically symmetric vacuum spacetime with a Robertson-Walker interior. The matching conditions show that the interior must be dust filled, the boundary must be comoving and the vacuum region must be polarized. We use a result of Thorne's to simplify the line element. We can then prove that the matching is impossible. This demonstrates the impossibility of generalising the Oppenheimer-Snyder model of gravitational collapse to the cylindrically symmetric case The second problem is an analysis of cylindncally symmetric spacetimes with self-similarity modelling gravitational collapse The field equations and regularity conditions are examined firstly for a vacuum spacetime and then for a dust filled spacetime. The vacuum case leads to an explicit solution but no solutions that are of relevance to gravitational collapse. In the dust case, the solution of the field equations reduces to the solution of a non-linear third-order ordinary differential equation. A dynamical systems approach is then adopted, and an autonomous three-dimensional system is obtained. A unique solution is found to emanate from the regular axis {r = 0, t 0}, which we define as Co The solution coming from Co has one parameter (a bifurcation has occurred) and propagates up to the future null cone, 3, through the scaling origin p,, where p, = {(r, t) = (0, 0)). We describe the physical invariants of the system and discuss the nature of such a spacetime in terms of its global structure.

Solutions for the Hyperbolic Dirac Equation on R1, m

In this article the hyperbolic unit ball in R m will be identified with the manifold of rays in the future null cone in R m+1. By means of the induced Clifford algebra structure there, one can introduce a definition of Dirac operators on sections of homogeneous line bundles. An infinite class of solutions for the resulting hyperbolic Dirac-equation will be constructed, in case of an odd dimension. In order to obtain these solutions a geometrical picture will be used, because each ray in the future cone will be identified with a point on a surface Σ in the future cone. The original Dirac-equation can then be rewritten in terms of the coordinates on this surface, and the resulting equation will be solved by means of Frobenius' method.

Clifford analysis on projective hyperbolic space II

AbstractIn this paper, we identify the hyperbolic unit ball with the manifold of rays within the future null cone. By means of theinduced Clifford algebra structure there we obtain the definition of Dirac operators on sections of homogeneous line bundles and a homogeneous version of the Borel–Pompeiu formula.Finally, we introduce for this projective model of hyperbolic space a scale of the so‐called Qp‐spaces and study some of its basic properties. Copyright © 2002 John Wiley & Sons, Ltd.

A spherical impulse gravity wave.

An exact solution of Einstein's vacuum field equations describing a spherical-fronted impulsive gravity wave is given in a coordinate system in which the metric tensor is continuous across the (future null cone) history of the wave front in Minkowskian space-time. A coordinate transformation is presented which reduces the line element to Minkowskian form in the future of the null cone (it is manifestly in Minkowskian form in the past of the null cone) and enables us to demonstrate how the solution has been constructed using the Penrose procedure of subdividing flat space-time into two halves with the future null cone as boundary and then reattaching the halves with a warp.

Bessel function expressions for Dirac fields

Penrose’s null initial data techniques are used to obtain a solution to the electromagnetism-free Dirac equation, involving Bessel functions. Each Dirac field is split up into an infinite number of elementary contributions whose densities satisfy Dirac-like distributional field density equations on the interior of the future cone of an origin of real Minkowski space. The initial data hypersurface for all the elementary fields is conveniently chosen to be the future null cone of the origin. It is observed that the field density equations are invariant under proper Lorentz transformations. The entire fields are then expressed as explicit scaling invariant convolutions taken over the initial data cone.

Conformal geodesics in general relativity

Conformal geodesics, space-time curves which are related to conformal structures in a similar way as geodesics are related to metric structures, are discussed. ‘Conformal normal coordinates’, ‘conformal Gauss systems’ and their associated ‘normal connections’, ‘normal frames’ and ‘normal metrics’ are introduced and used to study: (i) asymptotically simple solutions of Ric( ͠g ) Λ͠g near conformal infinity, (ii) asymptotically simple solutions of Ric( ͠g ) = 0 with a past null infinity, which can be represented as the future null cone of a point i - , past time-like infinity. In the first case we define an ∞-parameter family of (physical) Gauss systems near conformal infinity, in the second case a ten-parameter family of (physical) Gauss systems covering a neighbourhood of i - . The behaviour of physical geodesics can be analysed in a particularly simple way in these coordinate systems. Each of these systems allows an extremely simple transition from the conformal analysis to the physical description of space-time. For Λη 00 < 0 (De-Sitter type solutions) all solutions are characterized in terms of the physical space-time by their data on past time-like infinity. For Λ = 0 the conserved quantities of Newman and Penrose are characterized as the first non-trivial coefficient, given by the value of the rescaled Weyl tensor at i - , in an expansion of the physical field in a Gauss system of the type considered before.

On purely radiative space-times

The existence of space-times representing pure gravitational radiation which comes in from infinity and interacts with itself is discussed. They are characterized as solutions of Einstein's vacuum field equations possessing a smooth structure at past null infinity which forms the “future null cone at past timelike infinity with complete generators.” The “pure radiation problem” is analysed where “free initial data” for Einstein's field equations are prescribed on the null cone at past time-like infinity. It is demonstrated how the pure radiation problem can be formulated as a local initial value problem for the symmetric hyperbolic system of reduced conformal vacuum field equations. Its solutions are uniquely determined by the free data.

Light cone cuts of null infinity in Schwarzschild geometry

Light cone cuts of future null infinity in Schwarzschild geometry are studied here. The future null cone from an arbitrary apex in the space-time has been constructed, and its intersection with ℐ + is obtained. Knowledge of the cuts yields a great deal of information about the interior of the space-time. In particular, we use it to reconstruct the Schwarzschild metric up to a conformal factor.

Space-Time Code

It is known that the entire geometry of many relativistic space-times can be summed up in two concepts, a space-time measure $\ensuremath{\mu}$ and a space-time causal or chronological order relation $C$, defining a causal measure space. On grounds of finiteness, unity, and symmetry, we argue that macroscopic space-time may be the classical-geometrical limit of a causal quantum space. A tentative conceptual framework is provided. Mathematical individuals that naturally form causal spaces are symbol sets or words, taken in the order of their generation. The natural extension of this purely logical concept to quantum symbols is formulated. The problem is posed of giving finite quantum rules for the generation of quantum symbol sets such that the order of generation becomes, in the classical limit, the causal order of space-time---as it were, to break the space-time code. The causal quantum spaces of three simple codes are generated for comparison with reality. The singulary code (repetitions of one digit) gives a linearly ordered external world of one time dimension and a circular internal space. The binary code gives the future null cone of special relativity and a circular internal space. The causal quantum space of word pairs in the binary code gives the solid light cone $t>{({x}^{2}+{y}^{2}+{z}^{2})}^{\frac{1}{2}}$ of special relativity and an internal space $U(2,C)$ suitable for the description of charge and isospin. In the classical limit, there is full translational and proper Lorentz invariance except at the boundary of the light cone, where the classical-geometrical limit fails. Plausible consequences of this model for cosmology and elementary particles are discussed. There is a quantum of time $\ensuremath{\tau}\ensuremath{\lesssim}\frac{\ensuremath{\hbar}}{{m}_{\ensuremath{\mu}}{c}^{3}}$ and a space-time complementarity relation $\ensuremath{\Delta}t\ensuremath{\Delta}x\ensuremath{\Delta}y\ensuremath{\Delta}z\ensuremath{\gtrsim}{\ensuremath{\tau}}^{4}$.