Future Infinity Research Articles

On the hologram of de Sitter

A careful reduction of three-dimensional gravity to the Liouville description is performed, where all gauge fixing and on-shell conditions come from the definition of asymptotic de Sitter spaces. The roles of both past and future infinities are discussed and the conditions space-time evolution imposes on both Liouville fields are made explicit. Space-times which correspond to nonequivalent profiles of the Liouville field at ${\mathcal{I}}^{\ensuremath{-}}$ and ${\mathcal{I}}^{+}$ are shown to exist. The qualitative implications of this for any tentative dual theory are presented.

Graphical and kinematical approach to cosmological horizons

We study the apparition of event horizons in accelerated expanding cosmologies. We give a graphical and analytical representation of the horizons using proper distances to coordinate the events. Our analysis is mainly kinematical. We show that, independently of the dynamical equations, all the event horizons tend in the future infinity to a given expression depending on the scale factor that we call asymptotic horizon. We also encounter a subclass of accelerating models without horizon. When the ingoing null geodesics do not change concavity in its cosmic evolution we recover the de Sitter and quintessence-Friedmann-Robertson-Walker models.

Conformal vacua and entropy in de Sitter space

The dS/CFT correspondence is illuminated through an analysis of massive scalar field theory in d-dimensional de Sitter space. We consider a one-parameter family of dS-invariant vacua related by Bogolyubov transformations and compute the corresponding Green functions. It is shown that none of these Green functions correspond to the one obtained by analytic continuation from AdS. Among this family of vacua are in (out) vacua which have no incoming (outgoing) particles on past (future) infinity. Surprisingly, it is shown that in odd spacetime dimensions the in and out vacua are the same, implying the absence of particle production for this state. The correlators of the boundary CFT, as defined by the dS/CFT correspondence, are shown to depend on the choice of vacuum state - the correlators with all points on past infinity vanish in the in vacuum. For dS_3 we argue that this bulk vacuum dependence of the correlators is dual to a deformation of the boundary CFT_2 by a specific marginal operator. It is also shown that Witten's non-standard de Sitter inner product (slightly modified) reduces to the standard inner product of the boundary field theory. Next we consider a scalar field in the Kerr-dS_3 Euclidean vacuum. A density matrix is constructed by tracing out over modes which are causally inaccessible to a single geodesic observer. This is shown to be a thermal state at the Kerr-dS_3 temperature and angular potential. It is further shown that, assuming Cardy's formula, the microscopic entropy of such a thermal state in the boundary CFT precisely equals the Bekenstein-Hawking value of one quarter the area of the Kerr-dS_3 horizon.

Late-time evolution of a charged massless scalar field in the spacetime of a dilaton black hole

We investigate the power-law tails in the evolution of a charged massless scalar field around a fixed background of a dilaton black hole. Using both analytical and numerical methods we find the inverse power-law relaxation of charged fields at future timelike infinity, future null infinity, and along the outer horizon of the considered black hole. We invisage that a charged hair decays slower than neutral ones. The oscillatory inverse power law along the outer horizon of the dilaton black hole is of a great importance for a mass inflation scenario along the Cauchy horizon of a dynamically formed dilaton black hole.

Generalization of the model of Hawking radiation with modified high frequency dispersion relation

Hawking radiation is one of the most interesting phenomena predicted by the theory of quantum fields in curved space. The origin of Hawking radiation is closely related to the fact that a particle which marginally escapes from collapsing into a black hole is observed at future infinity with an infinitely large redshift. In other words, such a particle had a very high frequency when it was near the event horizon. Motivated by the possibility that the property of Hawking radiation may be altered by some unknown physics which may exist beyond some critical scale, Unruh proposed a model which has higher order spatial derivative terms. In his model, the effects of unknown physics are modeled so as to be suppressed for waves with a wavelength much longer than the critical scale ${k}_{0}^{\ensuremath{-}1}.$ Surprisingly, it was shown that the thermal spectrum is recovered for such modified models. To introduce such higher order spatial derivative terms, Lorentz invariance must be violated because one special direction needs to be chosen. In previous works, the rest frame of freely falling observers was employed as this special reference frame. Here we give an extension by allowing a more general choice of the reference frame. Developing the method taken by Corley, we show that the resulting spectrum of created particles again becomes the thermal one at the Hawking temperature even if the choice of the reference frame is generalized. Using the technique of the matched asymptotic expansion, we also show that the correction to the thermal radiation stays of order ${k}_{0}^{\ensuremath{-}2}$ or smaller when the spectrum of radiated particle around its peak is concerned.

Polyhomogeneity and zero-rest-mass fields with applications to Newman-Penrose constants

A discussion of polyhomogeneity (asymptotic expansions in terms of 1/r and lnr) for zero-rest-mass fields and gravity and its relation with the Newman-Penrose (NP) constants is given. It is shown that for spin-s zero-rest-mass fields propagating on Minkowski spacetime, the logarithmic terms in the asymptotic expansion appear naturally if the field does not obey the `peeling theorem'. The terms that give rise to the slower fall-off admit a natural interpretation in terms of an advanced field. The connection between such fields and the NP constants is also discussed. The case when the background spacetime is curved and polyhomogeneous (in general) is considered. The free fields have to be polyhomogeneous, but the logarithmic terms due to the connection appear at higher powers of 1/r. In the case of gravity, it is shown that it is possible to define a new auxiliary field, regular at null infinity, and containing some relevant information on the asymptotic behaviour of the spacetime. This auxiliary zero-rest-mass field `evaluated at future infinity (i+)' yields the logarithmic NP constants.

Late-time Evolution of the Yang-Mills Field in the Spherically Symmetric Gravitational Collapse

We investigate the late-time evolution of the Yang-Mills field in the self-gravitating backgrounds: Schwarzschild and Reissner-Nordstr\"om spacetimes. The late-time power-law tails develop in the three asymptotic regions: the future timelike infinity, the future null infinity and the black hole horizon. In these two backgrounds, however, the late-time evolution has quantitative and qualitative differences. In the Schwarzschild black hole background, the late-time tails of the Yang-Mills field are the same as those of the neutral massless scalar field with multipole moment l=1. The late-time evolution is dominated by the spacetime curvature. When the background is the Reissner-Nordstr\"om black hole, the late-time tails have not only a smaller power-law exponent, but also an oscillatory factor. The late-time evolution is dominated by the self-interacting term of the Yang-Mills field. The cause responsible for the differences is revealed.

Quantum global structure of de Sitter space

I study the global structure of de Sitter space in the semi-classical and one-loop approximations to quantum gravity. The creation and evaporation of neutral black holes causes the fragmentation of de Sitter space into disconnected daughter universes. If the black holes are stabilized by a charge, I find that the decay leads to a necklace of de Sitter universes (`beads') joined by near-extremal black hole throats. For sufficient charge, more and more beads keep forming on the necklace, so that an unbounded number of universes will be produced. In any case, future infinity will not be connected. This may have implications for a holographic description of quantum gravity in de Sitter space.

Late-time evolution of charged gravitational collapse and decay of charged scalar hair. II

We study analytically the initial value problem for a charged massless scalar-field on a Reissner-Nordstr\"om spacetime. Using the technique of spectral decomposition we extend recent results on this problem. Following the no-hair theorem we reveal the dynamical physical mechanism by which the charged hair is radiated away. We show that the charged perturbations decay according to an inverse power-law behaviour at future timelike infinity and along future null infinity. Along the future outer horizon we find an oscillatory inverse power-law relaxation of the charged fields. We find that a charged black hole becomes ``bald'' slower than a neutral one, due to the existence of charged perturbations. Our results are also important to the study of mass-inflation and the stability of Cauchy horizons during a dynamical gravitational collapse of charged matter in which a charged black-hole is formed.

Late-time evolution of charged gravitational collapse and decay of charged scalar hair. III. Nonlinear analysis

We study the nonlinear gravitational collapse of a charged massless scalar-field. We confirm the existence of oscillatory inverse power-law tails along future timelike infinity, future null infinity and along the future outer-horizon. The nonlinear dumping exponents are in excellent agreement with the analytically predicted ones. Our results prove the analytic conjecture according to which a charged hair decays slower than a neutral one and also suggest the occurrence of mass-inflation along the Cauchy horizon of a dynamically formed charged black-hole.

Late-time evolution of charged gravitational collapse and decay of charged scalar hair. I

We study analytically the asymptotic evolution of charged fields around a Reissner-Nordstr\"om black-hole. Following the no-hair theorem we focus attention on the dynamical mechanism by which the charged hair is radiated away. We find an inverse power-law relaxation of the charged fields at future timelike infinity, along future null infinity and an oscillatory inverse power-law relaxation along the future outer horizon. We show that a charged hair is shedd slower than a neutral one. Our results are also of importance to the study of mass inflation and the stability of Cauchy horizons during a dynamical gravitational collapse of charged matter to form a charged black-hole

Upper bound for entropy in asymptotically de Sitter space-time

We investigate the nature of asymptotically de Sitter space-times containing a black hole. We show that if the matter fields satisfy the dominant energy condition and cosmic censorship holds in the considered space-time, the area of the cosmological event horizon for an observer approaching a future timelike infinity does not decrease; i.e., the second law is satisfied. We also show under the same conditions that the total area of the black hole and the cosmological event horizon, a quarter of which is the total Bekenstein-Hawking entropy, is less than 12\ensuremath{\pi}/\ensuremath{\Lambda}, where \ensuremath{\Lambda} is the cosmological constant. The physical implications are also discussed.

Method for calculating the global structure of (singular) spacetimes.

A formalism and its numerical implementation is presented which allows one to calculate quantities determining the spacetime structure in the large directly. This is achieved by conformal techniques by which future null infinity (${\mathit{scrI}}^{+}$) and future timelike infinity (${\mathit{i}}^{+}$) are mapped to grid points on the numerical grid. The determination of the causal structure of singularities, the localization of event horizons, the extraction of radiation, and the avoidance of unphysical reflections at the outer boundary of the grid, are demonstrated with calculations of spherically symmetric models with a scalar field as matter and radiation model. \textcopyright{} 1996 The American Physical Society.

General relativistic scalar-field models and asymptotic flatness

For a class of scalar fields including the massless Klein--Gordon field the general relativistic hyperboloidal initial-value problems are equivalent in a certain sense. By using this equivalence and conformal techniques it is proved that the hyperboloidal initial-value problem for these scalar-fields has a unique solution which is weakly asymptotically flat. For data sufficiently close to data for flat spacetime there exists a smooth future null infinity and a regular future timelike infinity.

Scattering for the Yang-Mills Equations

We construct wave and scattering operators for the Yang-Mills equations on Minkowski space, ${{\mathbf {M}}_0} \cong {{\mathbf {R}}^4}$. Sufficiently regular solutions of the Yang-Mills equations on ${{\mathbf {M}}_0}$ are known to extend uniquely to solutions of the corresponding equations on the universal cover of its conformal compactification, $\tilde {\mathbf {M}} \cong {\mathbf {R}} \times {S^3}$. Moreover, the boundary of ${{\mathbf {M}}_0}$ as embedded in $\tilde {\mathbf {M}}$ is the union of "lightcones at future and past infinity", ${C_ \pm }$. We construct wave operators ${W_ \pm }$ as continuous maps from a space ${\mathbf {X}}$ of time-zero Cauchy data for the Yang-Mills equations to Hilbert spaces ${\mathbf {H}}({C_ \pm })$ of Goursat data on ${C_ \pm }$. The scattering operator is then a homeomorphism $S:{\mathbf {X}} \to {\mathbf {X}}$ such that $U{W_ + } = {W_ - }S$, where $U:{\mathbf {H}}({C_ + }) \to {\mathbf {H}}({C_ - })$ is the linear isomorphism arising from a conformal transformation of $\tilde {\mathbf {M}}$ mapping ${C_ - }$ onto ${C_ + }$. The maps ${W_ \pm }$ and $S$ are shown to be smooth in a certain sense.

Scattering for the Yang-Mills equations

We construct wave and scattering operators for the Yang-Mills equations on Minkowski space, M 0 ≅ R 4 {{\mathbf {M}}_0} \cong {{\mathbf {R}}^4} . Sufficiently regular solutions of the Yang-Mills equations on M 0 {{\mathbf {M}}_0} are known to extend uniquely to solutions of the corresponding equations on the universal cover of its conformal compactification, M ~ ≅ R × S 3 \tilde {\mathbf {M}} \cong {\mathbf {R}} \times {S^3} . Moreover, the boundary of M 0 {{\mathbf {M}}_0} as embedded in M ~ \tilde {\mathbf {M}} is the union of "lightcones at future and past infinity", C ± {C_ \pm } . We construct wave operators W ± {W_ \pm } as continuous maps from a space X {\mathbf {X}} of time-zero Cauchy data for the Yang-Mills equations to Hilbert spaces H ( C ± ) {\mathbf {H}}({C_ \pm }) of Goursat data on C ± {C_ \pm } . The scattering operator is then a homeomorphism S : X → X S:{\mathbf {X}} \to {\mathbf {X}} such that U W + = W − S U{W_ + } = {W_ - }S , where U : H ( C + ) → H ( C − ) U:{\mathbf {H}}({C_ + }) \to {\mathbf {H}}({C_ - }) is the linear isomorphism arising from a conformal transformation of M ~ \tilde {\mathbf {M}} mapping C − {C_ - } onto C + {C_ + } . The maps W ± {W_ \pm } and S S are shown to be smooth in a certain sense.

On the existence ofn-geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure

It is demonstrated that initial data sufficiently close to De-Sitter data develop into solutions of Einstein's equations Ric[g]=Λg with positive cosmological constant Λ, which are asymptotically simple in the past as well as in the future, whence null geodesically complete. Furthermore it is shown that hyperboloidal initial data (describing hypersurfaces which intersect future null infinity in a space-like two-sphere), which are sufficiently close to Minkowskian hyperboloidal data, develop into future asymptotically simple whence null geodesically future complete solutions of Einstein's equations Ric[g]=0, for which future null infinity forms a regular cone with vertexi+ that represents future time-like infinity.

The unpredictability of quantum gravity

Quantum gravity seems to introduce a new level of unpredictability into physics over and above that normally associated with the uncertainty principle. This is because the metric of spacetime can fluctuate from being globally hyperbolic. In other words, the evolution is not completely determined by Cauchy data at past or future infinity. I present a number of axioms that the asymptotic Green functions should obey in any reasonable theory of quantum gravity. These axioms are the same as for ordinary quantum field theory in flat spacetime, except that one axiom, that of asymptotic completeness, is omitted. This allows pure quantum states to decay into mixed states. Calculations with simple models of topologically non-trivial spacetime indicate that such loss of quantum coherence will occur but that the effect will be very small except for fundamental scalar particles, if any such exist.

Path-integral derivation of black-hole radiance

The Feynman path-integral method is applied to the quantum mechanics of a scalar particle moving in the background geometry of a Schwarzschild black hole. The amplitude for the black hole to emit a scalar particle in a particular mode is expressed as a sum over paths connecting the future singularity and infinity. By analytic continuation in the complexified Schwarzschild space this amplitude is related to that for a particle to propagate from the past singularity to infinity and hence by time reversal to the amplitude for the black hole to absorb a particle in the same mode. The form of the connection between the emission and absorption probabilities shows that a Schwarzschild black hole will emit scalar particles with a thermal spectrum characterized by a temperature which is related to its mass, $M$, by $T=\frac{\ensuremath{\hbar}{c}^{3}}{8\ensuremath{\pi}}\mathrm{GMk}$. Thereby a conceptually simple derivation of black-hole radiance is obtained. The extension of this result to other spin fields and other black-hole geometries is discussed.

Space-times with a future projective infinity

We use a projective structure to make precise the concept of ``future timelike infinity'' for certain space-times. We apply our definitions to many exact and approximate solutions of the Einstein equations. Some physical restrictions are necessary.