We investigate the problem of metric fluctuations in the presence of the vacuum fluctuations of matter fields and critically assess the usual assertion that vacuum energy implies a Planckian cosmological constant. A new stochastic classical approach to the quantum fluctuations of spacetime is developed. The work extends conceptually Boyer's random electrodynamics to a theory of random gravity but has a considerably richer structure for inheriting nonlinearity from general relativity. Attention is drawn to subtleties in choosing boundary conditions for metric fluctuations in relation to their dynamical consequences. Those compatible with the observed Lorentz invariance must allow for spontaneous conformal fluctuations, in addition to stochastic gravitational waves due to zero point gravitons. This is implemented through an effective metric defined in terms of the random spacetime metric modulo a fluctuating conformal factor. It satisfies an effective Einstein equation coupled to an effective stress-energy tensor incorporating gravitational self energy of metric fluctuations as well as matter fields. The effective Einstein equation is expanded up to second order nonlinearity. A UV-cutoff is introduced whose specific value, however, does not enter into the resulting description of random gravity. The averaged effective metric satisfies the empty space Einstein equation with an effective cosmological constant. This vanishes when only the massless matter fields are included. More generally, a finite effective cosmological constant compatible with the observational constraints can be obtained as long as the bare masses of the massive matter fields are nearly zero, or the conformal invariance of matter is restored at some high energy scale.
We study conformal gravity as an alternative theory of gravitation. For conformal gravity to be phenomenologically viable requires that the conformal symmetry is not manifest at the energy scales of the other known physical forces. Hence we require a mechanism for the spontaneous breaking of conformal invariance. In this paper we study the possibility that conformal invariance is spontaneously broken due to interactions with conformally coupled matter fields. The vacuum of the theory admits conformally non-invariant solutions corresponding to maximally symmetric space-times and variants thereo…
We explore the possibility of the spontaneous symmetry breaking in 5D conformally invariant gravity, whose action consists of a scalar field nonminimally coupled to the curvature with its potential. Performing dimensional reduction via ADM decomposition, we find that the model allows an exact solution giving rise to the 4D Minkowski vacuum. Exploiting the conformal invariance with Gaussian warp factor, we show that it also admits a solution which implement the spontaneous breaking of conformal symmetry. We investigate its stability by performing the tensor perturbation and find the resulting s…
We consider a multilevel hydrogen atom in interaction with the quantum electromagnetic field and separately calculate the contributions of the vacuum fluctuation and radiation reaction to the rate of change of the mean atomic energy of the atom for uniform acceleration. It is found that the acceleration disturbs the vacuum fluctuations in such a way that the delicate balance between the contributions of vacuum fluctuation and radiation reaction that exists for inertial atoms is broken, so that the transitions to higher-lying states from ground state are possible even in vacuum. In contrast to …
We elucidate the relation between the restricted Weyl symmetry and spontaneous symmetry breakdown of conformal symmetry. Using a scalar-tensor gravity, we show that the restricted Weyl symmetry leads to spontaneous symmetry breakdown of a global scale symmetry when the vacuum expectation value of a scalar field takes a non-zero value. It is then shown that this spontaneous symmetry breakdown induces spontaneous symmetry breakdown of special conformal symmetry in a flat Minkowski space-time, but the resultant Nambu-Goldstone boson is not an independent physical mode but expressed in terms of th…
We study the theory of Weyl conformal gravity with matter degrees of freedom in a conformally invariant interaction. Specifically, we consider a triplet of scalar fields and SO(3) non-abelian gauge fields, i.e. the Georgi-Glashow model conformally coupled to Weyl gravity. We show that the equations of motion admit solutions spontaneously breaking the conformal symmetry and the gauge symmetry, providing a mechanism for supplying a scale in the theory. The vacuum solution corresponds to anti-de-Sitter space-time, while localized soliton solutions correspond to magnetic monopoles in asymptoticall…
We study the spontaneous excitation of a uniformly accelerated two-level atom non-linearly coupled with vacuum Dirac field fluctuations using the formalism proposed by Dalibard, Dupont-Roc and Cohen-Tannoudji (DDC) and generalized to the present case in the current paper. We find that a cross term involving both vacuum fluctuations and radiation reaction appears, which is absent in the linear coupling cases such as an atom interacting with vacuum scalar or electromagnetic fluctuations. Furthermore, the contribution of this term actually dominates over that of radiation reaction. Thus, the mean…
We study, in the multipolar coupling scheme, a uniformly accelerated multilevel hydrogen atom in interaction with the quantum electromagnetic field near a conducting boundary and separately calculate the contributions of the vacuum fluctuation and radiation reaction to the rate of change of the mean atomic energy. It is found that the perfect balance between the contributions of vacuum fluctuations and radiation reaction that ensures the stability of ground-state atoms is disturbed, making spontaneous transition of ground-state atoms to excited states possible in vacuum with a conducting bound…
The generators of radial conformal symmetries in Minkowski space-time can be mapped to the generators of time evolution in conformal quantum mechanics. Within this correspondence we show that in conformal quantum mechanics the state associated to the inertial vacuum in Minkowski space-time has the structure of a thermofield double. Such state is built from a bipartite "vacuum state", the ground state of the generators of hyperbolic time evolution, which cover only part of the time domain. When time evolution is restricted to a finite time domain one obtains the temperature perceived by static …
This is the second of three papers on Conformal General Relativity (CGR). The conformal group is introduced here as the invariance group of the partial order of causal events in $n$D spacetime. Its general structure, discrete symmetries and field representations are described in detail. The spontaneous breakdown of conformal symmetry is then discussed and the role played by a ghost scalar field and a physical scalar field in 4D spacetime are evidenced. Kinematic--, conformal-- and proper--time hyperbolic coordinates are introduced in a negatively curved Milne spacetime for the purpose of provi…
We consider an atom in interaction with a massless scalar quantum field. We discuss the structure of the rate of variation of the atomic energy for an arbitrary stationary motion of the atom through the quantum vacuum. Our main intention is to identify and to analyze quantitatively the distinct contributions of vacuum fluctuations and radiation reaction to the spontaneous excitation of a uniformly accelerated atom in its ground state. This gives an understanding of the role of the different physical processes underlying the Unruh effect. The atom's evolution into equilibrium and the Einstein c…
We investigate several quantum phenomena related to quadratic gravity after rewriting the general fourth-order action in a more convenient form that is second-order in derivatives and produces only first-class constraints in phase space. We find that a Higgs mechanism may occur in the conformally invariant subset of the general quadratic action if the theory is conformally coupled to a scalar field that acquires a non-zero vacuum expectation value and spontaneously breaks the conformal symmetry. Then, in the broken phase, the originally massless spin-2 ghost may absorb both the scalar and vect…
Radiation from a mirror moving in vacuum electromagnetic fields is shown to vanish in the case of a uniformly accelerated motion. Such motions are related to conformal coordinate transformations, which preserve correlation functions characteristic of vacuum fluctuations. As a result, vacuum fluctuations remain invariant under reflection upon a uniformly accelerated mirror, which therefore does not radiate and experiences no radiation reaction force. Mechanical effects of vacuum fluctuations thus exhibit an invariance with respect to uniformly accelerated motions.
The spontaneous excitation of a two-level atom held static outside a four dimensional Schwarzschild black hole and in interaction with a massless scalar field in the Boulware, Unruh and Hartle-Hawking vacuum is investigated and the contributions of the vacuum fluctuations and radiation reaction to the rate of change of the mean atomic energy are calculated separately. We find that for the Boulware vacuum, the spontaneous excitation does not occur and the ground state atoms are stable, while the spontaneous emission rate for excited atoms in the Boulware vacuum, which is well-behaved at the eve…
The vacuum expectation values (VEVs) of the field squared and energy-momentum tensor for a massless scalar field are investigated in the Milne universe with general number of spatial dimensions. The vacuum state depends on the choice of the mode functions in the canonical quantization procedure and we assume that the field is prepared in the conformal vacuum. As the first step an integral representation for the difference of the Wightman functions corresponding to the conformal and Minkowski vacua is derived. The mean field squared and energy-momentum tensor are obtained in the coincidence lim…
In present paper the gravitational effect of spontaneous symmetry breaking vacuum energy density is investigated by subtracting the flat space-time contribution from the energy in the curved space-time. We found that the remain effective energy-momentum tensor is too small to cause the acceleration of the universe although it satisfies the characteristic of the dark energy. However it could provide a promising explanation to the puzzle why the gravitational effect produced by the huge symmetry breaking vacuum energy in the electroweak theory has not been observed, since it has a sufficient sma…
We discuss properties of conformal geodesics on general, vacuum, and warped product space-times and derive a system of conformal deviation equations. The results are used to show how to construct on the Schwarzschild-Kruskal space-time global conformal Gauss coordinates which extends smoothly and without degeneracy to future and past null infinity.
We study the spontaneous excitation of a radially polarized static multilevel atom outside a spherically symmetric black hole in multi-polar interaction with quantum electromagnetic fluctuations in the Boulware, Unruh and Hartle-Hawking vacuum states. We find that spontaneous excitation does not occur in the Boulware vacuum, and, in contrast to the scalar field case, spontaneous emission rate is not well-behaved at the event horizon as result of the blow-up of the proper acceleration of the static atom. However, spontaneous excitation can take place both in the Unruh and the Hartle-Hawking vac…
We point out that the (pseudo-)conformal Universe scenario may be realized as decay of conformally invariant, metastable vacuum, which proceeds via spontaneous nucleation and subsequent growth of a bubble of a putative new phase. We study perturbations about the bubble and show that their leading late-time properties coincide with those inherent in the original models with homogeneously rolling backgrounds. In particular, the perturbations of a spectator dimension-zero field have flat power spectrum.
We show that time-reflection symmetric, asymptotically flat, static vacuum data which admit a non-trivial conformal rescaling which leads again to such data must be axi-symmetric and admit a conformal Killing field. Moreover, it is shown that there exists a 3-parameter family of such data.
We discuss non-conformal non-supersymmetric large N gauge theories with vanishing vacuum energy density to all orders in perturbation theory. These gauge theories can be obtained via a field theory limit of Type IIB D3-branes embedded in orbifolded space-times. We also discuss gravity in this setup.
Motivated by constant-G theory, we introduce a one-parameter family of scalar-tensor theories as an extension of constant-G theory in which the conformal symmetry is a cosmological attractor. Since the model has the coupling function of negative curvature, we expect spontaneous scalarization occurs and the parameter is constrained by pulsar-timing measurements. Modeling neutron stars with realistic equation of states, we study the structure of neutron stars and calculate the effective scalar coupling with the neutron star in these theories. We find that within the parameter region where the ob…
Extending the results of an earlier article we give a complete description of the asymptotically flat, conformally non-flat, static vacuum data which admit non-trivial, asymptotically smooth conformal mappings onto other such data. These data form a 3-parameter family which decomposes into 1-parameter families of data which are conformal to each other. The data and the associated static vacuum solutions are given explicitly in terms of elliptic and, in a special case, elementary functions.
It is well-known that the $(1+1)$ dimensional Schwarzschild and spatially flat FLRW spacetimes are conformally flat. This work examines entanglement harvesting from the conformal field vacuums in these spacetimes between two Unruh-DeWitt detectors, moving along outgoing null trajectories. In $(1+1)$ dimensional Schwarzschild spacetime, we considered the Boulware and Unruh vacuums for our investigations. In this analysis, one observes that while entanglement harvesting is possible in $(1+1)$ dimensional Schwarzschild and $(1+3)$ dimensional de Sitter spacetimes, it is not possible in the $(1+1)…
In a series of recent papers Kaloper and Padilla proposed a mechanism to sequester standard model vacuum contributions to the cosmological constant. We study the consequences of embedding their proposal into a fully local quantum theory. In the original work, the bare cosmological constant $Λ$ and a scaling parameter $λ$ are introduced as global fields. We find that in the local case the resulting Lagrangian is that of a spontaneously broken conformal field theory where $λ$ plays the role of the dilaton. A vanishing or a small cosmological constant is thus a consequence of the underlying confo…
We suggest that spontaneous eternal inflation can provide a natural explanation for the thermodynamic arrow of time, and discuss the underlying assumptions and consequences of this view. In the absence of inflation, we argue that systems coupled to gravity usually evolve asymptotically to the vacuum, which is the only natural state in a thermodynamic sense. In the presence of a small positive vacuum energy and an appropriate inflaton field, the de Sitter vacuum is unstable to the spontaneous onset of inflation at a higher energy scale. Starting from de Sitter, inflation can increase the total …
It is shown that the notion of Conformal Mass can be defined within a given anti-de Sitter (AdS) branch of a Lovelock gravity theory as long as the corresponding vacuum is not degenerate. Indeed, conserved charges obtained by the addition of Kounterterms to the bulk action turn out to be proportional to the electric part of the Weyl tensor, when the fall-off of a generic solution in that AdS branch is considered. The factor of proportionality is the degeneracy condition for the vacua in the particular Lovelock AdS theory under study. This last feature explains the obstruction to define Conform…
Moitvated in part by [3], in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a timelike conformal Killing field X, then M must split as a metric product, and X must be Killing. This gives a partial proof of the Bartnik splitting conjecture in the vacuum setting.
In a recent paper we have established the form of the metric-torsional conformal gravitational field equations, and in the present paper we study their vacuum configurations; we will consider a specific situation that will enable us to look for the torsionless limit: two types of special exact solutions are found eventually. A discussion on general remarks will follow.
In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe metrics and with TT-tensor $σ$ = 0 have a non-trivial solution, and thus answer a question by D. Maxwell.
A free-theory vacuum state of an interacting field theory, e.g. quantum gravity, is unstable at tree level in general due to spontaneous emission of Fock-space particles in any spacetime with no global timelike Killing vectors, such as de Sitter spacetime, in the interaction picture. As an example, the rate of spontaneous emission of Fock-space particles is calculated in phi^4 theory in de Sitter spacetime. It is possible that this apparent spontaneous emission does not correspond to any physical processes because the states are not evolved by the true Hamiltonian in the interaction picture. N…
We give a quantum mechanical description of accelerated relativistic particles in the framework of Coherent States (CS) of the (3+1)-dimensional conformal group SU(2,2), with the role of accelerations played by special conformal transformations and with the role of (proper) time translations played by dilations. The accelerated ground state $\tildeϕ_0$ of first quantization is a CS of the conformal group. We compute the distribution function giving the occupation number of each energy level in $\tildeϕ_0$ and, with it, the partition function Z, mean energy E and entropy S, which resemble that …
A direct consequence of quantization of gravity would be quantum gravitational vacuum fluctuations which induce quadrupole moments in gravitationally polarizable atoms. In this paper, we study the spontaneous excitation of a gravitationally polarizable atom with a uniform acceleration $a$ in interaction with a bath of fluctuating quantum gravitational fields in vacuum, and compare the result with that of a static one in a thermal bath of gravitons at the Unruh temperature. We find that, under the fluctuations of spacetime itself, transitions to higher-lying excited states from the ground state…
We construct two classes of higher-derivative supergravity theories generalizing Einstein supergravity. We explore their dynamical content as well as their vacuum structure. The first class is found to be equivalent to Einstein supergravity coupled to a single chiral superfield. It has a unique stable vacuum solution except in a special case, when it becomes identical to a simple no-scale theory. The second class is found to be equivalent to Einstein supergravity coupled to two chiral superfields and has a richer vacuum structure. It is demonstrated that theories of the second class can posses…
Static vacuum metrics in (4 + 1) dimensions with Λ < 0 and squashed conformal infinity
School of Physics, Nankai University（南开大学物理学院）；Mathematical Sciences and STAG Research Centre, University of Southampton（南安普顿大学数学科学与STAG研究中心）；TU Wien, Institut für Analysis und Scientific Computing（维也纳工业大学分析与科学计算研究所）；Department of Mathematics, KTH Royal Institute of Technology（瑞典皇家理工学院数学系）；Beijing Institute of Mathematical Sciences and Applications（北京应用数学与交叉科学研究院）；Center for Theoretical Physics of the Polish Academy of Sciences（波兰科学院理论物理中心）
It is shown that Einstein gravity in four dimensions with small cosmological constant and small extra dimensions can be obtained by spontaneous compactification of Lovelock gravity in vacuum. Assuming that the extra dimensions are compact spaces of constant curvature, General Relativity is recovered within certain class of Lovelock theories possessing necessarily cubic or higher order terms in curvature. This bounds the higher dimension to be at least seven. Remarkably, the effective gauge coupling and Newton constant in four dimensions are not proportional to the gravitational constant in hig…
Using the formalism that separates the contributions of vacuum fluctuations and radiation reaction to the rate of change of the mean atomic energy, we show that a two-level atom in interaction with a quantum massless scalar field both in the Hartle-Hawking and Unruh vacuum in a 1+1 dimensional black hole background spontaneously excites as if there is thermal radiation at the Hawking temperature emanating from the black hole. Our calculation, therefore, ties the existence of Hawking radiation to the spontaneous excitation of a two-level atom placed in vacuum in the exterior of a black hole and…
We present a novel theory of the very early universe which addresses the traditional horizon and flatness problems of big bang cosmology and predicts a scale invariant spectrum of perturbations. Unlike inflation, this scenario requires no exponential accelerated expansion of space-time. Instead, the early universe is described by a conformal field theory minimally coupled to gravity. The conformal fields develop a time-dependent expectation value which breaks the flat space so(4,2) conformal symmetry down to so(4,1), the symmetries of de Sitter, giving perturbations a scale invariant spectrum.…
We consider the problem of picking a physically motivated vacuum state on a spherically symmetric spacetime with an extra conformal Killing vector, as opposed to an extra Killing vector as in the Schwarzschild case. Considering a conformal symmetry instead of a symmetry allows us to consider spacetimes that are dynamical and not static (like Schwarzschild). The extra conformal symmetry allows us to calculate the response of particle detectors however. We look at the specific example of a self-similar LTB spacetime that represents a spherically symmetric but inhomogeneous cosmology. We remark t…
Ibohal, Ishwarchandra and Singh proposed a class of exact, non-vacuum and conformally flat solutions of Einstein's equations whose stress tensor $T_{ab}$ has negative pressure. We show that $T_{ab}$ corresponds to an anisotropic fluid and the equation of state parameter seems not to be $ω= -1/2$. We consider the authors' constant cannot be the mass of a test particle but is related to a Rindler acceleration of a spherical distribution of uniformly accelerating observers.
The Bach equation, i.e., the vacuum field equation following from the Lagrangian L=C_{ijkl}C^{ijkl}, will be completely solved for the case that the metric is conformally related to the cartesian product of two 2-spaces; this covers the spherically and the plane symmetric space-times as special subcases. Contrary to other approaches, we make a covariant 2+2-decomposition of the field equation, and so we are able to apply results from 2-dimensional gravity. Finally, some cosmological solutions will be presented and discussed.
An analysis of conformal geodesics in the Schwarzschild-de Sitter and Schwarzschild-anti de Sitter families of spacetimes is given. For both families of spacetimes we show that initial data on a spacelike hypersurface can be given such that the congruence of conformal geodesics arising from this data cover the whole maximal extension of canonical conformal representations of the spacetimes without forming caustic points. For the Schwarzschild-de Sitter family, the resulting congruence can be used to obtain global conformal Gaussian systems of coordinates of the conformal representation. In the…
Using the implicit function theorem, we prove existence of solutions of the so-called conformally covariant split system on compact 3-dimensional Riemannian manifolds. They give rise to non-Constant Mean Curvature (non-CMC) vacuum initial data for the Einstein equations. We investigate the conformally covariant split system defined on compact manifolds with or without boundaries. In the former case, the boundary corresponds to an apparent horizon in the constructed initial data. The case with a cosmological constant is then considered separately. Finally, to demonstrate the applicability of th…
We construct second-quantized (field) theories on coset spaces of pseudo-unitary groups U(p,q)$. The existence of degenerated quantum vacua (coherent states of zero modes) leads to a breakdown of the original pseudo-unitary symmetry. The action of some spontaneously broken symmetry transformations destabilize the vacuum and make it to radiate. We study the structure of this thermal radiation for curved phase spaces of constant curvature: complex projective spaces CP^{N-1}=SU(N)/U(N-1) and open complex balls CD^{N-1}=SU(1,N-1)/U(N-1). Positive curvature is related to generalized Fermi-Dirac (FD…
We investigate neutrino flavor oscillations in vacuum within a general conformal coupling model. We first examine the flavor oscillations within the plane-wave description of neutrinos, and then we extend our analysis to the wavepacket-based formalism. In both cases, we derive the general formulas for the flavor transition probability in arbitrary static and spherically symmetric spacetimes. We thoroughly discuss and assess in both cases the different possible ways -- dictated by the presence of the conformal coupling -- of computing the flavor transition probability. We show that the conforma…
This is the first in a series of articles on the numerical solution of Friedrich's conformal field equations for Einstein's theory of gravity. We will discuss in this paper why one should be interested in applying the conformal method to physical problems and why there is good hope that this might even be a good idea from the numerical point of view. We describe in detail the derivation of the conformal field equations in the spinor formalism which we use for the implementation of the equations, and present all the equations as a reference for future work. Finally, we discuss the implications …
Motivated by recent work suggesting observably large spacetime fluctuations in the causal development of an empty region of flat space, we conjecture that these metric fluctuations can be quantitatively described in terms of a conformal field theory of near-horizon vacuum states. One consequence of this conjecture is that fluctuations in the modular Hamiltonian $ΔK$ of a causal diamond are equal to the entanglement entropy: $\langle ΔK^2 \rangle = \langle K \rangle = \frac{A(Σ_{d-2})}{4 G_d}$, where $A(Σ_{d-2})$ is the area of the entangling surface in $d$ dimensions. Our conjecture applies to…
For a vacuum initial data set of the Einstein field equations it is possible to carry out a conformal rescaling or conformal compactification of the data giving rise to an initial data set for the Friedrich vacuum conformal equations. When will the data development with respect to the conformal equations of this set be a conformal extension of a type D solution? In this work we provide a set of necessary and sufficient conditions on a set of initial data for the conformal equations that guarantees that the data development of the conformal equations has a subset that is conformal to a vacuum t…
Local scalar QFT (in Weyl algebraic approach) is constructed on degenerate semi-Riemannian manifolds corresponding to Killing horizons in spacetime. Covariance properties of the $C^*$-algebra of observables with respect to the conformal group $PSL(2,\bR)$ are studied.It is shown that, in addition to the state studied by Guido, Longo, Roberts and Verch for bifurcated Killing horizons, which is conformally invariant and KMS at Hawking temperature with respect to the Killing flow and defines a conformal net of von Neumann algebras, there is a further wide class of algebraic (coherent) states repr…
The null curvature condition (NCC) is the requirement that the Ricci curvature of a Lorentzian manifold be nonnegative along null directions, which ensures the focusing of null geodesic congruences. In this note, we show that the NCC together with the causal structure significantly constrain the metric. In particular, we prove that any conformal rescaling of a vacuum spacetime introduces either geodesic incompleteness or negative null curvature, provided the conformal factor is non-constant on at least one complete null geodesic. In the context of bulk reconstruction in AdS/CFT, our results co…
A cosmic potential which can relax the vacuum energy is proposed in a framework of scalar-tensor gravity. In the phase of the gravity scalar field around the evolution with an approximate emergent conformal symmetry, we have obtained a set of cosmological equations with the dark energy regulated to the order of a conformal anomaly parameter. Through a role of the cosmic potential, the vacuum energy which could be generated in matter Lagrangian does not contribute to the dark energy in the phase.
In the studies of quantum field theory in curved spacetime, the ambiguous concept of vacuum state and the particle content is a long-standing debatable aspect. So far it is well known to us that in the background of the curved spacetime, some privileged class of observers detect particle production in the suitably chosen vacuum states of the quantum matter fields. In this work we aim to study the characteristics behaviour of these produced particles in the background of the de-Sitter (dS) Friedmann-Lamaître-Robertson-Walker (FLRW) Universe (both for $(1+1)$ and $(3+1)$ dimensions) and $(1+1)$-…
In an earlier paper (Class. Quantum Grav. 19 (2002) p.259) the author wrote the homothetic equations for vacuum solutions in a first order formalism allowing for arbitrary alignment of the dyad. This paper generalises that method to conformal vectors in non-vacuum spaces. The method is applied to metrics admitting a three parameter motion group on non-null orbits.
Generalizing Riemannian theorems of Anderson-Herzlich and Biquard, we show that two $(n+1)$-dimensional stationary vacuum space-times (possibly with cosmological constant $Λ\in \R$) that coincide up to order one along a timelike hypersurface $\mycal T$ are isometric in a neighbourhood of $\mycal T$. We further prove that KIDS of $\partial M$ extend to Killing vectors near $\partial M$. In the AdS type setting, we show unique continuation near conformal infinity if the metrics have the same conformal infinity and the same undetermined term. Extension near $\partial M$ of conformal Killing vecto…
Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $σ\in W^{1,p}$ and a positive function $τ\in W^{1,p}$, $p > n$, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data $σ$ and $τ$. We show that if no solution exists then there is a non-trivial solution of another non-linear limit equation on $1$-forms. This last equation can be shown to be without solutions no solution in many situations. As a corollary, we get existence of solutions of the vacuum Einstein constraint equation under explic…
In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain…
