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 explicit assumptions which in particular hold on a dense set of metrics $g$ for the $C^0$-topology.
Let $M$ be a connected, simply connected, oriented, closed, smooth four-manifold which is spin (or equivalently having even intersection form) and put $M^\times:=M\setminus\{{\rm point}\}$.In this paper we prove that if $X^\times$ is a smooth four-manifold homeomorphic but not necessarily diffeomorphic to $M^\times$ (more precisely, it carries a smooth structure à la Gompf) then $X^\times$ can be equipped with a complete Ricci-flat Riemannian metric. As a byproduct of the construction it follows that this metric is self-dual as well consequently $X^\times$ with this metric is in fact a hyper-K…
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.
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.
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 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 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.
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.
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.
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.
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…
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）
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…
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…
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 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…
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.
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…
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.
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…
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…
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 …
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 …
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…
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…
We study the geometry of a general class of vacuum asymptotically Anti-de Sitter spacetimes near the conformal boundary. In particular, the spacetime is only assumed to have finite regularity, and it is allowed to have arbitrary boundary topology and geometry. For the main results, we derive limits at the conformal boundary of various geometric quantities, and we use these limits to construct partial Fefferman--Graham expansions from the boundary. The results of this article will be applied, in upcoming papers, toward proving symmetry extension and gravity--boundary correspondence theorems for…
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…
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.
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)…
It is shown that the Chern-Simons functional, built in the spinor representation from the initial data on spacelike hypersurfaces, is invariant with respect to infinitesimal conformal rescalings if and only if the vacuum Einstein equations are satisfied. As a consequence, we show that in the phase space the Hamiltonian constraint of vacuum general relativity is the Poisson bracket of the imaginary part of this Chern-Simons functional and Misner's time (essentially the 3-volume). Hence the vacuum Hamiltonian constraint is the condition on the canonical variables that the imaginary part of the C…
In this paper we consider a special case of vacuum non-linear electrodynamics with a stress-energy tensor conformal to the Maxwell theory. Distinctive features of this model are: the absence of dimensional parameter for non-linearity description and a very simple form of the dominant energy condition, which can be easily verified in an arbitrary pseudo-riemannian space-time with the consequent constrains on the model parameters. In this paper we analyse some properties of astrophysical compact objects coupled to conformal vacuum non-linear electrodynamics.
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…
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 …
We discuss the global properties of static, spherically symmetric configurations of a self-gravitating real scalar field $ϕ$ in general relativity (GR), scalar-tensor theories (STT) and high-order gravity ($L=f(R)$) in various dimensions. In GR, for fields with arbitrary potentials $V(ϕ)$, not necessarily positive-definite, it is shown that the list of all possible types of space-time causal structure in the models under consideration is the same as the one for $ϕ= const$. In particular, there are no regular black holes with any asymptotics. These features are extended to STT and $f(R)$ theori…
Conformal gravity can elegantly solve the problem of spacetime singularities present in Einstein's gravity. For every physical spacetime, there is an infinite family of conformally-equivalent singularity-free metrics. In the unbroken phase, every non-singular metric is equivalent and can be used to infer the physical properties of the spacetime. In the broken phase, a Higgs-like mechanism should select a certain vacuum, which thus becomes the physical one. However, in the absence of the complete theoretical framework we do not know how to select the right vacuum. In this paper, we study the en…
We make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. We consider Dirichlet boundary conditions for the evolution equatio…
We extend the conformal gluing construction of Isenberg-Mazzeo-Pollack [18] by establishing an analogous gluing result for field theories obtained by minimally coupling Einstein's gravitational theory with matter fields. We treat classical fields such as perfect fluids and the Yang-Mills equations as well as the Einstein-Vlasov system, which is an important example coming from kinetic theory. In carrying out these extensions, we extend the conformal gluing technique to higher dimensions and codify it in such a way as to make more transparent where it can, and can not, be applied. In particular…
We analyze Killing Initial Data on Cauchy surfaces in conformally rescaled vacuum space-times satisfying Friedrich's conformal field equations. As an application, we derive the KID equations on a spacelike $\mathcal{J}^-$.
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.
A maximally reduced system of equations corresponding to the twisting type N Einstein metrics is given. When the cosmological constant $λ\to 0$ they reduce to the standard equations for the vacuum twisting type N's. All the metrics which are conformally equivalent to the twisting type N metrics and which admit 3-dimensional conformal group of symmetries are presented. In the Feferman class of metrics an example is given of a twisting type N metric which satisfies Bach's equations but is not Einstein.
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 $(1+1)$-…
We introduce the hypothesis that the matter content of the universe can be a product of the decay of primordial vector bosons. The effect of the intensive cosmological creation of these primordial vector $W, ~Z $ bosons from the vacuum is studied in the framework of General Relativity and the Standard Model where the relative standard of measurement identifying conformal quantities with the measurable ones is accepted. The relative standard leads to the conformal cosmology with the z-history of masses with the constant temperature, instead of the conventional z-history of the temperature with …
The reduced vacuum Hamiltonian equations of conformal geometrodynamics of compact manifolds in extrinsic time are written. This is achieved by generalizing the theorem of implicit function derivative to the functional analysis.
We investigate the conformal invariant Lagrangian of the self-gravitating U(1) scalar-gauge field and find new features of the model on the time-dependent axially symmetric Bondi-Marder spacetime. By considering the conformal symmetry as exact at the level of the Lagrangian and broken in the vacuum, a consistent model is found with an exact solution of the vacuum Bondi-Marder spacetime $g_{μν}=ω^2 \bar g_{μν}$, where $ω$ is the conformal factor and $\bar g_{μν}$ the `un-physical` spacetime. If we try to match this vacuum solution onto the interior vortex solution of the coupled Einstein-scalar…
It was recently discovered that for a boundary system in the presence of a background magnetic field, the quantum fluctuation of the vacuum would create a non-uniform magnetization density for the vacuum and a magnetization current is induced in the vacuum. It was also shown that this "magnetic Casimir effect" of the vacuum is closely related to another quantum effect of the vacuum, the Weyl anomaly. Furthermore, the phenomena can be understood in terms of the holography of the boundary system. In this paper, we generalize this four dimensional effect to six dimensions. We use the AdS/BCFT hol…
The General Relativity formulated with the aid of the spin connection coefficients is considered in the finite space geometry of similarity with the Dirac scalar dilaton. We show that the redshift evolution of the General Relativity describes the vacuum creation of the matter in the empty Universe at the electroweak epoch and the dilaton vacuum energy plays a role of the dark energy.
We show that there is only one physically acceptable vacuum state for quantum fields in de Sitter space-time which is left invariant under the action of the de Sitter-Lorentz group $SO(1,d)$ and supply its physical interpretation in terms of the Poincare invariant quantum field theory (QFT) on one dimension higher Minkowski spacetime. We compute correlation functions of the generalized vertex operator $:e^{i\hat{S}(x)}:$, where $\hat{S}(x)$ is a massless scalar field, on the $d$-dimensional de Sitter space and demonstrate that their limiting values at timelike infinities on de Sitter space rep…
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 d…
A class of exact solutions of the gravitational field equations in the vacuum on the brane are obtained by assuming the existence of a conformal Killing vector field, with non-static and non-central symmetry. In this case the general solution of the field equations can be obtained in a parametric form in terms of the Bessel functions. The behavior of the basic physical parameters describing the non-local effects generated by the gravitational field of the bulk (dark radiation and dark pressure) is also considered in detail, and the equation of state satisfied at infinity by these quantities is…
Back-reaction of the massless scalar field vacuum to the Universe expansion is considered. Automatic renormalization procedure based on the equations of motion instead of the Friedman equation is used to avoid the cosmological constant problem. It is found, that the vacuum tends to decelerate an expansion of Universe if the conformal acceleration of Universe is equal to zero. In contract, the vacuum acts as the true cosmological constant in Universe expanding with the present day acceleration. Estimation for third derivative of the Universe scale factor with respect to conformal time is presen…
We consider general-symmetry higher spin fields in AdS_5 and derive expressions for their one-loop corrections to vacuum energy E and the associated 4d boundary conformal anomaly a-coefficient. We a propose a similar expression for the second conformal anomaly c-coefficient. We show that all the three quantities (E, a, c) computed for N=8 gauged 5d supergravity are -1/2 of the values for N=4 conformal 4d supergravity and also twice the values for N=4 Maxwell multiplet. This gives 5d derivation of the fact that the system of N=4 conformal supergravity and four N=4 Maxwell multiplets is anomaly …
