We study gravitational radiation for a positive value of the cosmological constant $Λ$. We rely on two battle-tested procedures: (i) We start from the same null coordinate system used by Bondi and Sachs for $Λ= 0$, but, introduce boundary conditions adapted to allow radiation when $Λ>0$. (ii) We determine the asymptotic symmetries by studying, à la Regge-Teitelboim, the surface integrals generated in the action by these boundary conditions. A crucial difference with the $Λ=0$ case is that the wave field does not vanish at large distances, but is of the same order as de Sitter space. This novel property causes no difficulty; on the contrary, it makes quantities finite at every step, without any regularization. A direct consequence is that the asymptotic symmetry algebra consists only of time translations and space rotations. Thus, it is not only finite-dimensional, but smaller than de Sitter algebra. We exhibit formulas for the energy and angular momentum and their fluxes. In the limit of $Λ$ tending to zero, these formulas go over continuously into those of Bondi, but the symmetry jumps to that of Bondi, Metzner and Sachs. The expressions are applied to exact solutions, with and without radiation present, and also to the linearized theory.
Using a 3+1 decomposition of spacetime, we derive a new formula to compute the gravitational light shifts as measured by two observers which are normal to the spacelike hypersurfaces defining the foliation. This formula is quite general and is also independent of the existence of Killing fields. Known examples are considered to illustrate the usefulness of the formula. In particular, we focus on the Sachs-Wolfe effect that arises in a perturbed Friedman-Robertson-Walker cosmology.
Exact and analytic solutions in Einstein-Aether scalar field theory with Kantowski-Sachs background space are determined. The theory of point symmetries is applied to determine the functional form of the unknown functions which defines the gravitational model. Conservation laws are applied to reduce the order of the field equations and write the analytic solution. Moreover, in order to understand the physical behaviour of the cosmological model a detailed analysis of the asymptotic behaviour for the gravitational field equations is performed.
We construct the gravitational energy-momentum of the Bondi-Sachs space-time, in the famework of the teleparallel equivalent of general relativity (TEGR). The Bondi-Sachs line element describes gravitational radiation in the asymptotic region of the space-time, and is determined by the mass aspect and by two functions, $c$ and $d$, that yield the news functions, which are interpreted as the radiating degrees of freedom of the gravitational field. The standard expression for the Bondi-Sachs energy-momentum is constructed in terms of the mass aspect only. The expression that we obtain in the con…
We study the Bondi-Sachs rockets with nonzero cosmological constant. We observe that the acceleration of the systems arises naturally in the asymptotic symmetries of (anti-) de Sitter spacetimes. Assuming the validity of the concepts of energy and mass previously introduced in asymptotically flat spacetimes, we find that the emission of pure radiation energy balances the loss of the Bondi mass in certain special families of the Bondi-Sachs rockets, so in these there is no gravitational radiation.
Although it is not a fundamental question, to determine exact and general solutions for a given theory has advantages over a numerical integration in many specific cases. Of course, respecting the peculiarities of the problem. Revisiting the integration of the General Relativity Theory field equations for the Kantowski-Sachs spacetime, that describes a homogeneous but anisotropic universe whose spatial section has the topology of $R \times S^2$, we integrate the equations for arbitrary curvature parameter, and writing the solutions considering the process of gravitational collapse. We took the…
Gravitational radiation plays an important role in astrophysics. Based on the fact that our universe is expanding, the gravitational radiation when a positive cosmological constant is presented has been studied along with two different ways recently, one is the Bondi-Sachs (BS) framework in which the result is shown by BS quantities in the asymptotic null structure, the other is the perturbation approach in which the result is presented by the quadrupoles of source. Therefore, it is worth to interpret the quantities in asymptotic null structure in terms of the information of the source. In thi…
The relation between gravitational memory effects and Bondi-Metzner-Sachs symmetry of the asymptotically flat spacetimes is studied in the scalar-tensor theory. For this purpose, the solutions to the equations of motion near the future null infinity are obtained in the generalized Bondi-Sachs coordinates with a suitable determinant condition. It turns out that the Bondi-Metzner-Sachs group is also a semi-direct product of an infinitesimal dimensional supertranslation group and the Lorentz group as in general relativity. There are also degenerate vacua in both the tensor and the scalar sectors …
We apply the Regge-Wheeler formalism to study axial and polar gravitational waves in Kantowski-Sachs universe. The background field equations and the linearised perturbation equations for the modes are derived in presence of matter. To find analytical solutions, we analyze the propagation of waves in vacuum spacetime. The background field equations in absence of matter are solved by assuming the expansion scalar to be proportional to the shear scalar. Using the method of separation of variables, the axial perturbation parameter $ h_0(t, r) $ is obtained from its wave equation. The other pertur…
The Bondi-Sachs formalism of General Relativity is a metric-based treatment of the Einstein equations in which the coordinates are adapted to the null geodesics of the spacetime. It provided the first convincing evidence that gravitational radiation is a nonlinear effect of general relativity and that the emission of gravitational waves from an isolated system is accompanied by a mass loss from the system. The asymptotic behaviour of the Bondi-Sachs metric revealed the existence of the symmetry group at null infinity, the Bondi-Metzner-Sachs group, which turned out to be larger than the Poinca…
The excitation of a black hole by infalling matter or radiation has been studied for a long time, mostly in linear perturbation theory. In this paper we study numerically the response of a Schwarzschild black hole to an incoming gravitational wave pulse. We present a numerically well-posed initial boundary value problem for the generalized conformal field equations in which a wave profile for the ingoing wave is specified at an outer time-like boundary which then hits an initially static and spherically symmetric black hole. The non-linear interaction of the black hole with the gravitational w…
Gravitational waves are messengers carrying valuable information about their sources. For sources at cosmological distances, the waves will contain also the imprint left by the intervening matter. The situation is in close analogy with cosmic microwave photons, for which the large-scale structures the photons traverse contribute to the observed temperature anisotropies, in a process known as the integrated Sachs-Wolfe effect. We derive the gravitational wave counterpart of this effect for waves propagating on a Friedman-Robertson-Walker background with scalar perturbations. We find that the ph…
A general treatment of vorticity-free, perfect fluid perturbations of Kantowski-Sachs models with a positive cosmological constant are considered within the framework of the 1+1+2 covariant decomposition of spacetime. The dynamics is encompassed in six evolution equations for six harmonic coefficients, describing gravito-magnetic, kinematic and matter perturbations, while a set of algebraic expressions determine the rest of the variables. The six equations further decouple into a set of four equations sourced by the perfect fluid, representing forced oscillations and two uncoupled damped oscil…
In a landscape of compactifications with different numbers of macroscopic dimensions, it is possible that our universe has nucleated from a vacuum where some of our four large dimensions were compact while other, now compact, directions were macroscopic. From our perspective, this shapeshifting can be perceived as an anisotropic background spacetime. As an example, we present a model where our universe emerged from a tunneling event which involves the decompactification of two dimensions compactified on the two-sphere. In this case, our universe is of the Kantowski-Sachs type and therefore hom…
We calculate the vorticity of world--lines of observers at rest in a Bondi--Sachs frame, produced by gravitational radiation, in a general Sachs metric. We claim that such an effect is related to the super--Poynting vector, in a similar way as the existence of the electromagnetic Poynting vector is related to the vorticity in stationary electrovacum spacetimes.
The nature of superluminal photon propagation in the gravitational field describing radiation from a time-dependent, isolated source (the Bondi-Sachs metric) is considered in an effective theory which includes interactions which violate the strong equivalence principle. Such interactions are, for example, generated by vacuum polarisation in conventional QED in curved spacetime. The relation of the resulting light-cone modifications to the Peeling Theorem for the Bondi-Sachs spacetime is explained.
In the gravitational context, Liouville theory is the two-dimensional conformal field theory that controls the boundary dynamics of asymptotically AdS_3 spacetimes at the classical level. By taking a suitable limit of the coupling constants of the Hamiltonian formulation of Liouville, we construct and analyze a BMS_3 invariant two-dimensional field theory that is likely to control the boundary dynamics at null infinity of three dimensional asymptotically flat gravity.
We compute the power in the lowest multipoles of CMB anisotropies in the presence of radiation-like solid, a hypothetical new kind of radiation with nonzero shear modulus. If only the ordinary Sachs-Wolfe effect is taken into account, the shear modulus to energy density ratio must be in absolute value of order $10^{-5}$ or less for the theory to be consistent with observations within cosmic variance. With the integrated Sachs-Wolfe effect switched on, the constraint is relaxed almost by two orders of magnitude.
We calculate the full second-order radiation transfer function for Cosmic Microwave Background anisotropies on large angular scales in a flat universe filled with matter and cosmological constant. It includes (i) the second-order generalization of the Sachs-Wolfe effect, and of (ii) both the early and late Integrated Sachs-Wolfe effects, (iii) the contribution of the second-order tensor modes, and is valid for a generic set of initial conditions specifying the level of primordial non-Gaussianity.
We present a numerical study of the evolution of a non-linearly disturbed black hole described by the Bondi--Sachs metric, for which the outgoing gravitational waves can readily be found using the news function. We compare the gravitational wave output obtained with the use of the news function in the Bondi--Sachs framework, with that obtained from the Weyl scalars, where the latter are evaluated in a quasi-Kinnersley tetrad. The latter method has the advantage of being applicable to any formulation of Einstein's equations---including the ADM formulation and its various descendants---in additi…
In the present work, we study the dynamical evolution of an homogeneous and anisotropic KS cosmological model, considering general relativity as the gravitational theory, such that there are three different perfect fluids in the matter sector. They are radiation, dust and phantom fluid. Our main motivation is determining if the present model tends to an homogeneous and isotropic FRW model, during its evolution. Also, we want to establish how the parameters and initial conditions of the model, quantitatively, influence the isotropization of the present model. In order to simplify our task, we u…
We construct general anisotropic cosmological scenarios governed by an $f(R)$ gravitational sector. Focusing then on Kantowski-Sachs geometries in the case of $R^n$-gravity, and modelling the matter content as a perfect fluid, we perform a detailed phase-space analysis. We find that at late times the universe can result to a state of accelerating expansion, and additionally, for a particular $n$-range ($2<n<3$) it exhibits phantom behavior. Furthermore, isotropization has been achieved independently of the initial anisotropy degree, showing in a natural way why the observable universe is so ho…
In this paper, we show that the peeling property still holds for Bondi-Sachs metrics with nonzero cosmological constant under the boundary condition given by Sommerfeld's radiation condition together with three nontrivial $Λ$-independent functions $B$, $a$, $b$. This should indicate the new boundary condition is natural. Moreover, we construct some nonstationary vacuum Bondi-Sachs metrics without Bondi news, which Newmann-Penrose quantities fall faster than usual. This provides a new feature of gravitational waves for nonzero cosmological constant.
This work is a pedagogical review dedicated to a modern description of the Bondi-Metzner-Sachs group. The curved space-times that will be taken into account are the ones that suitably approach, at infinity, Minkowski space-time. In particular we will focus on asymptotically flat space-times. In this work the concept of asymptotic symmetry group of those space-times will be studied. In the first two sections we derive the asymptotic group following the classical approach which was basically developed by Bondi, van den Burg, Metzner and Sachs. This is essentially the group of transformations bet…
In the weak field regime, gravitational waves can be considered as being made up of collisionless, relativistic tensor modes that travel along null geodesics of the perturbed background metric. We work in this geometric optics picture to calculate the anisotropies in gravitational wave backgrounds resulting from astrophysical and cosmological sources. Our formalism yields expressions for the angular power spectrum of the anisotropies. We show how the anisotropies are sourced by intrinsic, Doppler, Sachs-Wolfe, and Integrated Sachs-Wolfe terms in analogy with Cosmic Microwave Background photons…
Gravitational-wave memory effects are identified by their distinctive effects on families of freely falling observers: after a burst of waves pass by their locations, memory effects can cause lasting relative displacements of the observers. These effects are closely related to the infrared properties of gravity and other massless field theories, including their asymptotic symmetries and conserved quantities. In this paper, we investigate the connection between memory effects, symmetries, and conserved quantities in Brans-Dicke theory. We compute the field equations in Bondi coordinates, and we…
We linearize the Einstein equations when the metric is Bondi-Sachs, when the background is Schwarzschild or Minkowski, and when there is a matter source in the form of a thin shell whose density varies with time and angular position. By performing an eigenfunction decomposition, we reduce the problem to a system of linear ordinary differential equations which we are able to solve. The solutions are relevant to the characteristic formulation of numerical relativity: (a) as exact solutions against which computations of gravitational radiation can be compared; and (b) in formulating boundary cond…
We examine, from both a qualitative and a numerical point of view, the evolution of Kantowski--Sachs cosmological models whose source is a mixture of a gas of weakly interacting massive particles (WIMP's) and a radiative gas made up of a ``tightly coupled'' mixture of electrons, baryons and photons from the end of nucleosynthesis up to the duration of radiative interactions. In this cosmic era annihilation processes are negligible, while the WIMP's only interact gravitationally with the radiative gas and the latter behaves as a single dissipative fluid that can be studied within a hydrodynamic…
We point out that a pseudo-Newtonian interpretations of the $(1/3)$ factor in the Sachs-Wolfe effect, which relates the fluctuations in temperature and potential, $(δT/ T) = (1/ 3) δΦ$, is not supported by the General Relativistic analysis. Dividing the full gravitational effect into separate parts depends on the choice of time slicing (gauge) and there exist infinitely many different choices. More importantly, interpreting the parts as being due to the gravitational redshift and the time dilation is not justified in the rigorous relativistic perturbation theory. We suggest to regard the $(1/3…
There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at null infinity of an asymptotically flat spacetime in the Bondi-Sachs coordinates. The quasi-local mass leads to a local description of the radiation that is purely gravitational at null infinity. In particular, the quasi-local mass is evaluated in terms of the news function of the Bondi-Sachs coordinates.
Universality of Primordial Anisotropies in Gravitational Wave Background
School of Physics Science and Engineering, Tongji University（同济大学物理科学与工程学院）；Graduate School of Science and Engineering, Yamaguchi University（山口大学理工学研究科）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo（东京大学宇宙线研究所 Kavli宇宙物理数学研究所）；Institute for Advanced Study of Tongji University（同济大学高等研究院）；Center for Gravitation and Cosmology, Yangzhou University（扬州大学引力与宇宙学中心）
A new evolution algorithm for the characteristic initial value problem based upon an affine parameter rather than the areal radial coordinate used in the Bondi-Sachs formulation is applied in the spherically symmetric case to the gravitational collapse of a massless scalar field. The advantages over the Bondi-Sachs version are discussed, with particular emphasis on the application to critical collapse. Unexpected quadratures lead to a simple evolution algorithm based upon ordinary differential equations which can be integrated along the null rays. For collapse to a black hole in a Penrose comp…
The well-known treatment of asymptotically flat vacuum fields is adapted to pure radiation fields. In this approach we find a natural normalization of the radiation null vector. The energy balance at null infinity shows that the mass loss results from a linear superposition of the pure and the gravitational radiation parts. By transformation to Bondi-Sachs coordinates the Kinnersley photon rocket is found to be the only axisymmetric Robinson-Trautman pure radiation solution without gravitational radiation.
In this review paper we investigate the connection between gravity and electromagnetism from Faraday to the present day. The particular focus is on the connection between gravitational and electromagnetic radiation. We discuss electromagnetic radiation produced when a gravitational wave passes through a magnetic field. We then discuss the interaction of electromagnetic radiation with gravitational waves via Feynman diagrams of the process $graviton + graviton \to photon + photon$. Finally we review recent work on the vacuum production of counterpart electromagnetic radiation by gravitational w…
The details are presented of a new evolution algorithm for the characteristic initial-boundary value problem based upon an affine parameter rather than the areal radial coordinate used in the Bondi-Sachs formulation. The advantages over the Bondi-Sachs version are discussed, with particular emphasis on the application to the characteristic extraction of the gravitational waveform from Cauchy simulations of general relativistic astrophysical systems.
We study the behavior of light rays in perturbed Robertson-Walker cosmologies, calculating the redshift between an observer and the surface of last scattering to second order in the metric perturbation. At first order we recover the classic results of Sachs and Wolfe, and at second order we delineate the various new effects which appear; there is no {\it a priori} guarantee that these effects are significantly smaller than those at first order, since there are large length scales in the problem which could lead to sizable prefactors. We find that second order terms of potential observational i…
Consider compact objects --such as neutron star or black hole binaries-- in \emph{full, non-linear} general relativity. In the case with zero cosmological constant $Λ$, the gravitational radiation emitted by such systems is described by the well established, 50+ year old framework due to Bondi, Sachs, Penrose and others. However, so far we do not have a satisfactory extension of this framework to include a \emph{positive} cosmological constant --or, more generally, the dark energy responsible for the accelerated expansion of the universe. In particular, we do not yet have an adequate gauge inv…
Various works have suggested that the Bondi--Sachs--Penrose decay conditions on the gravitational field at null infinity are not generally representative of asymptotically flat space--times. We have made a detailed analysis of the constraint equations for ``asymptotically hyperboloidal'' initial data and find that log terms arise generically in asymptotic expansions. These terms are absent in the corresponding Bondi--Sachs--Penrose expansions, and can be related to explicit geometric quantities. We have nevertheless shown that there exists a large class of ``non--generic'' solutions of the con…
We study the method of calculating the Hawking radiation via gravitational anomalies in gravitational backgrounds of constant negative curvature. We apply the method to topological black holes and also to topological black holes conformally coupled to a scalar field.
The effective field theory of quantum gravity generically predicts non-locality to be present in the effective action, which results from the low-energy propagation of gravitons and massless matter. Working to second order in gravitational curvature, we reconsider the effects of quantum gravity on the gravitational radiation emitted from a binary system. In particular, we calculate for the first time the leading order quantum gravitational correction to the classical quadrupole radiation formula which appears at second order in Newton's constant.
We discuss an ambiguity of the derivation of the Hawking radiation through the gravitational anomaly method and propose modifications of this method such that it reproduces the correct thermal fluxes. In this modified gravitational anomaly method, we employ the two-dimensional conformal field theory technique.
We study the Hawking radiation from Rotating black holes from gravitational anomalies point of view. First, we show that the scalar field theory near the Kerr black hole horizon can be reduced to the 2-dimensional effective theory. Then, following Robinson and Wilczek, we derive the Hawking flux by requiring the cancellation of gravitational anomalies. We also apply this method to Hawking radiation from higher dimensional Myers-Perry black holes. In the Appendix, we present the trace anomaly derivation of Hawking radiation to argue the validity of the boundary condition at the horizon.
The connection between the Bondi-Sachs (BS) and the Newman-Penrose (NP) framework for the study of the asymptotics of the gravitational field is done. In particular the coordinate transformation relationg the BS luminosity parameter and the NP affine parameter is obtained. Using this coordinate transformation it is possible to express BS quantities in terms of NP quantities, and to show that if the Outgoing Radiation Condition is not satisfied then the spacetime will not decay in the way prescribed by the Peeling theorem.
Using the Bondi-Sachs formalism, the problem of a gravitational wave source surrounded by a spherical dust shell is considered. Using linearized perturbation theory, the geometry is found in the regions: in the shell, exterior to the shell, and interior to the shell. It is found that the dust shell causes the gravitational wave to be modified both in magnitude and phase, but without any energy being transferred to or from the dust.
The construction of exact linearized solutions to the Einstein equations within the Bondi-Sachs formalism is extended to the case of linearization about de Sitter spacetime. The gravitational wave field measured by distant observers is constructed, leading to a determination of the energy measured by such observers. It is found that gravitational wave energy conservation does not normally apply to inertial observers, but that it can be formulated for a class of accelerated observers, i.e. with worldlines that are timelike but not geodesic.
A recent refinement of Penrose's conformal framework for asymptotically flat space-times is summarized. The key idea concerns advanced and retarded conformal factors, which allow a rigid description of infinity as a locally metric light cone. In the new framework, the Bondi-Sachs energy-flux integrals of ingoing and outgoing gravitational radiation decay at spatial infinity such that the total radiated energy is finite, and the Bondi-Sachs energy-momentum has a unique limit at spatial infinity, coinciding with the uniquely rendered ADM energy-momentum.
