Searcharxiv⌕ Search

arXiv subjects

Robert M. Wald

Publications and source records attributed to Robert M. Wald.

At least 55 records · Page 3Linked to original sources
arXiv2015

Reflection Symmetry in Higher Dimensional Black Hole Spacetimes

In 4 spacetime dimensions there is a well known proof that for any asymptotically flat, stationary, and axisymmetric vacuum solution of Einstein's equation there exists a "$t$-$ϕ$" reflection isometry that reverses the direction of the timelike Killing vector field and the direction of the axial Killing vector field. However, this proof does not generalize to higher spacetime dimensions. Here we consider asymptotically flat, stationary, and axisymmetric (i.e., having one or more commuting rotational isometries) black hole spacetimes in vacuum general relativity in $d \geq 4$ spacetime dimensions such that the action of the isometry group is trivial. (Here "trivial" means that if the "axes"---i.e., the points where the axial Killing fields are linearly dependent---are removed, the action of the isometry group is that of a trivial principal fiber bundle. This excludes actions like that found in the Sorkin monopole.) We prove that there exists a "$t$-$ϕ$" reflection isometry that reverses the direction of the timelike Killing vector field and the direction of each axial Killing vector field. The proof relies in an essential way on the first law of black hole mechanics.

gr-qc↗
arXiv2014

How well is our universe described by an FLRW model?

Extremely well! In the $Λ$CDM model, the spacetime metric, $g_{ab}$, of our universe is approximated by an FLRW metric, $g_{ab}^{(0)}$, to about 1 part in $10^4$ or better on both large and small scales, except in the immediate vicinity of very strong field objects, such as black holes. However, derivatives of $g_{ab}$ are not close to derivatives of $g_{ab}^{(0)}$, so there can be significant differences in the behavior of geodesics and huge differences in curvature. Consequently, observable quantities in the actual universe may differ significantly from the corresponding observables in the FLRW model. Nevertheless, as we shall review here, we have proven general results showing that---within the framework of our approach to treating backreaction---the large matter inhomogeneities that occur on small scales cannot produce significant effects on large scales, so $g_{ab}^{(0)}$ satisfies Einstein's equation with the averaged stress-energy tensor of matter as its source. We discuss the flaws in some other approaches that have suggested that large backreaction effects may occur. As we also will review here, with a suitable "dictionary," Newtonian cosmologies provide excellent approximations to cosmological solutions to Einstein's equation (with dust and a cosmological constant) on all scales. Our results thereby provide strong justification for the mathematical consistency and validity of the $Λ$CDM model within the context of general relativistic cosmology.

gr-qc↗
arXiv2014

Quantum fields in curved spacetime

We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime. Our review emphasizes the conceptual issues arising in the formulation of the theory and presents known results in a mathematically precise way. Particular attention is paid to the distributional nature of quantum fields, to their local and covariant character, and to microlocal spectrum conditions satisfied by physically reasonable states. We review the Unruh and Hawking effects for free fields, as well as the behavior of free fields in deSitter spacetime and FLRW spacetimes with an exponential phase of expansion. We review how nonlinear observables of a free field, such as the stress-energy tensor, are defined, as well as time-ordered-products. The "renormalization ambiguities" involved in the definition of time-ordered products are fully characterized. Interacting fields are then perturbatively constructed. Our main focus is on the theory of a scalar field, but a brief discussion of gauge fields is included. We conclude with a brief discussion of a possible approach towards a nonperturbative formulation of quantum field theory in curved spacetime and some remarks on the formulation of quantum gravity.

gr-qc↗
arXiv2014

Examination of a simple example of gravitational wave memory

We examine a simple example of gravitational wave memory due to the decay of a point particle into two point particles. In the case where one of the decay products is null, there are two types of memory: a null memory due to the null particle and an ordinary memory due to the recoiling timelike particle. In the case where both decay products are timelike, there is only ordinary memory. However, this ordinary memory can mimic the null memory in the limit where one of the decay products has a large velocity.

gr-qc↗
arXiv2014

Retarded Fields of Null Particles and the Memory Effect

We consider the retarded solution to the scalar, electromagnetic, and linearized gravitational field equations in Minkowski spacetime, with source given by a particle moving on a null geodesic. In the scalar case and in the Lorenz gauge in the electromagnetic and gravitational cases, the retarded integral over the infinite past of the source does not converge as a distribution, so we cut off the null source suitably at a finite time $t_0$ and then consider two different limits: (i) the limit as the observation point goes to null infinity at fixed $t_0$, from which the ``$1/r$'' part of the fields can be extracted and (ii) the limit $t_0 \to - \infty$ at fixed ``observation point.'' The limit (i) gives rise to a ``velocity kick'' on distant test particles in the scalar and electromagnetic cases, and it gives rise to a ``memory effect'' (i.e., a permanent change in relative separation of two test particles) in the linearized gravitational case, in agreement with previous analyses. Although the second limit does not exist for the Lorenz gauge potentials in the electromagnetic and linearized gravitational cases, we obtain a well defined distributional limit for the electromagnetic field strength and for the linearized Riemann tensor. In the gravitational case, this limit agrees with the Aichelberg-Sexl solution, but there is no ``memory effect'' associated with this limiting solution. This strongly suggests that the memory effect---including nonlinear memory effect of Christodoulou---should not be interpreted as arising simply from the passage of (effective) null stress energy to null infinity but rather as arising from a ``burst of radiation'' associated with the creation of the null stress-energy (as in case (i) above) or, more generally, with radiation present in the spacetime that was not ``produced'' by the null stress-energy.

gr-qc↗
arXiv2013

Dynamic and Thermodynamic Stability of Relativistic, Perfect Fluid Stars

We consider perfect fluid bodies (stars) in general relativity, characterized by particle number density and entropy per particle. A star is said to be in dynamic equilibrium if it is a stationary, axisymmetric solution to the Einstein-fluid equations with circular flow. We prove that for a star in dynamic equilibrium, the necessary and sufficient condition for thermodynamic equilibrium (extremum of total entropy S) is constancy of angular velocity (Ω), redshifted temperature, and redshifted chemical potential. Friedman previously identified positivity of canonical energy, E, as a criterion for dynamic stability and argued that all rotating stars are dynamically unstable to sufficiently nonaxisymmetric perturbations (the CFS instability), so our main focus is on axisymmetric stability. We show that for a star in dynamic equilibrium, mode stability holds with respect to all axisymmetric perturbations if E is positive on a certain subspace, V, of axisymmetric Lagrangian perturbations that, in particular, have no Lagrangian change in angular momentum density. Conversely, if E fails to be positive on V, then there exist perturbations that can't become asymptotically stationary at late times. We further show that for a star in thermodynamic equilibrium, the canonical energy in the rotating frame, E_r, is related to second order changes in ADM mass, M, and angular momentum, J, by E_r = δ^2 M - Ωδ^2 J. Thus, positivity of E_r for perturbations with δJ = 0 is a necessary condition for thermodynamic stability (local maximum of S). For axisymmetric perturbations we have E = E_r, so a necessary condition for thermodynamic stability with respect to axisymmetric perturbations is positivity of E on all perturbations with δJ = 0, not merely on the perturbations in V. Many of our results are in close parallel to results of Hollands and Wald for the theory of black holes.

gr-qc↗
arXiv2013

Turning Point Instabilities for Relativistic Stars and Black Holes

In the light of recent results relating dynamic and thermodynamic stability of relativistic stars and black holes, we re-examine the relationship between "turning points"---i.e., extrema of thermodynamic variables along a one-parameter family of solutions---and instabilities. We give a proof of Sorkin's general result---showing the existence of a thermodynamic instability on one side of a turning point---that does not rely on heuristic arguments involving infinite dimensional manifold structure. We use the turning point results to prove the existence of a dynamic instability of black rings in 5 spacetime dimensions in the region where $c_J > 0$, in agreement with a result of Figueras, Murata, and Reall.

gr-qc↗
arXiv2013

Examples of backreaction of small scale inhomogeneities in cosmology

In previous work, we introduced a new framework to treat large scale backreaction effects due to small scale inhomogeneities in general relativity. We considered one-parameter families of spacetimes for which such backreaction effects can occur, and we proved that, provided the weak energy condition on matter is satisfied, the leading effect of small scale inhomogeneities on large scale dynamics is to produce a traceless effective stress-energy tensor that itself satisfies the weak energy condition. In this work, we illustrate the nature of our framework by providing two explicit examples of one-parameter families with backreaction. The first, based on previous work of Berger, is a family of polarized vacuum Gowdy spacetimes on a torus, which satisfies all of the assumptions of our framework. As the parameter approaches its limiting value, the metric uniformly approaches a smooth background metric, but spacetime derivatives of the deviation of the metric from the background metric do not converge uniformly to zero. The limiting metric has nontrivial backreaction from the small scale inhomogeneities, with an effective stress-energy that is traceless and satisfies the weak energy condition, in accord with our theorems. Our second one-parameter family consists of metrics which have a uniform Friedmann-Lemaitre-Robertson-Walker limit. This family satisfies all of our assumptions with the exception of the weak energy condition for matter. In this case, the limiting metric has an effective stress-energy tensor which is not traceless. We emphasize the importance of imposing energy conditions on matter in studies of backreaction.

gr-qc↗
arXiv2012

Stability of Black Holes and Black Branes

We establish a new criterion for the dynamical stability of black holes in $D \geq 4$ spacetime dimensions in general relativity with respect to axisymmetric perturbations: Dynamical stability is equivalent to the positivity of the canonical energy, $\E$, on a subspace, $\mathcal T$, of linearized solutions that have vanishing linearized ADM mass, momentum, and angular momentum at infinity and satisfy certain gauge conditions at the horizon. This is shown by proving that---apart from pure gauge perturbations and perturbations towards other stationary black holes---$\E$ is nondegenerate on $\mathcal T$ and that, for axisymmetric perturbations, $\E$ has positive flux properties at both infinity and the horizon. We further show that $\E$ is related to the second order variations of mass, angular momentum, and horizon area by $\E = δ^2 M - \sum_A Ω_A δ^2 J_A - \fracκ{8π} δ^2 A$, thereby establishing a close connection between dynamical stability and thermodynamic stability. Thermodynamic instability of a family of black holes need not imply dynamical instability because the perturbations towards other members of the family will not, in general, have vanishing linearized ADM mass and/or angular momentum. However, we prove that for any black brane corresponding to a thermodynamically unstable black hole, sufficiently long wavelength perturbations can be found with $\E < 0$ and vanishing linearized ADM quantities. Thus, all black branes corresponding to thermodynmically unstable black holes are dynamically unstable, as conjectured by Gubser and Mitra. We also prove that positivity of $\E$ on $\mathcal T$ is equivalent to the satisfaction of a "local Penrose inequality," thus showing that satisfaction of this local Penrose inequality is necessary and sufficient for dynamical stability.

gr-qc↗
arXiv2012

Measure and Probability in Cosmology

General relativity has a Hamiltonian formulation, which formally provides a canonical (Liouville) measure on the space of solutions. In ordinary statistical physics, the Liouville measure is used to compute probabilities of macrostates, and it would seem natural to use the similar measure arising in general relativity to compute probabilities in cosmology, such as the probability that the universe underwent an era of inflation. Indeed, a number of authors have used the restriction of this measure to the space of homogeneous and isotropic universes with scalar field matter (minisuperspace)---namely, the Gibbons-Hawking-Stewart measure---to make arguments about the likelihood of inflation. We argue here that there are at least four major difficulties with using the measure of general relativity to make probability arguments in cosmology: (1) Equilibration does not occur on cosmological length scales. (2) Even in the minisuperspace case, the measure of phase space is infinite and the computation of probabilities depends very strongly on how the infinity is regulated. (3) The inhomogeneous degrees of freedom must be taken into account (we illustrate how) even if one is interested only in universes that are very nearly homogeneous. The measure depends upon how the infinite number of degrees of freedom are truncated, and how one defines "nearly homogeneous." (4) In a universe where the second law of thermodynamics holds, one cannot make use of our knowledge of the present state of the universe to "retrodict" the likelihood of past conditions.

gr-qc↗
arXiv2012

Newtonian and Relativistic Cosmologies

Cosmological N-body simulations are now being performed using Newtonian gravity on scales larger than the Hubble radius. It is well known that a uniformly expanding, homogeneous ball of dust in Newtonian gravity satisfies the same equations as arise in relativistic FLRW cosmology, and it also is known that a correspondence between Newtonian and relativistic dust cosmologies continues to hold in linearized perturbation theory in the marginally bound/spatially flat case. Nevertheless, it is far from obvious that Newtonian gravity can provide a good global description of an inhomogeneous cosmology when there is significant nonlinear dynamical behavior at small scales. We investigate this issue in the light of a perturbative framework that we have recently developed, which allows for such nonlinearity at small scales. We propose a relatively straightforward "dictionary"---which is exact at the linearized level---that maps Newtonian dust cosmologies into general relativistic dust cosmologies, and we use our "ordering scheme" to determine the degree to which the resulting metric and matter distribution solve Einstein's equation. We find that Einstein's equation fails to hold at "order 1" at small scales and at "order $ε$" at large scales. We then find the additional corrections to the metric and matter distribution needed to satisfy Einstein's equation to these orders. While these corrections are of some interest in their own right, our main purpose in calculating them is that their smallness should provide a criterion for the validity of the original dictionary (as well as simplified versions of this dictionary). We expect that, in realistic Newtonian cosmologies, these additional corrections will be very small; if so, this should provide strong justification for the use of Newtonian simulations to describe relativistic cosmologies, even on scales larger than the Hubble radius.

gr-qc↗
arXiv2011

A Rigorous Derivation of Gravitational Self-force

There is general agreement that the MiSaTaQuWa equations should describe the motion of a "small body" in general relativity, taking into account the leading order self-force effects. However, previous derivations of these equations have made a number of ad hoc assumptions and/or contain a number of unsatisfactory features. For example, all previous derivations have invoked, without proper justification, the step of "Lorenz gauge relaxation", wherein the linearized Einstein equation is written down in the form appropriate to the Lorenz gauge, but the Lorenz gauge condition is then not imposed--thereby making the resulting equations for the metric perturbation inequivalent to the linearized Einstein equations. In this paper, we analyze the issue of "particle motion" in general relativity in a systematic and rigorous way by considering a one-parameter family of metrics, $g_{ab} (λ)$, corresponding to having a body (or black hole) that is "scaled down" to zero size and mass in an appropriate manner. We prove that the limiting worldline of such a one-parameter family must be a geodesic of the background metric, $g_{ab} (λ=0)$. Gravitational self-force--as well as the force due to coupling of the spin of the body to curvature--then arises as a first-order perturbative correction in $λ$ to this worldline. No assumptions are made in our analysis apart from the smoothness and limit properties of the one-parameter family of metrics. Our approach should provide a framework for systematically calculating higher order corrections to gravitational self-force, including higher multipole effects, although we do not attempt to go beyond first order calculations here. The status of the MiSaTaQuWa equations is explained.

gr-qc↗
arXiv2011

A new framework for analyzing the effects of small scale inhomogeneities in cosmology

We develop a new, mathematically precise framework for treating the effects of nonlinear phenomena occurring on small scales in general relativity. Our approach is an adaptation of Burnett's formulation of the "shortwave approximation", which we generalize to analyze the effects of matter inhomogeneities as well as gravitational radiation. Our framework requires the metric to be close to a "background metric", but allows arbitrarily large stress-energy fluctuations on small scales. We prove that, within our framework, if the matter stress-energy tensor satisfies the weak energy condition (i.e., positivity of energy density in all frames), then the only effect that small scale inhomogeneities can have on the dynamics of the background metric is to provide an "effective stress-energy tensor" that is traceless and has positive energy density---corresponding to the presence of gravitational radiation. In particular, nonlinear effects produced by small scale inhomogeneities cannot mimic the effects of dark energy. We also develop "perturbation theory" off of the background metric. We derive an equation for the "long-wavelength part" of the leading order deviation of the metric from the background metric, which contains the usual terms occurring in linearized perturbation theory plus additional contributions from the small-scale inhomogeneities. Under various assumptions concerning the absence of gravitational radiation and the non-relativistic behavior of the matter, we argue that the "short wavelength" deviations of the metric from the background metric near a point $x$ should be accurately described by Newtonian gravity, taking into account only the matter lying within a "homogeneity lengthscale" of $x$. Finally, we argue that our framework should provide an accurate description of the actual universe.

gr-qc↗
arXiv2010

Bobbing and Kicks in Electromagnetism and Gravity

We study systems analogous to binary black holes with spin in order to gain some insight into the origin and nature of "bobbing" motion and "kicks" that occur in this system. Our basic tool is a general formalism for describing the motion of extended test bodies in an external electromagnetic field in curved spacetime and possibly subject to other forces. We first show that bobbing of exactly the type as observed in numerical simulations of the binary black hole system occurs in a simple system consisting of two spinning balls connected by an elastic band in flat spacetime. This bobbing may be understood as arising from the difference between a spinning body's "lab frame centroid" and its true center of mass, and is purely "kinematical" in the sense that it will appear regardless of the forces holding two spinning bodies in orbit. Next, we develop precise rules for relating the motion of charged bodies in a stationary external electromagnetic field in flat spacetime with the motion of bodies in a weakly curved stationary spacetime. We then consider the system consisting of two orbiting charges with magnetic dipole moment and spin at a level of approximation corresponding to 1.5 post-Newtonian order. Here we find that considerable amounts of momentum are exchanged between the bodies and the electromagnetic field; however, the bodies store this momentum entirely as "hidden" mechanical momentum, so that the interchange does not give rise to any net bobbing. The net bobbing that does occur is due solely to the kinematical spin effect, and we therefore argue that the net bobbing of the electromagnetic binary is not associated with possible kicks. We believe that this conclusion holds in the gravitational case as well.

gr-qc↗
arXiv2009

A Rigorous Derivation of Electromagnetic Self-force

During the past century, there has been considerable discussion and analysis of the motion of a point charge, taking into account "self-force" effects due to the particle's own electromagnetic field. We analyze the issue of "particle motion" in classical electromagnetism in a rigorous and systematic way by considering a one-parameter family of solutions to the coupled Maxwell and matter equations corresponding to having a body whose charge-current density $J^a(λ)$ and stress-energy tensor $T_{ab} (λ)$ scale to zero size in an asymptotically self-similar manner about a worldline $γ$ as $λ\to 0$. In this limit, the charge, $q$, and total mass, $m$, of the body go to zero, and $q/m$ goes to a well defined limit. The Maxwell field $F_{ab}(λ)$ is assumed to be the retarded solution associated with $J^a(λ)$ plus a homogeneous solution (the "external field") that varies smoothly with $λ$. We prove that the worldline $γ$ must be a solution to the Lorentz force equations of motion in the external field $F_{ab}(λ=0)$. We then obtain self-force, dipole forces, and spin force as first order perturbative corrections to the center of mass motion of the body. We believe that this is the first rigorous derivation of the complete first order correction to Lorentz force motion. We also address the issue of obtaining a self-consistent perturbative equation of motion associated with our perturbative result, and argue that the self-force equations of motion that have previously been written down in conjunction with the "reduction of order" procedure should provide accurate equations of motion for a sufficiently small charged body with negligible dipole moments and spin. There is no corresponding justification for the non-reduced-order equations.

gr-qc↗
arXiv2009

Introduction to Gravitational Self-Force

The motion of sufficiently small body in general relativity should be accurately described by a geodesic. However, there should be ``gravitational self-force'' corrections to geodesic motion, analogous to the ``radiation reaction forces'' that occur in electrodynamics. It is of considerable importance to be able to calculate these self-force corrections in order to be able to determine such effects as inspiral motion in the extreme mass ratio limit. However, severe difficulties arise if one attempts to consider point particles in the context of general relativity. This article describes these difficulties and how they have been dealt with.

gr-qc↗
arXiv2009

Derivation of Gravitational Self-Force

We analyze the issue of ``particle motion'' in general relativity in a systematic and rigorous way by considering a one-parameter family of metrics corresponding to having a body (or black hole) that is ``scaled down'' to zero size and mass in an appropriate manner. We prove that the limiting worldline of such a one-parameter family must be a geodesic of the background metric and obtain the leading order perturbative corrections, which include gravitational self-force, spin force, and geodesic deviation effects. The status the MiSaTaQuWa equation is explained as a candidate ``self-consistent perturbative equation'' associated with our rigorous perturbative result

gr-qc↗