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Robert M. Wald

Bio: Robert M. Wald is an academic researcher from University of Chicago. The author has contributed to research in topics: Black hole & General relativity. The author has an hindex of 77, co-authored 252 publications receiving 28717 citations. Previous affiliations of Robert M. Wald include University of Maryland, College Park.


Papers
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B. P. Abbott1, Richard J. Abbott1, T. D. Abbott2, Fausto Acernese3  +1062 moreInstitutions (115)
TL;DR: The magnitude of modifications to the gravitational-wave dispersion relation is constrain, the graviton mass is bound to m_{g}≤7.7×10^{-23} eV/c^{2} and null tests of general relativity are performed, finding that GW170104 is consistent with general relativity.
Abstract: We describe the observation of GW170104, a gravitational-wave signal produced by the coalescence of a pair of stellar-mass black holes. The signal was measured on January 4, 2017 at 10∶11:58.6 UTC by the twin advanced detectors of the Laser Interferometer Gravitational-Wave Observatory during their second observing run, with a network signal-to-noise ratio of 13 and a false alarm rate less than 1 in 70 000 years. The inferred component black hole masses are 31.2^(8.4) _(−6.0)M_⊙ and 19.4^(5.3)_( −5.9)M_⊙ (at the 90% credible level). The black hole spins are best constrained through measurement of the effective inspiral spin parameter, a mass-weighted combination of the spin components perpendicular to the orbital plane, χ_(eff) = −0.12^(0.21)_( −0.30). This result implies that spin configurations with both component spins positively aligned with the orbital angular momentum are disfavored. The source luminosity distance is 880^(450)_(−390) Mpc corresponding to a redshift of z = 0.18^(0.08)_( −0.07) . We constrain the magnitude of modifications to the gravitational-wave dispersion relation and perform null tests of general relativity. Assuming that gravitons are dispersed in vacuum like massive particles, we bound the graviton mass to m_g ≤ 7.7 × 10^(−23) eV/c^2. In all cases, we find that GW170104 is consistent with general relativity.

2,569 citations

Journal ArticleDOI
TL;DR: The results show that the validity of the "second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory.
Abstract: We consider a general, classical theory of gravity in $n$ dimensions, arising from a diffeomorphism-invariant Lagrangian. In any such theory, to each vector field ${\ensuremath{\xi}}^{a}$ on spacetime one can associate a local symmetry and, hence, a Noether current ($n\ensuremath{-}1$)-form j and (for solutions to the field equations) a Noether charge ($n\ensuremath{-}2$)-form Q, both of which are locally constructed from ${\ensuremath{\xi}}^{a}$ and the fields appearing in the Lagrangian. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon (with bifurcation surface $\ensuremath{\Sigma}$), and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply $2\ensuremath{\pi}$ times the integral over $\ensuremath{\Sigma}$ of the Noether charge ($n\ensuremath{-}2$)-form associated with the horizon Killing field. Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the "second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via "Euclidean methods" also is explained.

2,538 citations

Journal ArticleDOI
TL;DR: It is proved that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole, and a local, geometrical prescription is proposed for the entropy, $S_{dyn}$, of a dynamical black hole.
Abstract: We consider a general, classical theory of gravity with arbitrary matter fields in n dimensions, arising from a diffeomorphism-invariant Lagrangian L. We first show that L alwasy can be written in a ``manifestly covariant'' form. We then show that the symplectic potential current (n-1)-form FTHETA and the symplectic current (n-1)-form \ensuremath{\omega} for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current (n-1)-form J and corresponding Noether charge (n-2)-form Q. We derive a general ``decomposition formula'' for Q. Using this formula for the Noether charge, we prove that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole. (For higher derivative theories, previous arguments had established this law only for stationary perturbations.) Finally, we propose a local, geometrical prescription for the entropy ${\mathit{S}}_{\mathrm{dyn}}$ of a dynamical black hole. This prescription agrees with the Noether charge formula for stationary black holes and their perturbations, and is independent of all ambiguities associated with the choices of L, FTHETA, and Q. However, the issue of whether this dynamical entropy in general obeys a ``second law'' of black hole mechanics remains open. In an appendix, we apply some of our results to theories with a nondynamical metric and also briefly develop the theory of stress-energy pseudotensors.

2,321 citations

Book
01 Jan 1994
TL;DR: In this paper, a pedagogical introduction to the formulation of quantum field theory in curved spacetime is provided, and a comprehensive, up-to-date account of the Unruh effect, the Hawking effect, and some of its ramifications is provided.
Abstract: In this book, Robert Wald provides a pedagogical introduction to the formulation of quantum field theory in curved spacetime. He begins with a treatment of the ordinary one-dimensional quantum harmonic oscillator, progresses through the construction of quantum field theory in flat spacetime to possible constructions of quantum field theory in curved spacetime, and, ultimately, to an algebraic formulation of the theory. In his presentation, Wald disentangles essential features of the theory from inessential ones (such as a particle interpretation) and clarifies relationships between various approaches to the formulation of the theory. He also provides a comprehensive, up-to-date account of the Unruh effect, the Hawking effect, and some of its ramifications. In particular, the subject of black hole thermodynamics, which remains an active area of research, is treated in depth. This book is intended for students and researchers who have had introductory courses in general relativity and quantum field theory, and should be of interest to scientists in general relativity and related fields.

1,793 citations

Journal ArticleDOI
TL;DR: In this paper, a general definition of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang-Mills theory and general relativity.
Abstract: The general relationship between local symmetries occurring in a Lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied. First, a prescription—applicable to an arbitrary Lagrangian field theory—for the construction of phase space from the manifold of field configurations on space‐time is given. Next, a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang–Mills theory and general relativity. Local symmetries on phase space are then defined via projection from field configuration space. It is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space. Moreover, the constraints thereby obtained are always first class, and the Poisson bracket algebra of the constraint functions is isomorphic to the Lie bracket algebra of the local symmetries on the constraint submanifold of phase space. The differences that occur in the structure of constraints in Yang–Mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space: In Yang–Mills theory all the ‘‘field‐independent’’ local symmetries project to all of phase space, whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are ‘‘field dependent.’’ As by‐products of the present work, definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary Lagrangian field theory, and the Noether current density associated with an arbitrary local symmetry. A number of properties of these currents are established and some relationships between them are obtained.

833 citations


Cited by
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[...]

08 Dec 2001-BMJ
TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

33,785 citations

Journal ArticleDOI
TL;DR: In this paper, the mass density, Omega_M, and cosmological-constant energy density of the universe were measured using the analysis of 42 Type Ia supernovae discovered by the Supernova Cosmology project.
Abstract: We report measurements of the mass density, Omega_M, and cosmological-constant energy density, Omega_Lambda, of the universe based on the analysis of 42 Type Ia supernovae discovered by the Supernova Cosmology Project. The magnitude-redshift data for these SNe, at redshifts between 0.18 and 0.83, are fit jointly with a set of SNe from the Calan/Tololo Supernova Survey, at redshifts below 0.1, to yield values for the cosmological parameters. All SN peak magnitudes are standardized using a SN Ia lightcurve width-luminosity relation. The measurement yields a joint probability distribution of the cosmological parameters that is approximated by the relation 0.8 Omega_M - 0.6 Omega_Lambda ~= -0.2 +/- 0.1 in the region of interest (Omega_M <~ 1.5). For a flat (Omega_M + Omega_Lambda = 1) cosmology we find Omega_M = 0.28{+0.09,-0.08} (1 sigma statistical) {+0.05,-0.04} (identified systematics). The data are strongly inconsistent with a Lambda = 0 flat cosmology, the simplest inflationary universe model. An open, Lambda = 0 cosmology also does not fit the data well: the data indicate that the cosmological constant is non-zero and positive, with a confidence of P(Lambda > 0) = 99%, including the identified systematic uncertainties. The best-fit age of the universe relative to the Hubble time is t_0 = 14.9{+1.4,-1.1} (0.63/h) Gyr for a flat cosmology. The size of our sample allows us to perform a variety of statistical tests to check for possible systematic errors and biases. We find no significant differences in either the host reddening distribution or Malmquist bias between the low-redshift Calan/Tololo sample and our high-redshift sample. The conclusions are robust whether or not a width-luminosity relation is used to standardize the SN peak magnitudes.

16,838 citations

Journal ArticleDOI
B. P. Abbott1, Richard J. Abbott1, T. D. Abbott2, Fausto Acernese3  +1131 moreInstitutions (123)
TL;DR: The association of GRB 170817A, detected by Fermi-GBM 1.7 s after the coalescence, corroborates the hypothesis of a neutron star merger and provides the first direct evidence of a link between these mergers and short γ-ray bursts.
Abstract: On August 17, 2017 at 12∶41:04 UTC the Advanced LIGO and Advanced Virgo gravitational-wave detectors made their first observation of a binary neutron star inspiral. The signal, GW170817, was detected with a combined signal-to-noise ratio of 32.4 and a false-alarm-rate estimate of less than one per 8.0×10^{4} years. We infer the component masses of the binary to be between 0.86 and 2.26 M_{⊙}, in agreement with masses of known neutron stars. Restricting the component spins to the range inferred in binary neutron stars, we find the component masses to be in the range 1.17-1.60 M_{⊙}, with the total mass of the system 2.74_{-0.01}^{+0.04}M_{⊙}. The source was localized within a sky region of 28 deg^{2} (90% probability) and had a luminosity distance of 40_{-14}^{+8} Mpc, the closest and most precisely localized gravitational-wave signal yet. The association with the γ-ray burst GRB 170817A, detected by Fermi-GBM 1.7 s after the coalescence, corroborates the hypothesis of a neutron star merger and provides the first direct evidence of a link between these mergers and short γ-ray bursts. Subsequent identification of transient counterparts across the electromagnetic spectrum in the same location further supports the interpretation of this event as a neutron star merger. This unprecedented joint gravitational and electromagnetic observation provides insight into astrophysics, dense matter, gravitation, and cosmology.

7,327 citations

Journal ArticleDOI
TL;DR: In this article, the authors review the observational evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence and tachyon.
Abstract: We review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating universe. In particular we discuss the arguments for and recent progress made towards understanding the nature of dark energy. We review the observational evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence, tachyon, phantom and dilatonic models. The importance of cosmological scaling solutions is emphasized when studying the dynamical system of scalar fields including coupled dark energy. We study the evolution of cosmological perturbations allowing us to confront them with the observation of the Cosmic Microwave Background and Large Scale Structure and demonstrate how it is possible in principle to reconstruct the equation of state of dark energy by also using Supernovae Ia observational data. We also discuss in detail the nature of tracking solutions in cosmology, particle physics and braneworld models of dark energy, the nature of possible future singularities, the effect of higher order curvature terms to avoid a Big Rip singularity, and approaches to modifying gravity which leads to a late-time accelerated expansion without recourse to a new form of dark energy.

5,954 citations

Journal ArticleDOI
TL;DR: In this paper, the holographic correspondence between field theories and string/M theory is discussed, focusing on the relation between compactifications of string theory on anti-de Sitter spaces and conformal field theories.

5,610 citations