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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 always can be written in a "manifestly covariant" form. We then show that the symplectic potential current (n − 1)-form, Θ, and the symplectic current (n − 1)-form, ω, 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, S 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, Θ, 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. 1Recently, many authors have investigated the validity of the first law of black hole mechanics and the definition of the entropy of a black hole in a wide class of theories derivable from a Hamiltonian or Lagrangian [1]- [10]. In particular, in [6] the first law was proven to hold in an arbitrary theory of gravity derived from a diffeomorphism invariant Lagrangian, and the quantity playing the role of the entropy of the black hole was identified as the integral over the horizon of the Noether charge associated with the horizon Killing vector field. Although some key issues concerning the validity of the first law and the definition of black hole entropy in a general theory of gravity were thereby resolved, the analysis of [6], nevertheless, was deficient in the following ways: (1) It was not recognized that a diffeomorphism covariant choice of the symplectic potential current form always can be made. Consequently, several steps in the arguments were made in an unnecessarily awkward manner.(2) While a completely general proof of the first law of black hole mechanics was given for perturbations to nearby stationary black holes, a proof of the first law for non-stationary perturbations was given only for theories in which the Noether charge takes a particular, simple form. (3) A proposal was made for defining the entropy of a dynamical black hole. However, this proposal made use of a rather arbitrary choice of algorithm for defining the symplectic potential current form, and it turns out to possess the undesireable feature that the addition of an exact form to the Lagrangian (which has no effect upon th...

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, ξ a , on spacetime one can associate a local symmetry and, hence, a Noether current (n − 1)-form, j, and (for solutions to the field equations) a Noether charge (n − 2)form, Q, both of which are locally constructed from ξ a and the the fields appearing in the Lagrangian. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon (with bifurcation surface Σ), 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 2π times the integral over Σ of the Noether charge (n − 2)-form associated with the horizon Killing field (i.e., the Killing field which vanishes on Σ), normalized so as to have unit surface gravity. 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 1 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.

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. −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.

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In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define "conserved quantities" in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry. (This fact is closely related to the fact that the desired "conserved quantities" are not, in general, conserved!) In this paper we give a prescription for defining "conserved quantities" by proposing a modification of the equation that must be satisfied by a Hamiltonian. Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, 1 although we have not investigated existence and uniqueness issues in the most general contexts. In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations. 2

The detection of gravitational waves by Advanced LIGO and Advanced Virgo provides an opportunity to test general relativity in a regime that is inaccessible to traditional astronomical observations and laboratory tests. We present four tests of the consistency of the data with binary black hole gravitational waveforms predicted by general relativity. One test subtracts the best-fit waveform from the data and checks the consistency of the residual with detector noise. The second test checks the consistency of the low-and high-frequency parts of the observed signals. The third test checks that phenomenological deviations introduced in the waveform model (including in the post-Newtonian coefficients) are consistent with 0. The fourth test constrains modifications to the propagation of gravitational waves due to a modified dispersion relation, including that from a massive graviton. We present results both for individual events and also results obtained by combining together particularly strong events from the first and second observing runs of Advanced LIGO and Advanced Virgo, as collected in the catalog GWTC-1. We do not find any inconsistency of the data with the predictions of general relativity and improve our previously presented combined constraints by factors of 1.1 to 2.5. In particular, we bound the mass of the graviton to be m g ≤ 4.7 × 10 −23 eV=c 2 (90% credible level), an improvement of a factor of 1.6 over our previously presented results. Additionally, we check that the four gravitational-wave events published for the first time in GWTC-1 do not lead to stronger constraints on alternative polarizations than those published previously.

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.