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We present thought experiments to measure the Arnowitt-Deser-Misner and Bondi-Sachs energy of isolated systems in general relativity. The expression of the Bondi-Sachs energy used in the protocol is likely to have other applications. In particular, it is well-suited to to be promoted to an operator in non-perturbative loop quantum gravity., Comment: Written in honor of Jorge Pullin on the occasion of his 60th birthday. 21 pages, 1 figure

We introduce a Hamiltonian framework tailored to degrees of freedom (DOF) of field theories that reside in suitable 3-dimensional open regions, and then apply it to the gravitational DOF of general relativity. Specifically, these DOF now refer to open regions of null infinity, and of black hole (and cosmological) horizons representing equilibrium situations. At null infinity the new Hamiltonian framework yields the well-known BMS fluxes and charges. By contrast, all fluxes vanish identically at black hole (and cosmological) horizons just as one would physically expect. In a companion paper we showed that, somewhat surprisingly, the geometry and symmetries of these two physical configurations descend from a common framework. This paper reinforces that theme: Very different physics emerges in the two cases from a common Hamiltonian framework because of the difference in the nature of degrees of freedom. Finally, we compare and contrast this Hamiltonian approach with those available in the literature., Comment: 35 pages. At referee's suggestion version 1 of arXiv:2402.17977 was split into two papers. This paper contains slightly expanded versions of the material that was in sections IV and V and Appendix A and B of that submission. Both papers are at press at Phy. Rev. D

We propose a new set of BMS charges at null infinity, characterized by a super-translation flux that contains only the `hard' term. This is achieved with a specific corner improvement of the symplectic 2-form, and we spell the conditions under which it is unique. The charges are associated to a Wald-Zoupas symplectic potential, and satisfy all standard criteria: they are covariant, provide a center-less realization of the symmetry algebra, have vanishing flux in non-radiative spacetimes, and vanish in Minkowski. We use them to define a notion of spatially localized energy density of gravitational waves, and explain how it can be measured doing experiments which are purely local in space and over an extended period of time., Comment: 12 pages. V2: some clarifications and minor improvements, matches published version. Long elapsed time from v1 due to reviewing at PRL which did not lead to publication

We show that when the Wald-Zoupas prescription is implemented, the resulting charges realize the BMS symmetry algebra without any 2-cocycle nor central extension, at any cut of future null infinity. We refine the covariance prescription for application to the charge aspects, and introduce a new aspect for Geroch's super-momentum with better covariance properties. For the extended BMS symmetry with singular conformal Killing vectors we find that a Wald-Zoupas symplectic potential exists, if one is willing to modify the symplectic structure by a corner term. The resulting algebra of Noether currents between two arbitrary cuts is center-less. The charge algebra at a given cut has a residual field-dependent 2-cocycle, but time-independent and non-radiative. More precisely, super-rotation fluxes act covariantly, but super-rotation charges act covariantly only on global translations. The take home message is that in any situation where 2-cocycles appears in the literature, covariance has likely been lost in the charge prescription, and that the criterium of covariance is a powerful one to reduce ambiguities in the charges, and can be used also for ambiguities in the charge aspects., Comment: 55 pages. v2: minor amendments, matches published version

In general relativity as well as gauge theories, non-trivial symmetries can appear at boundaries. In the presence of radiation some of the symmetries are not Hamiltonian vector fields, hence the definition of charges for the symmetries becomes delicate. It can lead to the problem of field-dependent 2-cocycles in the charge algebra, as opposed to the central extensions allowed in standard classical mechanics. We show that if the Wald-Zoupas prescription is implemented, its covariance requirement guarantees that the algebra of Noether currents is free of field-dependent 2-cocycles, and its stationarity requirement further removes central extensions. Therefore the charge algebra admits at most a time-independent field-dependent 2-cocycle, whose existence depends on the boundary conditions. We report on new results for asymptotic symmetries at future null infinity that can be derived with this approach., Comment: 10 pages

Null infinity arises as a boundary of the Penrose conformal completion of an asymptotically flat physical space-time. We first note that null infinity is a weakly isolated horizon (WIH), and then show that its familiar properties can be derived from the general WIH framework. This seems quite surprising because physics associated with black hole (and cosmological) WIHs is very different from that extracted at null infinity. We show that these differences can be directly traced back to the fact that null infinity is a WIH in the conformal completion rather than the physical space-time. In particular, the BMS group at null infinity stems from the symmetry group of WIHs. In a companion paper, we obtain fluxes and charges associated with symmetries associated with both null infinity and black hole (and cosmological) horizons using a new Hamiltonian framework. The fact that is there is a single mathematical framework underlying these different situations paves the way to explore the relation between horizon dynamics in the strong field region and waveforms at infinity. It should also be useful in the analysis of black hole evaporation in quantum gravity., Comment: 21 pages. At referee's suggestion version 1 of this paper was split into two papers. Consequently the current version v2 contains only the first three sections of v1. The remaining sections appear in a paper entitled "Null Infinity and Horizons: New Approach to Fluxes and Charges', arXiv:2407.03254. Both papers are at press at Phy. Rev. D

Black hole horizons in equilibrium and null infinity of asymptotically flat space-times are null 3-manifolds but have very different physical connotations. We first show that they share a large number of geometric properties, making them both weakly isolated horizons. We then use this new unified perspective to unravel the origin of the drastic differences in the physics they contain. Interestingly, the themes are woven together in a manner reminiscent of voices in a fugue., Comment: 9 pages. v2: minor amendments, one table added. Version to appear in Phys Rev D (Lett)

Motivated by issues in the context of asymptotically flat spacetimes at null infinity, we discuss in the simplest example of a massless scalar field in two dimensions several subtleties that arise when setting up the canonical formulation on a single or on two intersecting null hyperplanes with a special emphasis on the infinite-dimensional global and conformal symmetries and their canonical generators, the free data, a consistent treatment of zero modes, matching conditions, and implications for quantization of massless versus massive fields., Comment: 52 pages, 3 figures, cosmetic changes

Spinfoams provide a framework for the dynamics of loop quantum gravity that is manifestly covariant under the full four-dimensional diffeomorphism symmetry group of general relativity. In this way they complete the ideal of three-dimensional diffeomorphism covariance that consistently motivates loop quantum gravity at every step. Specifically, spinfoam models aim to provide a projector onto, and a physical inner product on, the simultaneous kernel of all of the constraints of loop quantum gravity by means of a discretization of the gravitational path integral. In the limit of small Planck constant, they are closely related to the path integral for Regge calculus, while at the same time retaining all of the tools of a canonical quantum theory of gravity. They may also be understood as generalizations of well-understood state sum models for topological quantum field theories. In this chapter, we review all of these aspects of spinfoams, as well as review in detail the derivation of the currently most used spinfoam model, the EPRL model, calculational tools for it, and the various extensions of it in the literature. We additionally summarize some of the successes and open problems in the field., Comment: 43 pages, 1 figure, Invited Chapter for the Handbook of Quantum Gravity (Eds. Bambi, Modesto and Shapiro, Springer 2023)

We perform a detailed study of the covariance properties of the symplectic potential of general relativity on a null hypersurface, and of the different polarizations that can be used to study conservative as well as leaky boundary conditions. This allows us to identify a one-parameter family of covariant symplectic potentials. We compute the charges and fluxes for the most general phase space with arbitrary variations. We study five symmetry groups that arise when different restrictions on the variations are included, two of which are new. Requiring stationarity as in the original Wald-Zoupas prescription selects a unique member of the family of symplectic potentials, the one of Chandrasekaran, Flanagan and Prabhu. The associated charges are all conserved on non-expanding horizons, but not on flat spacetime. We show that it is possible to require a weaker notion of stationarity which selects another symplectic potential, again in a unique way, and whose charges are conserved on both non-expanding horizons and flat light-cones. Furthermore, the flux of future-pointing diffeomorphisms at leading-order around an outgoing flat light-cone is positive and reproduces a tidal heating plus a memory term. We also study the conformal conservative boundary conditions suggested by the alternative polarization and identify under which conditions they define a non-ambiguous variational principle. Our results have applications for dynamical notions of entropy, and are useful to clarify the interplay between different boundary conditions, charge prescriptions, and symmetry groups that can be associated with a null boundary., Comment: 54 pages. v2: Improved text, minor corrections, references added

We show that the Wald-Zoupas prescription for gravitational charges is valid in the presence of anomalies and field-dependent diffeomorphism, but only if these are related to one another in a specific way. The geometric interpretation of the allowed anomalies is exposed looking at the example of BMS symmetries: They correspond to soft terms in the charges. We determine if the Wald-Zoupas prescription coincides with an improved Noether charge. The necessary condition is a certain differential equation, and when it is satisfied, the boundary Lagrangian of the resulting improved Noether charge contains in general a non-trivial corner term that can be identified a priori from a condition of anomaly-freeness. Our results explain why the Wald-Zoupas prescription works in spite of the anomalous behaviour of BMS transformations, and should be helpful to relate different branches of the literature on surface charges., Comment: 19 pages plus Appendix. V2: many improvements to the text, clarifications added, improved comparison with the results in the literature. More general analysis of the WZ covariance requirement, leading to a simpler discussion of some results at future null infinity. V3: minor amendments, matches published version

Non-singular black holes models can be described by modified classical equations motivated by loop quantum gravity. We investigate what happens when the sine function typically used in the modification is replaced by an arbitrary bounded function, a generalization meant to study the effect of ambiguities such as the choice of representation of the holonomy. A number of features can be determined without committing to a specific choice of functions. We find generic singularity resolution. The presence and number of horizons is determined by global features of the function regularizing the angular components of the connection, and the presence and number of bounces by global features of the function regularizing the time component. The trapping or anti-trapping nature of regions inside horizons depends on the relative location with respect to eventual bounces. We use these results to comment on some of the ambiguities of polymer black hole models., Comment: 32 pages, many figures

We compute the Hamiltonian surface charges of gravity for a family of conservative boundary conditions, that include Dirichlet, Neumann, and York's mixed boundary conditions defined by holding fixed the conformal induced metric and the trace of the extrinsic curvature. We show that for all boundary conditions considered, canonical methods give the same answer as covariant phase space methods improved by a boundary Lagrangian, a prescription recently developed in the literature and thus supported by our results. The procedure also suggests a new integrable charge for the Einstein-Hilbert Lagrangian, different from the Komar charge for non-Killing and non-tangential diffeomorphisms. We study how the energy depends on the choice of boundary conditions, showing that both the quasi-local and the asymptotic expressions are affected. Finally, we generalize the analysis to non-orthogonal corners, confirm the matching between the covariant and canonical results without any change in the prescription, and discuss the subtleties associated with this case., Comment: v2: Revised discussion of the case with non-orthogonal corners, the matching between covariant and canonical formulas is now obtained thanks to the Legendre transform at the corner. Further amendments. Matches published version

We develop the covariant phase space formalism allowing for non-vanishing flux, anomalies and field dependence in the vector field generators. We construct a charge bracket that generalizes the one introduced by Barnich and Troessaert and includes contributions from the Lagrangian and its anomaly. This bracket is uniquely determined by the choice of Lagrangian representative of the theory. We then extend the notion of corner symmetry algebra to include the surface translation symmetries and prove that the charge bracket provides a canonical representation of the extended corner symmetry algebra. This representation property is shown to be equivalent to the projection of the gravitational equations of motion on the corner, providing us with an encoding of the bulk dynamics in a locally holographic manner., Comment: 27 pages + Appendix, 2 figures; v3 published version

We propose an extension of the BMS group, which we refer to as Weyl BMS or BMSW for short, that includes, besides super-translations, local Weyl rescalings and arbitrary diffeomorphisms of the 2d sphere metric. After generalizing the Barnich-Troessaert bracket, we show that the Noether charges of the BMSW group provide a centerless representation of the BMSW Lie algebra at every cross section of null infinity. This result is tantamount to proving that the flux-balance laws for the Noether charges imply the validity of the asymptotic Einstein's equations at null infinity. The extension requires a holographic renormalization procedure, which we construct without any dependence on background fields. The renormalized phase space of null infinity reveals new pairs of conjugate variables. Finally, we show that BMSW group elements label the gravitational vacua., Comment: 53 pages + appendix, 1 figure; v3 minor revision to match published version

Minimal varying $\Lambda$ theories are defined by an action built from the Einstein-Cartan-Holst first order action for gravity with the cosmological constant $\Lambda$ as an independent scalar field, and supplemented by the Euler and Pontryagin densities multiplied by $1/\Lambda$. We identify the canonical structure of these theories which turn out to represent an example of irregular systems. We find five degrees of freedom on generic backgrounds and for generic values of parameters, whereas if the parameters satisfy a certain condition (which includes the most commonly considered Euler case) only three degrees of freedom remain. On de Sitter-like backgrounds the canonical structure changes, and due to an emergent conformal symmetry one degree of freedom drops from the spectrum. We also analyze the self-dual case with an holomorphic action depending only on the self-dual part of the connection. In this case we find two (complex) degrees of freedom, and further discuss the Kodama state, the restriction to de Sitter background and the effect of reality conditions., Comment: 38 pages

We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of the principal series, corresponding to generalized Clebsch-Gordan coefficients with $n\geq3$ legs. We find critical configurations of the quantum labels with a power-law decay of the invariants. They describe 3d polygons that can be deformed into one another via a Lorentz transformation. This is defined viewing the edge vectors of the polygons are the electric part of bivectors satisfying a (frame-dependent) relation between their electric and magnetic parts known as $\gamma$-simplicity in the loop quantum gravity literature. The frame depends on the SU(2) spin labelling the basis elements of the invariants. We compute a saddle point approximation using the critical points and provide a leading-order approximation of the invariants. The power-law is universal if the SU(2) spins have their lowest value, and $n$-dependent otherwise. As a side result, we provide a compact formula for $\gamma$-simplicity in arbitrary frames. The results have applications to the current EPRL model, but also to future research aiming at going beyond the use of fixed time gauge in spin foam models., Comment: 29 pages and 3 figures. v2:minor corrections and a subsection added to match published version

Dual gravitational charges have been recently computed from the Holst term in tetrad variables using covariant phase space methods. We highlight that they originate from an exact 3-form in the tetrad symplectic potential that has no analogue in metric variables. Hence there exists a choice of the tetrad symplectic potential that sets the dual charges to zero. This observation relies on the ambiguity of the covariant phase space methods. To shed more light on the dual contributions, we use the Kosmann variation to compute (quasi-local) Hamiltonian charges for arbitrary diffeomorphisms. We obtain a formula that illustrates comprehensively why the dual contribution to the Hamiltonian charges: (i) vanishes for exact isometries and asymptotic symmetries at spatial infinity; (ii) persists for asymptotic symmetries at future null infinity, in addition to the usual BMS contribution. Finally, we point out that dual gravitational charges can be equally derived using the Barnich-Brandt prescription based on cohomological methods, and that the same considerations on asymptotic symmetries apply., Comment: 18 pages; v2: improved version, refs added, matches version accepted for publication

We introduce a new family of coherent states for loop quantum gravity, inspired by the twisted geometry parametrization. We compute their peakedness properties and compare them with the heat-kernel coherent states. They show similar features for the area and the holonomy operators, but improved peakedness in the direction of the flux. At the gauge-invariant level, the new family is built from tensor products of coherent intertwiners. To study the peakedness of the holonomy operator, we introduce a new shift operator based on the harmonic oscillator representation associated with the twisted geometry parametrization. The new shift operator captures the components of the holonomy relevant to disentangle its action into a simple positive shift of the spins., Comment: 21 pages + Appendices. Various color figures. v2: minor corrections, matches published version

We describe a technique to study the asymptotics of SL(2,C) invariant tensors associated to graphs, with unitary irreps and lowest SU(2) spins, and apply it to the Lorentzian EPRL-KKL (Engle, Pereira, Rovelli, Livine; Kaminski, Kieselowski, Lewandowski) model of quantum gravity. We reproduce the known asymptotics of the 4-simplex graph with a different perspective on the geometric variables and introduce an algorithm valid for any graph. On general grounds, we find that critical configurations are not just Regge geometries, but a larger set corresponding to conformal twisted geometries. These can be either Euclidean or Lorentzian, and include curved and flat 4d polytopes as subsets. For modular graphs, we show that multiple pairs of critical points exist, and there exist critical configurations of mixed signature, Euclidean and Lorentzian in different subgraphs, with no 4d embedding possible., Comment: 40 Pages + 5 Appendices. 11 Figures. v2: Refined presentation of the general algorithm, additional minor amendments. v3: paragraph added in section 5 about curved embeddings

By allowing torsion into the gravitational dynamics one can promote the cosmological constant, $\Lambda$, to a dynamical variable in a class of quasi-topological theories. In this paper we perform a mini-superspace quantization of these theories in the connection representation. If $\Lambda$ is kept fixed, the solution is a delta-normalizable version of the Chern-Simons (CS) state, which is the dual of the Hartle and Hawking and Vilenkin wave-functions. We find that the CS state solves the Wheeler-DeWitt equation also if $\Lambda$ is rendered dynamical by an Euler quasi-topological invariant, {\it in the parity-even branch of the theory}. In the absence of an infra-red (IR) cut-off, the CS state suggests the marginal probability $P(\Lambda)=\delta(\Lambda)$. Should there be an IR cutoff (for whatever reason) the probability is sharply peaked at the cut off. In the parity-odd branch, however, we can still find the CS state as a particular (but not most general) solution, but further work is needed to sharpen the predictions. For the theory based on the Pontryagin invariant (which only has a parity-odd branch) the CS wave function no longer is a solution to the constraints. We find the most general solution in this case, which again leaves room for a range of predictions for $\Lambda$., Comment: Version to be published in Phys.Rev.D

Varying the gravitational Lagrangian produces a boundary contribution that has various physical applications. It determines the right boundary terms to be added to the action once boundary conditions are specified, and defines the symplectic structure of covariant phase space methods. We study general boundary variations using tetrads instead of the metric. This choice streamlines many calculations, especially in the case of null hypersurfaces with arbitrary coordinates, where we show that the spin-1 momentum coincides with the rotational 1-form of isolated horizons. The additional gauge symmetry of internal Lorentz transformations leaves however an imprint: the boundary variation differs from the metric one by an exact 3-form. On the one hand, this difference helps in the variational principle: gluing hypersurfaces to determine the action boundary terms for given boundary conditions is simpler, including the most general case of non-orthogonal corners. On the other hand, it affects the construction of Hamiltonian surface charges with covariant phase space methods, which end up being generically different from the metric ones, in both first and second-order formalisms. This situation is treated in the literature gauge-fixing the tetrad to be adapted to the hypersurface or introducing a fine-tuned internal Lorentz transformation depending non-linearly on the fields. We point out and explore the alternative approach of dressing the bare symplectic potential to recover the value of all metric charges, and not just for isometries. Surface charges can also be constructed using a cohomological prescription: in this case we find that the exact 3-form mismatch plays no role, and tetrad and metric charges are equal. This prescription leads however to different charges whether one uses a first-order or second-order Lagrangian, and only for isometries one recovers the same charges., Comment: 47 pages, 1 figure; v2: improved text, updated refs, Sec.(VI.E) rewritten and extended with Kosmann derivative, Eq.(VII.5) corrected

We analyze a bi-gravity model based on the first order formalism, having as fundamental variables two tetrads but only one Lorentz connection. We show that on a large class of backgrounds its linearization agrees with general relativity. At the non-linear level, additional degrees of freedom appear, and we reveal the mechanism hiding them around the special backgrounds. We further argue that they do not contain a massive graviton, nor the Boulware-Deser ghost. The model thus propagates only one graviton, whereas the nature of the additional degrees of freedom remains to be investigated. We also present a foliation-preserving deformation of the model, which keeps all symmetries except time diffeomorphisms and has three degrees of freedom., Comment: 29 pages

The Lorentzian EPRL spin foam amplitude for loop quantum gravity is a multi-dimensional non-compact integral of highly oscillating functions. Using a method based on the decomposition of Clebsch-Gordan coefficients for the unitary infinite-dimensional representations of SL(2,C) in terms of those of SU(2), we are able to provide for the first time numerical evaluations of the vertex amplitude. The values obtained support the asymptotic formula obtained by Barrett and collaborators with a saddle point approximation, showing, in particular, a power-law decay and oscillations related to the Regge action. The comparison offers a test of the efficiency of the method. Truncating the decomposition to the first few terms provides a qualitative matching of the power-law decay and oscillations. For vector and Euclidean Regge boundary data, a qualitative matching is obtained with just the first term, which corresponds to the simplified EPRL model. We comment on future developments for the numerics and extension to higher vertices. We complete our work with some analytic results: We provide an algorithm and explicit configurations for the different geometries that can arise as boundary data, and explain the geometric consequences of the decomposition used., Comment: 31 pages + 13 pages of Appendix. 18 Figures. The numerical code used in this work is public and available at the address: bitbucket.org/giorgiosarno/sl2cfoam-asym-1.0 v2: some amendments, matches published version V3: few typos corrected, small additions and numerical factors corrected in one table

We show how the manifold $T^*SU(2, 2)$ arises as a symplectic reduction from eight copies of the twistor space. Some of the constraints in the twistor space correspond to an octahedral configuration of twelve complex light rays in the Minkowski space. We discuss a mechanism to break the conformal symmetry down to the twistorial parametrisation of $T^*SL(2, C)$ used in loop quantum gravity., Comment: 10 pages, 4 figures. v2: expanded introduction and conclusions

We study null geodesic congruences (NGCs) in the presence of spacetime torsion, recovering and extending results in the literature. Only the highest spin irreducible component of torsion gives a proper acceleration with respect to metric NGCs, but at the same time obstructs abreastness of the geodesics. This means that it is necessary to follow the evolution of the drift term in the optical equations, and not just shear, twist and expansion. We show how the optical equations depend on the non-Riemannian components of the curvature, and how they reduce to the metric ones when the highest spin component of torsion vanishes., Comment: v2: improved text, a few equations and a subsection added. v3: typo in one equation corrected, improved conclusions and other minor amendments

We prove that a conserved effective energy-momentum tensor for Einstein-Cartan theory can be identified from the Noether identities of the matter Lagrangian, using the torsion field equations relating them. More precisely, a one-parameter family labelled by the Immirzi parameter. We use this result and the contorsion description to show that Jacobson's thermodynamical derivation of the Einstein equations follows as in the metric theory, namely from the equilibrium Clausius relation and the fact that a Killing horizon is metric-geodetic. Our derivation works for an arbitrary torsion field. In the course of our discussion we review the laws of black hole mechanics and their dependence on torsion., Comment: 21 pages, 2 figures

We identify a symplectic potential for general relativity in tetrad and connection variables that is fully gauge-invariant, using the freedom to add surface terms. When torsion vanishes, it does not lead to surface charges associated with the internal Lorentz transformations, and reduces exactly to the symplectic potential given by the Einstein-Hilbert action. In particular, it reproduces the Komar form when the variation is a Lie derivative, and the geometric expression in terms of extrinsic curvature and 2d corner data for a general variation. The additional surface term vanishes at spatial infinity for asymptotically flat spacetimes, thus the usual Poincare charges are obtained. We prove that the first law of black hole mechanics follows from the Noether identity associated with the covariant Lie derivative, and that it is independent of the ambiguities in the symplectic potential provided one takes into account the presence of non-trivial Lorentz charges that these ambiguities can introduce., Comment: 19 pages. V2: some amendments to the text, typos corrected, updated references and more details in Section 5. Matches published version