Covariant Phase Space Formalism Research Articles

Notes on thermodynamics of Schwarzschild-like bumblebee black hole

In this work, we revisited the thermodynamics of Schwarzschild-like bumblebee black hole using Iyer–Wald covariant phase space formalism. With a non zero vacuum expectation value, the bumblebee field is responsible for spontaneously breaking the Lorentz symmetry. As the bumblebee field is non-minimally coupled to gravity, we showed that the thermodynamic variables will be different from the counterpart in Einstein gravity. Especially, by using Iyer–Wald formalism, we found that the black hole entropy also differs from the result obtained from Wald entropy formula. Like Horndeski gravity, this mismatch is due to the divergence of bumblebee one-form field at the horizon. After figuring out the thermodynamics, we also briefly discussed the evaporation behavior of Schwarzschild like bumblebee black hole. We found that although bumblebee field has no influence on the critical impact factor, it can influence the black hole evaporation time.

Thermodynamics of accelerating AdS4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_4$$\\end{document} black holes from the covariant phase space

We study the charges and first law of thermodynamics for accelerating, non-rotating black holes with dyonic charges in AdS4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_4$$\\end{document} using the covariant phase space formalism. In order to apply the formalism to these solutions (which are asymptotically locally AdS and admit a non-smooth conformal boundary I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {I}}$$\\end{document}) we make two key improvements: (1) We relax the requirement to impose Dirichlet boundary conditions and demand merely a well-posed variational problem. (2) We keep careful track of the codimension-2 corner term induced by the holographic counterterms, a necessary requirement due to the presence of “cosmic strings” piercing I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {I}}$$\\end{document}. Using these improvements we are able to match the holographic Noether charges to the Wald Hamiltonians of the covariant phase space and derive the first law of black hole thermodynamics with the correct “thermodynamic length” terms arising from the strings. We investigate the relationship between the charges imposed by supersymmetry and show that our first law can be consistently applied to various classes of non-supersymmetric solutions for which the cross-sections of the horizon are spindles.

No logarithmic corrections to entropy in shift-symmetric Gauss-Bonnet gravity

Employing the covariant phase space formalism, we discuss black hole thermodynamics in four-dimensional scalar-tensor Einstein-Gauss-Bonnet gravity. We argue that logarithmic corrections to Wald entropy previously reported in this theory do not appear, due to the symmetry of the theory under constant shifts of the scalar field. Instead, we obtain the standard Bekenstein entropy of general relativity. Then, to satisfy the first law of black hole mechanics, the Hawking temperature must be modified. It has been proposed that such temperature modifications occur generically in scalar-tensor theories, due to different propagation speeds of gravitons and photons. We show that the temperature modifications also emerge in the Euclidean canonical ensemble approach to black hole thermodynamics. Notably, the boundary terms of the type we consider here can be considered in any scalar-tensor gravitational theories. Hence, we illustrate that adding a suitable boundary term to action may drastically affect black hole thermodynamics, changing both the entropy and the temperature.

Phase space renormalization and finite BMS charges in six dimensions

We perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions — those that are analytic near I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}+ — admit a non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS) group which contains infinite-dimensional supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff×Weyl transformations of the celestial S4. Using the covariant phase space formalism and a new technique which we develop in this paper (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are local and covariant. The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. Finally, we show that the semi-classical Ward identities for supertranslations and superrotations are precisely the leading and subleading soft-graviton theorems respectively.

Finite charges from the bulk action

Constructing charges in the covariant phase space formalism often leads to formally divergent expressions, even when the fields satisfy physically acceptable fall-off conditions. These expressions can be rendered finite by corner ambiguities in the definition of the presymplectic potential, which in some cases may be motivated by arguments involving boundary Lagrangians. We show that the necessary corner terms are already present in the variation of the bulk action and can be extracted in a straightforward way. Once these corner terms are included in the presymplectic potential, charges derived from an associated codimension-2 form are automatically finite. We illustrate the procedure with examples in two and three dimensions, working in Bondi gauge and obtaining integrable charges. As a by-product, actions are derived for these theories that admit a well-defined variational principle when the fields satisfy boundary conditions on a timelike surface with corners. An interesting feature of our analysis is that the fields are not required to be fully on-shell.

Systematics of boundary actions in gauge theory and gravity

We undertake a general study of the boundary (or edge) modes that arise in gauge and gravitational theories defined on a space with boundary, either asymptotic or at finite distance, focusing on efficient techniques for computing the corresponding boundary action. Such actions capture all the dynamics of the system that are implied by its asymptotic symmetry group, such as correlation functions of the corresponding conserved currents. Working in the covariant phase space formalism, we develop a collection of approaches for isolating the boundary modes and their dynamics, and illustrate with various examples, notably AdS3 gravity (with and without a gravitational Chern-Simons terms) subject to assorted boundary conditions.

Asymptotic symmetries and memories of gauge theories in FLRW spacetimes

In this paper, we investigate the asymptotic structure of gauge theories in decelerating and spatially flat Friedmann-Lemaître-Robertson-Walker universes. Firstly, we thoroughly explore the asymptotic symmetries of electrodynamics in this background, which reveals a major inconsistency already present in the flat case. Taking advantage of this treatment, we derive the associated memory effects, discussing their regime of validity and differences with respect to their flat counterparts. Next, we extend our analysis to non-Abelian Yang-Mills, coupling it dynamically and simultaneously to a Dirac spinor and a complex scalar field. Within this novel setting, we examine the possibility of constructing Poisson superbrackets based on the covariant phase space formalism.

Magnetic charges in supergravity

In this paper we study the dual charges of mathcal{N} = 1 supergravity in asymptotically flat space-time. The action considered is the usual supergravity action with a topological contribution. This is the Nieh-Yan term and the magnetic term of the free Rarita-Schwinger field. Through methods of the covariant phase space formalism we construct the charges conjugate to supersymmetry, diffeomorphism and Lorentz transformations. The additional term in the action will lead to new, non-vanishing contributions to these charges. The magnetic diffeomorphism charges are equivalent to the ones previously found for gravity, while the dual supersymmetric charges are new and do not appear for the free Rarita-Schwinger field. The dual Lorentz charges serve to regularize the previous two. We find that the asymptotic symmetry group for supergravity can only include globally well-defined super-rotations.

Embeddings and Integrable Charges for Extended Corner Symmetry.

We revisit the problem of extending the phase space of diffeomorphism-invariant theories to account for embeddings associated with the boundary of subregions. We do so by emphasizing the importance of a careful treatment of embeddings in all aspects of the covariant phase space formalism. In so doing we introduce a new notion of the extension of field space associated with the embeddings which has the important feature that the Noether charges associated with all extended corner symmetries are in fact integrable, but not necessarily conserved. We give an intuitive understanding of this description. We then show that the charges give a representation of the extended corner symmetry via the Poisson bracket, without central extension.

Edge modes as reference frames and boundary actions from post-selection

We introduce a general framework realizing edge modes in (classical) gauge field theory as dynamical reference frames, an often suggested interpretation that we make entirely explicit. We focus on a bounded region M with a co-dimension one time-like boundary Γ, which we embed in a global spacetime. Taking as input a variational principle at the global level, we develop a systematic formalism inducing consistent variational principles (and in particular, boundary actions) for the subregion M. This relies on a post-selection procedure on Γ, which isolates the subsector of the global theory compatible with a general choice of gauge-invariant boundary conditions for the dynamics in M. Crucially, the latter relate the configuration fields on Γ to a dynamical frame field carrying information about the spacetime complement of M; as such, they may be equivalently interpreted as frame-dressed or relational observables. Generically, the external frame field keeps an imprint on the ensuing dynamics for subregion M, where it materializes itself as a local field on the time-like boundary Γ; in other words, an edge mode. We identify boundary symmetries as frame reorientations and show that they divide into three types, depending on the boundary conditions, that affect the physical status of the edge modes. Our construction relies on the covariant phase space formalism, and is in principle applicable to any gauge (field) theory. We illustrate it on three standard examples: Maxwell, Abelian Chern-Simons and non-Abelian Yang-Mills theories. In complement, we also analyze a mechanical toy-model to connect our work with recent efforts on (quantum) reference frames.

Holographically description of Kerr-Bolt black hole in terms of the warped conformal field theory

Recently it has been speculated that a set of diffeomorphisms exist which act non-trivially on the horizon of some black holes such as Kerr and Kerr-Newman black hole. Using this symmetry in covariant phase space formalism one can obtain conserved charges as surface integrals on the horizon. Kerr-Bolt spacetime is well-known for its asymptotic topology and has been studied widely in recent years. In this work we are going to study Kerr-Bolt black hole and provide a holographic description of it in term of warped conformal field theory.

Soft thermodynamics of gravitational shock wave

The gravitational shock waves have provided crucial insights into entanglement structures of black holes in the AdS/CFT correspondence. Recent progress on the soft hair physics suggests that these developments from holography may also be applicable to geometries beyond negatively curved spacetime. In this work, we derive a remarkably simple thermodynamic relation which relates the gravitational shock wave to a microscopic area deformation. Our treatment is based on the covariant phase space formalism and is applicable to any Killing horizon in generic static spacetime which is governed by arbitrary covariant theory of gravity. The central idea is to probe the gravitational shock wave, which shifts the horizon in the $u$ direction, by the Noether charge constructed from a vector field which shifts the horizon in the $v$ direction. As an application, we illustrate its use for the Gauss-Bonnet gravity. We also derive a simplified form of the gravitational scattering unitary matrix and show that its leading-order contribution is nothing but the exponential of the horizon area: $\mathcal{U}=\exp(i \text{Area})$.

Extended corner symmetry, charge bracket and Einstein\u2019s equations

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.

Black hole thermodynamics in the presence of nonlinear electromagnetic fields

As the interaction between the black holes and highly energetic infalling charged matter receives quantum corrections, the basic laws of black hole mechanics have to be carefully rederived. Using the covariant phase space formalism, we generalize the first law of black hole mechanics, both "equilibrium state" and "physical process" versions, in the presence of nonlinear electrodynamics fields, defined by Lagrangians depending on both quadratic electromagnetic invariants, $F_{ab}F^{ab}$ and $F_{ab}\,{\star F}^{ab}$. Derivation of this law demands a specific treatment of the Lagrangian parameters, similar to embedding of the cosmological constant into thermodynamic context. Furthermore, we discuss the validity of energy conditions, several complementing proofs of the zeroth law of black hole electrodynamics and some aspects of the recently generalized Smarr formula, its (non-)linearity and relation to the first law.

Accelerating black hole thermodynamics with boost time

We derive a thermodynamic first law for the electrically charged C-metric with vanishing cosmological constant. This spacetime describes a pair of identical accelerating black holes each pulled by a cosmic string. Treating the ‘boost time’ of this spacetime as the canonical time, we find a thermodynamic first law in which every term has an unambiguous physical meaning. We then show how this first law may be derived using Noetherian methods in the covariant phase space formalism. We argue that the area of the acceleration horizon contributes to the entropy and that the appropriate notion of energy of this spacetime is a ‘boost mass’ which vanishes identically. The recovery of the Reissner–Nordstrom first law in the limit of small string tension is also demonstrated. Finally, we compute the action of the Euclidean section of the C-metric and show it agrees with the thermodynamic grand potential, providing an independent confirmation of the validity of our first law. We also briefly speculate on the significance of firewalls in this spacetime.

Covariant phase space and soft factorization in non-Abelian gauge theories

We perform a careful study of the infrared sector of massless non-abelian gauge theories in four-dimensional Minkowski spacetime using the covariant phase space formalism, taking into account the boundary contributions arising from the gauge sector of the theory. Upon quantization, we show that the boundary contributions lead to an infinite degeneracy of the vacua. The Hilbert space of the vacuum sector is not only shown to be remarkably simple, but also universal. We derive a Ward identity that relates the n-point amplitude between two generic in- and out-vacuum states to the one computed in standard QFT. In addition, we demonstrate that the familiar single soft gluon theorem and multiple consecutive soft gluon theorem are consequences of the Ward identity.

Most general theory of 3d gravity: covariant phase space, dual diffeomorphisms, and more

We show that the phase space of three-dimensional gravity contains two layers of dualities: between diffeomorphisms and a notion of “dual diffeomorphisms” on the one hand, and between first order curvature and torsion on the other hand. This is most elegantly revealed and understood when studying the most general Lorentz-invariant first order theory in connection and triad variables, described by the so-called Mielke-Baekler Lagrangian. By analyzing the quasi-local symmetries of this theory in the covariant phase space formalism, we show that in each sector of the torsion/curvature duality there exists a well-defined notion of dual diffeomorphism, which furthermore follows uniquely from the Sugawara construction. Together with the usual diffeomorphisms, these duals form at finite distance, without any boundary conditions, and for any sign of the cosmological constant, a centreless double Virasoro algebra which in the flat case reduces to the BMS3 algebra. These algebras can then be centrally-extended via the twisted Sugawara construction. This shows that the celebrated results about asymptotic symmetry algebras are actually generic features of three-dimensional gravity at any finite distance. They are however only revealed when working in first order connection and triad variables, and a priori inaccessible from Chern-Simons theory. As a bonus, we study the second order equations of motion of the Mielke-Baekler model, as well as the on-shell Lagrangian. This reveals the duality between Riemannian metric and teleparallel gravity, and a new candidate theory for three-dimensional massive gravity which we call teleparallel topologically massive gravity.

Anomalies in gravitational charge algebras of null boundaries and black hole entropy

We revisit the covariant phase space formalism applied to gravitational theories with null boundaries, utilizing the most general boundary conditions consistent with a fixed null normal. To fix the ambiguity inherent in the Wald-Zoupas definition of quasilocal charges, we propose a new principle, based on holographic reasoning, that the flux be of Dirichlet form. This also produces an expression for the analog of the Brown-York stress tensor on the null surface. Defining the algebra of charges using the Barnich-Troessaert bracket for open subsystems, we give a general formula for the central — or more generally, abelian — extensions that appear in terms of the anomalous transformation of the boundary term in the gravitational action. This anomaly arises from having fixed a frame for the null normal, and we draw parallels between it and the holographic Weyl anomaly that occurs in AdS/CFT. As an application of this formalism, we analyze the near-horizon Virasoro symmetry considered by Haco, Hawking, Perry, and Strominger, and perform a systematic derivation of the fluxes and central charges. Applying the Cardy formula to the result yields an entropy that is twice the Bekenstein-Hawking entropy of the horizon. Motivated by the extended Hilbert space construction, we interpret this in terms of a pair of entangled CFTs associated with edge modes on either side of the bifurcation surface.

Covariant phase space with boundaries

The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically treat total derivatives and boundary terms, which has led to some confusion about how exactly to apply the formalism in the presence of boundaries. In particular the original construction of the canonical Hamiltonian relies on the assumed existence of a certain boundary quantity “B”, whose physical interpretation has not been clear. We here give an algorithmic procedure for applying the covariant phase space formalism to field theories with spatial boundaries, from which the term in the Hamiltonian involving B emerges naturally. Our procedure also produces an additional boundary term, which was not present in the original literature and which so far has only appeared implicitly in specific examples, and which is already nonvanishing even in general relativity with sufficiently permissive boundary conditions. The only requirement we impose is that at solutions of the equations of motion the action is stationary modulo future/past boundary terms under arbitrary variations obeying the spatial boundary conditions; from this the symplectic structure and the Hamiltonian for any diffeomorphism that preserves the theory are unambiguously constructed. We show in examples that the Hamiltonian so constructed agrees with previous results. We also show that the Poisson bracket on covariant phase space directly coincides with the Peierls bracket, without any need for non-covariant intermediate steps, and we discuss possible implications for the entropy of dynamical black hole horizons.

Kerr–Bolt black hole entropy and soft hair

Recently it has been speculated that a set of infinitesimal [Formula: see text] diffeomorphisms exist which act nontrivially on the horizon of some black holes such as Kerr and Kerr–Newman black holes.[Formula: see text] Having applied this symmetry in covariant phase space formalism, one can obtain Virasoro charges as surface integrals on the horizon. Kerr–Bolt space–time is well known for its asymptotically topology and has been studied widely in recent years. In this work, we are interested to find conserved charge associated to the Virasoro symmetry of Kerr–Bolt geometry using covariant phase space formalism. We will show right and left central charge are [Formula: see text], respectively. Our results also show good agreement with Kerr space–time in the limiting behavior.