Diffeomorphism Invariant Theories Research Articles

Symmetry restoration in transverse diffeomorphism invariant scalar field theories

Symmetry restoration in transverse diffeomorphism invariant scalar field theories

Higher-spin self-dual General Relativity: 6d and 4d pictures, covariant vs. lightcone

We study the higher-spin extension of self-dual General Relativity (GR) with cosmological constant, proposed by Krasnov, Skvortsov and Tran. We show that this theory is actually a gauge-fixing of a 6d diffeomorphism-invariant Abelian theory, living on (4d spacetime)×(2d spinor space) modulo a finite group. On the other hand, we point out that the theory respects the 4d geometry of a self-dual GR solution, with no backreaction from the higher-spin fields. We also present a lightcone ansatz that reduces the covariant fields to one scalar field for each helicity. The field equations governing these scalars have only cubic vertices. We compare our lightcone ansatz to Metsaev’s lightcone formalism. We conclude with a new perspective on the lightcone formalism in (A)dS spacetime: not merely a complication of its Minkowski-space cousin, it has a built-in Lorentz covariance, and is closely related to Vasiliev’s concept of unfolding.

Iyer-Wald ambiguities and gauge covariance of Entropy current in Higher derivative theories of gravity

In [1, 2] [arXiv:2105.06455, arXiv:2206.04538], the authors have been able to argue for an ultra-local version of the second law of black hole mechanics, for arbitrary diffeomorphism invariant theories of gravity non-minimally coupled to matter fields, by constructing an entropy current on the dynamical horizon with manifestly positive divergence. This has been achieved by working in the horizon-adapted coordinate system. In this work, we show that the local entropy production through the divergence of the entropy current is covariant under affine reparametrizations that leave the gauge of horizon-adapted coordinates invariant. We explicitly derive a formula for how the entropy current transforms under such coordinate transformations. This extends the analysis of [3] [arXiv:2204.08447] for arbitrary diffeomorphism invariant theories of gravity non-minimally coupled to matter fields. We also study the Iyer-Wald ambiguities of the covariant phase formalism that generically plague the components of the entropy current.

Unified transverse diffeomorphism invariant field theory for the dark sector

Unified transverse diffeomorphism invariant field theory for the dark sector

Bootstrapping gravity and its extension to metric-affine theories

In this work we study diffeomorphism-invariant metric-affine theories of gravity from the point of view of self-interacting field theories on top of Minkowski spacetime (or other background). We revise how standard metric theories couple to their own energy-momentum tensor, and discuss the generalization of these ideas when torsion and nonmetricity are also present. We review the computation of the corresponding currents through the Hilbert and canonical (Noether) prescriptions, emphasizing the potential ambiguities arising from both. We also provide the extension of this consistent self-coupling procedure to the vielbein formalism, so that fermions can be included in the matter sector. In addition, we clarify some subtle issues regarding previous discussions on the self-coupling problem for metric theories, both General Relativity and its higher derivative generalizations. We also suggest a connection between Lovelock theorem and the ambiguities in the bootstrapping procedure arising from those in the definition of conserved currents.

Extended phase space in general gauge theories

In a recent paper, it was shown that in diffeomorphism-invariant theories, the symplectic vector fields induced by spacetime diffeomorphisms are integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs (φ,Φ), where φ is a Lie algebroid morphism and Φ is a field configuration in the non-extended sense. Starting from this data, we outline a very powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons gauge theories and display its usefulness by re-computing the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories.

Entropy-current for dynamical black holes in Chern-Simons theories of gravity

We construct an entropy current and establish a local version of the classical second law of thermodynamics for dynamical black holes in Chern-Simons (CS) theories of gravity. We work in a chosen set of Gaussian null coordinates and assume the dynamics to be small perturbations around the Killing horizon. In explicit examples of both purely gravitational and mixed gauge gravity CS theories in (2 + 1) and (4 + 1)-dimensions, the entropy current is obtained by studying the off-shell structure of the equations of motion evaluated on the horizon. For the CS theory in (2 + 1) dimensions, we argue that the second law holds to quadratic order in perturbations by considering it as a low energy effective field theory with the leading piece given by Einstein gravity. In all such examples, we show that the construction of entropy current is invariant under the reparameterization of the null horizon coordinates. Finally, extending an existing formalism for diffeomorphism invariant theories, we construct an abstract proof for the linearised second law in arbitrary Chern-Simons theories in any given odd dimensions by studying the off-shell equations of motion. As a check of consistency, we verify that the outcome of this algorithmic proof matches precisely with the results obtained in explicit examples.

Geometrical trinity of unimodular gravity

We construct a Weyl transverse diffeomorphism invariant theory of teleparallel gravity by employing the Weyl compensator formalism. The low-energy dynamics has a single spin two gravition without a scalar degree of freedom. By construction, it is equivalent to unimodular gravity (as well as Einstein’s general relativity with an adjustable cosmological constant) at the non-linear level. Combined with our earlier construction of a Weyl transverse diffeomorphism invariant theory of symmetric teleparallel gravity, unimodular gravity is represented in three alternative ways.

A Stationary Black Hole Must be Axisymmetric in Effective Field Theory

The black hole rigidity theorem asserts that a rotating stationary black hole must be axisymmetric. This theorem holds for General Relativity with suitable matter fields, in four or more dimensions. We show that the theorem can be extended to any diffeomorphism invariant theory of vacuum gravity, assuming that this is interpreted in the sense of effective field theory, with coupling constants determined in terms of a “UV scale”, and that the black hole solution can locally be expanded as a power series in this scale.

Gravitational-wave energy and other fluxes in ghost-free bigravity

One of the key ingredients for making binary waveform predictions in a beyond-general-relativity (GR) theory of gravity is understanding the energy and angular momentum carried by gravitational waves and any other radiated fields. Identifying the appropriate energy functional is unclear in Hassan-Rosen bigravity, a ghost-free theory with one massive and one massless graviton. The difficulty arises from the new degrees of freedom and length scales which are not present in GR, rendering an Isaacson-style averaging calculation ambiguous. In this article we compute the energy carried by gravitational waves in bigravity starting from the action, using the canonical current formalism. The canonical current agrees with other common energy calculations in GR, and is unambiguous (modulo boundary terms), making it a convenient choice for quantifying the energy of gravitational waves in bigravity or any diffeomorphism-invariant theories of gravity. This calculation opens the door for future waveform modeling in bigravity to correctly include backreaction due to emission of gravitational waves.

Non-minimal coupling of scalar and gauge fields with gravity: an entropy current and linearized second law

This work extends the proof of a local version of the linearized second law involving an entropy current with non-negative divergence by including the arbitrary non-minimal coupling of scalar and U(1) gauge fields with gravity. In recent works, the construction of entropy current to prove the linearized second law rested on an important assumption about the possible matter couplings to gravity: the corresponding matter stress tensor was assumed to satisfy the null energy conditions. However, the null energy condition can be violated, even classically, when the non-minimal coupling of matter fields to gravity is considered. Considering small dynamical perturbations around stationary black holes in diffeomorphism invariant theories of gravity with non-minimal coupling to scalar or gauge fields, we prove that an entropy current with non-negative divergence can still be constructed. The additional non-minimal couplings that we have incorporated contribute to the entropy current, which may even survive in the equilibrium limit. We also obtain a spatial current on the horizon apart from the entropy density in out-of-equilibrium situations. We achieve this by using a boost symmetry of the near horizon geometry, which constraints the off-shell structure of a specific component of the equations of motion with newer terms due to the non-minimal couplings. The final expression for the entropy current is U(1) gauge-invariant for gauge fields coupled to gravity. We explicitly check that the entropy current obtained from our abstract arguments is consistent with the expressions already available in the literature for specific model theories involving non-minimal coupling of matter with higher derivative theories of gravity. Finally, we also argue that the physical process version of the first law holds for these theories with arbitrary non-minimal matter couplings.

Pole Skipping in Holographic Theories with Bosonic Fields.

We study pole skipping in holographic conformal field theories dual to diffeomorphism invariant theories containing an arbitrary number of bosonic fields in the large N limit. Defining a weight to organize the bulk equations of motion, a set of general pole skipping conditions are derived. In particular, the frequencies simply follow from general covariance and weight matching. In the presence of higher-spin fields, we find that the imaginary frequency for the highest-weight pole skipping point equals the higher-spin Lyapunov exponent which lies outside of the chaos bound. Without higher-spin fields, we show that the energy density Green's function has its highest-weight pole skipping happening at a location related to the out-of-time-order correlator for arbitrary higher-derivative gravity, with a Lyapunov exponent saturating the chaos bound and a butterfly velocity matching that extracted from a shockwave calculation. We also suggest an explanation for this matching at the metric level by obtaining the on-shell shockwave solution from a regularized limit of the metric perturbation at the skipped pole.

The zeroth law of black hole thermodynamics in arbitrary higher derivative theories of gravity

We consider diffeomorphism invariant theories of gravity with arbitrary higher derivative terms in the Lagrangian as corrections to the leading two derivative theory of Einstein’s general relativity. We construct a proof of the zeroth law of black hole thermo-dynamics in such theories. We assume that a stationary black hole solution in an arbitrary higher derivative theory can be obtained by starting with the corresponding stationary solution in general relativity and correcting it order by order in a perturbative expansion in the coupling constants of the higher derivative Lagrangian. We prove that surface gravity remains constant on its horizon when computed for such stationary black holes, which is the zeroth law. We argue that the constancy of surface gravity on the horizon is related to specific components of the equations of motion in such theories. We further use a specific boost symmetry of the near horizon space-time of the stationary black hole to constrain the off-shell structure of the equations of motion. Our proof for the zeroth law is valid up to arbitrary order in the expansion in the higher derivative couplings.

Noether charge formalism for Weyl invariant theories of gravity

Gravitational theories invariant under transverse diffeomorphisms and Weyl transformations have the same classical solutions as the corresponding fully diffeomorphism invariant theories. However, they solve some of the problems related to the cosmological constant and in principle allow local energy non-conservation. In the present work, we obtain the Noether charge formalism for these theories. We first derive expressions for the Noether currents and charges corresponding to transverse diffeomorphisms and Weyl transformations, showing that the latter vanish identically. We then use these results to obtain an expression for a perturbation of a Hamiltonian corresponding to evolution along a transverse diffeomorphism generator. From this expression, we derive the first law of black hole mechanics, identifying the total energy, the total angular momentum, the Wald entropy, and the contributions of the cosmological constant perturbations and energy non-conservation. Lastly, we extend our formalism to derive the first law of causal diamonds.

Weyl transverse diffeomorphism invariant theory of symmetric teleparallel gravity

We construct a Weyl transverse diffeomorphism invariant theory of symmetric teleparallel gravity by employing the Weyl compensator formalism. The low-energy dynamics has a single spin two gravitation without a scalar degree of freedom. By construction, it is equivalent to the unimodular gravity (as well as the Einstein gravity) at the non-linear level.

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.

Thermodynamics of spacetime and unimodular gravity

In this review, we discuss the emergence of unimodular gravity (or, more precisely, Weyl transverse gravity) from the thermodynamics of spacetime. By analyzing three different ways to obtain gravitational equations of motion by thermodynamic arguments, we show that the results point to unimodular rather than fully diffeomorphism invariant theories and that this is true even for modified gravity. The unimodular character of dynamics is especially evident from the status of cosmological constant and energy-momentum conservation.

Nilpotent Symmetries of a Model of 2D Diffeomorphism Invariant Theory: BRST Approach

Within the framework of the Becchi-Rouet-Stora-Tyutin (BRST) formalism, we discuss the full set of proper BRST and anti-BRST transformations for a 2D diffeomorphism invariant theory which is described by the Lagrangian density of a standard bosonic string. The above (anti-)BRST transformations are off-shell nilpotent and absolutely anticommuting. The latter property is valid on a submanifold of the space of quantum fields where the 2D version of the universal (anti-)BRST invariant Curci-Ferrari (CF) type of restrictions is satisfied. We derive the precise forms of the BRST and anti-BRST invariant Lagrangian densities as well as the exact expressions for the conserved (anti-)BRST and ghost charges. The lucid derivation of the proper anti-BRST symmetry transformations and the emergence of the CF-type restrictions are completely novel results for our present bosonic string which has already been discussed earlier in literature where only the BRST symmetry transformations have been pointed out. We briefly mention the derivation of the CF-type restrictions from the modified version of the Bonora-Tonin superfield approach, too.

Can quantum fluctuations differentiate between standard and unimodular gravity?

We formally prove the existence of a quantization procedure that makes the path integral of a general diffeomorphism-invariant theory of gravity, with fixed total spacetime volume, equivalent to that of its unimodular version. This is achieved by means of a partial gauge fixing of diffeomorphisms together with a careful definition of the unimodular measure. The statement holds also in the presence of matter. As an explicit example, we consider scalar-tensor theories and compute the corresponding logarithmic divergences in both settings. In spite of significant differences in the coupling of the scalar field to gravity, the results are equivalent for all couplings, including non-minimal ones.

Cosmology from pregeometry

We discuss cosmological solutions for a diffeomorphism invariant gauge theory of the non-compact Lorentz group $SO(1,3)$. Besides the gauge bosons our model of pregeometry contains a vector field in the vector representation of $SO(1,3)$ and a scalar singlet. General relativity and variable gravity emerge as effective theories for large distances and times in Planck units. We propose an approximation to the effective action with up to two derivatives. For a suitable range of parameters the universe approaches for large times stable Minkowski space. For late cosmology the model predicts dynamical dark energy and provides for a candidate for dark matter. Early cosmology is characterized by an inflationary epoch. The beginning of the universe in the infinite past is great emptiness, corresponding to an ultraviolet fixed point with the associated quantum scale symmetry. The beginning universe is a vacuum state with vanishing expectation values and finite non-vanishing correlation functions for the fluctuations of all fields. There is no physical big bang singularity.