Boundary terms, variational principles, and higher derivative modified gravity

The Palatini formalism, which assumes the metric and the affine connection as independent variables, is developed for gravitational theories in flat geometries. We focus on two particularly interesting scenarios. First, we fix the connection to be metric compatible, as done in the usual teleparallel theories, but we follow a completely covariant approach by imposing the constraints with suitable Lagrange multipliers. For a general quadratic theory we show how torsion naturally propagates and we reproduce the Teleparallel Equivalent of General Relativity as a particular quadratic action that features an additional Lorentz symmetry. We then study the much less explored theories formulated in a geometry with neither curvature nor torsion, so that all the geometrical information is encoded in the non-metricity. We discuss how this geometrical framework leads to a purely inertial connection that can thus be completely removed by a coordinate gauge choice, the coincident gauge. From the quadratic theory we recover a simpler formulation of General Relativity in the form of the Einstein action, which enjoys an enhanced symmetry that reduces to a second linearised diffeomorphism at linear order. More general theories in both geometries can be formulated consistently by taking into account the inertial connection and the associated additional degrees of freedom. As immediate applications, the new cosmological equations and their Newtonian limit are considered, where the role of the lapse in the consistency of the equations is clarified, and the Schwarzschild black hole entropy is computed by evaluating the corresponding Euclidean action. We discuss how the boundary terms in the usual formulation of General Relativity are related to different choices of coordinates in its coincident version and show that in isotropic coordinates the Euclidean action is finite without the need to introduce boundary or normalisation terms. Finally, we discuss the double-copy structure of the gravity amplitudes and the bootstrapping of gravity within the framework of coincident General Relativity.

In this article we will analyze the possibility of a nontrivial central\nextension of the Poisson algebra of the diffeomorphism generators, which\nrespect certain boundary conditions on the black hole bifurcation. The origin\nof a possible central extension in the algebra is due to the existence of\nboundary terms in the in the canonical generators. The existence of such\nboundary terms depend on the exact boundary conditions one takes. We will check\ntwo possible boundary conditions i.e. fixed bolt metric and fixed surface\ngravity. In the case of fixed metric the the action acquires a boundary term\nassociated to the bifurcation but this is canceled in the Legendre\ntransformation and so absent in the canonical generator and so in this case the\npossibility of a nontrivial central extension is ruled out. In the case of\nfixed surface gravity the boundary term in the action is absent but present in\nthe Hamiltonian. Also in this case we will see that there is no nontrivial\ncentral extension, also if there exist a boundary term in the generator.\n

In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. By construing change as essential time dependence (lack of a time-like Killing vector field), one can find change locally in vacuum GR in the Hamiltonian formulation just where it should be. But what if spinors are present? This paper is motivated by the tendency in space-time philosophy tends to slight fermionic/spinorial matter, the tendency in Hamiltonian GR to misplace changes of time coordinate, and the tendency in treatments of the Einstein-Dirac equation to include a gratuitous local Lorentz gauge symmetry along with the physically significant coordinate freedom. Spatial dependence is dropped in most of the paper, both restricting the physical situation and largely fixing the spatial coordinates. In the interest of including all and only the coordinate freedom, the Einstein-Dirac equation is investigated using the Schwinger time gauge and Kibble-Deser symmetric triad condition are employed as a 3+1 version of the DeWitt-Ogievetsky-Polubarinov nonlinear group realization formalism that dispenses with a tetrad and local Lorentz gauge freedom. Change is the lack of a time-like stronger-than-Killing field for which the Lie derivative of the metric-spinor complex vanishes. An appropriate 3+1-friendly form of the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first class-constraints, is shown to change the canonical Lagrangian by a total derivative, implying the preservation of Hamilton’s equations. Given the essential presence of second-class constraints with spinors and their lack of resemblance to a gauge theory (unlike, say, massive photons), it is useful to have an explicit physically interesting example. This gauge generator implements changes of time coordinate for solutions of the equations of motion, showing that the gauge generator makes sense even with spinors.

Electromagnetic duality of Maxwell theory is a symmetry of equations but not of the action. The usual application of the “complexity = action” conjecture would thus lose this duality. It was recently proposed in arXivid:1901.00014 that the duality can be restored by adding some appropriate boundary term, at the price of introducing the mixed boundary condition in the variation principle. We present universal such a term in both first-order and second-order formalism for a general theory of a minimally-coupled Maxwell field. The first-order formalism has the advantage that the variation principle involves only the Dirichlet boundary condition. Including this term, we compute the on-shell actions in the Wheeler-De Witt patch and find that the duality is preserved in these actions for a variety of theories, including Einstein-Maxwell, Einstein-Maxwell-Dilaton, Einstein-Born-Infeld and Einstein-Horndeski-Maxwell theories.

Generalized variational principles with relaxed continuity requirements for certain nonlinear problems, with an application to nonlinear elasticity

The extended scalar-tensor and vector-tensor theories admit black hole solutions with the nontrivial profiles of the scalar and vector fields, respectively. The disformal transformation maps a solution in a class of the scalar-tensor or vector-tensor theories to that in another class, and hence it can be a useful tool to construct a new nontrivial solution from the known one. First, we investigate how the stationary and axisymmetric solutions in the vector-tensor theories without and with the $U(1)$ gauge symmetry are disformally transformed. We start from a stationary and axisymmetric solution satisfying the circularity conditions, and show that in both the cases the metric of the disformed solution in general does not satisfy the circularity conditions. Using the fact that a solution in a class of the vector-tensor theories with the vanishing field strength is mapped to that in a class of the shift-symmetric scalar-tensor theories, we derive the disformed stationary and axisymmetric solutions in a class of these theories, and show that the metric of the disformed solutions does not satisfy the circularity conditions if the scalar field depends on the time or azimuthal coordinate. We also confirm that in the scalar-tensor theories without the shift symmetry, the disformed stationary and axisymmetric solutions satisfy the circularity conditions. Second, we investigate the disformal transformations of the stationary and axisymmetric black hole solutions in the generalized Proca theory with the nonminimal coupling to the Einstein tensor, the shift-symmetric scalar-tensor theory with the nonminimal derivative coupling to the Einstein tensor, the Einstein-Maxwell theory, and the Einstein-conformally coupled scalar field theory. We show that the disformal transformations modify the causal properties of the spacetime.

A definite form for the boundary term that produces the finiteness of both the conserved quantities and Euclidean action for any Lovelock gravity with AdS asymptotics is presented. This prescription merely tells even from odd bulk dimensions, regardless the particular theory considered, what is valid even for Einstein-Hilbert and Einstein-Gauss-Bonnet AdS gravity. The boundary term is a given polynomial of the boundary extrinsic and intrinsic curvatures (also referred to as Kounterterms series). Only the coupling constant of the boundary term changes accordingly, such that it always preserves a well-posed variational principle for boundary conditions suitable for asymptotically AdS spaces. The background-independent conserved charges associated to asymptotic symmetries are found. In odd bulk dimensions, this regularization produces a generalized formula for the vacuum energy in Lovelock AdS gravity. The standard entropy for asymptotically AdS black holes is recovered directly from the regularization of the Euclidean action, and not only from the first law of thermodynamics associated to the conserved quantities.

For the field or Eulerian description of heat conduction, a method is discussed associated with description of thermal fields by a variational principle involving suitably constructed potentials rather than original physical variables. The considered processes are: simple hyperbolic heat transfer and coupled parabolic transfer of heat, mass and electric charge. By using various gradient or nongradient representations of original physical fields in terms of potentials, which are quantities of similar sort as those which Clebsch used in his representation of hydrodynamic velocity, suitable action-type criteria can be found, and corresponding Lagrangian and Hamiltonian formalisms can be developed. Symmetry principles can also be considered and components of energy-momentum tensor can be evaluated for a gauged lagrangian. The limiting reversible case appears as a suitable reference frame. The results suggest that thermodynamic irreversibility does not change neither kinetic potential nor action functional; it only complicates potential representations of physical fields in comparison with those describing the reversible evolution.

It is widely accepted that a variational principle cannot be constructed for an arbitrary differential equation; a rigorous mathematical condition shows which equations can have a variational formulation. On the other hand, the importance for variational principles in various fields of physics resulted in several methods to circumvent this condition and to construct another type of variational principles for any differential equation. In this paper the common origin of the considered methods is investigated, and a generalized Hamiltonian formalism is formulated. Additionally, constructive algorithms are given by different methods to construct variational principles. Simple examples are presented to make construction methods more transparent: several Lagrangians are constructed for the different forms of the Maxwell equations and for the extended heat conduction equation.

Previous explorations of the Asymptotic Safety scenario in Quantum Einstein Gravity (QEG) by means of the effective average action and its associated functional renormalization group (RG) equation assumed spacetime manifolds which have no boundaries. Here we take a first step towards a generalization for non-trivial boundaries, restricting ourselves to action functionals which are at most of second order in the derivatives acting on the metric. We analyze two examples of truncated actions with running boundary terms: full fledged QEG within the single-metric Einstein-Hilbert (EH) truncation, augmented by a scale dependent Gibbons-Hawking (GH) surface term, and a bi-metric truncation for gravity coupled to scalar matter fields. The latter contains 17 running couplings, related to both bulk and boundary terms, whose beta-functions are computed in the induced gravity approximation. We find that the bulk and the boundary Newton constant, pertaining to the EH and GH term, respectively, show opposite RG running; proposing a scale dependent variant of the ADM mass we argue that the running of both couplings is consistent with gravitational anti-screening. We describe a simple device for counting the number of field modes integrated out between the infrared cutoff scale and the ultraviolet. This method makes it manifest that, in an asymptotically safe theory, there are effectively no field modes integrated out while the RG trajectory stays in the scaling regime of the underlying fixed point. As an application, we investigate how the semiclassical theory of Black Hole Thermodynamics gets modified by quantum gravity effects and compare the new picture to older work on `RG-improved black holes' which incorporated the running of the bulk Newton constant only. We find, for instance, that the black hole's entropy vanishes and its specific heat capacity turns positive at Planckian scales.

We obtain the classical holographic relation for the general Lovelock gravity and decompose the full Lagrangian into the bulk term and the surface term, expressed as a total derivative $\partial_\mu J^\mu$. By classical holographic relation, we mean that $J^\mu$ is determined completely by the bulk term. We find that the bulk term is not degenerate, or first-order in this foliation-independent approach. We then consider the Arnowitt-Deser-Misner (ADM) formalism where the foliation coordinate $w$ is treated as special. We obtain the classical holographic-degenerate relation with the first-order bulk term that does not involve higher than one derivative of $w$. For Einstein gravity, the two approaches lead to the same bulk term, but different ones for higher-order Lovelock gravities. The classical holographic-degenerate formulation in the ADM approach allows us to consider alternative boundary conditions in the variation principle with different Myers terms. We show in the semiclassical approximation that the black hole entropy in all cases is the same as the one obtained under the standard Dirichlet boundary condition. We also generalize the formalism to general $f(L_{\rm Lovelock}^{(k)})$-gravity.

Quantum corrections are studied for a charged black hole in a two-dimensional (2D) model obtained by spherisymmetric reduction of the 4D Einstein-Maxwell theory. The classical (tree-level) thermodynamics is reformulated in the framework of the off-shell approach, considering systems at arbitrary temperature. This implies a conical singularity at the horizon and modifies the gravitational action by terms defined on the horizon. A consistent variational procedure for the action functional is formulated. It is shown that the free energy reaches an extremum on the regular manifold with $T={T}_{H}$. The one-loop contribution to the action in the Liouville-Polyakov from is reexamined. All the boundary terms are taken into account and the dependence on the state of the quantum field is established. The modification of the Liouville-Polyakov term for a 2D space with a conical defect is derived. The back reaction of the Hawking radiation on the geometry is studied and the quantum-corrected black hole metric is calculated perturbatively. Within the off-shell approach the one-loop thermodynamical quantities, energy, and entropy, are found. They are shown to contain a part due to hot gas surrounding the black hole and a part due to the hole itself. It is noted that the contribution of the hot gas can be eliminated by appropriate choice of the (generally, nonflat) reference geometry. The deviation of the "entropy - horizon area" relation for the quantum-corrected black hole from the classical law is discovered and possible physical consequences are discussed.

In this work, the cosmological inflationary parameters in the correspondence of teleparallel gravity for the scalar–tensor theory are investigated. After the review of [Formula: see text] and [Formula: see text] gravity cosmology, we use the slow-roll approximations to study the behavior of the inflationary parameters namely the spectral index [Formula: see text] and tensor-to-scalar ratio [Formula: see text], and a comparison with observational data for different paradigmatic [Formula: see text] gravity models such as exponential, Linder and power-law models is considered. We also consider the boundary term [Formula: see text] associated with these three models. The obtained behavior of the parameters under consideration shows that it is possible to constrain [Formula: see text] and [Formula: see text] models based on observational data.

In this paper, special features of the variational principle in Hamiltonian mechanics (the problem of covariant formulation and boundary conditions) are analyzed; the difference between the variational principles in the Lagrangian and Hamiltonian formalisms is emphasized. Moreover, four version of the covariant variational principle are suggested. 1. Determining Hamiltonian mechanics requires specifying a function (Hamiltonian) on an even-dimensional manifold M 2 n endowed with a symplectic struc

Although the Wald entropy is commonly suitable for the first law of black hole thermodynamics, whether it satisfies the second law has not been examined sufficiently. For any high-order curvature gravity, the Wald entropy has been shown not to obey the linearized second law, and the general expression of entropy of black holes that always satisfies the linearized second law is further obtained. For gravity with nonminimal coupling matter fields, however, whether the Wald entropy of black holes satisfies the second law has not been studied enough. Recently, a general second-order scalar-tensor gravitational theory has been proposed. The Lagrangian of the gravitational theory is a linear combination of four components, and the Wald entropy can also be written as four parts. We should investigate each part of the expression separately to examine whether the Wald entropy satisfies the linearized second law. For the first and fourth parts contributed by two nonminimal coupling terms in Lagrangian contained in Horndeski gravity and Gauss-Bonnet gravity, the entropy of black holes is not expressed as the Wald entropy because the entropy in Gauss-Bonnet gravity is Jacobson-Myers entropy. The third part contributed by the Lagrangian in Einstein-Hilbert action is Bekenstein-Hawking entropy, which obeys the second law automatically. Therefore, to obtain the entropy of black holes that meets the linearized second law, we only need to study the second part of the Wald entropy. According to the null energy condition and the Raychaudhuri equation, one can show that the second part of the Wald entropy with correction terms will increase monotonically constrained by the null energy condition during the perturbation process. The second part of the Wald entropy should be modified to satisfy the linearized second law, and the expression of the entropy of black holes, which always obeys the linearized second law, is obtained in the general second-order scalar-tensor gravity.