General relativity with boundary terms: collapse without final singularity

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.

The Einstein-Hilbert action for general relativity is not well posed in terms of the metric $g_{ab}$ as a dynamical variable. There have been many proposals to obtain an well posed action principle for general relativity, e.g., addition of the Gibbons-Hawking-York boundary term to the Einstein-Hilbert action. These boundary terms are dependent on what one fixes on the boundary and in particular on spacetime dimensions as well. Following recent works of Padmanabhan we will introduce two new variables to describe general relativity and the action principle with these new dynamical variables will turn out to be well posed. Then we will connect these dynamical variables and boundary term obtained thereof to existing literature and shall comment on a few properties of Einstein-Hilbert action which might have been unnoticed earlier in the literature. Before concluding with future prospects and discussions, we will perform a general analysis of the boundary term of Einstein-Hilbert action for null surfaces as well.

This paper has three main aims: first, to give a pedagogical introduction to Noether's two theorems and their implications for energy conservation in general relativity, which was a central point of discussion between Hilbert, Klein, Noether and Einstein. Second, it introduces and compares two proposals for gravitational energy and momentum, one of which is very influential in physics: and, so far as I know, neither of the two has been discussed in the philosophical literature. Third, it assesses these proposals in connection with recent philosophical discussions of energy and momentum in general relativity. After briefly reviewing the debates about energy conservation between Hilbert, Klein, Noether and Einstein, I give Noether's two theorems. I show that Einstein's gravitational energy-momentum pseudo-tensor, including its superpotential, is fixed, through Noether's theorem, by the boundary terms in the action. That is, the freedom to add an arbitrary superpotential to the gravitational pseudo-tensor corresponds to the freedom to add boundary terms to the action without changing the equations of motion. This freedom is fixed in the same way for both problems. I also review two proposals for energy and momentum in GR, of which one is a quasi-local alternative to the local expressions, and the other builds on Einstein's local pseudo-tensor approach. I discuss the recent philosophical literature on the conservation of energy and momentum in general relativity, and I assess and compare the two proposals in the light of this literature: especially, in light of questions about diffeomorphism invariance and background-independence.

In this paper, we intend to establish a rigorous mathematical relation between the Gibbons–Hawking–York (GHY) boundary term and the conserved symmetric current which arises due to Noether’s second theorem. The GHY term is necessary to ensure that the equations of motion in general relativity are well-posed, and it is a key ingredient in the thermodynamics of black holes and the derivation of the Bekenstein–Hawking entropy. Noether’s second theorem on the other hand relates the symmetries of the action to conserved quantities of the system. In this paper, we prove that the connection between the GHY term and Noether’s second theorem can be established by considering a diffeomorphism that preserves the boundary condition of the surface integral. Additionally, we also derive the associated conserved current and charge.

The metric-affine variational principle is applied to generate teleparallel and symmetric teleparallel theories of gravity. From the latter is discovered an exceptional class which is consistent with a vanishing affine connection. Based on this remarkable property, this work proposes a simpler geometrical formulation of General Relativity that is oblivious to the affine spacetime structure, thus fundamentally depriving gravity of any inertial character. The resulting theory is described by the Hilbert action purged from the boundary term and is more robustly underpinned by the spin-2 field theory, where an extra symmetry is now manifest, possibly related to the double copy structure of the gravity amplitudes. This construction also provides a novel starting point for modified gravity theories, and the paper presents new and simple generalisations where analytical self-accelerating cosmological solutions arise naturally in the early and late time universe.

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.

In this paper, we find new scalarized black holes by coupling a scalar field with the Gauss-Bonnet invariant in teleparallel gravity. The teleparallel formulation of this theory uses torsion instead of curvature to describe the gravitational interaction, and it turns out that, in this language, the usual Gauss-Bonnet term in four dimensions decays in two distinct boundary terms, the teleparallel Gauss-Bonnet invariants. Both can be coupled individually or in any combination to a scalar field, to obtain a teleparallel Gauss-Bonnet extension of the teleparallel equivalent of general relativity. The theory we study contains the familiar Riemannian Einstein-Gauss-Bonnet gravity theory as a particular limit and offers a natural extension, in which scalarization is triggered by torsion and with new interesting phenomenology. We demonstrate numerically the existence of asymptotically flat scalarized black hole solutions and show that, depending on the choice of coupling of the boundary terms, they can have a distinct behavior compared to the ones known from the usual Einstein-Gauss-Bonnet case. More specifically, nonmonotonicity of the metric functions and the scalar field can be present, a feature that was not observed until now for static scalarized black hole solutions.

Big Bang nucleosynthes describes a significant process in the early universe, responsible for the formation of light nuclei a while after the Big Bang explosion. Its sensitivity to the expansion rate makes BBN an effective probe for testing deviations from standard cosmology, particularly within the framework of modified theories of gravity. In this article, we investigate the evolution of the Universe through Big Bang nucleosynthesis within the framework of f ( T , B , T G , B G ) gravity. Here, T , B , T G and B G correspond to the torsion, boundary term, teleparallel Gauss-Bonnet term, and Gauss-Bonnet boundary term, respectively. To explore the deviation from the teleparallel equivalent of general relativity, we apply the transformation f ( T , B , T G , B G ) = − T + F ( T , B , T G , B G ) . For an in-depth analysis of the Big Bang nucleosynthesis, we choose five distinct models from gravity and compared the results with the observational limits on | Δ T f T f | to examine restrictions on the free parameters associated with these models. Our outcomes indicate that f ( T , B , T G , B G ) gravity can agree with the constraint of big bang nucleosynthesis, and hence build a valid view of the universe. Moreover, we take the limiting cases of the assumed theory, that is, f ( T , B ) and f ( T ) theories, which also ensure consistency with BBN data.

Constraining [formula omitted] gravity by dynamical system analysis

A proof of the positive energy theorem of general relativity is given. In this proof the gravitational Lagrangian is identified with that of Lau and is equivalent to the teleparallel Lagrangian modulo, a boundary term. The approach adopted in this proof uses the two-spinor method and the extended Witten identities and then combines the Brown-York and the Nester-Witten approaches. At the same time the proof is extended to the case involving the contribution of angular momentum by choosing a special shift vector.

Ashtekar's complex variables in general relativity and its teleparallelism equivalent

So far, general relativity has been viewed from the four-dimensional Lagrangian perspective. This chapter introduces the (3+1)-dimensional Hamiltonian formalism, starting with the ADM form of the metric and extrinsic curvature. The Hamiltonian form of the action is served, and the nature of the constraints—and, more generally, of constraints and gauge invariance in Hamiltonian systems—is discussed. The formalism is used to count the physical degrees of freedom of the gravitational field. The chapter ends with a discussion of boundary terms and the ADM energy.

A well-defined variational principle for gravitational actions typically requires to cancel boundary terms produced by the variation of the bulk action with a suitable set of boundary counterterms. This can be achieved by carefully balancing the coefficients multiplying the bulk operators with those multiplying the boundary ones. A typical example of this construction is the Gibbons-Hawking-York boundary action that needs to be added to the Einstein-Hilbert one in order to have a well-defined metric variation for General Relativity with Dirichlet boundary conditions. Quantum fluctuations of matter fields lead to a renormalization of these coefficients which may or may not preserve this balance. Indeed, already at the level of General Relativity, the resilience of the matching between bulk and boundary constants is far from obvious and it is anyway incomplete given that matter generically induces quadratic curvature operators. We investigate here the resilience of the matching of higher-order couplings upon renormalization by a non-minimally coupled scalar field and show that a problem is present. Even though we do not completely solve the latter, we show that it can be greatly ameliorated by a wise splitting between dynamical and topological contributions. Doing so, we find that the bulk-boundary matching is preserved up to a universal term (present for any Weyl invariant matter field content), whose nature and possible cancellation we shall discuss in the end.

General Relativity allows for a cosmological constant ($\Lambda$) which has inspired models of cosmic Inflation and Dark Energy. We show instead that $r_\Lambda = \sqrt{3/\Lambda}$ corresponds to an event horizon: a causal boundary term in the action. Our Universe is expanding inside its Schwarzschild radius $r_S=\rL=2GM$, which could have originated from a uniform free falling cloud of mass $M$ that collapsed as a Black Hole (BH) 25 Gyrs ago. Such a BH Universe allows for large-scale structure formation without the need of Inflation or Dark Energy.

Using the Gödel metric, we obtain some relevant solutions compatible with spatiotemporal jumps for the geodesic equations, by using an extension of General Relativity with nonzero boundary terms, which are described on an extended manifold generated by the connections delta Gamma ^{mu }_{alpha beta } = b,U^{mu },g_{alpha beta }. These terms are given by a flow of velocities with components U^{nu }: 3,b^2,nabla _{nu }U^{nu }=g^{alpha beta }, delta R_{alpha beta } = lambda left[ sleft( x^{alpha }right) right] ,g^{alpha beta }, delta g_{alpha beta } in the varied Einstein–Hilbert action. The solutions are valid for an arbitrary equation of state with ordinary matter: Omega =P/(c^2,rho ) = frac{left( frac{omega }{c}right) ^2-lambda (s)}{left( frac{omega }{c}right) ^{2}+lambda (s)}.