Boundary Terms of the Einstein–Hilbert Action

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 Hamiltonian for dynamic geometry generates the evolution of a spatial region along a vector field. It includes a boundary term which determines both the value of the Hamiltonian and the boundary conditions. The value gives the quasi-local quantities: energy-momentum, angular-momentum and center-of-mass. The boundary term depends not only on the dynamical variables but also on their reference values; the latter determine the ground state (having vanishing quasi-local quantities). For our preferred boundary term for Einstein's GR we propose 4D isometric matching and extremizing the energy to determine the reference metric and connection values.

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $\omega$. Together with Minkowski metric, $e$ will define a metric $g$ on the manifold. Denote $A_S(e)$ as the area of $S$, for a given choice of $e$. The Einstein-Hilbert action $S(e,\omega)$ is defined on $e$ and $\omega$. We will quantize the area of the surface $S$ by integrating $A_S(e)$ against a holonomy operator of a hyperlink $L$, disjoint from $S$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connections $\omega$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between $L$ and $S$. By assigning an irreducible representation of $\mathfrak{su}(2)\times\mathfrak{su}(2)$ to each component of $L$, the area operator gives the total net momentum impact on the surface $S$.

Gravitational waves and magnetic monopoles during inflation with Weitzenböck torsion

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.

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.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

I discuss the relation between arbitrarily high-order theories of gravity and scalar--tensor gravity at the level of the field equations and the action. I show that (2n+4)-order gravity is dynamically equivalent to Brans--Dicke gravity with an interaction potential for the Brans--Dicke field and n further scalar fields. This scalar--tensor action is then conformally equivalent to the Einstein--Hilbert action with n+1 scalar fields. This clarifies the nature and extent of the conformal equivalence between extended gravity theories and general relativity with many scalar fields.

Joel Franklin's textbook `Advanced Mechanics and General Relativity'comprises two partially overlapping, partially complementaryintroductory paths into general relativity at advanced undergraduatelevel.Path I starts with the Lagrangian and Hamiltonian formulations ofNewtonian point particle motion, emphasising the action principle andthe connection between symmetries and conservation laws. The conceptsare then adapted to point particle motion in Minkowski space,introducing Lorentz transformations as symmetries of the action.There follows a focused development of tensor calculus, paralleltransport and curvature, using examples from Newtonian mechanics andspecial relativity, culminating in the field equations of generalrelativity. The Schwarzschild solution is analysed, including adetailed discussion of the tidal forces on a radially infallingobserver. Basics of gravitational radiation are examined, highlightingthe similarities to and differences from electromagneticradiation. The final topics in Path I are equatorial geodesics in Kerrand the motion of a relativistic string in Minkowski space.Path II starts by introducing scalar field theory on Minkowski spaceas a limit of point masses connected by springs, emphasising theaction principle, conservation laws and the energy–momentumtensor. The action principle for electromagnetism is introduced, andthe coupling of electromagnetism to a complex scalar field isdeveloped in a detailed and pedagogical fashion. A free symmetricsecond-rank tensor field on Minkowski space is introduced, and theaction principle of general relativity is recovered from coupling thesecond-rank tensor to its own energy-momentum tensor.Path II then merges with Path I and, supplanted with judicious earlyselections from Path I, can proceed to the Schwarzschild solution.The choice of material in each path is logical and focused. A notableexample in Path I is that Lorentz transformations in Minkowki spaceare introduced efficiently and with a minimum of fuss, as symmetriesof a geodesic action principle. Another example is a similarlyefficient and hands-on introduction of Killing vectors. A consequenceof this focus is that some perhaps traditional material is omitted. For example, Lorentz contraction appears briefly in the incompatibility discussion of special relativity and Newtoniangravity but is not introduced in a more systematic manner. The style is informal and very readable, with detailed explanations,frequent summaries of what has been achieved and pointers to what isabout to follow. There are plenty of examples and some 150 well-chosenexercises, and the author's website hosts relevant Maple samplescripts for tensor manipulations and variational problems. The textconveys an enthusiasm for explaining the subject, frequentlyreminiscent of the Feynman lectures.The presentation emphasises explicit calculations and examples,largely avoiding technical definitions of abstract mathematicalconcepts. The author negotiates the challenge between readability andtechnical accuracy with admirable skill, striking a balance that willbe much appreciated by the target audience. For example, the notion ofspherical symmetry in curved spacetime is introduced informally as ageneralisation of a spherically symmetric vector field in Minkowskispace, and spherically symmetric vacuum and electrovacuum solutionsare then carefully discussed so that a formal definition of sphericalsymmetry is not required. A rare instance that may border onoversimplification is the brief discussion of curvature scalars versusspacetime singularities.Towards the end of the book, the text mentions with increasing explicitness that inserting a gauge condition or an ansatzin an action before varying may not always give the correctequations of motion. It would be useful to be more explicit about thispoint already earlier in the book. In particular, the text refers tothe reparametrisation-invariant square root action of a relativisticpoint particle as being `in proper time parametrisation', while theactual calculations of course impose the proper time condition only inthe equation of motion after the action has been varied.Two presentational conventions surprised me. First, the speed of lightis throughout kept explicitly as c: might advanced undergraduatesappreciate being trusted with geometric units, reinstating c bydimensional analysis when desired? Second, in Minkowski space fieldtheory, the overall coefficient in the action is chosen so that thetime derivative term is negative, with the consequence that theHamiltonian is negative (as explicitly noted in an exercise) and thedefinition of the energy-momentum tensor must include a minus sign toachieve the usual choice T00 > 0. This convention eliminates someminus signs in the computations with the spin two field: does thiscomputational saving outweigh the adjustment awaiting those whocontinue with the topic at graduate level?Overall, Franklin's book is an excellent addition to the literature,and its readability and explicitness will be appreciated by the targetaudience. Should I be teaching an introductory undergraduate class ingeneral relativity in the near future, I would seriously consider this book for the main class text.

The topic of this chapter is the debate between adherents to respectively relationalism and substantivalism in the settings of relativity theory. In special relativity theory one may consistently hold that spacetime intervals are mere attributes of pairs of material objects. Even though spacetime often is treated as an object in relativity theory, there is no reason to think that it is a physical object; we may very well accept it as a mathematical object without concluding that this mathematical object has a counterpart in the physical world. By the same argument one may conclude that even if string theory is a nearly correct theory, the fact that there are 10 (or 11) spacetime dimensions in this theory doesn’t say anything about the number of spacetime dimensions in the real world.When we go to general relativity we have no obvious way to distinguish between object and attribute. Einstein’s equation, the fundamental principle in general relativity, tells us that a function of the spacetime metric is equal to the matter-energy tensor. The equation merely relates two quantities to each other. One is then inclined to ask which object are these quantities attributes of and the answer is that general relativity doesn’t make any distinction between object and attribute. One may say that either we can treat matter-energy distribution as a description of what exists and the function of the spacetime metric an attribute of that object, or reverse and hold that spacetime with its metric is what exists. The structure of general relativity transcends the structure of our common sense thinking in terms of objects with attributes.

The post-Minkowskian expansion of Einstein’s general theory of relativity has received much attention in recent years due to the possibility of harnessing the computational power of modern amplitude calculations in such a classical context. In this brief review, we focus on the post-Minkowskian expansion as applied to the two-body problem in general relativity without spin, and we describe how relativistic quantum field theory can be used to greatly simplify analytical calculations based on the Einstein–Hilbert action. Subtleties related to the extraction of classical physics from such quantum mechanical calculations highlight the care which must be taken when both positive and negative powers of Planck’s constant are at play. In the process of obtaining classical results in both Einstein gravity and supergravity, one learns new aspects of quantum field theory that are obscured when using units in which Planck’s constant is set to unity. The scattering amplitude approach provides a self-contained framework for deriving the two-body scattering valid in all regimes of energy. There is hope that the full impact of amplitude computations in this field may significantly alter the way in which gravitational wave predictions will advance in the coming years.

The generalization of the black hole in three-dimensional spacetime to include an electric charge Q in addition to the mass M and the angular momentum J is given. The field equations are first solved explicitly when Q is small and the general form of the field at large distances is established. The total “hairs ” M, J and Q are exhibited as boundary terms at infinity. It is found that the inner horizon of the rotating uncharged black hole is unstable under the addition of a small electric charge. Next it is shown that when Q = 0 the spinning black hole may be obtained from the one with J = 0 by a Lorentz boost in the ϕ − t plane. This boost is an “illegitimate coordinate transformation ” because it changes the physical parameters of the solution. The extreme black hole appears as the analog of a particle moving with the speed of light. The same boost may be used when Q ̸ = 0 to generate a solution with angular momentum from that with J = 0, 1 although the geometrical meaning of the transformation is much less transparent since in the charged case the black holes are not obtained by identifying points in anti-de Sitter space. The metric is given explicitly in terms of three parameters, ˜ M, ˜ Q and ω which are the “rest mass ” and “rest charge ” and the angular velocity of the boost. These parameters are related to M, J and Q through the solution of an algebraic cubic equation. Altogether, even without angular momentum, the electrically charged 2+1 black hole is somewhat pathological since (i) it exists for arbitrarily negative values of the mass, and (ii) there is no upper bound on the electric charge. 2 1

Which action for brane worlds?

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.

General relativity with boundary terms: collapse without final singularity