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Generalized Kerr-Schild nilpotent deformations of an arbitrary background spacetime $\overline{g}_{\m\n}(x)$ are considered. Those are of the form $\overline{g}_{\m\n}+l_\m l_\n$ with $l^2=0$. The relationship between the Ricci tensor of the background metric and the Ricci tensor of the deformed metric is found exactly. It consists of two terms, one is essentially the Fierz-Pauli operator, and the other is new. When the background is flat, the Kerr-Schild family is recovered. Novel examples for more general backgrounds (even including some simple sources) are discussed., Comment: Latex 13 pages,comments added, title changed

Linear superposition of gravitational fields is shown to be possible for a large class of spacetimes, in some specific coordinates. Explicit examples are presented., Comment: 11 pages, LaTeX, more details given

Gravitational plane waves (when Ricci flat) belong to the VSI family. The achronym VSI stands for vanishing scalar invariants, meaning that all scalar invariants built out of Riemann tensor and its derivatives vanish, although the Riemann tensor itself does not. In the particular case of plane waves many interesting phenomena have been uncovered for strings propagating in this background. Here we comment on gravitons propagating in such a spacetime, which itself presumably consists of an Avogadro number of such gravitons., Comment: 17 pages, LaTeX

It is well-known that in unimodular gravity the cosmological constant is not sourced by a constant energy density, but rather appears as some sort of integration constant. In this work we try to flesh this out by studying in some detail a couple of examples, one from cosmology and the other from gravitational collapse., Comment: LaTeX, 17 pages, cosmetic changes

In this paper we study the behavior of a scalar field coupled to gravitons on the Janis-Newman-Winicour background, which somewhat interpolates between Minkowski and Schwarzschild space-times. The most important physical effect we find is that there is a 17-dimensional position-dependent mass matrix YABpxq which happens to be non-diagonal in the basis in which the kinetic energy term is diagonal. There is a different basis with a mixing between the scalar field and the graviton trace in which the mass matrix is diagonal, but this basis fails to diagonalize the kinetic energy piece. This is at variance with what happens in the Standard Model with the quark mixing, and is of course due to the fact that the mass matrix here is position dependent and thus it does not commute with the kinetic energy operator, so that both operators cannot be diagonalized simultaneously., Comment: 22 pages, LaTeX

When implementing a non-linear constraint in quantum field theory by means of a Lagrange multiplier, $\l(x)$, it is often the case that quantum dynamics induce quadratic and even higher order terms in $\l(x)$, which then does not enforce the constraint anymore. This is illustrated in the case of Unimodular Gravity, where the constraint is that the metric tensor has to be unimodular ($g(x)\equiv \det\,g_{\m\n}(x)=-1$)., Comment: 12 pages, 2 figures. A change in the Acknowledgemwnts section

Unimodularity can be implemented in different ways. In this paper we consider only the formulation of Unimodular Gravity in which the unimodular metric is obtained out of an unrestricted one as $\g_{\m\n}=|g|^{-{1\over n}} g_{\m\n}$. This procedure induces an extra Weyl symmetry. Some physical implications of this symmetry on the conserved currents are discussed. Finally the results are illustrated for the Painlev\'e-Gullstrand extension of Schwarzschild spacetime., Comment: LateX, 19 pages, figure included, revised

The effect of modifying General Relativity with the addition of some higher dimensional operators, generalizations of the Goroff-Sagnotti operator, is discussed. We determine in particular, the general solution of the classical equations of motion, assuming it to be spherically symmetric, not necessarily static. Even in the non-spherically symmetric case, we present a necessary condition for an algebraically generic spacetime to solve the corresponding equations of motion. Some examples of an application of said condition are explicitly worked out., Comment: 12 pages

This chapter provides an introduction to Unimodular Gravity both at the classical and quantum level, discussing the r\^ole it might play in the partial solution of the Cosmological Constant problem. The main objective of this work is to serve as a conceptual introduction to Unimodular gravity without disregarding computational detail. In that sense, techniques used at the research level are presented and applied in detail., Comment: 40 pages, Invited chapter for "Handbook of Quantum Gravity" (Eds. C. Bambi, L. Modesto, and I. L. Shapiro, Springer 2023). Minor typos corrected

We have studied a lagrangian in which the Einstein-Hilbert term is deformed by the Weyl cube operator, which is the lowest-dimension operator that is non-vanishing on shell and appears as a two-loop counterterm. There is a tension between the Schwarzschild de Sitter (SdS) spacetime and this operator, which we study in some detail., Comment: 29 pages, expanded; references added

In this paper some techniques useful to perform quantum field theory computations in a covariant manner are reviewed. In particular the background field gauge, the zeta function regularization and the heat kernel approach are highlighted. Some detailed calculations of the Schwinger-de Witt coefficients of the small proper time expansion of the heat kernel are also repeated in detail. This work reports lectures given by Enrique \'Alvarez at the IFT-UAM-CSIC in Madrid., Comment: 96 pages, LaTeX, lecture notes given by Enrique \'Alvarez at the IFT-UAM-CSIC in Madrid. New section added. References added. Typos corrected

The divergences coming from a particular sector of gravitational fluctuations around a generic background in general theories of quadratic gravity are analyzed. They can be summarized in a particular type of scalar model, whose properties are analyzed., Comment: 23 pages. No figures. References and comments added

It is claimed that in the unimodular gravity framework the observational fact of exponential expansion of the universe cannot be taken as evidence for the presence for a cosmological constant or similar quintessence., Comment: 17 pages

The full one-loop (scalar) effective action is computed for both hyperbolic and elliptic spacetimes., Comment: LaTeX, 13 pages, references added

We discuss the weight of vacuum energy in various contexts. First, we compute the vacuum energy for flat spacetimes of the form $\mathbb{T}^3 \times \mathbb{R}$, where $\mathbb{T}^3$ stands for a general 3-torus. We discover a quite simple relationship between energy at radius $R$ and energy at radius $\frac{l_s^2}{ R}$. Then we consider quantum gravity effects in the vacuum energy of a scalar field in $\mathbb{M}_3 \times S^1$ where $\mathbb{M}_3$ is a general curved spacetime, and the circle $S^1$ refers to a spacelike coordinate. We compute it for General Relativity and generic transverse {\em TDiff} theories. In the particular case of Unimodular Gravity vacuum energy does not gravitate., Comment: 32 pages. Minor corrections

This paper is concerned with the structural stability of spherical horizons. By this we mean stability with respect to variations of the second member of the corresponding differential equations, corresponding to the inclusion of the contribution of operators quadratic in curvature. This we do both in the usual second order approach (in which the independent variable is the spacetime metric) and in the first order one (where the independent variables are the spacetime metric and the connection field). In second order, it is claimed that the generic solution in the asymptotic regime (large radius) can be matched not only with the usual solutions with horizons (like Schwarzschild-de Sitter) but also with a more generic (in the sense that it depends on more arbitrary parameters) horizonless family of solutions. It is however remarkable that these horizonless solutions are absent in the {\em restricted} (that is, when the background connection is the metric one) first order approach., Comment: 42 pages, references added

These are notes for a graduate course in quantum gravity planned at IFT-UAM/CSIC for the spring term of 2020, but delayed for an indefinite period of time (although a shortened version was given in the Escuela de Fisica (Institute of Physics) in the University of Costa Rica.). The aim of the course was to highlight the most important conceptual problems in the field and to summarize some of the most imaginative solutions to them that have been proposed in the literature. The course can best be characterized as idiosyncratic rather than encyclopedic., Comment: 73 pages; 7 figures. Lectures on quantum gravity. Small changes; references added

A new class of conformal field theories is presented, where the background gravitational field is conformally flat. Conformally flat (CF) spacetimes enjoy conformal properties quite similar to the ones of flat spacetime. The conformal isometry group is of maximal rank and the conformal Killing vectors in conformally flat coordinates are {\em exactly} the same as the ones of flat spacetime. In this work, a new concept of distance is introduced, the {\em conformal distance}, which transforms covariantly under all conformal isometries of the CF space. It is shown that precisely for CF spacetimes, an adequate power of the said conformal distance is a solution of the non-minimal d'Alembert equation., Comment: Minor changes and some clarifications added. Published version. 16 pages

The generalization of scale invariance when gravitational effects are considered is Weyl invariance, namely, invariance under (global or local) rescalings of the metric. In this work, we discuss in some details the implications of the fact that the value of the anomaly for the global Weyl invariant coupling of scalar fields to gravity is sensitive to the dynamics (or absence thereof) of the gravitational field., Comment: 24 pages, streamlined, some errata fixed

The most general lagrangian describing spin 2 particles in flat spacetime and containing operators up to (mass) dimension 6 is carefully analyzed, determining the precise conditions for it to be invariant under linearized (transverse) diffeomorphisms, linearized Weyl rescalings, and conformal transformations., Comment: 24 pages

A definition of quasi-local energy in a gravitational field based upon its embedding into flat space is discussed. The outcome is not satisfactory from many points of view., Comment: 12 pages

A ghost free massive deformation of unimodular gravity (UG), in the spirit of {\em mimetic massive gravity}, is shown to exist. This construction avoids the no-go theorem for a Fierz-Pauli type of mass term in UG by giving up on Lorentz invariance. In our framework, the mimetic degree of freedom vanishes on-shell., Comment: Several clarifications added. Published version. 11 pages

Based on the quasi-local energy definition of Brown and York, we compute the integral of the trace of the extrinsic curvature over a codimension-2 hypersurface. In particular, we study the difference between the uncompactified Minkowski spacetime and the toroidal Kaluza-Klein compactification. For the latter, we find that this quantity interpolates between zero and the value for the uncompactified spacetime, as the size of the compact dimension increases. Thus, the quasi-local energy is able to discriminate between the two spacetimes., Comment: 11 pages, 2 figures

We have recently undergone an analysis of gravitational theories as defined in first order formalism, where the metric and the connection are treated as independent fields. The physical meaning of the connection field has historically been somewhat elusive. In this paper, a complete spin analysis of the torsionless connection field is performed, and its consequences are explored. The main properties of a hypothetical consistent truncation of the theory are discussed as well., Comment: Appendix added. References added

We consider the most general action for gravity which is quadratic in curvature. In this case first order and second order formalisms are not equivalent. This framework is a good candidate for a unitary and renormalizable theory of the gravitational field; in particular, there are no propagators falling down faster than $\tfrac{1}{p^2}$. The UV regime is in a conformal invariant phase; only when Weyl invariance is broken through the coupling to matter can an Einstein-Hilbert term (and its corresponding Planck mass scale) be generated., Comment: 4 pages, presented by E.A at the 18th Lomonosov conference

The unimodular version of the ghost-free higher derivative gravity is obtained. It is the unimodular reduction of some particular lagrangians quadratic in curvature., Comment: 11 pages. LaTeX, some mistakes corrected; reference added. Verdio to be published in PRD

One-loop counterterms are computed in the first order formalism for the Einstein-Hilbert action with a minimally coupled scalar field using the background field method and the heat kernel technique. The {\em off-shell} divergent piece in the harmonic gauge is {\em exactly} the same as the one first found by 't Hooft and Veltman., Comment: Some misprints have been correct

We consider the most general action for gravity which is quadratic in curvature. In this case first order and second order formalisms are not equivalent. This framework is a good candidate for a unitary and renormalizable theory of the gravitational field; in particular, there are no propagators falling down faster than $\tfrac{1}{p^2}$. Calculations are somewhat involved when all monomials are considered, but we have intended to laid down the general case. The interaction between external sources is analyzed; this interaction is conveyed mainly by the three-index connection field. The theory as it stands, is in the conformal invariant phase; only when Weyl invariance is broken through the coupling to matter can an Einstein-Hilbert term (and its corresponding Planck mass scale) be generated by quantum corrections., Comment: 50 pages, additions and new section. Version to be published in JCAP; Spotted errata corrected

The on shell equivalence of first order and second order formalisms for the Einstein-Hilbert action does not hold for those actions quadratic in curvature. It would seem that by considering the connection and the metric as independent dynamical variables, there are no quartic propagators for any dynamical variable. This suggests that it is possible to get both renormalizability and unitarity along these lines. We have studied a particular instance of those theories, namely Weyl gravity. Although the ground state of this system is difficult to analyze, we have been able to study the physical effects of some external sources., Comment: 14 pages

The maximally helicity violating (MHV) tree level scattering amplitudes involving three, four or five gravitons are worked out in Unimodular Gravity. They are found to coincide with the corresponding amplitudes in General Relativity. This a remarkable result, insofar as both the propagators and the vertices are quite different in both theories., Comment: 20 pages, 5 figures

A theory with a mass scale and yet Weyl invariant is presented. The theory is not invariant under all diffeomorphisms but only under transverse ones. This is the reason why Weyl invariance does not imply scale invariance in a free falling frame. Physical implications of this framework are discussed., Comment: 3 pages

The symmetries of Unimodular Gravity are clarified somewhat., Comment: 4 pages, v2: acknowledgments corrected

When quantizing Conformal Dilaton Gravity there is a conformal anomaly which starts at two loop order. This anomaly stems from evanescent operators on the divergent parts of the effective action. The general form of the finite counterterm which is necessary in order to insure cancellation of the Weyl anomaly to every order in perturbation theory has been determined using only conformal invariance . Those finite counterterms do not have any inverse power of any mass scale in front of them (precisely because of conformal invariance) and then they are not negligible in the low energy deep infrared limit. The general form of the ensuing modifications to the scalar field equation of motion has been determined and some physical consequences extracted., Comment: 9 pages, 4 figures

First order lagrangians for the Weyl invariant formulation of Unimodular Gravity are proposed. Several alternatives are examined; in some of them first and second order are equivalent in a certain gauge only., Comment: 4 pages

The problem of the cosmological constant appears in a new light in Unimodular Gravity. In particular, the zero momentum piece of the potential (that is, the constant piece independent of the matter fields) does not automatically produce a cosmological constant proportional to it. The aim of this paper is to give some details on a calculation showing that quantum corrections do not renormalize the classical value of this observable., Comment: 34 pages

We examine some Weyl invariant cosmological models in the framework of generalized dilaton gravity, in which the action is made of a set of $N$ conformally coupled scalar fields. It will be shown that when the FRW ansatz for the spacetime metric is assumed, the Ward identity for conformal invariance guarantees that the gravitational equations hold whenever the scalar fields EM do so. It follows that any scale factor can solve the theory provided a non-trivial profile for a dilaton field. In particular, accelerated expansion is a natural solution to the full set of equations., Comment: 8 pages, explanation extended, a particular example has been added. This version matches the one to be published in JCAP

The quantum dynamics of the gravitational field non-minimally coupled to an (also dynamical) scalar field is studied in the {\em broken phase}. For a particular value of the coupling the system is classically conformal, and can actually be understood as the group averaging of Einstein-Hilbert's action under conformal transformations. Conformal invariance implies a simple Ward identity asserting that the trace of the equation of motion for the graviton is the equation of motion of the scalar field. We perform an explicit one-loop computation to show that the DeWitt effective action is not UV divergent {\em on shell} and to find that the Weyl symmetry Ward identity is preserved {\em on shell} at that level. We also discuss the fate of this Ward identity at the two-loop level --under the assumption that the two-loop UV divergent part of the effective action can be retrieved from the Goroff-Sagnotti counterterm-- and show that its preservation in the renormalized theory requires the introduction of counterterms which exhibit a logarithmic dependence on the dilaton field., Comment: LateX, 50 pages. Several points clarified; references added. New section on Weyl invariant renormalisation added

A Weyl invariant extension of Einstein gravity is studied. It simply consists in the group averaging of Einstein's action under Weyl transformations. Contradicting cherished beliefs, a conformal anomaly is found in the trace of the equations of motion if diffeomorphism invariance is to be a symmetry of the quantum theory. This anomaly however vanishes on shell, which signals the existence of another gauge in which the anomaly is absent. It can be made also to vanish off-shell at the price of losing diffeomorphism invariance, Comment: 6 pages, to be published in the proceedings of "Windows on the universe" (Rencontres du Vietnam, 2013)

Some cosmological solutions of the unimodular theory of gravity are studied. These solutions can always be mapped to solutions of Einstein's general relativity in an appropiate frame. Constant scalar potentials however do not gravitate. Some physical consequences are pointed out., Comment: 14 pages, LaTeX. Withdrawn. Its contents will be incorporated in a forthcoming preprint

The conformal invariance of unimodular gravity survives quantum corrections, even in the presence of conformal matter. Unimodular gravity can actually be understood as a certain truncation of the full Einstein-Hilbert theory, where in the Einstein frame the metric tensor enjoys unit determinant. Our result is compatible with the idea that the corresponding restriction in the functional integral is consistent as well., Comment: 20 pages; misprints corrected

The only allowed source of the gravitational field in the unimodular theory, invariant under area-preserving (transverse) diffeomorphisms as well as Weyl transformations, is just the traceless piece of the energy-momentum tensor. This fact notwithstanding, the free energy produced by arbitrary sources (not only static ones) is identical to the one predicted by general relativity. This encompasses all weak field tests of gravitation., Comment: 14 pages. Extended introduction; typos corrected; cosmetic changes

Einstein's traceless 1919 gravitational theory is analyzed from a variational viewpoint. It is shown to be equivalent to a transverse (invariant only under diffeomorphisms that preserve the Lebesgue measure) theory, with an additional Weyl symmetry, in which the gauge is partially fixed so that the metric becomes unimodular. In spite of the fact that this symmetry forbids direct coupling of the potential energy with the gravitational sector, the equivalence principle is recovered in the unimodular gauge owing to Bianchi's identities., Comment: LaTeX, 11 pages

The approximate late conformal time wavefunction for self-interacting scalar quantum fields in de Sitter space is computed. It dominates for conformal times much larger than a certain critical value, $\eta=\eta_{\text{crit}}$ which depends on the dimension and on other characteristics of the matter fields. Its self-consistency (i.e., whether the back reaction is consistent with the de Sitter symmetry) is analyzed., Comment: 22 pages. Withdrawn. The conclusion seems to be in error. Some parts will be incorporated in a forthcoming preprint

The approximate late conformal time wavefunctional for self-interacting scalar quantum fields in Friedmann-Robertson-Walker spaces is computed. It is a good candidate for the {\em vacuum wavefunctional} in such spaces., Comment: 7 pages, small changes. Contents incorporated in a later preprint

The problem of the vacuum energy decay is studied through the analysis of the vacuum survival amplitude ${\mathcal A}(z, z')$. Transition amplitudes are computed for finite time-span, $Z\equiv z^\prime-z$, and their {\em late time} behavior is discussed up to first order in the coupling constant, $\l$., Comment: 4 pages, 1 figure, to appear in the proceedings of the 15th Lomonosov Conference on Elementary Particle Physics

The problem of the vacuum energy decay is studied for both signs of the cosmological constant, through the analysis of the vacuum survival amplitude, defined in terms of the {\em conformal time}, $z$, by ${\mathcal A}(z,z^\prime)\equiv <\text{vac}\,z|\text{vac}\,z^\prime>$. Transition amplitudes are computed for finite time-span, $Z\equiv z^\prime-z$, and their {\em late time} behavior (directly related to the putative decay width of the state) as well as the transients are discussed up to first order in the coupling constant, $\lambda$., Comment: 40 pages, 1 figure

A general introduction is given to what can be predicated about quantum gravity once the lessons from the standard model of particle physics are taken into account. In particular, the effective lagrangian point of view is briefly commented upon ., Comment: Contribution to ERE2010 (Granada, Spain)

Contents included in arXiv:1004.4867, Comment: This preprint has been withdrawn. The contents of this work have been included in arXiv:1004.4867

The {\em instability of vacuum energy} in de Sitter space as discussed recently by Polyakov is argued to be a generic feature when external gravitational fields are present. It is related to the existence of {\em forbidden} (by momentum conservation) decays derived in some detail by Bros, Epstein and Moschella. Some calculations are discussed in a conformally invariant setting, and a flat space model is presented., Comment: 25 pages, 1 figures; significant changes; added content from another preprint

Scale invariant (transverse) gravitational theories are introduced. They are invariant under pure metric rescalings (i.e. the matter fields are inert under those). This symmetry forbids the presence of a cosmological constant. Those theories are not invariant under the full set of diffeomorphisms, but only with respect to those locally characterized by the fact that their generator is transverse $\pd_\a \xi^\a=0$., Comment: 11 pages