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We make a conceptual overview of a particular approach to the initial-value problem in canonical gauge theories. We stress how the first-class phase-space constraints may be relaxed if we interpret them as fixing the values of new degrees of freedom. This idea goes back to Fock and Stueckelberg, leading to restrictions of the gauge symmetry of a theory, and it corresponds, in certain cases, to promoting constants of Nature to physical fields. Recently, different versions of this formulation have gained considerable attention in the literature, with several independent iterations, particularly in classical and quantum descriptions of gravity, cosmology, and electromagnetism. In particular, in the case of canonical quantum gravity, the Fock--Stueckelberg approach is relevant to the so-called problem of time. Our overview recalls the work of Fock and Stueckelberg and its physical interpretation with the aim of conceptually unifying the different iterations of the idea that appear in the literature and of motivating further research.

We study the quantum hair associated with coherent states describing slowly rotating black holes and show how it can be naturally related with the Bekenstein-Hawking entropy and with 1-loop quantum corrections of the metric for the (effectively) non-rotating case. We also estimate corrections induced by such quantum hair to the temperature of the Hawking radiation through the tunnelling method., Comment: Plain LaTeX, 19 pages, clarifications added

We present a simple analytical model for studying the collapse of an ultracompact stellar object (regular black hole mimicker with infinite redshift surface) to form a (integrable) black hole, in the framework of General Relativity. Both initial and final configurations have the same ADM mass, so that the transition represents an internal redistribution of matter without emission of energy. The model, despite being quite idealized, can be viewed as a good starting point to investigate near-horizon quantum physics, Comment: 7 pages, 4 figures

We show how to reliably calculate quantum gravitational corrections to cosmological models using the unique effective action formalism for quantum gravity. Our calculations are model independent and apply to any ultra-violet complete theory of quantum gravity that admit general relativity as a low energy theory. We show that it is important to consider the full effective action to obtain renormalization group invariant solutions. We investigate the validity range of our techniques within simple cosmological models., Comment: 11 pages, 2 figures

We study cosmological models based on the interior of the revisited Schwarzschild black hole recently reported in [Phys.~Rev.~D{\bf 109} (2024) 104032]. We find that these solutions describe a non-trivial Kantowski-Sachs universe, for which we provide an explicit analytical example with all the details and describe some general features of the singularity., Comment: 7 pages

We consider quantum gravitational corrections to the Oppenheimer-Snyder metric describing time-dependent dust ball collapse. The interior metric also describes Friedmann-Lemaitre-Robertson-Walker cosmology and our results are interpreted in that context. The exterior corrections are an example of quantum hair, and are shown to persist throughout the collapse. Our results show the quantum hair survives throughout the horizon formation and that the internal state of the resulting black hole is accessible to outside observers., Comment: 13 pages

We describe the ground state for a gravitationally collapsed ball of dust as the direct product of wavefunctions for dust particles distributed over an arbitrary number of nested layers. This allows us to estimate the expectation value of the global radius as well as the effective energy density and pressures for the dust core of quantum black holes. In particular, the size of the quantum core does not depend on the number of layers and the mass function is shown to grow linearly with the areal radius up to the outermost layer., Comment: 9 pages, 2 figures. Version to appear in PLB

We consider black holes generically sourced by quantum matter described by regular wavefunctions. This allows for integrable effective energy densities and the removal of Cauchy horizons in spherically symmetric configurations. Moreover, we identify the ultrarigid rotation of the Kerr spacetime as causing the existence of an inner horizon in rotating systems, and describe general properties for quantum matter cores at the centre of rotating black holes with integrable singularities and no Cauchy horizon., Comment: LaTeX, 12 pages, 1 figure, final version to appear in JHEP

The Hawking-Penrose theorem is not covariant under field redefinitions. Should the invariance under such transformations be a true principle in Nature, spacetime singularities become dubious objects. We here review the concept of covariant singularities, that is, singularities that are invariant under both spacetime diffeomorphisms and field redefinitions., Comment: Accepted for publication in Mod. Phys. Lett. A

The Hawking-Penrose theorem is not covariant under field redefinitions. Should the invariance under such transformations be a true principle in Nature, spacetime singularities become dubious objects. We here review the concept of covariant singularities, that is, singularities that are invariant under both spacetime diffeomorphisms and field redefinitions., Comment: Accepted for publication in Mod. Phys. Lett. A

We study the effect of rotation on the spectrum of bound states for dust cores that source (quantum) black holes found in Eur. Phys. J. C 82 (2022) 10. The dust ball is assumed to spin rigidly with sufficiently slow angular velocity that perturbation theory can be applied. Like the total mass, the total angular momentum is shown to be quantised in Planck units, hence so is the horizon area. For sufficiently small fraction of mass in the outermost layer, the model admits ground states which can spin fast enough within the perturbative regime so as to describe regular rotating objects rather than black holes., Comment: 9 pages, 6 graphs in 2 figures

The notions of minimum geometrical length and minimum length scale are discussed with reference to correlation functions obtained from in-in and in-out amplitudes in quantum field theory. Whereas the in-in propagator for metric perturbations does not admit the former, the in-out Feynman propagator shows the emergence of the latter. A connection between the Feynman propagator of quantum field theories of gravity and the deformation parameter $\delta_0$ of the generalised uncertainty principle (GUP) is then exhibited, which allows to determine an exact expression for $\delta_0$ in terms of the residues of the causal propagator. A correspondence between the non-renormalisability of (some) theories (of gravity) and the existence of a minimum length scale is then conjectured to support the idea that non-renormalisable theories are self-complete and finite. The role played by the sign of the deformation parameter is further discussed by considering an implementation of the GUP on the lattice., Comment: LaTeX, 12 pages, no figures, final version to appear in PLB

It is generally believed that a full-fledged theory of quantum gravity should exhibit background independence and diffeomorphism invariance. In its most general form, the latter comprises field redefinitions, which are diffeomorphisms in configuration space. We show that any path-integral approach to quantum gravity leads to a tension between these properties, such that they cannot hold simultaneously., Comment: matches version accepted for publication

We explore the relationship between the quantum state of a compact matter source and of its asymptotic graviton field. For a matter source in an energy eigenstate, the graviton state is determined at leading order by the energy eigenvalue. Insofar as there are no accidental energy degeneracies there is a one to one map between graviton states on the boundary of spacetime and the matter source states. A typical semiclassical matter source results in an entangled asymptotic graviton state. We exhibit a purely quantum gravitational effect which causes the subleading asymptotic behavior of the graviton state to depend on the internal structure of the source. These observations establish the existence of ubiquitous quantum hair due to gravitational effects., Comment: 12 pages

We consider the quantization of matter fields in a background described by the teleparallel equivalent to general relativity. The presence of local Lorentz and gauge symmetries gives rise to different coupling prescriptions, which we analyse separately. As expected, quantum matter fields produce divergences that cannot be absorbed by terms in the background action of teleparallel equivalent to general relativity. Nonetheless, the formulation of teleparallel gravity allows one to find out the source of the problem. By imposing local Lorentz invariance after quantization, we show that a modified teleparallel gravity, in which the coefficients in the action are replaced by free parameters, can be renormalized at one-loop order without introducing higher-order terms. This precludes the appearance of ghosts in the theory.

We consider the quantum description of a toy model universe in which the Schwarzschild-de Sitter geometry emerges from the coherent state of a massless scalar field. Although highly idealised, this simple model allows us to find clear hints supporting the conclusion that the reaction of the de Sitter background to the presence of matter sources induces i) a modified Newtonian dynamics at galactic scales and ii) different values measured for the present Hubble parameter. Both effects stem from the conditions required to have a normalisable quantum state., Comment: LaTeX, 12 pages, no figures

We present a simple quantum description of the gravitational collapse of a ball of dust which excludes those states whose width is arbitrarily smaller than the gravitational radius of the matter source and supports the conclusion that black holes are macroscopic extended objects. We also comment briefly on the relevance of this result for the ultraviolet self-completion of gravity and the corpuscular picture of black holes., Comment: 8 pages, LaTeX, one figure, references and comments added

It is rather well-known that spacetime singularities are not covariant under field redefinitions. A manifestly covariant approach to singularities in classical gravity was proposed in arXiv:2008.09387. In this paper, we start to extend this analysis to the quantum realm. We identify two types of covariant singularities in field space corresponding to geodesic incompleteness and ill-defined path integrals (hereby dubbed functional singularities). We argue that the former might not be harmful after all, whilst the latter makes all observables undefined. We show that the path-integral measure is regular in any four-dimensional theory of gravity without matter or in any theory in which gravity is either absent or treated semi-classically. This might suggest the absence of functional singularities in these cases, however it can only be confirmed with a thorough analysis, case by case, of the path integral. We provide a topological and model-independent classification of functional singularities using homotopy groups and we discuss examples of theories with and without such singularities., Comment: Accepted for publication in Nucl. Phys. B

We present a quantum description of black holes given by coherent states of gravitons sourced by a matter core. The expected behaviour in the weak-field region outside the horizon is recovered, with arbitrarily good approximation, but the classical central singularity is not resolved because the coherent states may not contain modes of arbitrarily short wavelength and the matter core must therefore have finite size. Ensuing quantum corrections both in the interior and exterior are also estimated by assuming the mean-field approximation holds everywhere. These deviations from the classical black hole geometry can be viewed as quantum hair and lead to a quantum corrected horizon radius and thermodynamics., Comment: LaTeX, 16 pages, 6 graphs in 4 figures

Bootstrapped Newtonian gravity is a non-linear version of Newton's law which can be lifted to a fully geometric theory of gravity starting from a modified potential. Here, we study geodesics in the bootstrapped Newtonian effective metric in vacuum and obtain bounds on a free parameter from Solar System data and $S$-star orbits near our Galaxy centre. These bounds make {\em vacuum} bootstrapped Newtonian gravity experimentally indistinguishable from General Relativity., Comment: REVTeX 4, 11 pages, 5 figures, 5 tables

The aim of this thesis is to entertain the possibility of a quantum departure from the general relativistic description of compact sources in strong field regime and claim that a quantum understanding of the classical background could be necessary. We therefore develop an effective field theory providing a simplified framework to address the effects of non-linearities in strong gravitational backgrounds. Starting from a massless Fierz- Pauli-type lagrangian for the Newtonian potential and introducing the self-coupling terms, we arrive at a non-linear equation describing the effective gravitational potential of an arbitrarily compact homogeneous source. Unlike the general relativistic solutions no Buchdahl limit is found as the solutions display a regular behaviour in any compactness regime. Moreover, we provide a quantum interpretation of these results in terms of a quantum coherent state formalism. Such an approach proves to be widely capable of accounting for classical field configurations as well as providing some collective properties of the constituent soft quanta. The latter show a good agreement with some of the crucial relations of the corpuscular model. We do not interpret this approach as a model of phenomenological relevance but better as a simplified picture aimed at capturing novel quantum feature of black holes physics.

We consider the quantization of matter fields in a background described by the teleparallel equivalent to general relativity. The presence of local Lorentz and gauge symmetries gives rise to different coupling prescriptions, which we analyse separately. As expected, quantum matter fields produce divergences that cannot be absorbed by terms in the background action of teleparallel equivalent to general relativity. Nonetheless, the formulation of teleparallel gravity allows one to find out the source of the problem. By imposing local Lorentz invariance after quantization, we show that a modified teleparallel gravity, in which the coefficients in the action are replaced by free parameters, can be renormalized at one-loop order without introducing higher-order terms. This precludes the appearance of ghosts in the theory.

We explore the relationship between the quantum state of a compact matter source and of its asymptotic graviton field. For a matter source in an energy eigenstate, the graviton state is determined at leading order by the energy eigenvalue. Insofar as there are no accidental energy degeneracies there is a one to one map between graviton states on the boundary of spacetime and the matter source states. A typical semiclassical matter source results in an entangled asymptotic graviton state. We exhibit a purely quantum gravitational effect which causes the subleading asymptotic behavior of the graviton state to depend on the internal structure of the source. These observations establish the existence of ubiquitous quantum hair due to gravitational effects., Comment: 12 pages

We consider the quantum description of a toy model universe in which the Schwarzschild-de Sitter geometry emerges from the coherent state of a massless scalar field. Although highly idealised, this simple model allows us to find clear hints supporting the conclusion that the reaction of the de Sitter background to the presence of matter sources induces i) a modified Newtonian dynamics at galactic scales and ii) different values measured for the present Hubble parameter. Both effects stem from the conditions required to have a normalisable quantum state., Comment: LaTeX, 12 pages, no figures

Bootstrapped Newtonian gravity is a non-linear version of Newton's law which can be lifted to a fully geometric theory of gravity starting from a modified potential. Here, we study geodesics in the bootstrapped Newtonian effective metric in vacuum and obtain bounds on a free parameter from Solar System data and S-star orbits near our Galaxy centre. These bounds make vacuum bootstrapped Newtonian gravity experimentally indistinguishable from General Relativity., Comment: REVTeX 4, 11 pages, 5 figures, 5 tables. Version to appear in PRD

We determine the complete space-time metric from the bootstrapped Newtonian potential generated by a static spherically symmetric source in the surrounding vacuum. This metric contains post-Newtonian parameters which can be further used to constrain the complete underlying dynamical theory. For values of the post-Newtonian parameters within experimental bounds, the reconstructed metric appears very close to the Schwarzschild solution of General Relativity in the whole region outside the event horizon. The latter is however larger in size for the same value of the mass compared to the Schwarzschild case., Comment: 27 pages, 3 figures, minor changes to match the accepted version in PRD

The aim of this thesis is to entertain the possibility of a quantum departure from the general relativistic description of compact sources in strong field regime and claim that a quantum understanding of the classical background could be necessary. We therefore develop an effective field theory providing a simplified framework to address the effects of non-linearities in strong gravitational backgrounds. Starting from a massless Fierz- Pauli-type lagrangian for the Newtonian potential and introducing the self-coupling terms, we arrive at a non-linear equation describing the effective gravitational potential of an arbitrarily compact homogeneous source. Unlike the general relativistic solutions no Buchdahl limit is found as the solutions display a regular behaviour in any compactness regime. Moreover, we provide a quantum interpretation of these results in terms of a quantum coherent state formalism. Such an approach proves to be widely capable of accounting for classical field configurations as well as providing some collective properties of the constituent soft quanta. The latter show a good agreement with some of the crucial relations of the corpuscular model. We do not interpret this approach as a model of phenomenological relevance but better as a simplified picture aimed at capturing novel quantum feature of black holes physics.

The assumptions of the Hawking-Penrose singularity theorem are not covariant under field redefinitions. Thus we propose to study singularities in field space, where the spacetime metric is treated as a coordinate along with any other fields. We show that the field-space Kretschmann scalar for a certain choice of the DeWitt field-space metric is everywhere finite. This fact could be interpreted as an indication that no singularities actually exist in pure gravity for any gravitational action. In particular, all vacuum singularities of General Relativity result from an unhappy choice of field variables. The extension to the case in which matter fields are present, as required by singularity theorems, is left for future development., Comment: Matches version to be published at IJMPD

Bootstrapped Newtonian gravity was developed with the purpose of estimating the impact of quantum physics in the nonlinear regime of the gravitational interaction, akin to corpuscular models of black holes and inflation. In this work, we set the ground for extending the bootstrapped Newtonian picture to cosmological spaces. We further discuss how such models of quantum cosmology can lead to a natural solution of the cosmological constant problem., Comment: 11 pages, no figures, to appear in Symmetry

In this paper we derive the leading quantum gravitational corrections to the geodesics and the equations of motion for a scalar field in the spacetime containing a constant density star. It is shown that these corrections can be calculated in quantum gravity reliably and in a model independent way. Furthermore, we find that quantum gravity gives rise to an additional redshift that results from the gradient instead of the amplitude of the density profile., Comment: 13 pages

The existence of a minimum length in quantum gravity is investigated by computing the in-in expectation value of the proper distance in the Schwinger-Keldysh formalism. No minimum geometrical length is found for arbitrary gravitational theories to all orders in perturbation theory. Using non-perturbative techniques, we also show that neither the conformal sector of general relativity nor higher-derivative gravity features a minimum length. A minimum length scale, on the other hand, seems to always be present when one considers in-out amplitudes, from which one could extract the energy scale of scattering processes., Comment: Version accepted for publication

We compute quantum corrections for the gravitational potential obtained by including a derivative self-coupling in its classical dynamics as a toy model for analysing quantum gravity in the strong field regime. In particular, we focus on quantum corrections to the classical solutions in the vacuum outside localised matter sources., Comment: LaTeX, 13 pages, no figures

In this paper we consider a static and regular fluid generating a locally spherically symmetric and time-independent space-time and calculate the leading quantum corrections to the metric to first order in curvature. Starting from a singularity free classical solution of general relativity, we show that singularities can be introduced in the curvature invariants by quantum gravitational corrections calculated using an effective field theory approach to quantum gravity. We identify non-trivial conditions that ensure that curvature invariants remain singularity free to leading order in the curvature expansion of the effective action., Comment: 10 pages

In this paper we consider a static and regular fluid generating a locally spherically symmetric and time-independent space-time and calculate the leading quantum corrections to the metric to first order in curvature. Starting from a singularity free classical solution of general relativity, we show that singularities can be introduced in the curvature invariants by quantum gravitational corrections calculated using an effective field theory approach to quantum gravity. We identify non-trivial conditions that ensure that curvature invariants remain singularity free to leading order in the curvature expansion of the effective action., Comment: 10 pages

We compute quantum corrections for the gravitational potential obtained by including a derivative self-coupling in its classical dynamics as a toy model for analysing quantum gravity in the strong field regime. In particular, we focus on quantum corrections to the classical solutions in the vacuum outside localised matter sources., Comment: LaTeX, 13 pages, no figures

The purpose of this work is to investigate the consequences of quantum gravity for the singularity problem. We study the higher-derivative terms that invariably appear in any quantum field theoretical model of gravity, handling them both non-perturbatively and perturbatively. In the former case, by computing the contributions of the additional degrees of freedom to a congruence of geodesics, we show that the appearance of singularities is no longer a necessity. In the latter, which corresponds to treating the quantised general relativity as an effective field theory, we generalise the Hawking-Penrose theorems to include one-loop corrections of both massless matter and graviton fluctuations.

In this paper we consider the full set of quantum gravitational corrections to a star metric to second order in curvature. As we use an effective field theoretical approach, these corrections apply to any model of quantum gravity that is based on general coordinate invariance. We then discuss the black hole limit and identify an interesting phenomenon which could shed some light on the nature of astrophysical black holes: while star metrics receive corrections at second order in curvature, vacuum solutions such as black hole metrics do not. What happens to these corrections when a star collapses?

We show that the bootstrapped Newtonian potential generated by a uniform and isotropic source does not depend on the one-loop correction for the matter coupling to gravity. The latter however affects the relation between the proper mass and the ADM mass and, consequently, the pressure needed to keep the configuration stable., Comment: 11 pages, 7 figures

It is shown that the recent corpuscular description of gravity generically excludes de Sitter spacetime in any semiclassical version of f(R) gravity. A phantom phenomenology of the cosmic dynamics is also naturally excluded., Comment: 9 pages, no figures

We study equilibrium configurations of a homogenous ball of matter in a bootstrapped description of gravity which includes a gravitational self-interaction term beyond the Newtonian coupling. Both matter density and pressure are accounted for as sources of the gravitational potential for test particles. Unlike the general relativistic case, no Buchdahl limit is found and the pressure can in principle support a star of arbitrarily large compactness. By defining the horizon as the location where the escape velocity of test particles equals the speed of light, like in Newtonian gravity, we find a minimum value of the compactness for which this occurs. The solutions for the gravitational potential here found could effectively describe the interior of macroscopic black holes in the quantum theory, as well as predict consequent deviations from general relativity in the strong field regime of very compact objects., Comment: 27 pages, 17 figures. Version accepted for publication in EPJ C

We try to shed some light on the role of matter in the final stages of black hole evaporation from the fundamental frameworks of classicalization and the black-to-white hole bouncing scenario. Despite being based on very different grounds, these two approaches attempt at going beyond the background field method and treat black holes as fully quantum systems rather than considering quantum field theory on the corresponding classical manifolds. They also lead to the common prediction that the semiclassical description of black hole evaporation should break down and the system be disrupted by internal quantum pressure, but they both arrive at this conclusion neglecting the matter that formed the black hole. We instead estimate this pressure from the bootstrapped description of black holes, which allows us to express the total Arnowitt-Deser-Misner mass in terms of the baryonic mass still present inside the black hole. We conclude that, although these two scenarios provide qualitatively similar predictions for the final stages, the corpuscular model does not seem to suggest any sizeable deviation from the semiclassical time scale at which the disruption should occur, unlike the black-to-white hole bouncing scenario. This, in turn, makes the phenomenology of corpuscular black holes more subtle from an astrophysical perspective., Comment: 5 pages, no figure

A new class of relativistic stellar structure equations which include the effects of an energy-dependent Newton coupling is presented. Significant modifications in the mass-radius relation for neutron stars are possible only if the running of the Newton coupling due to quantum gravity occurs at low energies. A new Buchdahl limit is derived and its physical implications are discussed. In particular, sub-Planckian self-gravitating objects with arbitrarily small radii are possible., Comment: 19 pages, 3 figures. V2: minor changes, references added. Matches published version

The purpose of this work is to investigate the consequences of quantum gravity for the singularity problem. We study the higher-derivative terms that invariably appear in any quantum field theoretical model of gravity, handling them both non-perturbatively and perturbatively. In the former case, by computing the contributions of the additional degrees of freedom to a congruence of geodesics, we show that the appearance of singularities is no longer a necessity. In the latter, which corresponds to treating the quantised general relativity as an effective field theory, we generalise the Hawking-Penrose theorems to include one-loop corrections of both massless matter and graviton fluctuations.

We show that the bootstrapped Newtonian potential generated by a uniform and isotropic source does not depend on the one-loop correction for the matter coupling to gravity. The latter however affects the relation between the proper mass and the ADM mass and, consequently, the pressure needed to keep the configuration stable., Comment: 11 pages, 7 figures

A new class of relativistic stellar structure equations which include the effects of an energy-dependent Newton coupling is presented. Significant modifications in the mass-radius relation for neutron stars are possible only if the running of the Newton coupling due to quantum gravity occurs at low energies. A new Buchdahl limit is derived and its physical implications are discussed. In particular, sub-Planckian self-gravitating objects with arbitrarily small radii are possible., Comment: 19 pages, 3 figures. V2: minor changes, references added. Matches published version

In this paper we consider the full set of quantum gravitational corrections to a star metric to second order in curvature. As we use an effective field theoretical approach, these corrections apply to any model of quantum gravity that is based on general coordinate invariance. We then discuss the black hole limit and identify an interesting phenomenon which could shed some light on the nature of astrophysical black holes: while star metrics receive corrections at second order in curvature, vacuum solutions such as black hole metrics do not. What happens to these corrections when a star collapses?

We try to shed some light on the role of matter in the final stages of black hole evaporation from the fundamental frameworks of classicalization and the black-to-white hole bouncing scenario. Despite being based on very different grounds, these two approaches attempt at going beyond the background field method and treat black holes as fully quantum systems rather than considering quantum field theory on the corresponding classical manifolds. They also lead to the common prediction that the semiclassical description of black hole evaporation should break down and the system be disrupted by internal quantum pressure, but they both arrive at this conclusion neglecting the matter that formed the black hole. We instead estimate this pressure from the bootstrapped description of black holes, which allows us to express the total Arnowitt-Deser-Misner mass in terms of the baryonic mass still present inside the black hole. We conclude that, although these two scenarios provide qualitatively similar predictions for the final stages, the corpuscular model does not seem to suggest any sizeable deviation from the semiclassical time scale at which the disruption should occur, unlike the black-to-white hole bouncing scenario. This, in turn, makes the phenomenology of corpuscular black holes more subtle from an astrophysical perspective., Comment: 5 pages, no figure

We study equilibrium configurations of a homogenous ball of matter in a bootstrapped description of gravity which includes a gravitational self-interaction term beyond the Newtonian coupling. Both matter density and pressure are accounted for as sources of the gravitational potential for test particles. Unlike the general relativistic case, no Buchdahl limit is found and the pressure can in principle support a star of arbitrarily large compactness. By defining the horizon as the location where the escape velocity of test particles equals the speed of light, like in Newtonian gravity, we find a minimum value of the compactness for which this occurs. The solutions for the gravitational potential here found could effectively describe the interior of macroscopic black holes in the quantum theory, as well as predict consequent deviations from general relativity in the strong field regime of very compact objects., Comment: 27 pages, 17 figures. Version accepted for publication in EPJ C

It is shown that the recent corpuscular description of gravity generically excludes de Sitter spacetime in any semiclassical version of f(R) gravity. A phantom phenomenology of the cosmic dynamics is also naturally excluded., Comment: 9 pages, no figures

A new class of relativistic stellar structure equations which include the effects of an energy-dependent Newton coupling is presented. Significant modifications in the mass-radius relation for neutron stars are possible only if the running of the Newton coupling due to quantum gravity occurs at low energies. A new Buchdahl limit is derived and its physical implications are discussed. In particular, sub-Planckian self-gravitating objects with arbitrarily small radii are possible., Comment: 19 pages, 3 figures. V2: minor changes, references added. Matches published version