Massive Spin-2 Modes Research Articles

Black holes in multimetric gravity

We construct a wide class of black hole solutions to the general theory of ghost-free multimetric gravity in arbitrary spacetime dimension, extending and generalizing the known results in four-dimensional dRGT massive gravity and bigravity. The solutions are split into three generic classes based on whether the metrics can be simultaneously diagonalized—one of which does not exist in dRGT massive gravity nor bigravity, and is only possible when one has more than two interacting metric fields. We also linearize the general multimetric theory to determine the dynamics of the massive spin-2 modes, including examples where this can be done analytically, and use the linear theory to discuss the stability of the four-dimensional multi-Schwarzschild and multi-Kerr solutions. We explain how the instabilities that plague these solutions in dRGT massive gravity and bigravity carry across to the general multimetric theory, touching upon ideas of dimensional deconstruction to make sense of the results. Published by the American Physical Society 2024

Nonlocal Massive Gravity from Einstein Gravity.

We present a top-down construction of a three-dimensional nonlocal theory of massive gravity. This "nonlocal massive gravity" (NLMG) is obtained as the gravitational theory induced by Einstein gravity on a brane inserted in anti-de Sitter space modified by an overall minus sign. The theory involves an infinite series of increasingly complicated higher-derivative corrections to the Einstein-Hilbert action, with the quadratic term coinciding with new massive gravity. We obtain an analytic formula for the quadratic action of NLMG and show that its linearized spectrum consists of an infinite tower of positive-energy massive spin-2 modes. We compute the Newtonian potential and show that the introduction of the infinite series of terms makes it behave as ∼1/r at short distances, as opposed to the logarithmic behavior encountered when the series is truncated at any finite order. We use this and input from brane-world holography to argue that the theory may contain asymptotically flat black hole solutions.

The dominating mode of two competing massive modes of quadratic gravity

Over the last two decades, motivations for modified gravity have emerged from both theoretical and observational levels. f(R) and Chern-Simons gravity have received more attention as they are the simplest generalization. However, f(R) and Chern-Simons gravity contain only an additional scalar (spin-0) degree of freedom and, as a result, do not include other modes of modified theories of gravity. In contrast, quadratic gravity (also referred to as Stelle gravity) is the most general second-order modification to 4-D general relativity and contains a massive spin-2 mode that is not present in f(R) and Chern-Simons gravity. Using two different physical settings—the gravitational wave energy-flux measured by the detectors and the backreaction of the emitted gravitational radiation on the spacetime of the remnant black hole—we demonstrate that massive spin-2 mode carries more energy than the spin-0 mode. Our analysis shows that the effects are pronounced for intermediate-mass black holes, which are prime targets for LISA.

Gravitational waves from inspiraling black holes in quadratic gravity

We perform a study of gravitational waves emitted by inspiraling black holes in the context of quadratic gravity. By linearizing the field equations around a flat background, we demonstrate that all degrees of freedom satisfy wavelike equations. These degrees of freedom split into three modes: a massive spin-2 mode, a massive spin-0 mode, and the expected massless spin-2 mode. We construct the energy-momentum tensor of gravitational waves and show that, due to the massive spin-2 mode, it presents the Ostrogradsky instability. We also show how to deal with this possible pathology and obtain consistent physical interpretations for the system. Using the energy-momentum tensor, we study the influence of each massive mode in the orbital dynamics and compare it with the standard result of general relativity. Moreover, we present two methods to constrain the parameter $\ensuremath{\alpha}$ associated with the massive spin-2 contribution. From the first method, using the combined waveform for the spin-2 modes, we obtain the constraint $\ensuremath{\alpha}\ensuremath{\lesssim}1.1\ifmmode\times\else\texttimes\fi{}{10}^{21}\text{ }\text{ }{\mathrm{m}}^{2}$. In the second method, using the coalescence time, we get the constraint $\ensuremath{\alpha}\ensuremath{\lesssim}1.1\ifmmode\times\else\texttimes\fi{}{10}^{13}\text{ }\text{ }{\mathrm{m}}^{2}$.

Minimal Massive Supergravity.

Minimal massive gravity in three dimensions propagates a single massive spin-2 mode around an anti-de Sitter vacuum. It is distinguished by allowing for vacua with positive central charges of the asymptotic conformal algebra and a bulk graviton of positive energy. We present a new action for the model (and its higher order extensions) in terms of a dreibein and an independent spin connection. From this, we construct its supersymmetric extension. Surprisingly, all vacua complying with bulk and boundary unitarity appear to break supersymmetry spontaneously. In contrast, all supersymmetric vacua have a negative central charge whenever the bulk graviton has positive energy.

Nonlinearly ghost-free higher curvature gravity

We find unitary and local theories of higher curvature gravity in the vielbein formalism, known as Poincar\'e gauge theory, by utilizing the equivalence to ghost-free massive bigravity. We especially focus on three and four dimensions, but extensions into a higher-dimensional spacetime are straightforward. In three dimensions, quadratic gravity $\mathcal{L}=R+{T}^{2}+{R}^{2}$, where $R$ is the curvature and $T$ is the torsion with indices omitted, is shown to be equivalent to zwei-dreibein gravity and free from the ghost at fully nonlinear orders. In a special limit, new massive gravity is recovered. When the model is applied to the $\mathrm{AdS}/\mathrm{CFT}$ correspondence, unitarity both in the bulk theory and in the boundary theory implies that the torsion must not vanish. On the other hand, in four dimensions, the absence of a ghost at nonlinear order requires an infinite number of higher curvature terms, and these terms can be given by a schematic form $R(1+R/\ensuremath{\alpha}{m}^{2}{)}^{\ensuremath{-}1}R$, where $m$ is the mass of the massive spin-2 mode originating from the higher curvature terms and $\ensuremath{\alpha}$ is an additional parameter that determines the amplitude of the torsion. We also provide another four-dimensional ghost-free higher curvature theory that contains a massive spin-0 mode as well as a massive spin-2 mode.

Consistent inflationary cosmology from quadratic gravity with dynamical torsion

The idea of gauge theories of gravity predicts that there should exist not only the massless graviton but also massive particles carrying the gravitational force. We study the cosmology in a quadratic gravity with dynamical torsion where gravity may be interpreted as a gauge force associated with the Poincaré group. In addition to the massless spin-2 graviton, the model contains four non-ghost massive particle species: a couple of spin-0, a spin-1 and a spin-2. Supposing the restoration of the local Weyl invariance in the UV limit and the parity invariance, we find the most general minisuperspace action describing a homogeneous and isotropic universe with a flat spatial geometry. We then transform the minisuperspace action to a quasi-Einstein frame in which the field space is a hyperboloid and the field potential is a combination of those of a Starobinsky-like inflation and a natural inflation. Remarkably, thanks to the multi-field dynamics, the Starobinsky-like inflationary trajectory can be realized even if the initial condition is away from the top of the Starobinsky-like potential. We also study linear tensor perturbations and find qualitatively different features than the Starobinsky inflation, spontaneous parity violation and mixing of the massless and massive spin-2 modes, which might reveal the underlying nature of gravity through inflationary observables.

Spin-2 fields and the weak gravity conjecture

Recently, it has been argued that application of the Weak Gravity Conjecture (WGC) to spin-2 fields implies a universal upper bound on the cutoff of the effective theory for a single spin-2 field. We point out here that these arguments are largely spurious, because of the absence of states carrying spin-2 Stuckelberg $U(1)$ charge, and because of incorrect scaling assumptions. Known examples such as Kaluza-Klein theory that respect the usual WGC do so because of the existence of a genuine $U(1)$ field under which states are charged, as in the case of the Stuckelberg formulation of spin-1 theories, for which there is an unambiguously defined $U(1)$ charge. Theories of bigravity naturally satisfy a naive formulation of the WGC, $M_W< M_{\rm Pl}$, since the force of the massless graviton is always weaker than the massive spin-2 modes. It also follows that theories of massive gravity trivially satisfies this form of the WGC. We also point out that the identification of a massive spin-2 state in a truncated higher derivative theory, such as Einstein-Weyl-squared or its supergravity extension, bears no relationship with massive spin-2 states in the UV completion, contrary to previous statements in the literature. We also discuss the conjecture from a swampland perspective and show how the emergence of a universal upper bound on the cutoff relies on strong assumptions on the scale of the couplings between the spin-2 and other fields, an assumption which is known to be violated in explicit examples.

Unitary Extension of Exotic Massive 3D Gravity from Bigravity.

We obtain a new 3D gravity model from two copies of parity-odd Einstein-Cartan theories. Using Hamiltonian analysis, we demonstrate that the only local degrees of freedom are two massive spin-2 modes. Unitarity of the model in anti-deSitter and Minkowski backgrounds can be satisfied for vast choices of the parameters without fine-tuning. The recent "exotic massive 3D gravity" model arises as a limiting case of the new model. We also show that there exist trajectories on the parameter space of the new model which cross the boundary between unitary and nonunitary regions. At the crossing point, one massive graviton decouples resulting in a unitary model with just one bulk degree of freedom but two positive central charges at odds with the usual expectation that the critical model has at least one vanishing central charge. Given the fact that a suitable nonrelativistic version of bigravity has been used as an effective theory for gapped spin-2 fractional quantum Hall states, our model may have interesting applications in condensed matter physics.

Noether–Wald energy in Critical Gravity

Criticality represents a specific point in the parameter space of a higher-derivative gravity theory, where the linearized field equations become degenerate. In 4D Critical Gravity, the Lagrangian contains a Weyl-squared term, which does not modify the asymptotic form of the curvature. The Weyl2 coupling is chosen such that it eliminates the massive scalar mode and it renders the massive spin-2 mode massless. In doing so, the theory turns consistent around the critical point.Here, we employ the Noether–Wald method to derive the conserved quantities for the action of Critical Gravity. It is manifest from this energy definition that, at the critical point, the mass is identically zero for Einstein spacetimes, what is a defining property of the theory. As the entropy is obtained from the Noether–Wald charges at the horizon, it is evident that it also vanishes for any Einstein black hole.

Exotic massive 3D gravity

The linearized equations of “New Massive Gravity” propagate a parity doublet of massive spin-2 modes in 3D Minkowski spacetime, but a different non-linear extension is made possible by “third-way” consistency. There is a “Chern-Simons-like” action, as for other 3D massive gravity models, but the new theory is “exotic”: its action is parity odd. This “Exotic Massive Gravity” is the next-to-simplest case in an infinite sequence of third-way consistent 3D gravity theories, the simplest being the “Minimal Massive Gravity” alternative to “Topologically Massive Gravity”.

Noether-Wald Charges in Critical Gravity

Critical Gravity theory is defined by a particular combination of quadratic couplings in the curvature added on top of 4D Einstein-Hilbert action with negative cosmological constant. As the Lagrangian is given by a Weyl-squared term, the asymptotic form of the curvature is not modified. The coupling of the W eyl2 term is such the massive scalar mode is eliminated and the massive spin-2 mode become massless, rendering the theory consistent around the critical point.In the present work, we construct the Noether-Wald charges for the action of Critical Gravity. Such construction makes manifest a defining property of this theory: both the energy and entropy for Einstein black holes vanish identically.

Dark matter scenarios with multiple spin-2 fields

We study ghost-free multimetric theories for (N+1) tensor fields with a coupling to matter and maximal global symmetry group SN×(Z2)N. Their mass spectra contain a massless mode, the graviton, and N massive spin-2 modes. One of the massive modes is distinct by being the heaviest, the remaining (N−1) massive modes are simply identical copies of each other. All relevant physics can therefore be understood from the case N=2. Focussing on this case, we compute the full perturbative action up to cubic order and derive several features that hold to all orders in perturbation theory. The lighter massive mode does not couple to matter and neither of the massive modes decay into massless gravitons. We propose the lighter massive particle as a candidate for dark matter and investigate its phenomenology in the parameter region where the matter coupling is dominated by the massless graviton. The relic density of massive spin-2 can originate from a freeze-in mechanism or from gravitational particle production, giving rise to two different dark matter scenarios. The allowed parameter regions are very different from those in scenarios with only one massive spin-2 field and more accessible to experiments.

On complexity for F (R) and critical gravity

Using “complexity=action” proposal we study complexity growth of certain gravitational theories containing higher derivative terms. These include critical gravity in diverse dimensions. One observes that the complexity growth for neutral black holes saturates the proposed bound when the results are written in terms of physical quantities of the model. We will also study effects of shock wave to the complexity growth where we find that the presence of massive spin-2 mode slows down the rate of growth.

Zwei-Dreibein Gravity: A Two-Frame-Field Model of 3D Massive Gravity

We present a generally covariant and parity-invariant two-frame field ("zwei-dreibein") action for gravity in three space-time dimensions that propagates two massive spin-2 modes, unitarily, and we use Hamiltonian methods to confirm the absence of unphysical degrees of freedom. We show how zwei-dreibein gravity unifies previous "3D massive gravity" models and extends them, in the context of the AdS/CFT correspondence, to allow for a positive central charge consistent with bulk unitarity.

Counterterms, critical gravity, and holography

We consider counterterms for odd dimensional holographic CFTs. These counterterms are derived by demanding cut-off independence of the CFT partition function on $S^d$ and $S^1 \times S^{d-1}$. The same choice of counterterms leads to a cut-off independent Schwarzschild black hole entropy. When treated as independent actions, these counterterm actions resemble critical theories of gravity, i.e., higher curvature gravity theories where the additional massive spin-2 modes become massless. Equivalently, in the context of AdS/CFT, these are theories where at least one of the central charges associated with the trace anomaly vanishes. Connections between these theories and logarithmic CFTs are discussed. For a specific choice of parameters, the theories arising from counterterms are non-dynamical and resemble a DBI generalization of gravity. For even dimensional CFTs, analogous counterterms cancel log-independent cut-off dependence.

ON BLACK HOLE STABILITY IN CRITICAL GRAVITIES

We consider extended cosmological gravities with Ricci tensor and scalar squared terms in diverse dimensions. These theories admit solutions of Einstein metrics, including the Schwarzschild–Tangherlini AdS black holes, whose mass and entropy vanish at the critical point. We perform linearized analysis around the black holes and show that in general the spectrum consists of the usual spin-2 massless and ghost massive modes. We demonstrate that there is no exponentially-growing tachyon mode in the black holes. At the critical point, the massless spin-2 modes have zero energy whilst the massive spin-2 modes are replaced by the log modes. There always exist certain linear combination of massless and log modes that has negative energy. Thus the stability of the black holes requires that the log modes to be truncated out by the boundary condition.

Conformal gravity and extensions of critical gravity

Higher-order curvature corrections involving the conformally invariant Weyl-squared action have played a role in two recent investigations of four-dimensional gravity: in critical gravity, where they are added to the standard cosmological Einstein-Hilbert action with a coefficient tuned to make the massive ghostlike spin-2 excitations massless, and in a pure Weyl-squared action considered by Maldacena, where the massive spin-2 modes are removed by the imposition of boundary conditions. We exhibit the connections between the two approaches, and we also generalize critical gravity to a wider class of Weyl-squared modifications to cosmological Einstein gravity where one can eliminate the massive ghostlike spin-2 modes by means of boundary conditions. The cosmological constant plays a crucial role in the discussion, since there is then a ``window'' of negative mass-squared spin-2 modes around ${\mathrm{AdS}}_{4}$ that are not tachyonic. We also construct analogous conformal and nonconformal gravities in six dimensions.

Modes of log gravity

The physical modes of a recently proposed $D$-dimensional ``critical gravity'', linearized about its anti-de Sitter vacuum, are investigated. All ``log mode'' solutions, which we categorize as ``spin-2'' or ``Proca'', arise as limits of the massive spin-2 modes of the noncritical theory. The linearized Einstein tensor of a spin-2 log mode is itself a `nongauge' solution of the linearized Einstein equations whereas the linearized Einstein tensor of a Proca mode takes the form of a linearized general coordinate transformation. Our results suggest the existence of a holographically dual logarithmic conformal field theory.

Critical Gravity in Four Dimensions

We study four-dimensional gravity theories that are rendered renormalizable by the inclusion of curvature-squared terms to the usual Einstein action with a cosmological constant. By choosing the parameters appropriately, the massive scalar mode can be eliminated and the massive spin-2 mode can become massless. This "critical" theory may be viewed as a four-dimensional analogue of chiral topologically massive gravity, or of critical "new massive gravity" with a cosmological constant, in three dimensions. We find that the on-shell energy for the remaining massless gravitons vanishes. There are also logarithmic spin-2 modes, which have positive energy. The mass and entropy of standard Schwarzschild-type black holes vanish. The critical theory might provide a consistent toy model for quantum gravity in four dimensions.