Narrow Results

The almost-scale-invariant spectrum for the stochastic background wave (in primordial Universe) is a firm prediction of inflationary scenarios. In this work, to study of primordial Gravitational Waves, one-dimensional toy model in generalizations of the Einstein–Hilbert frame, described by second-order curvature invariant (called pure Quadratic Gravity) is considered. Solutions to the primordial perturbations are more varied than Einstein–Hilbert frame and include simple incoming and outcoming waves. By examining the spectrum diagram of the perturbation solutions, it can be seen that the oscillating solutions are ruled out due to not having the necessary spectral power, but the solutions as damping exponential have the ability to produce a scale-invariant spectrum (confirmed by CMB data).

In this paper, we investigate the relationship between quadratic gravity and a restricted Weyl symmetry where a gauge parameter $\mathrm{\Omega }(x)$ of Weyl transformation satisfies a constraint $\square \mathrm{\Omega }=0$ in a curved spacetime. First, we briefly review a model with a restricted gauge symmetry on the basis of QED, where a $U(1)$ gauge parameter $𝜃(x)$ obeys a similar constraint $\square 𝜃=0$ in a flat Minkowski spacetime, and explain that the restricted gauge symmetry removes one on-shell mode of gauge field, which together with the Feynman gauge leaves only two transverse polarizations as physical states. Next, it is shown that the restricted Weyl symmetry also eliminates one component of a dipole field in quadratic gravity around a flat Minkowski background, leaving only a single scalar state. Finally, we show that the restricted Weyl symmetry cannot remove any dynamical degrees of freedom in static background metrics by using the zero-energy theorem of quadratic gravity. This fact also holds for the Euclidean background metrics without imposing the static condition.

In [Phys. Rev. D 107, 104008 (2023)], we reported a novel exact closed-form solution which describes asymptotically flat spacetimes in pure ${{ℛ}}^2$ gravity. The solution is Ricci scalar flat, viz. ${ℛ}\equiv 0$ everywhere. Whereas any metric with a null Ricci scalar would trivially satisfy the ${{ℛ}}^2$ vacuo field equation, ${ℛ}({{ℛ}}_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }{ℛ})+g_{\mu \nu }\square {ℛ}-{\nabla }_{\mu }{\nabla }_{\nu }{ℛ}=0$, in this paper, we shall show that our solution satisfies a “stronger” version of the ${{ℛ}}^2$ vacuo field equation, viz. ${{ℛ}}_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }{ℛ}+{{ℛ}}^{-1}(g_{\mu \nu }\square {ℛ}-{\nabla }_{\mu }{\nabla }_{\nu }{ℛ})=0$, despite the term ${{ℛ}}^{-1}$ being singular. Even though ${ℛ}$ identically vanishes, for our solution, the combinations ${{ℛ}}^{-1}{\nabla }_{\mu }{\nabla }_{\nu }{ℛ}$ and ${{ℛ}}^{-1}\square {ℛ}$ are free of singularity. This exceptional property sets our solution apart from the set of null-Ricci-scalar metrics and makes it a genuinely nontrivial solution. We further demonstrate that, as a member of a larger class of asymptotically de Sitter metrics, our solution is resilient against perturbations in the scalar curvature at largest distances, making it relevant for physical situations where the background deviates from asymptotic flatness.

More than three decades ago quadratic gravity was found to present a perturbative, renormalizable and asymptotically free theory of quantum gravity. Unfortunately, the theory appeared to have problems with a spin-2 ghost. In this essay, we revisit quadratic gravity in a different light by considering the case that the asymptotically free interaction flows to a strongly interacting regime. This occurs when the coefficient of the Einstein–Hilbert term is smaller than the scale ${\mathrm{\Lambda }}_{\mathrm{\text{QG}}}$ where the quadratic couplings grow strong. Here quantum chromodynamics (QCD) provides some useful insights. By pushing the analogy with QCD, we conjecture that the nonperturbative effects can remove the naive spin-2 ghost and lead to the emergence of general relativity (GR) in the IR.

Black holes constitute some of the most fascinating objects in our universe. According to Einstein’s theory of general relativity, they are also deceivingly simple: Schwarzschild black holes are completely determined by their mass. Moreover, the singularity theorems by Penrose and Hawking indicate that they host a curvature singularity within their event horizon. The presence of the latter invites the question whether these dead-end points of spacetime can be made regular by considering (quantum) corrections to the classical field equations. In this light, we use the Frobenius method to investigate the phase space of asymptotically flat, static, and spherically symmetric black hole solutions in quadratic gravity. We argue that the only asymptotically flat black hole solution visible in this approach is the Schwarzschild solution.

Using the principle of least action, the motion equations for a singular hypersurface of arbitrary type in quadratic gravity are derived. Equations containing the “external pressure” and the “external flow” components of the surface energy–momentum tensor together with the Lichnerowicz conditions serve to find the hypersurface itself, while the remaining ones define arbitrary functions that arise due to the implicit presence of the delta function derivative. It turns out that neither double layers nor thin shells exist for the quadratic Gauss–Bonnet term. It is shown that there is no “external pressure” for null singular hypersurfaces. The Lichnerowicz conditions imply the continuity of the scalar curvature in the case of spherically symmetric null singular hypersurfaces. These hypersurfaces must be thin shells if the Lichnerowicz conditions are necessary. It is shown that for this particular case the Lichnerowicz conditions can be completely removed therefore a spherically symmetric null double layer exists. Spherically symmetric null singular hypersurfaces in conformal gravity are explored as application.

Circa 1962 Hans A. Buchdahl pioneered a program — and made significant progress — seeking vacuo configurations in pure ${{ℛ}}^2$ gravity (H. A. Buchdahl, Nuovo Cimento 23 (1962) 141). Unfortunately, he deemed the final step in his calculations impassable and prematurely suspended his pursuit. Since then, his achievements on this front have faded into dormancy. Unbeknownst to Buchdahl himself, the ${{ℛ}}^2$ vacua that he sought were within his striking distance. In our recent three-paper body of work published in Nguyen (Phys. Rev. D 106 (2022) 104004; Phys. Rev. D 107 (2023) 104008; Phys. Rev. D 107 (2023) 104009), we broke this six-decades-old impasse and accomplished his goal: A novel class of metrics, describing non-Schwarzschild spacetimes in quadratic gravity and carrying a hallmark of higher-derivative characteristic. Intriguing properties of Buchdahl-inspired spacetimes and their associated Morris–Thorne–Buchdahl wormholes — summarized herein — embody a new branch of phenomenology that transcends the Einstein–Hilbert paradigm.

May this unsung legacy of Hans A. Buchdahl

long “forsaken” and “forgotten”

finally live on

Our essay is a passageway bridging the past and the future. The past is rooted in a research program originated but left unfinished by Buchdahl going back threescore years ago. The future is the new horizons his program brings forth. In between is our labor — summarized in a three-installment “Beyond Schwarzschild–de Sitter spacetimes” series^[1–3] — to advance the Buchdahl program and achieve his ultimate goal.

It is believed that soon after the Planck time, Einstein's general relativity theory should be corrected to an effective quadratic theory. In this work we present the 3 + 1 decomposition for the zero vorticity case for arbitrary spatially homogenous spaces. We specialize for the particular Bianchi I diagonal case. The 3- curvature can be understood as a generalized potential, and the Bianchi I case is a limiting case where this potential is negligible to the dynamics. The spirit should be analogous, in some sense to the BKL solution. In this sense, a better understanding of the Bianchi I case could shed some light into the general Bianchi case.

In our contribution we present spherically symmetric solutions in four dimensions to the Einstein-Weyl (and also general quadratic) gravity admitting arbitrary value of the cosmological constant. The main attention is paid to the black hole spacetimes representing one parameter extension of the well-known Schwarzschild-(anti-)de Sitter geometry of Einstein’s general relativity. This additional parameter corresponds to the non-vanishing value of the Bach tensor at the black hole horizon. Our work thus generalises recent paper by Lü et al. (Phys. Rev. Lett. 114 (2015) 171) to any value of the cosmological constant. Moreover, using more convenient metric ansatz, the field equations form an autonomous system and their solution is explicitly obtained in an exact form of power series. The physical interpretation of these geometries is also discussed, namely we investigate specific tidal effects on free test particles or their basic thermodynamic properties.

We study static, spherically symmetric spacetimes in quadratic gravity. We show that using a conformal-to-Kundt ansatz leads to a considerable simplification of the vacuum field equations. This simplification allows us to find Schwarzschild-Bach black hole in the form of a power series expansion with coefficients given by a recurrent formula. The Schwarzschild-Bach solution is specified by two parameters, the horizon position and an additional “non-Schwarzschild parameter” b that encodes the value of the Bach tensor on the horizon. For vanishing “Bach” parameter, the Bach tensor vanishes as well and the Schwarzschild metric is recovered. The new form of the metric enables us to investigate the geometrical and physical properties of these black holes, such as tidal effects on test particles and thermodynamical quantities.

Quadratic Gravity is one of the modified theories of gravity; its action contains additional terms quadratic in Riemann tensor and its contractions. We study Kundt metrics, defined geometrically by admitting non-expanding, non-twisting and non-shearing null geodesic congruence, in the framework of Quadratic Gravity. We examine the special cases, namely the pp-waves and VSI spacetimes. Equations of geodesic deviation are calculated to provide means of physical interpretation.

Many quantum gravity theories predict several interesting phenomenological features such as minimal length scales and maximal momenta. Generalized uncertainty principles (GUPs), which are extensions of the standard Heisenberg uncertainty principle, have proven very useful in modelling the effects of such features on physics at sub-Planck energy scales. In this talk, we use a GUP modelling maximal momentum to establish a correspondence between the GUP modified dynamics of a massless spin 2 field and Stelle gravity with suitably constrained parameters. Thus, Stelle gravity can be regarded as the classical manifestation of the imposition of a momentum cutoff at the quantum gravity level. We then study the applications of Stelle gravity to cosmology. Specifically, we analytically show that Stelle gravity, when applied to a homogeneous, isotropic background, leads to inflation with exit. Lastly, using numerical simulations and data from CMB observations, we obtain strong bounds on the GUP parameter. Unlike previous works which fixed only upper bounds for GUP parameters, we show that we can bound the GUP parameter from above and from below.

We study exact vacuum solutions to Einstein gravity and quadratic gravity (QG) of the Weyl types N and III. We show that in an arbitrary dimension all Einstein spacetimes of the Weyl type N and a subclass of Einstein type III spacetimes are exact solutions to QG and we refer to explicitly known metrics within these classes. We also study a wider class of spacetimes admitting a pure radiation term in the Ricci tensor. In contrast to the Einstein case, the field equations of generic QG determine optical properties of the geometry and restrict such exact solutions to the Kundt class.

Near the singularity, gravity should be modified to an effective theory, in the same sense as with the Euler-Heisenberg electrodynamics. This effective gravity surmounts to higher derivative theory, and as is well known, a much more reacher theory concerning the solution space. On the other hand, as a highly non linear theory, the understanding of this solution space must go beyond the linearized approach. In this talk we will present some results previously published by collaborators and myself, concerning solutions for vacuum spatially homogenous cases of Bianchi types I and VII_A. These are the anisotropic generalizations of the cosmological spatially “flat”, and “open” models respectively. The solutions present isotropisation in a weak sense depending on the initial condition. Also, depending on the initial condition, singular solutions are obtained.