There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0×10(-21). It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203,000 years, equivalent to a significance greater than 5.1σ. The source lies at a luminosity distance of 410(-180)(+160)  Mpc corresponding to a redshift z=0.09(-0.04)(+0.03). In the source frame, the initial black hole masses are 36(-4)(+5)M⊙ and 29(-4)(+4)M⊙, and the final black hole mass is 62(-4)(+4)M⊙, with 3.0(-0.5)(+0.5)M⊙c(2) radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.

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The Observation of Gravitational Waves from a Binary Black Hole Merger

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Gravitation And Cosmology

Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies

Teleparallel gravity and its popular generalization gravity can be formulated as fully invariant (under both coordinate transformations and local Lorentz transformations) theories of gravity. Several misconceptions about teleparallel gravity and its generalizations can be found in the literature, especially regarding their local Lorentz invariance. We describe how these misunderstandings may have arisen and attempt to clarify the situation. In particular, the central point of confusion in the literature appears to be related to the inertial spin connection in teleparallel gravity models. While inertial spin connections are commonplace in special relativity, and not something inherent to teleparallel gravity, the role of the inertial spin connection in removing the spurious inertial effects within a given frame of reference is emphasized here. The careful consideration of the inertial spin connection leads to the construction of a fully invariant theory of teleparallel gravity and its generalizations. Indeed, it is the nature of the spin connection that differentiates the relationship between what have been called good tetrads and bad tetrads and clearly shows that, in principle, any tetrad can be utilized. The field equations for the fully invariant formulation of teleparallel gravity and its generalizations are presented and a number of examples using different assumptions on the frame and spin connection are displayed to illustrate the covariant procedure. Various modified teleparallel gravity models are also briefly reviewed.

https://iopscience.iop.org/article/10.1088/1361-6382/ab2e1f/pdf

Teleparallel theories of gravity: illuminating a fully invariant approach

We investigate the gravitational waves and their properties in various modified teleparallel theories, such as $f(T)$, $f(T,B)$, and $f(T,{T}_{G})$ gravities. We perform the perturbation analysis both around a Minkowski background and in the case where a cosmological constant is present, and for clarity we use both the metric and the tetrad languages. For $f(T)$ gravity we verify the result that no further polarization modes comparing to general relativity are present at first-order perturbation level, and we show that in order to see extra modes one should look at third-order perturbations. For nontrivial $f(T,B)$ gravity, by examining the geodesic deviation equations, we show that extra polarization models, namely the longitudinal and breathing modes, do appear at first-order perturbation level, and the reason for this behavior is the fact that although the first-order perturbation does not have any effect on $T$, it does affect the boundary term $B$. Finally, for $f(T,{T}_{G})$ gravity we show that at first-order perturbations the gravitational waves exhibit the same behavior as those of $f(T)$ gravity. Since different modified teleparallel theories exhibit different gravitational wave properties, the advancing gravitational-wave astronomy would help to alleviate the degeneracy not only between curvature and torsional modified gravity but also between different subclasses of modified teleparallel gravities.

Gravitational waves in modified teleparallel theories

We investigate the four solar system tests of gravity---perihelion precession, light bending, Shapiro time delay, gravitational redshift---in $f(T)$ gravity In particular, we investigate the solution derived by Ruggiero and Radicella, Phys Rev D 91, 104014 (2015) for a nondiagonal vierbein field for a polynomial $f(T)=T+\ensuremath{\alpha}{T}^{n}$, where $\ensuremath{\alpha}$ is a constant and $|n|\ensuremath{\ne}1$ In this paper, we derive the solutions for each test, in which Weinberg's, Bodenner and Will's, Cattani et al, and Rindler and Ishak's methods are applied, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972); Am J Phys 71 (2003); Phys Rev D 87, 047503 (2013); Phys Rev D 76, 043006 (2007) We set a constraint on $\ensuremath{\alpha}$ for $n=2$, 3 by using data available from literature

Solar System tests in f(T) gravity

In this paper, we investigate the stability of the flat Friedmann-Lemaitre-Robertson-Walker metric in $f(T)$ gravity. This is achieved by analyzing the small perturbations, $\ensuremath{\delta}$ about the Hubble parameter and the matter energy density, ${\ensuremath{\delta}}_{\mathrm{m}}$. We find that $\ensuremath{\delta}\ensuremath{\propto}\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{H}/H$ and ${\ensuremath{\delta}}_{\mathrm{m}}\ensuremath{\propto}H$. Since the Hubble parameter depends on the function $f(T)$, two models were considered (A) the power-law model $f(T)=\ensuremath{\alpha}(\ensuremath{-}T{)}^{n}$, and (B) the exponential model $f(T)=\ensuremath{\alpha}{T}_{0}(1\ensuremath{-}\mathrm{exp}[\ensuremath{-}p\sqrt{\frac{T}{{T}_{0}}}])$, where the parameters $n$ and $p$ were chosen to give comparable physical results. For the parameters considered, it was found that the solutions are stable with vanishing $\ensuremath{\delta}$ and decaying then constant ${\ensuremath{\delta}}_{\mathrm{m}}$, meaning that the matter perturbations persist during late times.

Stability of the flat FLRW metric in f(T) gravity

Symmetric teleparallel gravity (STG) offers an interesting avenue to formulate a theory of gravitation that relies neither on curvature nor torsion but only on non-metricity Q. Given the growing number of observations of gravitational waves (GWs) and their use to explore gravitational theories, in this work we investigate the GWs in various extensions of STG, focusing on their speed and polarization. For the simple STG, for theories that arise from the generalized irreducible decomposition of STG, and for f(Q) gravity, we obtain the same speed and polarizations with general relativity. For scalar - non-metricity theories, where a scalar field is nonminimally coupled to f(Q), we find that GWs propagate in general with a speed different than the one of light. Finally, for the case of f(Q,B) gravity we find that new polarizations do appear.

Polarization of gravitational waves in symmetric teleparallel theories of gravity and their modifications

Teleparallel gravity has significantly increased in popularity in recent decades, bringing attention to Einstein's other theory of gravity. In this Review, we relate this form of geometry to the broader metric-affine approach to forming gravitational theories where we describe a systematic way of constructing consistent teleparallel theories that respect certain physical conditions such as local Lorentz invariance. We first use teleparallel gravity to formulate a teleparallel equivalent of general relativity which is dynamically equivalent to general relativity but which may have different behaviors for other scenarios, such as quantum gravity. After setting this foundation, we describe the plethora of modified teleparallel theories of gravity that have been proposed in the literature. In the second part of the Review, we first survey works in teleparallel astrophysics literature where we focus on the open questions in this regime of physics. We then discuss the cosmological consequences for the various formulations of teleparallel gravity. We do this at background level by exploring works using various approaches ranging from dynamical systems to Noether symmetries, and more. Naturally, we then discuss perturbation theory, firstly by giving a concise approach in which this can be applied in teleparallel gravity theories and then apply it to a number of important theories in the literature. Finally, we examine works in observational and precision cosmology across the plethora of proposal theories. This is done using some of the latest observations and is used to tackle cosmological tensions which may be alleviated in teleparallel cosmology. We also introduce a number of recent works in the application of machine learning to gravity, we do this through deep learning and Gaussian processes, together with discussions about other approaches in the literature.

Gabriel Farrugia

Papers

Gravitational waves in modified teleparallel theories

Solar System tests in f(T) gravity

Stability of the flat FLRW metric in f(T) gravity

Polarization of gravitational waves in symmetric teleparallel theories of gravity and their modifications

Teleparallel Gravity: From Theory to Cosmology