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

We report the observation of a gravitational-wave signal produced by the coalescence of two stellar-mass black holes. The signal, GW151226, was observed by the twin detectors of the Laser Interferometer Gravitational-Wave Observatory (LIGO) on December 26, 2015 at 03:38:53 UTC. The signal was initially identified within 70 s by an online matched-filter search targeting binary coalescences. Subsequent off-line analyses recovered GW151226 with a network signal-to-noise ratio of 13 and a significance greater than 5 σ. The signal persisted in the LIGO frequency band for approximately 1 s, increasing in frequency and amplitude over about 55 cycles from 35 to 450 Hz, and reached a peak gravitational strain of 3.4+0.7−0.9×10−22. The inferred source-frame initial black hole masses are 14.2+8.3−3.7M⊙ and 7.5+2.3−2.3M⊙ and the final black hole mass is 20.8+6.1−1.7M⊙. We find that at least one of the component black holes has spin greater than 0.2. This source is located at a luminosity distance of 440+180−190 Mpc corresponding to a redshift 0.09+0.03−0.04. All uncertainties define a 90 % credible interval. This second gravitational-wave observation provides improved constraints on stellar populations and on deviations from general relativity.

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GW151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence

The status of experimental tests of general relativity and of theoretical frameworks for analyzing them is reviewed and updated. Einstein’s equivalence principle (EEP) is well supported by experiments such as the Eotvos experiment, tests of local Lorentz invariance and clock experiments. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse-Taylor binary pulsar, and a growing family of other binary pulsar systems is yielding new tests, especially of strong-field effects. Current and future tests of relativity will center on strong gravity and gravitational waves.

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The Confrontation Between General Relativity and Experiment

QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

We describe early success in the evolution of binary black-hole spacetimes with a numerical code based on a generalization of harmonic coordinates. Indications are that with sufficient resolution this scheme is capable of evolving binary systems for enough time to extract information about the orbit, merger, and gravitational waves emitted during the event. As an example we show results from the evolution of a binary composed of two equal mass, nonspinning black holes, through a single plunge orbit, merger, and ringdown. The resultant black hole is estimated to be a Kerr black hole with angular momentum parameter a[approximate]0.70. At present, lack of resolution far from the binary prevents an accurate estimate of the energy emitted, though a rough calculation suggests on the order of 5% of the initial rest mass of the system is radiated as gravitational waves during the final orbit and ringdown.

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Evolution of binary black-hole spacetimes.

The recent gravitational wave observations GW150914 and GW151226 reported by the LIGO and Virgo collaborations confirmed a key prediction of general relativity (GR). In this recent Physical Review D paper, the authors ask whether these events are also compatible with a variety of modifications of GR and alternative theories of gravity. In this wide-ranging and thorough study, the authors find that the recent observations put severe constraints on these non-standard theories.

Theoretical physics implications of the binary black-hole mergers GW150914 and GW151226

We consider the concept of fundamental bias in gravitational wave astrophysics as the assumption that general relativity is the correct theory of gravity during the entire wave-generation and propagation regime. Such an assumption is valid in the weak field, as verified by precision experiments and observations, but it need not hold in the dynamical strong-field regime where tests are lacking. Fundamental bias can cause systematic errors in the detection and parameter estimation of signals, which can lead to a mischaracterization of the Universe through incorrect inferences about source event rates and populations. We propose a remedy through the introduction of the parametrized post-Einsteinian framework, which consists of the enhancement of waveform templates via the inclusion of post-Einsteinian parameters. These parameters would ostensibly be designed to interpolate between templates constructed in general relativity and well-motivated alternative theories of gravity, and also include extrapolations that follow sound theoretical principles, such as consistency with conservation laws and symmetries. As an example, we construct parametrized post-Einsteinian templates for the binary coalescence of equal-mass, nonspinning compact objects in a quasicircular inspiral. The parametrized post-Einsteinian framework should allow matched filtered data to select a specific set of post-Einsteinian parameters without a priori assuming the validity of the former, thus either verifying general relativity or pointing to possible dynamical strong-field deviations.

Fundamental theoretical bias in gravitational wave astrophysics and the parametrized post-Einsteinian framework

The low-energy limit of string theory contains an anomaly-canceling correction to the Einstein-Hilbert action, which defines an effective theory: Chern-Simons (CS) modified gravity. The CS correction consists of the product of a scalar field with the Pontryagin density, where the former can be treated as a background field (nondynamical formulation) or as an evolving field (dynamical formulation). Many solutions of general relativity persist in the modified theory; a notable exception is the Kerr metric, which has sparked a search for rotating black hole solutions. Here, for the first time, we find a solution describing a rotating black hole within the dynamical framework, and in the small-coupling/slow-rotation limit. The solution is axisymmetric and stationary, constituting a deformation of the Kerr metric with dipole scalar ``hair,'' whose effect on geodesic motion is to weaken the frame-dragging effect and shift the location of the innermost stable circular orbit outwards (inwards) relative to Kerr for corotating (counterrotating) geodesics. We further show that the correction to the metric scales inversely with the fourth power of the radial distance to the black hole, suggesting it will escape any meaningful bounds from weak-field experiments. For example, using binary pulsar data we can only place an initial bound on the magnitude of the dynamical coupling constant of ${\ensuremath{\xi}}^{1/4}\ensuremath{\lesssim}{10}^{4}\text{ }\text{ }\mathrm{km}$. More stringent bounds will require observations of inherently strong-field phenomena.

Dynamical Chern-Simons modified gravity: Spinning black holes in the slow-rotation approximation

We describe the behavior of 5-dimensional black strings, subject to the Gregory-Laflamme instability. Beyond the linear level, the evolving strings exhibit a rich dynamics, where at intermediate stages the horizon can be described as a sequence of 3-dimensional spherical black holes joined by black string segments. These segments are themselves subject to a Gregory-Laflamme instability, resulting in a self-similar cascade, where ever-smaller satellite black holes form connected by ever-thinner string segments. This behavior is akin to satellite formation in low-viscosity fluid streams subject to the Rayleigh-Plateau instability. The simulation results imply that the string segments will reach zero radius in finite asymptotic time, whence the classical space-time terminates in a naked singularity. Since no fine-tuning is required to excite the instability, this constitutes a generic violation of cosmic censorship.

Frans Pretorius

Papers

Evolution of binary black-hole spacetimes.

Theoretical physics implications of the binary black-hole mergers GW150914 and GW151226

Fundamental theoretical bias in gravitational wave astrophysics and the parametrized post-Einsteinian framework

Dynamical Chern-Simons modified gravity: Spinning black holes in the slow-rotation approximation

Black strings, low viscosity fluids, and violation of cosmic censorship.