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

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

인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

Let L be a finite-dimensional vector space over a field &#x1d542; of characteristic zero, [−,−] : L × L → L an anti-commutative product on L and ω : L × L → &#x1d542; a bilinear form on L. The triple (L, [−,−], ω) is called an ω-Lie algebra if [[y,z],x] + [[z,x],y] + [[x,y],z] = ω(y,z)x + ω(z,x)y + ω(x,y)z (graded ω-Jacobi identity) for all x,y,z ∈ L. In this paper, we introduce the notion of an ω-Lie superalgebra. We study elementary properties and representations of ω-Lie superalgebras. We classify all 3- and 4-dimensional ω-Lie superalgebras over the field of complex numbers.

On ω-Lie superalgebras

https://www.fuw.edu.pl/~nurowski/prace/confode.pdf

Differential equations and conformal structures

Obstructions to conformally Einstein metrics in n dimensions

The equivalence problem for second-order ordinary differential equations (ODEs) given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate three-dimensional Cauchy–Riemann structures. This approach enables an analogue of the Fefferman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second-order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs.

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Three-dimensional Cauchy?Riemann structures and second-order ordinary differential equations

We present a covariant definition of inertial forces in general relativity (gravitational, centrifugal, Euler and Coriolis-Lense-Thirring) which is valid in all spacetimes including ones with no symmetry.

https://iopscience.iop.org/article/10.1088/0264-9381/10/10/001/pdf

Covariant definition of inertial forces

A Lorentzian manifold is defined here as a smooth pseudo-Riemannian manifold with a metric tensor of signature (2n+1,1). A Robinson manifold is a Lorentzian manifold M of dimension ⩾4 with a subbundle N of the complexification of TM such that the fibers of N→M are maximal totally null (isotropic) and [SecN,SecN]⊂SecN. Robinson manifolds are close analogs of the proper Riemannian, Hermite manifolds. In dimension 4, they correspond to space-times of general relativity, foliated by a family of null geodesics without shear. Such space-times, introduced in the 1950s by Ivor Robinson, played an important role in the study of solutions of Einstein's equations: plane and sphere-fronted waves, the Godel universe, the Kerr solution, and their generalizations, are among them. In this survey article, the analogies between Hermite and Robinson manifolds are presented in considerable detail.

Pawel Nurowski

Papers

Differential equations and conformal structures

Obstructions to conformally Einstein metrics in n dimensions

Three-dimensional Cauchy?Riemann structures and second-order ordinary differential equations

Covariant definition of inertial forces

Robinson manifolds as the Lorentzian analogs of Hermite manifolds