Study note about Binary Black Hole mergers (1)

The main references of this study note are Phys. Rev. Lett. 116, 061102 and Annalen der Physik 529, 1-2. Here I focus on the first Gravitaional-Wave (GW) event GW150914 detected by the LIGO Hanford, WA, and Livingston, LA, observatories on September 14, 2015 at 09:50:45 UTC to illustrate the Binary Black Hole (BBH) mergers.

GW150914 is produced by a Binary Black Hole system which is one kind of typical GW signal sources. As such a system comprises two black holes, it is a real 2-body system. To better describe the BBH system, here I begin with the basic features of its main component - the black holes. In this note, I mainly introduce the properties of the Black holes.

Black Hole

Black Hole is a region of space-time where the gravitational field is so strong that nothing, not even light and other electromagnetic waves, can escape. The boundary of no escape is called the event horizon. No-hair theorem states that a stationary black hole can be completely characterized by only three independent externally observable classical parameters: mass, electric charge, and angular momentum.

One kind of typical black holes is the static black hole described by Schwarzschild solution. Such a static black hole has neither electric charge nor angular momentum. Hence, according to no-hair theorem, mass is the only identity of the static black hole. If there are two Schwarzschild Black Holes with the same mass, but the first black hole was made by collapsing ordinary matter whereas the second was made out of antimatter; the no-hair theorem indicates that they will be completely indistinguishable to an observer outside the event horizon situated at the Schwarzschild radius ($r_{s}$). Schwarzschild radius associated with the mass $m$ is mathematically expressed as:

$$r_{s}(m) = \frac{2Gm}{c^{2}} = 2.95(\frac{m}{M_{\odot}})\text{ km}$$

where $M_{\odot} = 1.99 \times 10^{30}\text{ kg}$ is the solar mass, $G = 6.67 \times 10^{-11} \text{ m}^{3}/\text{kg s}$ is Newton’s gravitational constant, and $c = 2.998 \times 10^{8} \text{ m/s}$ is the speed of light. (Phys. Rev. Lett. 116, 061102) According to the hoop conjecture, if a non-spinning mass is compressed to within its corresponding Schwarzschild radius, then it must form a black hole. Once the black hole is formed, any object that comes within this radius can no longer escape out of it.