7 January 2005

Black Holes and Gravitational Waves

Scott Little

Editor's note: This article originally appeared in the 3 December 2004 issue of The Citizen Scientist. George Hrabovsky has kindly formatted the equations in Mathematica and made editorial revisions approved by author Scott Little.

Introduction

Black holes are created when a star whose mass is greater 1.2 Solar Masses (called the Chandrasekhar Limit) collapses and forms a region with a gravitational force so strong that not even light can escape. Black holes apparently radiate as Stephen Hawking showed in papers published in 1974 and 1975.  This radiation bears his name, Hawking Radiation.  According to Hawking black holes radiate like black bodies with a temperature given by,

where   and h is Planck's constant and has the value , is the acceleration due to gravity, is the speed of light in a vacuum and has the value of , and is the so-called Schwarzschild radius.  The Schwarzschild radius is a function of the mass of the black hole and the universal constant of gravitation ().

Gravitational Waves

Einstein first predicted the existence of gravitational waves in 1918. They were thought to exist due to the fact that the presence of mass and/or energy warps spacetime. This warping can collide with another warping to produce a ripple effect. This effect is similar to throwing two rocks into a pond and watching the water ripples collide with each other. In 1961 Joseph Weber developed a device consisting of an aluminum cylinder two meters long oriented broadside to the gravitational waves. The waves were predicted to first compress, and then stretch the bar's ends to cause it to resonate like a tuning fork. The resonant frequency was predicted to be below about 10,000 Hertz (cycles/second).

If the bar resonated at this frequency without being effected by outside forces, Weber could infer the existence of gravitational waves. The sensor used to measure this tiny perturbation was made of a piezoelectric crystal.

In 1999,work was completed on The Laser Interferometer Gravitational-Wave Observatory (LIGO). The project was a joint effort between scientists at The California Institute of Technology and Massachusetts Institute of Technology.

Figure 1. Diagram of the LIGO (courtesy of www.ligo.caltech.edu/LIGO_web/about/factsheet.html).

The detectors are looking for distortions of the tubes on a very small scale, centimeters (or 0.1 Fermi units). This change is to be measured over the four-km length of the tube. These distortions, should they be observed, will be caused by disturbances in spacetime that will shrink the detector on the order of a tenth of a Fermi as noted above. These distortions will be detectable by an outside observer, because the gravitational wave will propagate at the speed of light and will not cause this effect in both arms at the same time. They will compare arms and note any disparity.

At least two detectors are required to rule out interference from local disturbances, such as seismic activity. An identical signal at both detectors indicates a gravity wave, while a signal at only one detector can be ignored. Ideally, LIGO observations will be compared with results from gravitational wave detectors now under construction in several other countries.

Black Holes and Gravitational Waves

The causes of measurable gravitational waves are thought to be events that cause ripples in the fabric of spacetime. A typical event would be either an explosion of a massive star (supernova) or the collision of two compact objects (such as neutron stars and black holes). One black hole is not sufficient to produce measurable gravitational waves, because the all fluctuations are radiated away completely when the black hole forms.

However, if two black holes were to collide, measurable ripples in spacetime could propagate outwards; much like the previous mentioned ripples on the surface of a pond. This is illustrated in Fig. 2.

Figure 2. Diagram illustrating hypothetical propagation of ripples in the space-time
fabric caused by the collision of two black holes. Diagram by Scott Little.

The force of these waves may be increased by more collisions with subsequent waves, as when ocean waves bounce off a dock and collide with incoming waves. This is known as the Wedge Effect. [Editor's note: I am not aware of anyone using this terminology in gravitational research.]

The curvature of space-time is the underlying reason that gravitational waves are possible. The tensor field equation that defines General Relativity and the curvature due to gravity is:

Where = the Ricci curvature tensor, = the Ricci curvature scalar,   = the metric tensor, = the stress-energy tensor that represents the presence of matter and energy in a volume of spacetime.  The is the cosmological constant derived by Einstein,

The symbol represents the energy density in the vacuum.

The basic idea behind the field equation is that the right hand side of the equation is the presence of matter and energy, on the left hand side there is the curvature of spacetime.  The equation tells us that the presence of matter and energy causes the curvature of spacetime, while the curvature of spacetime causes the density of matter and energy to behave in a special way.  In an oversimplified way it states, "curvature tells matter how to move, and matter tells space how to curve (4).

Have gravitational waves been detected?

While LIGO scientists have established noise limits for their system, so far they have not unequivocally detected gravitational waves. Because detectable gravitational wave events are believed to be rare, years of simultaneous observations by the LIGO detectors may be required before a gravitational wave event is detected. Meanwhile, LIGO scientists have proposed an improved detector with ten times the sensitivity of the current LIGO.

For more information about the current status of LIGO, see the LIGO web site www.ligo.caltech.edu/LIGO_web/about/factsheet.html. For more information about gravitational waves, consider searching on relevant key words using a search engine such as www.google.com.

Resources

1. www.physicstoday.org/pt/vol-54/iss-7/p74.html
2. http://casa.Colorado.edu/~ajsh/hawk.html
3. www.ligo.caltech.edu/LIGO_web/about/factsheet.html
4. http://en.wikipedia.org/wiki/General_relativity

Created by Mathematica  (December 3, 2004)