Gravitational Waves

This page provides a brief overview of gravitational waves within the theory of General Relativity. For a more detailed discussion we direct the reader to the standard texts such as K.S. Thorne, in Three Hundred Years of Gravitation, edited by S.W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1987).

The theory of General Relativity predicts that gravitational waves will be produced by a time varying quadrupole moment. The wave itself creates a time varying tidal force that propagates at the speed of light. The field of a gravitational wave is often described by a time dependent strain tensor. The size of the strain tensor will indicate how strongly the gravitational wave will curve spacetime.

A gravitational wave will induce a stress on an extended body. For a bar antenna, the stress from a gravitational wave will cause the ends of the bar to contract and expand. The force that a gravitational wave exerts on the bar depends on both the density of the bar's material as well as the bar's length. By increasing the mass of the bar we increase the force a gravitational wave exerts on the bar, thus increasing the bar's cross-section.

Figure 1: The distortion of a ring of test particles during one cycle of a gravitational wave traveling in the z direction. The effect of both linear polarizations is shown.

The image at right shows the effect on a ring of test masses from a polarized wave traveling in the z direction. The wave will compress the ring in one direction, while expanding it in the other. The two polarizations are equivalent except for a 45 degree rotation about the propagation axis.

Gravitational waves from astrophysical sources can be divided into three classes: bursts, periodic waves, and stochastic waves. Bursts are emissions that last for a very short time, only a few cycles. Potential sources of bursts include: the collapse of a star to a neutron star or black hole, coalescence of compact binaries, and the fall of stars and small black holes into supermassive black holes. Sources of periodic waves include: rotating neutron stars and binary stars. Stochastic waves are a potential stationary random background of gravitational waves. A stochastic background might come from primordial gravitational waves or from the superposition of the radiation from a large population of binary stars in our galaxy and other galaxies.

Burst sources are the most likely to have large amplitudes at higher frequency; therefore, they are the best candidates for detection by resonant mass detectors. Burst sources must be very violent events. One candidate is the gravitational collapse of a massive star to form a neutron star. The strength of emission depends on the degree of non-sphericity in the collapse and also on the speed of the collapse. A perfectly spherical collapse will produce no waves, whereas a highly antisymmetric collapse will produce strong waves. The burst of gravitational waves will cover a large frequency bandwidth, however the newly created neutron star is expected to have quadrupole modes that resonate on the order of 2 kHz, creating gravitational waves at that frequency. Supernovae are thought to occur at a rate of about one per 40 years in our galaxy and at a rate of several per year at a distance out to the center of the Virgo cluster.

A gravitational collapse might also form a black hole instead of a neutron star. Again, the greater the non-sphericity the collapse, the stronger the gravitational waves emitted. The rate for this type of collapse is predicted to be about 1/3 the rate for collapse to a neutron star.

A third source for bursts is the coalescence of compact binaries. These are close binary systems containing neutron stars or black holes. The binary pulsar PSR1913+16 is an example of a coalescing binary; it is predicted to coalesce in 3.5e8 years.

Although there has not been any confirmed direct evidence of gravitational waves, there is strong indirect evidence of their existence. In 1975 Hulse and Taylor observed the binary pulsar PSR1913+16, whose period changes at a rate consistent with General Relativistic predictions of gravitational wave emissions. For the past 20 years astronomers have continued to observe the pulsar and to this day, the orbital decay remains consistent with the predictions of General Relativity.

This page is a modified excerpt of Stephen Merkowitz's Dissertation.