Gravitational waves and the need for numerical relativity

In the previous page we have formulated the Einstein field equations in a neat and compact form. This should not hide the fact that the Einstein tensor $ \hbox{\vec G}_{\alpha \beta}$ is in fact a complicated function of the metric $ \hbox{\vec g}_{\alpha \beta}$ and its first and second derivatives. Due to the symmetry of the Einstein tensor and the energy momentum tensor the field equations represent 10 coupled, non-linear partial differential equations, which written explicitly may contain of the order of 100,000 terms in the general case.

It therefore came as quite a surprise when Karl Schwarzschild found a non-trivial, analytic solution to these equations just some months after their publication. Since then many analytic solutions have been found and a whole branch of the studies of general relativity is concerned with their classification. Enormous insight into the structure of general relativity has been gained from these analytic solutions, but due to the complexity of the field equations these solutions are normally idealized and restricted by symmetry assumptions. In order to obtain accurate descriptions of astrophysically relevant scenarios one may therefore have to go beyond purely analytic studies.

A particularly important area of research connected with general relativity that has emerged in recent years concerns the detection of gravitational waves. In analogy to the prediction of electromagnetic waves by the Maxwell equations of electrodynamics, the Einstein field equations admit radiative solutions with a characteristic propagation speed given by the speed of light. In contrast to electromagnetic waves, who will only exert a force on charged particles, gravitational waves will have an effect on any kind of matter. This effect is best illustrated by considering a set of freely moving test particles arranged in a circle as in the figure below. If a gravitational wave falls onto the plane spanned by these test particles (i.e. coming out of the picture), it will cause the particles to move. Gravitational waves are known to possess two states of polarization called plus (+) and cross (X) and the induced motion of the particles will depend on the polarization as demonstrated in the animation.

[+ polarization] [X polarization]
+ polarization (top), X polarization (bottom): Click to animate and leave cursor in image (70 kB each)


Due to the weak coupling constant of the gravitational interaction, which is a factor of $ 10^{40}$ smaller than the electromagnetic coupling constant, the particle motion induced by gravitational waves will have an extremely small amplitude in reality. If one considers for example a bar of a length of several kilometres, estimates have shown that the detection of gravitational waves requires one to measure changes in length orders of magnitude smaller than the diameter of an atomic nucleus. Even though attempts to detect gravitational radiation go back to the work of Joe Weber in the early sixties, it is only the recent advance of computer and laser technology that provides scientists with a realistic chance of success. The current generation of gravitational wave detectors GEO-600, LIGO, TAMA and VIRGO that have been constructed for this purpose are complex multi-national collaborations and have recently gone online or are expected to go online in the near future. The following image shows an aerial view over the LIGO detector in Livingston, Lousiana.


These detectors essentially consist of two bars with a length of 100s of metres or kilometres arranged in the shape of a 'L'. Variations in length of these bars are measured by lasers to very high accuracy. Due to the extreme smallness of the signals, however, the accumulation of data over several years will be necessary to improve the chances of a positive identification of gravitational wave signals from extra-galactic sources.

Confidence in the existence of gravitational waves has been significantly boosted by the Nobel prize winning discovery of the binary neutron star system PSR1913+16 (sometimes referred to as the "Hulse and Taylor pulsar") The spin-down of this system has been found to agree remarkably well with the energy-loss predicted by general relativity due to the emission of gravitational waves and is generally accepted as indirect proof of the existence of gravitational radiation.

In order to simplify the enormous task of detecting gravitational waves, it is vital to obtain information about the structure of the signals one is looking for. It is necessary for this purpose to accurately model the astrophysical scenarios that are considered likely sources of gravitational waves and extract the corresponding signals from these models. According to Birkhoff's theorem the Schwarzschild solution, which describes a static, spherically symmetric vacuum spacetime, is the only spherically symmetric, asymptotically flat solution to the Einstein vacuum field equations. As a consequence a spherically symmetric spacetime, even if it contains a radially pulsating object, will necessarily have an exterior static region and be non-radiating. It is necessary, therefore, to use less restrictive symmetry assumptions in the modelling of astrophysical sources of gravitational waves. In fact the most promising sources of gravitational waves currently under consideration are the in-spiralling and merger of two compact bodies (neutron stars or black holes) and complicated oscillation modes of neutron stars that increase in amplitude due to the emission of gravitational waves by extracting energy from the rotation of the star. Even though a great deal of information about these scenarios has been gained from approximative studies, such as the post-Newtonian formalism or the use of perturbative techniques, a detailed simulation will require the solution of the Einstein equations in three dimensions. The complicated structure of the corresponding models in combination with the enormous advance in computer technology has given rise to numerical relativity, the computer based generation of solutions to Einstein's field equations.