Gravitational Waves

Type of physical science: Relativity

Field of study: General relativity

The general theory of relativity predicts oscillatory distortions in the geometry of space-time that propagate, at the speed of light, away from any mass that accelerates or changes shape. Binary pulsars exhibit changes in their orbits that are ascribed to energy carried away in these waves. Direct laboratory measurements of gravitational waves will be an important confirmation of theoretical expectations and will provide a valuable new tool for astronomy.

Overview

Gravitational waves, changes in the geometry of space-time predicted by Albert Einstein's general theory of relativity, should travel outward at the speed of light from any mass that experiences acceleration or change of shape. The theoretical properties of gravitational waves have been known for many years, but their direct detection in laboratory experiments remains an extremely difficult task primarily because of the extreme weakness of gravitation compared to the other long-range interaction in nature associated with the propagation of waves: electromagnetism. Nevertheless, many of the qualitative properties that gravitational waves are expected to have can be understood in terms of their similarity to or difference from these more familiar electromagnetic waves.

Whenever an electric charge experiences oscillatory motion, the electric and magnetic fields caused by the charge and its current also oscillate, so that energy is carried from regions near the charge to regions farther away. This propagation of the variations in the fields occurs at the speed of light and is an electromagnetic wave. The electrical field outside any spherically symmetrical distribution of charge is the same as if all the charge were concentrated at the center of the sphere, so an isolated charge (electric monopole) cannot produce waves merely by oscillating in size. The simplest system that can be a source of electromagnetic waves is a pair of equal charges of opposite sign separated by a displacement that varies in time. Such a system is called an electric dipole. It is characterized by a dipole moment vector, which is the product of the magnitude of one of the charges and the displacement between them. Dipole radiation is produced whenever the dipole moment of a collection of charges is accelerated. The amplitude of the electric or magnetic field of the wave is proportional to the acceleration (second time rate of change) of the dipole moment, and the intensity (power across a unit area perpendicular to the motion of these transverse waves) is proportional to the square of the field amplitude. A ring of identical charges in a plane crossed by such a wave can be used as a detector. At any particular moment, all the charges will be displaced in the same direction by the electric field vector; as time passes, the ring shape will not be distorted, but its center will oscillate along a line. More complex oscillation patterns are possible, but the power in such higher multipole moment waves decreases more rapidly with distance from the source than the power in dipole waves.

In Sir Isaac Newton's theory, gravity may be described by a field analogous to the electrostatic field that obeys Coulomb's law. Newtonian gravity theory, however, which has no feature analogous to the magnetic field, predicts that when masses oscillate, their associated gravitational fields readjust instantaneously everywhere in space. Such instantaneous action at a distance is incompatible with the relativistic postulate of the invariant speed of light, which is the basis for the unification of space and time into a four-dimensional description that treats them on an equal footing. Thus, Einstein was motivated, in creating his general theory of relativity, to extend the unification of space and time for observers in inertial frames of reference to all observers, including those in accelerated frames of reference. In general relativity, the gravitational field is identified with the metric tensor, the mathematical entity that describes the geometry of four-dimensional space-time. Thus, changes in the spatial part of the gravitational field are intimately related to the dependence of the gravitational field on time. In effect, the tensor gravitational field incorporates aspects analogous to both electric and magnetic fields.

Thus, solutions of Einstein's field equations include gravitational waves traveling in empty space away from their sources (accelerated masses), just as solutions of James Clerk Maxwell's field equations include electromagnetic waves traveling in empty space away from their sources (accelerated charges). As in Newtonian theory, according to general relativity, the gravitational field outside any spherically symmetrical distribution of mass is the same as if all the mass were concentrated at the center, so an isolated mass (gravitational monopole) cannot produce waves merely by oscillating in size. Since all mass has the same sign, gravitational dipoles are impossible. Therefore, the simplest source of gravitational waves is a quadrupole moment that varies with time. An extended object whose shape oscillates between prolate (cigar-shaped) and oblate (pancake-shaped) forms is one example. Another example is a binary system, two bodies in orbit around their common center of mass. A completely general analysis of gravitational waves is complicated by the fact that Einstein's equations are not linear, so that in principle, linear superposition does not apply. This is the mathematical expression of the concept that gravitational energy is equivalent to mass, which is the source of gravitational fields: Gravity gravitates. If the gravitational waves are sufficiently weak that they may be treated as a linearized perturbation of the background gravitational field on which they travel, however, then the amplitude of the gravitational waves is proportional to the third time rate of change (sometimes called the jerk) of the quadrupole moment. For high frequencies (wavelengths that are short compared to the length scale of substantial change in the background field), the gravitational wave energy can be localized, and it is found that the intensity of the gravitational wave at any point is proportional to the square of its amplitude. For use as a detector, consider a ring of identical masses in a plane crossed by such a wave. At any particular moment, masses at various points are displaced by varying amounts in different directions, so that as time passes the shape of the ring oscillates, though its center of mass does not move. If these masses are elastically coupled into a structure with a natural frequency matching that of the gravitational waves, it should be possible for them to detect these waves through resonance. This is the basic idea for the first gravitational wave detectors that were constructed, which also presents another approach to detection, comments on expected astrophysical sources of gravitational waves, and discusses the effects on observable features of these sources.

Applications

The quest to develop gravitational wave detectors has produced numerous advances in high-precision measurement technology. The first gravitational wave antennae were a pair of aluminum cylinders, each about 2 meters long and 1 meter in diameter and weighing about 1,300 kilograms. Piezoelectric transducers, which produce electric currents when subjected to strain, were glued to these cylinders and constantly produced signals, as numerous random disturbances made the cylinders vibrate. The effects at the sensitivity threshold corresponded to changes in the length of these cylinders that were comparable to the diameter of the nucleus of an atom. One antenna was set up at the University of Maryland, near Washington, D.C., and the other was set up at the Argonne National Laboratory, near Chicago, Illinois, and attention was focused on coincidences. It was argued that the probability of simultaneous identical random effects at both locations was extremely small, while a gravitational wave reaching the earth from an astrophysical source would make both cylinders vibrate in the same way at the same time. By 1968, after years of refinement of the apparatus and analysis, such coincidences, which were seen daily, displayed a rough correlation with the times when the cylinders' crude directional sensitivity was aligned toward the center of the galaxy. Assuming that these coincidences were caused by events near the center of the galaxy that produced isotropic bursts of gravitational waves, however, led to amazingly large estimates for the total power carried by such radiation.

Furthermore, attempts to replicate these results by several other groups using more sensitive detectors were fruitless. Thus, it is now generally agreed that the coincidences recorded in this pioneering experiment were not caused by gravitational waves.

Since a primary limitation to the sensitivity of these early detectors was thermal noise, several groups are constructing antennae that will be monitored when cooled to liquid helium temperatures. This is expected to increase amplitude sensitivity by about three orders of magnitude (powers of ten), and perhaps two more orders of magnitude could be achieved by cooling to the millidegree range, the limit of present technology. Detection through resonant oscillation, however, can succeed only if the frequency of the signal matches that of the detector.

This means that such detectors can respond only to one component of a burst of radiation.

Another approach is under development that should, in principle, be capable of recording the entire wave-form of an incoming gravitational wave. Several groups have begun construction of laser interferometer detectors, in which a gravitational wave would change the lengths of perpendicular interferometer arms and measurement would involve the observation of the resulting fringe shift in the interference pattern. Technology indicates the feasibility of interferometers larger and more precise than any yet completed, which might achieve sensitivities even greater than the limit on resonant cylinders at millidegree temperatures. In all, about sixteen groups around the world have begun the construction of detectors for gravitational waves.

Although experimental attempts to observe gravitational waves in laboratories have not yet produced incontrovertible evidence of their detection, the study of a binary pulsar conclusively indicates that gravitational waves can have significant effects on the behavior of astronomical systems. The binary pulsar PSR 1913+16, discovered in 1974, is a pulsar orbiting an unseen companion with a period of about 7.75 hours and carrying its own clock (the neutron star's rotation) with a period accurate to twelve significant figures. By applying relativistic formulas to the extremely precise Doppler shift measurements, it was possible to determine, among other things, accurate values for the masses of the pulsar and its compact companion as well as the distance between them. These three parameters determine the quadrupole moment of the system, whose time rates of change are related to the orbital period. Continuous observation since discovery has shown that the orbital period of this binary system is decreasing at a rate of 76 microseconds per year. This is in excellent agreement with the rate, predicted by the quadrupole formula, at which it should be changing as a result of energy being carried away by gravitational waves.

Context

Albert Einstein predicted gravitational waves as soon as he had created the general theory of relativity. In 1916, he published a paper in which he derived a formula to determine the rate at which these waves would carry energy away from a source. Although the mathematical assumptions he made to simplify the calculation were not completely valid, and there was a trivial arithmetic error (quickly pointed out by Sir Arthur Stanley Eddington) that made his answer two times too large, the basic concept was correct: Gravitational waves are a consequence of a field theory of gravity consistent with the principle of relativity. Because of the nonlinearity of Einstein's field equations, it has not been possible to derive exact solutions that display the interaction of gravitational waves and their sources in general relativity. This fact, however, remained for a long time merely an arcane mathematical problem of interest only to theoreticians. The extreme weakness predicted for these waves by the application of the quadrupole formula to then known astronomical sources indicated that there was no hope of direct detection with the laboratory apparatus of that time, so little more was done on this topic for more than forty years. For example, the gravitational waves produced by the orbital motion of the earth around the sun are calculated to have a total radiated power of only 200 watts. It was not until the late 1950's that Joseph Weber built the first gravity wave antenna. Although his eventual claim of possible detection was not corroborated, his pioneering efforts stimulated vigorous experimental activity by many groups. Meanwhile, the first discovery of a binary pulsar by Russell Hulse and Joseph Taylor in 1974 provided an astrophysical laboratory in which to test several predictions of the general theory of relativity, including the decay of a binary orbit caused by energy being carried away by gravitational waves. The remarkably precise agreement between theory and observation in this case emphasized the importance of gravitational waves in cosmic processes. Bursts of gravitational waves are expected to accompany such cataclysmic events as supernova explosions, the formation of neutron stars and black holes in space by gravitational collapse, and the fusion of such compact objects at the end of their evolution in binary systems.

The pulses of radiation from such processes are expected to be of short duration but to include components of many frequencies. The universe should also provide continuous waves of long duration from such sources as oscillating neutron stars or binary stars and pulsars. These gravitational waves should have rather narrow frequency spectra, whose detailed study would be a source of information about the internal structures of their sources. A comparison of the sensitivity estimates for gravitational wave detectors under construction with the expected amplitudes and frequencies from the tremendously varied list of possible astrophysical sources indicates that one should expect to witness direct laboratory measurement of the properties of gravitational waves in the near future. Further development of technology will open the rich new field of gravitational wave astronomy. Since gravitational waves are not absorbed by matter in the same way as electromagnetic radiation, this new tool will facilitate the detailed exploration of many regions of the universe, such as the centers of galaxies, which are now hidden from view.

Principal terms

BINARY PULSAR: a double star system, one of whose components is a rapidly rotating neutron star that emits radio waves

METRIC: a quantity that combines particular coordinate values into measures of geometrical properties independent of choice of description

QUADRUPOLE MOMENT: a measure of how much the shape of an object or system departs from spherical symmetry

SPACE-TIME: in relativity theory, a four-dimensional space unifying various observers' measurements of three dimensions of physical space and one dimension of time

TENSOR: a quantity associated with a number of directions, called its rank; the metric is a second-rank tensor, while vectors are first-rank tensors.

Bibliography

Bartels, Meghan, et. al. “What is the Theory of General Relativity?” Space.com, 14 May 2023, www.space.com/17661-theory-general-relativity.html. Accessed 28 Sep. 2023.

Boyd, Padi. “The Invisible World of Gravitational Waves.” NASA's Curious Universe, National Aeronautics and Space Institute, 28 Feb. 2023, www.nasa.gov/mediacast/the-invisible-world-of-gravitational-waves. Accessed 28 Sep. 2023.

“What are Gravitational Waves?” LIGO, California Institute of Technology, www.ligo.caltech.edu/page/what-are-gw. Accessed 28 Sep. 2023.