General relativity is the flagship of applied mathematics. Although from its inception this has been regarded as an extraordinarily difficult theory, it is in fact the simplest theory to consummate the union of special relativity and Newtonian gravity. Einstein's ‘popular articles’ set a high standard which is now emulated by many in the range of introductory textbooks. Having mastered one of these the new reader is recommended to move next to one of the more specialized monographs, e.g. Chandrasekhar, 1983, Kramer et al., 1980, before considering review anthologies such as Einstein (centenary), Hawking and Israel, 1979, Held, 1980 and Newton (tercentenary), Hawking and Israel, 1987. As plausible gravitational wave detectors come on line in the next decade (or two) interest will focus on gravitational radiation from isolated sources, e.g., a collapsing star or a binary system including one, and I have therefore chosen to concentrate in this book on the theoretical background to this topic.

The material for the first three chapters is based on my lecture courses for graduate students. The first chapter of this book presents an account of local differential geometry for the benefit of the beginner and as a reminder of notation for more experienced readers. Chapter 2 is devoted to two-component spinors which give a representation of the Lorentz group appropriate for the description of gravitational radiation. (The relationship to the more common Dirac four-component spinors is discussed in an appendix.) Far from an isolated gravitating object one might expect spacetime to become asymptotically Minkowskian, so that the description of the gravitational field would be especially simple.

Introduction

Why good broadband seismic isolation is an essential design feature for laser interferometric antennas

One of the key features of laser interferometric detectors is the potential wideband nature of their operation. Proposed long baseline detectors are intended to achieve sensitivities in the region h ∼ 10–21 to 10–22 or better over a range of frequencies f from a few tens of hertz (possibly as low as 10 Hz) to a few kilohertz in a bandwidth Δf ≈ f / 2. If the performance of such detectors is limited by photon shot noise in the output light, for constant light power the effect of this noise source decreases towards lower frequencies for a constant light intensity, when the detectors are operated in searches for burst sources or a stochastic background. However, other sources of noise have spectra which rise towards lower frequencies. These include thermal noise from the pendulum suspensions of the masses, and, more particularly, seismic noise. In fact it is likely that the extent to which these detectors can be operated with reasonable sensitivity at the lower end of the frequency spectrum will depend crucially on the level of seismic and mechanical isolation achievable. Since there are interesting sources of gravitational waves in the region of ten to a few hundred hertz, such as fast pulsars and coalescing compact binary systems, it is advantageous to incorporate as much seismic isolation as practicably possible into the design of these detectors.

The detection of gravitational radiation will not only be a milestone in scientific achievement; it will also be of immense cultural and philosophical significance. It will perhaps complete the process by which Western culture has gradually been forced to let go of its absolutist heresy. The heresy goes back to Aristotle and beyond. It is intimately tied up with the Judeo-Christian prejudice of an unchanging homocentric universe. It is epitomised by the ancient belief in a heavenly crystalline celestial sphere rigidly rotating and unchanging above us.

This heretical edifice has been tumbling slowly under the onslaught of scientific investigation. Newton gave us absolute space, but contributed to the demolition of the geocentric universe brought about by Galileo, Tycho, Kepler and Copernicus. Darwin discovered the impermanence of species; the plate tectonic theory gave us impermanent continents. Einstein demolished Newtonian absolute space and time, and gave us both spacetime curvature and the theory of gravitational radiation. The observation of gravitational radiation will demonstrate that spacetime not only curves predictably in the presence of matter, but is also subject to unpredictable perturbations as gravitational waves ripple through the universe.

Absolutism is surely connected with prejudice. The absolutist prejudice has led to a lingering battle in the case of Darwinism, and most relativists suffer minor irritations from the Einstein-was-wrong brigade. Tycho Brahe wrote of ‘his’ supernova in 1572:

Introduction

Laser-interferometric gravitational wave antennas face one of the most formidable data handling problems in all of physics. The problem is compounded of several parts: the data will be taken at reasonably high data rates (of the order of 20 kHz of 16 bit data); they may be accompanied by twice as much ‘housekeeping” data to ensure that the system is working appropriately; the data will be collected 24 hours a day for many years; the data need to be searched in real time for a variety of rare, weak events of short duration (one second or less); the data need to be searched for pulsar signals; the data from two or more detectors should be cross-correlated with each other; and the data need to be archived in searchable form in case later information makes a re-analysis desirable. One detector might generate 400 Mbytes of data each hour. Even using optical discs or digital magnetic tapes with a capacity of 3 Gbytes, a network of four interferometers would generate almost 5000 discs or tapes per year. The gathering, exchange, analysis, and storage of these data will require international agreements on standards and protocols. The object of all of this effort will of course be to make astronomical observations. Because the detectors are nearly omni-directional, a network of at least three and preferably more detectors will be necessary to reconstruct a gravitational wave event completely, from which the astronomical information can be inferred.

Introduction

To be able to detect gravitational radiation, resonant mass antennae must achieve a dimensionless strain sensitivity of ∼10–19–10–20 (Thorne, 1987). Such a high sensitivity can only be obtained by the use of well isolated, massive, high acoustic Q antennae which are cooled to liquid-helium temperatures, and use ‘quantum limited” transducers to read out the antenna's vibrations. Modern resonant mass antennae generally consist of a high Q cylindrical bar to which is attached one or more smaller masses which are resonant at the antenna frequency, to form a two-mode or multi-mode antenna (see Richard and Folkner's chapter 7). The coupled resonators mechanically amplify the bar's vibrations thereby reducing the effect of transducer wideband noise. It is important that the acoustic Q of the entire antenna be high so as to minimise noise due to Brownian motion of the masses.

Initially, resonant mass antennae used passive PZT crystal transducers which were mounted near or around the girth of the bar. These were subsequently superseded by passive, modulated inductance and capacitance transducers which have proven to be much more sensitive, and are still being developed (see chapter 7). More recently, several groups have started to investigate another class of transducers: the parametric or active transducer (Bordoni et al., 1986; Braginsky, Panov and Popel'nyuk, 1981; Oelfke and Hamilton, 1983; Tsubono, Ohashi and Hirakawa, 1986; Veitch et al., 1987). This type of transducer differs from passive transducers in that it requires an external power source (a pump oscillator), and it has intrinsic power gain.

Introduction

Resonant-bar detectors are designed to measure the acoustic signal induced in a massive bar due to its coupling to a gravitational wave. The large amplitude of thermal vibration in the bar normally considerably exceeds the amplitudes expected from astrophysical sources, and without methods to suppress this noise the principle of detection by resonant masses would be impossible. Weber's key contribution was the realisation that in a high Q antenna-one with a low acoustic loss – the effective noise energy is reduced by a factor ∽τi/τa where τi is the effective measurement integration time, and τa is the antenna ring down time. The advantage from using a low acoustic loss antenna is a direct result of the fluctuation-dissipation theorem. A high Q antenna approaches an ideal harmonic oscillator, whose motion is exactly predictable at a time in the future from the observed amplitude, frequency and phase at an earlier time.

In this chapter we will examine the key concepts of resonant-bar detectors, and provide the framework for the following chapters on different aspects of resonant-bar technology.

Intrinsic noise in resonant-mass antennas

In 1971, Gibbons and Hawking gave an analysis of resonant-mass antennas which led to improved techniques and better understanding of the noise sources. They noted that Weber had monitored the energy or RMS amplitude of the fundamental mode of his antennas.

Introduction

The detection of gravitational waves with frequencies less than 1 kHz appears to be impossible on earth, due to the magnitude of the earth's seismic noise at these frequencies. These waves, therefore, will only be seen in space-based detectors.

A simple gravitational wave detector in space can be created by setting up two free masses and using an electromagnetic signal passing from one to the other as a probe of the spacetime curvature of the region between them. This is the fundamental idea involved in several gravitational wave detectors in space, including pulsar timing, two-way Doppler tracking of interplanetary spacecraft, and spaceborne interferometers. In this article we will discuss the theory and practice of such detectors.

The outline of this chapter is as follows. In section 17.2, we will briefly discuss the sources for the gravitational waves that are to be the targets of the space-based detectors. Then, in section 17.3, the effect of a plane gravitational wave on the arrival time of electromagnetic signals is derived. Our derivation follows that of Hellings (1983) and gives the same result as that first found by Estabrook and Wahlquist (1975). In sections 17.4 and 17.5, these results are used to discuss existing results from pulsar timing experiments and spacecraft Doppler tracking experiments. Finally, prospects for space interferometers are discussed in section 17.6.

Generally, gravitational radiation (GR) is divided into three classes according to its nature: burst, continuous and stochastic waves. Continuous waves can be described as a sinusoidal stationary train of metric perturbation for a sufficiently long time, in contrast with burst waves, which are characterized by their short duration. The third type of GR, stochastic waves, is characterized by its random nature of statistics of arrival, regardless of wave form.

In searching for continuous waves with resonant antennas, various kinds of detecting methods and signal analyses are employed which are different from those used in the detection of burst waves. For example, a resonant antenna should be tuned precisely to the frequency of the source in order to obtain the best sensitivity. Also, long-time integration of the signal output from the detector is a necessary technique for distinguishing a coherent signal buried in a noise. Under these circumstances, usually, the sensitivity of a detector for continuous waves is determined by the level of Brownian motion of the antenna. These features are not common with the case of burst events.

Continuous sources, such as pulsars and binaries, have rather low frequencies, except for rapid pulsars (Backer et al., 1982) or new-born pulsars. Since the pioneering work of J. Weber (1969) bars have been widely used as resonant antennas in detecting burst waves.