Spherical Gravitational Wave Detectors Krishna Venkateswara April 16 2007 Abstract Resonant bar gravitational wave detectors have improved greatly over the last 40 years but are currently completely outperformed by laser interferometer detectors. Spherical detector configuration has several advantages over conventional bar detectors and can be considered as the next generation of resonant gravitational wave antenna. In this report, I will give a brief introduction to gravitational waves and detectors and then review the major research done on spherical detectors. 1. Introduction 1.1 Gravitational waves Gravitational waves (GW) are a prediction of Einstein's General Relativity (GR). They are predicted to be transverse and spin 2 waves [1]. There has been indirect evidence for the existence of gravitational waves in the form certain theoretical prediction (based on GR) of inspiral velocity as measured in the binary neutron star system PSR1913. The observations match the theoretical value nearly perfectly and provide convincing evidence for the existence of gravitational waves. However, any attempts to directly detect gravitational waves have not been successful yet. According to GR, gravitational waves are emitted by time varying mass quadrupole moments (accelerated massive objects). The relative difference in the strengths of gravitational interactions as compared to electromagnetic interactions also ensures that gravitational waves are extremely weakly interacting while electro-magnetic waves are easily scattered and absorbed by dust and other matter between the object and the observer. Thus the detection of gravitational waves will reveal a new and different view of the universe. In particular, it might lead to new insights in cosmology, strong field gravity by observing black hole signatures, large-scale nuclear matter (neutron stars) and the inner processes of supernova explosions. Far away from the source one can use the weak field approximation to solve the Einstein field equations in vacuum yielding a normal wave equation. Using the transverse-traceless gauge its general solutions can be written as …(1) where z is the direction of propagation and h+ and h are the two polarizations and hµν is the hh tzch tzch tzch tzcµν=− −− − − + ×× +0 0 0 00 00 00 0 0 0( ) ( )( ) ( )small perturbation to the flat Minkowski metric gµν. A gravitational wave passing through an object, say a cylindrical rod, will exert a force on it which will cause the object to stretch and squeeze in directions perpendicular to the propagation of the wave. The strength of a gravitational wave can be expressed as a dimension-less quantity, the strain h which measures the relative length change ∆L / L. Let us try to calculate an order of magnitude estimate for the strain h from typical astrophysical sources: …(2) If one sets the value of the non-symmetric part of the kinetic energy to be of the order of a solar mass, and r is assumed to be of the order of distance to Virgo Cluster and Hubble distance respectively, one obtains, Clearly, gravitational waves are very hard to detect! 1.2 Gravitational wave detectors The search for gravitational waves started with Joseph Weber about forty years ago, using what are now called as “Bar detectors”. Since then, various methods to detect gravitational waves have been proposed and many types of GW detectors have been built all over the world. They fall into broadly two categories: a) Resonant mass detectors and b) Laser interferometer detectors Resonant mass detectors use the principle that a passing gravitational wave will deposit energy in the vibrational mode of a spring mass system, which can be detected by measuring its motion. Bar detectors use the fundamental longitudinal mode of a large cylindrical bar (usually made of Aluminum). Spherical detectors on the other hand attempt to use the quadrupole modes of a solid sphere. Transducers amplify and convert the vibrational motion into an electrical signal and can be of several types: capacitive transducers, resonant inductive transducers, optical transducers or even piezoelectric crystals. Laser interferometer detectors like the LIGO (Laser Interferometer Gravitational wave Observatory) sense the motion of two lightly suspended mirrors at the two ends of a Michelson interferometer to detect any change in the length of the interferometer arms when a GW passes through it. They have very high strain sensitivity of the order of 10-23 Hz-1/2. Also, these detectors have a wide bandwidth, as the two test masses can be considered free in the direction of the interferometer arm and are limited by noise sources, chiefly seismic noise, shot noise and radiation pressure noise. hG QcrG Ecrkinns≈ ≈( )4 4handh≈≈−−101021232. Principle of a Spherical GW Detector A sphere is a natural shape for a resonant antenna for GW as it has 5 degenerate quadrupole modes, which interact strongly with the wave [8]. It has the advantage of offering uniform cross-section and hence a full-sky coverage. By measuring the amplitude of the 5 quadrupolar modes excited, it is possible to determine both source direction and polarization of the gravitational wave. As there are 5 mode amplitudes and only 4 gravitational wave unknowns, one can use the extra information to veto non-gravitational wave disturbances [8]. Also, as the mass of the sphere would be much larger than a bar antenna of the same fundamental frequency, the sphere has a much better cross-section as compared to a bar antenna. 2.1 The TIGA Detector When coupled with transducers in arbitrary locations on the sphere, the quadrupole modes split unequally and it becomes extremely complicated to deconvolve the signal. Johnson and Merkowitz [2] came up with the idea of locating the resonators on the face-centers of a truncated icosahedron (TIGA, Truncated Icosohedral Gravitational Antenna). This symmetric arrangement splits the modes evenly and makes it much easier to deconvolve the GW signal. They estimate that the sensitivity in energy would be about 56