Difference between revisions of "Generation and detection of gravitational waves"

Intensity of gravitational radiation, as well as the electromagnetic one, is quadratic in charge of the source, e.g. for gravitation it is quadratic in mass $M^2$, while the total energy reserve ($Mc^2$) is proportional to mass in the first power. This means that with increase of mass with the same accelerations the effectiveness of gravitational wave generation should grow.

The two main reasons for the low effectiveness of energy conversion from mechanical into gravitational waves are the following. First, weakness of gravitational interaction -- the gravitational constant $G$ is small. Second, gravitational radiation in the first non-vanishing order is quadrupole, as opposed to electromagnetic radiation, which can be produced already in the dipole approximation.

Nothing. Nothing at all, and this actually follows from the principle of equivalence. Also in the first (linear) order by perturbation the geodesic equation in the TT gauge is $du^\mu /dt =0$. The wave only reveals itself when acting on a system of particles, by varying distances between them; the geodesic deviation is non-zero in the linear theory.

In General Relativity, the effects of both the stationary field and gravitational radiation are described by the tidal forces they produce on free test masses. In other words, single geodesics alone cannot detect gravity or gravitational radiation; we need at least a pair of geodesics.
While the stationary tidal force due to the Newtonian potential $\phi$ of a self-gravitating source at a distance $r$ falls off as $\nabla \nabla \phi \sim {{r}^{-3}}$, the tidal force due to the gravitational wave of amplitude $h$ that it emits at wavelength $\lambda $ decreases as $\nabla \nabla h\sim r^{-1}\lambda^{-2}$. Therefore, the stationary Coulomb gravitational potential is the dominant tidal force close to the gravitating body (in the near zone, where $r\le \lambda $). However, in the far zone ($r\gg \lambda $) the tidal effect of the waves is much stronger.
The stationary part of the tidal field is a DC effect, and simply adds to the stationary tidal forces of all other objects in the universe. It is not possible to discriminate one source from another. Gravitational waves carry time-dependent tidal forces, and so they can be discriminated from the stationary field if one knows what kind of time dependence to look for. Interferometers are ideal detectors in this respect because they sense only changes in the position of an interference fringe, which makes them insensitive to the DC part of the tidal field.

The high frequency band of gravitational wave, $1Hz\le f\le {10}^{4}Hz$, $M_{\odot }\le M\le {10}^{4}M_{\odot}$ , is the band targeted by the modern generation of ground-based laser interferometric detectors, such as LIGO. It also corresponds roughly to the audio band of the human ear: when converted to sound, LIGO sources are human audible without any frequency scaling.
The low frequency end of this band is set by the fact that it is extremely difficult to isolate against ground vibrations at low frequencies, and probably impossible to isolate against gravitational coupling to ground vibrations, human activity, and atmospheric motions. The high end of the band is set by the fact that it is unlikely any interesting gravitational-wave source radiate at frequencies higher than a few kilohertz.

In a laser interferometer like LIGO a carefully prepared laser state is split at the beamsplitter and sent into the Fabry-Perot arm cavities of the detector. The reflectivities of the mirrors in these cavities are chosen such that the light bounces roughly 100 times before exiting the arm cavity (that is, the finesse $F$ of the cavity is roughly 100). This corresponds to about half a cycle of a 100 Hz gravitational wave. The phase shift acquired by the light during those 100 round trips is \[\Delta \varphi_{GW} \sim 100\times 2\times \Delta L\times 2\pi /\lambda \sim 10^{-9}\]
This phase shift can be measured provided that the shot noise at the photodiode, $\Delta {{\varphi }_{shot}}\sim 1/\sqrt{N}$ is less than $\Delta {{\varphi }_{GW}}$. $N$ is the number of photons accumulated over the measurement; $1/\sqrt{N}$ is the phase fluctuation in a quantum mechanical coherent state that describes a laser. We therefore must accumulate $\sim {10}^{18}$ photons over the roughly 0.01 second measurement, translating to a laser power of about 100 watts.

Gravitational waves are detectable because the atomic vibrations are random and incoherent. The $\sim 7cm$ wide laser beam averages over about ${10}^{17}$ atoms and at least ${10}^{11}$ vibrations in a typical measurement. These atomic vibrations cancel out, becoming irrelevant compared to the coherent effect of a gravitational waves.
Other thermal vibrations, however, end up dominating the detectors' noise spectra in certain frequency bands. For example, the test masses' normal modes are thermally excited. The typical frequency of these modes is $\omega \sim {10}^{5}s^{-1}$, and they have mass $m\sim 10kg$, so $\delta l_{mass}\sim {10}^{-14}cm$. This, again, is much larger than the effect we wish to observe. However, the modes are very high frequency, and so can be averaged away provided the test mass mode is confined to a very narrow band near $\omega$, and thus doesn't leak into the band we want to use for measurements.