Gravitational Waves from Spinning Neutron Stars
Gravitational waves (GWs), i.e. small deformations of spacetime traveling at the speed of light, are a fundamental consequence of Einstein's general theory of relativity. There has been no direct observation of GWs so far, although first indirect evidence
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Gravitational Waves from Spinning Neutron Stars Reinhard Prix (for the LIGO Scientific Collaboration)
24.1 Introduction Gravitational waves (GWs), i.e. small deformations of spacetime traveling at the speed of light, are a fundamental consequence of Einstein’s general theory of relativity. There has been no direct observation of GWs so far, although first indirect evidence was found in the observed inspiral of the binary pulsar PSR 1913+16, which agreed to within 1% with the predictions of general relativity [75, 88]. Similar measurements on the recently discovered “double pulsar” system have allowed to improve these experimental tests of General Relativity to the level of 0.05%. GWs are purely transverse waves, characterized by two polarization states (denoted as ‘+’ and ‘×’, respectively). These two polarization bases differ by a rotation of 45◦ around the propagation axis, corresponding to the quadrupolar (spin-2) nature of the gravitational field. In contrast, the two polarization bases of electromagnetic waves differ by a rotation of 90◦ , reflecting the dipolar (spin-1) nature of the electromagnetic field. Any likely sources of detectable GWs will be at astrophysical distances, thus the signals reaching Earth have very small amplitudes and are nearly plane waves. A linearized version of general relativity (e.g., see [57]) can therefore be used to describe GWs in terms of a small metric perturbation hμν , i.e. one can write the metric as gμν = ημν + hμν , where |hμν | 1 is the gravitational wave and ημν is the Minkowski metric of the unperturbed flat spacetime. One can then show that the Einstein field equations in vacuum reduce to the familiar wave-equation for a perturbation hμν propagating (at the speed of light) through flat spacetime ημν , i.e. σρ 2hTT ∂σ ∂ρ hTT μν = η μν = 0 ,
(24.1)
R. Prix Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut, Am M¨uhlenberg 1, 14476 Golm, Germany e-mail: [email protected] W. Becker (ed.), Neutron Stars and Pulsars, Astrophysics and Space Science Library 357, c Springer-Verlag Berlin Heidelberg 2009
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where hTT μν is the tensor hμν expressed in the transverse–traceless (TT) gauge. This is a choice of coordinates, {t, x, y, z} say, corresponding to an inertial (Lorentz) frame in the unperturbed flat background, which makes explicit that the perturbation is transverse, i.e. orthogonal to the direction of propagation, and trace-less, namely the perturbation does not ‘compress’ or ‘expand’ elements of spacetime, but induces a (volume-preserving) ‘strain’ only. In this gauge a plane gravitational wave propagating along the z-axis can be written as ⎛ ⎞ 0 0 0 0 ⎜0 h+ h× 0⎟ TT ⎜ ⎟ (24.2) hTT μν (t, z) = hμν (t − z/c) = ⎝0 h −h 0⎠ , × + 0 0 0 0 where c is the speed of light and h+,× (t − z/c) are the two polarizations of the wave. The effect of such a GW on two freely falling test-masses is a time-dependent change δ l in their spatial distance l, which can be monitored using laser interferometry. This is the principle behind inte
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