Space-borne gravitational-wave detectors as time-delayed differential dynamometers

The basic constituent of many space-borne gravitational missions, in particular, for interferometric gravitational-wave detectors, is the so-called “link” made out of a satellite sending an electromagnetic beam to a second satellite. We illustrate how, by measuring the time derivative of the frequency of the received beam, the link behaves as a differential, time-delayed dynamometer in which the effect of gravity is exactly equivalent to an effective differential force applied to the two satellites. We also show that this differential force gives an integrated measurement of curvature along the beam. Finally, we discuss how this approach can be implemented to benefit the data analysis of gravitational-wave detectors.

Figure 1

Space-time sketch of the congruence. The lines marked $e$ and $r$ represent the emitter and receiver space-time geodesics. The solid line joining them represents the geodesic of the beam leaving the emitter at its proper time ${\tau }_e$ and reaching the receiver when its proper time is ${\tau }_r$. The dashed line represents the beam geodesics emitted at ${\tau }_e+d{\tau }_e$ and received at ${\tau }_r+d{\tau }_r$. A dashed arrow joins a point on the first beam geodesic with coordinates ${\xi }^{\mu }(\lambda ,{\tau }_r)$ to a point on the displaced beam geodesic with coordinates ${\xi }^{\mu }(\lambda ,{\tau }_r+d{\tau }_r)$. Also shown are the 4-velocities of emitter and receivers $v_e^{\mu }$ and $v_r^{\mu }$, the beam wave vector $k^{\mu }$, and $u^{\mu }=\partial {\xi }^{\mu }/\partial {\tau }_r$.