Measuring distances with light

The light is reflected back, and a similar argument gives the total time for the return trip:

Note that coordinate time t in the TT coordinates is proper time, so that this equation gives a value that can be measured.

What does this equation tell us? First, suppose that L is actually rather small compared to the gravitational wavelength, or equivalently that the return time is small compared to the period of the wave, so that h+ is effectively constant during the flight of the photon. Then the return time is just proportional to the proper distance to L as measured by this metric. This should not be surprising: for a small separation we could set up a local inertial frame in free fall with the particles, and in this frame all experiments should come out as they do in special relativity. In SR we know that radar ranging gives the correct proper distance, so it must do so here as well.

More generally, how do we use this equation to measure the metric of the wave? The simplest way to use it is to differentiate treturn with respect to tstart, i.e. to monitor the rate of change of the return time as the wave passes. Since the only way that tstart enters the integrals in Eq. (9.61) is as the argument tstart + x of h+, a derivative of the integral operates only as a derivative of h+ with respect to its argument. Then the integration with respect to x is an integral of the derivative of h+ over its argument, which simply produces h+ again. The result of this is the vastly simpler expression

This is rather a remarkable result: the rate of change of the return time depends only on the metric of the wave at the time the wave was emitted and when it was received back at the origin. In particular, the wave amplitude when the photon reflected off the distant object plays no role.

Now, if the signal sent out from the origin is not a single photon but a continuous electromagnetic wave with some frequency ν, then each ‘crest’ of the wave can be thought of as another null ray or another photon being sent out and reflected back. The derivative of the time it takes these rays to return is nothing more than the change in the frequency of the electromagnetic wave:

dtreturn = νreturn . dtstart νstart

So if we monitor changes in the redshift of the returning wave, we can relate that directly to changes in the amplitude of the gravitational waves.

So far we have used a rather special arrangement of objects and wave: the wave was traveling in a direction perpendicular to the separation of the objects. If instead the wave is traveling at an angle θ to the z axis in the x − −z plane, the return time derivative does involve the wave amplitude at the reflection time:

This three-term relation is the starting point for analyzing the response of beam detectors, as we shall see next. For its derivation see, Exer. 27, § 9.7.

Beam detectors

The simplest beam detector is spacecraft tracking (Armstrong 2006). Interplanetary spacecraft carry transponders, which are radio receivers that amplify and return the signals they receive from the ground tracking station. A measurement of the return time of the signal tells the space agency how far away the spacecraft is. If the measurement is accurate enough, small changes in the return time might be caused by gravitational waves. In practice, this is a difficult measurement to make, because fluctuations might also be caused by changes in the index of refraction of the thin interplanetary plasma or of the ionosphere, both of which the signals must pass through. These effects can be discriminated from a true gravitational wave by using the three-term relation, Eq. (9.63). The detected waveform has to appear in three different places in the data, once with the opposite sign. Random fluctuations are unlikely to do this.

Plasma fluctuations can also be suppressed by using multiple transponding frequencies (allowing them to be measured) or by using higher-frequency transponding, even using infrared laser light, which is hardly affected by plasma at all. But even then there will be another limit on the accuracy: the stability of the clock at the tracing station that measures the elapsed time treturn. Even the best clocks are, at present, not stable at the 10−19 level. It follows that a beam detector of this type could not expect to measure gravitational waves with amplitudes of 10−20 or below. Unfortunately, as we shall see below, this is where we expect almost all amplitudes to lie.

The remedy is to use an interferometer. In an interferometer, light from a stable laser passes through a beam splitter, which sends half the light down one arm and the other half down a perpendicular arm. The two beams of light have correlated phases. When they return after reflecting off mirrors at the ends of the arms, they are brought back into interference (see Fig. 9.3). The interference measures the difference in the arm-lengths of the two arms. If this difference changes, say because a gravitational wave passes through, then the interference pattern changes and the wave can in principle be detected.

Now, an interferometer can usefully be thought of as two beam detectors laid perpendicular to one another. The two beams are correlated in phase: any given ‘crest’ of the light wave starts out down both arms at the same time. If the arms have the same proper length, then their beams will return in phase, interfering constructively. If the arms differ, say, by half the wavelength of light, then they will return out of phase and they will destructively interfere. An interferometer solves the problem that a beam detector needs a stable clock. The ‘clock’ in this case is one of the arms. The time-travel of the light in one arm essentially provides a reference for the return time of the light in the other arm. The technology of lasers, mirrors, and other systems is such that this reference can be used to measure changes in the return travel time of the other arm much more stably than the best atomic clocks would be able to do.