Staff Emeritus

Science Advisor

Homework Helper

cianfa72 said:

TL;DR: About the analysis of LIGO measurements in TT gauge coordinates

I'd like to consider again what discussed in this Gravitational wave propagation in GR.

The analysis of GWs' LIGO measurements is performed in Transverse-Traceless (TT) Gauge coordinates. Basically, in the linearized gravity model, the metric tensor field ##g_{ab}## is assumed to be in the form ##g_{ab} = \eta_{ab} + h_{ab}## (note the use of abstract index notation).

As far as I can tell, the TT gauge defines a chart in which the EFEs in vacuum transform into the wave equation ##\Box \bar h_{ab}## for the tensor field ##\bar h_{ab}##. The latter is defined as $$\bar h_{ab} = h_{ab} - \frac {1} {2} \eta_{ab}h^c_c$$ So far so good.

Matematically, the wave equation above has for instance plane wave solutions. They are wave solutions in TT gauge coordinates.

As far as I understood, in LIGO measurement analysis, the arms of the interferometer are taken as "at rest" in TT coordinates (i.e. the worldlines of the arm's worldsheets have constant spacelike coordinates and varying timelike coordiante).

My questions: Why the above holds true and is the solution in wave form an invariant fact ?

My understanding is that the test masses in Ligo essentially follow space-time geodesics (aka are in free fall), ignoring some of the complexities regarding the fact that they need to be suspended against the Earth's gravity. This is done with multiple isolating pendulums. Of course there is still unavoidable noise caused by the suspension system especially at low frequencies.

So what Ligo measures is essentially the change in the separation of test masses following the geodesics. They're not actually freely floating in a vacuum, but the suspension mechanism is designed so that we can treat them as if they were.

This distance measurement is done with what they call a "Fabry Perot" interferometer, which bounces the light back and forth between the mirrors attached to the test masses ~300 times to amplify the phase shift / increase the effective optical path length of the arms.

Are you also interested in the respopnse of the Earth as a whole to the gravitational waves? I gather it's resonant frequency is well below the band Ligo measures, basically if you view the Earth as a distributed spring-mass system, the "spring" parts are pretty weak.

So you can think of the Earth and the test mases as moving in a similar manner as far as Ligo is concerned. The "springiness" of the Earth isn't high enough to affect the motion of the ground much over a gravitational wave cycle in Ligo's bandwidth.