Gravitational wave propagation in GR - follow up

[cianfa72]

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 ?

 

[Paul Colby]

In the TT gauge (coordinates), the metric stain, $h_{ab}$, only has spatial components, $h_{xx}, h_{xy},\cdots$, are non-zero and are time dependent. One way to view LIGO is in terms of the proper lengths of the interferometer arms. These lengths are time dependent as the wave passes. “At rest” in your post I assume means the spatial locations (coordinate values) of the interferometers mirrors are not time dependent in the TT frame.

 

[pervect]

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.

 

[cianfa72]

In the TT gauge (coordinates), the metric stain, $h_{ab}$, only has spatial components, $h_{xx}, h_{xy},\cdots$, are non-zero and are time dependent. One way to view LIGO is in terms of the proper lengths of the interferometer arms. These lengths are time dependent as the wave passes. “At rest” in your post I assume means the spatial locations (coordinate values) of the interferometers mirrors are not time dependent in the TT frame.

Ok, so the timelike congruence "at rest" in TT gauge coordinates (i.e. the congruence described by timelike worldlines with fixed spacelike TT coordinates) is hypersurface orthogonal even though it isn't stationary. This congruence should be geodesic, so it describes freely-falling masses.

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.

Ok, so such test masses can be taken as free-falling for the purposes of measurements made with LIGO's interferometers.

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.

Ok, so in a spacetime diagram we can consider the light back and forth path between the mirrors attached to the free-falling test masses. Of course the proper time along both mirror's worldlines for the round-trip journey of the light beam changes in the presence of a GW wave.