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.

 

[Paul Colby]

so it describes freely-falling masses.

Yes. The $\Gamma^\mu_{\nu\alpha}$ terms in the geodesic equation only sprout space components in the TT gauge. Basically, $\Gamma^x_{tt}=\Gamma^y_{tt}=\Gamma^z_{tt} = 0$. A point particle at rest in this frame remains at rest as the wave passes because all the $\frac{dx^\mu}{ds}$ terms start and remain 0 as the wave passes. The distance between two point masses, however, does change in time as the wave passes.

I’ve found the TT gauge very useful in visualizing how a Weber bar detector converts GWs into vibrational energy.

 

[cianfa72]

Yes. The $\Gamma^\mu_{\nu\alpha}$ terms in the geodesic equation only sprout space components in the TT gauge. Basically, $\Gamma^x_{tt}=\Gamma^y_{tt}=\Gamma^z_{tt} = 0$. A point particle at rest in this frame remains at rest as the wave passes because all the $\frac{dx^\mu}{ds}$ terms start and remain 0 as the wave passes.

Ok, this because for a worldline at rest in TT gauge coordinates, i.e. $(t, x_0, y_0,z_0)$, the geodesic equation becomes: $$ \frac {d^2x^{\mu}} {ds^2} = - \Gamma^{\mu}_{tt} = 0$$ and $(t, x_0, y_0,z_0)$ indeed solves it.

The distance between two point masses, however, does change in time as the wave passes.

You mean the spacelike distance between them evaluated on spacelike hypersurfaces of constant coordinate time $t$ (i.e. the path integral along the geodesic of the induced metric connecting the intersections of point masses' worldlines on each of those spacelike hypersurfaces).

Note, however, that the measurements performed by laser interferometers involve the spacetime path taken for the round-trip journey of the light beams.

 

[Paul Colby]

You mean the spacelike distance between them evaluated on spacelike hypersurfaces of constant coordinate time t (i.e. the path integral along the geodesic of the induced metric connecting the intersections of point masses' worldlines on each of those spacelike hypersurfaces).

I mean proper distance changes with $s$, the proper time. These are frame independent quantities. In the TT coordinates, $s=t$.

The actual measurement occurs at the point where the beams from each arm are combined. What’s measured is the time variation of the light intensity caused by the phase difference between the two beams. This phase difference is proportional to the proper length difference of the arms.

 

[cianfa72]

I mean proper distance changes with $s$, the proper time. These are frame independent quantities. In the TT coordinates, $s=t$.

ok yes.

The actual measurement occurs at the point where the beams from each arm are combined. What’s measured is the time variation of the light intensity caused by the phase difference between the two beams. This phase difference is proportional to the proper length difference of the arms.

Yes. Just to be clear: the metric isn't stationary (in the spacetime region where the GW passes). The proper length of each arm is defined as the length along a spacelike curve orthogonal to the arm's worldsheet (assuming it has only one spatial dimension). So, what you can get from the round-trip journey of light beams is not exactly the arm's proper length (although what is relevant is the phase difference between the two arms of the interferometer).

 

[Paul Colby]

So, what you can get from the round-trip journey of light beams is not exactly the arm's proper length (although what is relevant is the phase difference between the two arms of the interferometer).

At this point I find it instructive to consider some numbers. A round trip time for light in a 4km long interferometer is,
$$
\frac{8\times 10 ^3}{3\times 10^8} = 27 us
$$
Gravitational events detected are in 500ms to 1s duration range. Of course the number of light transversals for LIGO is much higher than one because the interferometer finesse (Q value) is quite high. In the end the rate of change of the lengths are much much slower than the light transversal times.

 

[Ibix]

So, what you can get from the round-trip journey of light beams is not exactly the arm's proper length

Note that LIGO doesn't measure arm length. It's an interferometer, so what it measures is differences in flight time between the two arms, not the flight times themselves. This gives you the difference in the arm lengths as a function of time, but not the lengths themselves.

 

[pervect]

At this point I find it instructive to consider some numbers. A round trip time for light in a 4km long interferometer is,
$$
\frac{8\times 10 ^3}{3\times 10^8} = 27 us
$$
Gravitational events detected are in 500ms to 1s duration range. Of course the number of light transversals for LIGO is much higher than one because the interferometer finesse (Q value) is quite high. In the end the rate of change of the lengths are much much slower than the light transversal times.

Because of the interferometer design, the beam bounces back and forth multiple times, making the effective length (and the storage time) much longer. The web is suggesting the actual storage time is in the 1-3 millisecond range, due to what they call the "Finesse" of the Fabry-Perot interferometer.

 

[pervect]

Possibly of interest. https://arxiv.org/abs/1409.4648 "Fermi-Normal, optical, and wave-syncrhonus coordinates for spaceetime with a plane gravitational wave".

Why it might be of interest- Fermi Normal coordinates are generally the closest one can come to an inertial frame of reference. There is of course no difficulty with using generalized coordinates if they are more convenient to work withy, and while I haven't read this particular paper, in general Fermi Normal coordinates aren't all that convenient to work with.