I Gravitational wave propagation in GR

[Ibix]

Then one can write down the null geodesic equation that describes the round-trip laser path along an arm and from the proper time elapsed along the timelike path representing the "corner" endpoint of "L shape" between the sending and receiving back event of the laser beam, evaluate whether the "arm length" has changed or not.

Yeah, but Peter's point is that the arm length also changes when a mouse sneezes somewhere near LIGO. In an ideal world, flight time change implies a GW. In the real world it just implies something caused a flight time change, and there is a lot of signal processing that goes on to pull GW signals out of the noise.

 

[cianfa72]

Then one can write down the null geodesic equation that describes the round-trip laser path along an arm and from the proper time elapsed along the timelike path representing the "corner" endpoint of "L shape" between the sending and receiving back event of the laser beam, evaluate whether the "arm length" has changed or not.

Or, to put it the other way around, whether there are not GWs hitting LIGO arms, then one can assume flat metric and arms "at rest" in a inertial coordinate chart (the timelike congruence given from each arm worldtube's worldlines is geodesic). Then the laser beam round-trip travel time actually "measures" the "proper distance" between each arm's endpoints (in this case there is a natural definition of "proper distance" for them).

 

[PeterDonis]

whether there are not GWs hitting LIGO arms, then one can assume the metric is flat

I think you mean when there are not GWs hitting LIGO arms.

Then the laser beam round-trip travel time actually "measures" the "proper distance" between each arm's endpoints (in this case there is a natural definition of "proper distance").

You are confusing yourself.

The issue that there is no invariant way to define the "proper distance" between the arms (or, to put it another way, to define the "distance the light travels" on its round trip) is an issue whether there is a GW passing or not. But we can ignore that issue because LIGO does not measure the proper distance. It measures the difference in round-trip light travel time between the two arms. That is an invariant independent of any issues about how "proper distance" is defined.

The question is, supposing that a single LIGO apparatus observes a difference in the round-trip light travel time between the two arms, what caused that difference? A GW is not the only possible cause. A GW does it by changing the spacetime geometry in the arms, while leaving the reflectors at the ends of the arms in free fall (actually they're not because they are being suspended in Earth's gravitational field, but we can ignore that and only look at horizontal motion and consider them to be in free fall horizontally). Other causes do it by moving the reflectors at the ends of the arms, i.e., by pushing them so they are not in free fall.

But a single LIGO apparatus has no way of telling which of those two things happened to cause a difference in the round-trip light travel time between the two arms. The only thing LIGO can do is have multiple detectors at widely different locations and look to see if the same signal appears in both of them. That is what would be expected to be the case only if the signal is due to a GW: other causes pushing on the reflectors at the ends of the arms would not be expected to cause the same signal in both detectors at widely different locations.

 

[cianfa72]

A GW does it by changing the spacetime geometry in the arms, while leaving the reflectors at the ends of the arms in free fall (actually they're not because they are being suspended in Earth's gravitational field, but we can ignore that and only look at horizontal motion and consider them to be in free fall horizontally).

Ok, so a GW acts by changing the spacetime geometry in the future lightcone of some "explosion" in spacetime. As you pointed out it leaves the reflectors at the ends of the arms in free-fall (their timelike geodesic worldlines are "at rest" in Lorenz TT gauge coordinates).

From the above it follows that the "frame-invariant" laser beam round-trip travel time (between reflectors at the ends of each arm) would indeed change, right?

 

[PeterDonis]

a GW acts by changing the spacetime geometry in the future lightcone of some "explosion" in spacetime.

"Changing" it from what it would have been if the explosion had not taken place, yes.

it leaves the reflectors at the ends of the arms in free-fall

Yes.

From the above it follows that the "frame-invariant" laser beam round-trip travel time (between reflectors at the ends of each arm) would indeed change, right?

Isn't that what I've already said, more than once?

 

[cianfa72]

Ok, got it

 

[cianfa72]

But a single LIGO apparatus has no way of telling which of those two things happened to cause a difference in the round-trip light travel time between the two arms. The only thing LIGO can do is have multiple detectors at widely different locations and look to see if the same signal appears in both of them. That is what would be expected to be the lcase only if the signal is due to a GW: other causes pushing on the reflectors at the ends of the arms would not be expected to cause the same signal in both detectors at widely different locations.

I'm aware of LIGO is actually a distribuited Observatory consisting of two sites in U.S. When you say LIGO has multiple detectors at widely different locations, are you referring to those two different "site locations" or to multiple detectors installed in the same site ?

 

[PeterDonis]

When you say LIGO has multiple detectors at widely different locations, are you referring to those two different "site locations"

Yes. But note that there are other interferometer GW detectors besides the LIGO ones as well, and they all share data:

https://en.wikipedia.org/wiki/Gravitational-wave_observatory#Interferometers

 

[cianfa72]

Ah ok, so there is one (Advanced Virgo) here in Italy too

 

[PeterDonis]

Ah ok, so there is one (Advanced Virgo) here in Italy too

Yes, and one in Japan, and some others as well.