Detecting Gravitational Waves w/ Interferometers: Explained

[cianfa72]

TL;DR Summary

Fundamentals about physical processes involved in the detection of gravitational waves using interferometers (e.g. LIGO)

Hi,

I would like to ask for some clarification about the physics involved in the gravitational waves detection using interferometers.

Starting from this thread Light speed and the LIGO experiment I'm aware of the two ends of an arm of the interferometer (e.g. LIGO) can be taken as the worldlines of two inertial (free-falling) bodies from the point of view of the experiment (or if you prefer imagine the same experiment done in free-falling in deep space).

From my understanding the goal is to detect a geodesic deviation (tidal gravity) between those two worldlines due to the presence of gravitational waves. To do that we use "null paths" i.e. light beams exchanged between the ends of the two arms of the interferometer.

My doubt is that the "length" of a whatever null path is actually zero. Can we employ it however to detect geodesic deviations ? Thanks.

 

[vanhees71]

I think it's easier to think about this when remembering that a gravitational wave is just a time-dependent gravitational field, leading to oszillations of the LIGO mirrors, allowing for measuring the transients ("wave forms") of the GW signal with all the information about its source.

What's measured is not the "null path" but the relative phase shift of the light going different paths. In principle it's just an utmost accurate Michelson interferometer using some quantum properties of light (squeezed states) to reach the accuracy enabling all these great observations on black-hole/neutron star mergers of the recent years.

 

[cianfa72]

What's measured is not the "null path" but the relative phase shift of the light going different paths.

ok, but forget for a moment the GW detection, can we still use null paths to detect geodesic deviation between the two timelike geodesics ?

 

[martinbn]

But the detectors don't have just one arm!

 

[cianfa72]

But the detectors don't have just one arm!

As we said in that thread (see below) there are 2 arms: for each of them one endpoint is the beam emitter/detector and the other is the mirror at the end of the arm.

The worldlines I was referring to are the timelike worldlines of the endpoints between which the light signals are moving--in LIGO, they would be the worldline of the beam emitter/phase shift detector and that of the mirror at the end of one of the arms.

 

[PeterDonis]

My doubt is that the "length" of a whatever null path is actually zero. Can we employ it however to detect geodesic deviations ?

Yes. Why not?

can we still use null paths to detect geodesic deviation between the two timelike geodesics ?

Yes. Why not?

 

[cianfa72]

Yes. Why not?

ok, as said before in the other thread Is acceleration absolute or relative - follow up we can actually employ the invariant definition of "constant distance" as constant round-trip light travel time. If such round-trip light travel time (measured by wristwatches along the timelike worldlines) is not constant then geodesic deviation between those timelike worldlines is not null.

Does that definition of "constant distance" for the evaluation of geodesic deviation actually make sense only for specific type of spacetime (e.g. stationary or static) ?

 

[PeterDonis]

Does that definition of "constant distance" for the evaluation of geodesic deviation actually make sense only for specific type of spacetime (e.g. stationary or static) ?

No. It works in any spacetime.

 

[pervect]

Summary:: Fundamentals about physical processes involved in the detection of gravitational waves using interferometers (e.g. LIGO)

Hi,

I would like to ask for some clarification about the physics involved in the gravitational waves detection using interferometers.

Starting from this thread Light speed and the LIGO experiment I'm aware of the two ends of an arm of the interferometer (e.g. LIGO) can be taken as the worldlines of two inertial (free-falling) bodies from the point of view of the experiment (or if you prefer imagine the same experiment done in free-falling in deep space).

From my understanding the goal is to detect a geodesic deviation (tidal gravity) between those two worldlines due to the presence of gravitational waves. To do that we use "null paths" i.e. light beams exchanged between the ends of the two arms of the interferometer.

My doubt is that the "length" of a whatever null path is actually zero. Can we employ it however to detect geodesic deviations ? Thanks.

The null paths do not directly measure the geodesic deviation, which can be formally thought of as arising from the geodesic deviation equation, which is.

$$\frac{d^2 x^a}{d \tau^2} = -R^a{}_{bcd} u^b u^d x^c$$

The test masses follow geodesics, which in the above equation are represented be the 4-velocites of the geodesics $u^b$ and $u^d$.

The distance between the geodesics, $x^a$, is defined and measured as being perpendicularly to the worldlines of the geodesics. The null paths of the light rays are not perpendicular to the geodesics, though the distances can be computed from the null paths.

The procedure is basically the same procedure as one uses to compute the distance between two objects / worldlines using radar, or in this case optical radar (lidar) in flat space-time. This procedure is basically that the round-trip time of the light gives the distance (for the limit of short distances). The worldlines of the light pulses are always null, so the lorentz interval along the light beams is always zero. However, the distance is not null, not zero, it is computed from the round-trip time.

For this to work as described, the round trip time has to be short enough that the velocities and distances don't change too much over the round trip time. This complicates the analysis slightly, but it's no more complicated than using radar to calculate the distance between a stationary base station and an accelerating object in flat space-time. The difference here is that we can regard both objects as being stationary (in the sense that they are following geodesics). Nonetheless, because of the changing curvature of space-time, the distances between these two "stationary" objects / worldlines varies with time.

ps - the Latex doesn't seem to be working right, though I don't see my error (if any).

 

[cianfa72]

The distance between the geodesics, $x^a$, is defined and measured as being perpendicularly to the worldlines of the geodesics. The null paths of the light rays are not perpendicular to the geodesics, though the distances can be computed from the null paths.

The procedure is basically the same procedure as one uses to compute the distance between two objects / worldlines using radar, or in this case optical radar (lidar) in flat space-time. This procedure is basically that the round-trip time of the light gives the distance (for the limit of short distances). The worldlines of the light pulses are always null, so the lorentz interval along the light beams is always zero. However, the distance is not null, not zero, it is computed from the round-trip time.

For this to work as described, the round trip time has to be short enough that the velocities and distances don't change too much over the round trip time. This complicates the analysis slightly, but it's no more complicated than using radar to calculate the distance between a stationary base station and an accelerating object in flat space-time. The difference here is that we can regard both objects as being stationary (in the sense that they are following geodesics). Nonetheless, because of the changing curvature of space-time, the distances between these two "stationary" objects / worldlines varies with time.

ok, so the "distance" $x^a$ in the geodesic deviation equation is actually the spacetime length along a curve orthogonal to both geodesics. From my understanding that actually means:

From the tangent space at each point (event) along one of the two (timelike) geodesics take its tangent vector (i.e. its 4-velocity at the given point). Starting from it (let's call it the fiducial geodesic) take the orthogonal vector and "build" a curve from that point that turns out to be orthogonal to the second geodesic's 4-velocity at the point where they intersect.

I'm not sure that such curve turns out to be a geodesic itself; it should be in a limited region of spacetime around the two geodesics we started from.

 

[pervect]

ok, so the "distance" $x^a$ in the geodesic deviation equation is actually the spacetime length along a curve orthogonal to both geodesics. From my understanding that actually means:

From the tangent space at each point (event) along one of the two (timelike) geodesics take its tangent vector (i.e. its 4-velocity at the given point). Starting from it (let's call it the fiducial geodesic) take the orthogonal vector and "build" a curve from that point that turns out to be orthogonal to the second geodesic's 4-velocity at the point where they intersect.

I'm not sure that such curve turns out to be a geodesic itself; it should be in a limited region of spacetime around the two geodesics we started from.

I believe these concerns are reasonable, though I haven't worked out the specifics of your particular concern. In general, I would say that the geodesic equation technically applies only in the limit as the separation between geodesics approaches zero. As the separation approaches zero, for a continuously differentiable manifold, the curve perpendicular to one geodesic will be perpendicular to the second as one takes the aforementioned limit.

Applying the geodesic deviation equation for a finite separation is an approximation, but it's usually a good one for "small" separations. For more insight, I'd suggest applying the geodesic deviation on a purely spatial sphere. Another more complex example might be to consider two timelike geodesics in the Schwarzschild space-time (physically, these would be two orbiting bodies), and using the geodesic equation to calculate the rate of change of the separation of orbits. Notge that I haven't actually worked out these cases, but I think they'd be illuminating.

What I'd expect is that one would find in the first case that the geodesic equation worked as long as the separation was much smaller than the radius of the sphere, in the second case that the equation would work over timescales that were a small fraction of an orbital period.

It would also be useful to consider a one-parameter family of geodesics, one can then take the appropriate limit of infinitesimal separation between the nearby geodesics by taking the limit as the parameter approaches zero. The process of applying this to a separation over a finite distance would be a process of integration.

 

[cianfa72]

I would say that the geodesic equation technically applies only in the limit as the separation between geodesics approaches zero. As the separation approaches zero, for a continuously differentiable manifold, the curve perpendicular to one geodesic will be perpendicular to the second as one takes the aforementioned limit.

I'm not sure if that follows somehow from the topic discussed in Wald in section 3.3 (i.e. Gaussian normal coordinates or synchronous coordinates).

 

[pervect]

I'm not sure if that follows somehow from the topic discussed in Wald in section 3.3 (i.e. Gaussian normal coordinates or synchronous coordinates).

I was thinking about the spherical case some more. Consider a one-parameter family of geodesics (great circles) starting at the equator, and heading north. These will be curves of constant longitude, and the parameter of our group will be the longitude of the curve. These curves will be further parameterize them by an affine parameter $\tau$. Eventually, they all cross at the north pole, diverge again, then continue on.

The distance between these geodesics along a curve perpendicular everywhere to this one-parameter family will be what we are calling the "distance between geodesics". At the equator, the curve we measure the distance along this will be the equator, which is a geodesic, but at higher or lower lattitudes, this curve along which we measure the "distance" (with regard to this particular one-parameter group) will be a circle of constant lattitude, which is not a geodesic because it's not a great circle (except at the equator).

At the equator, the initial rate of separation of geodesics, $dx/d\tau$, x being the "distance" as we've defined it, will be zero, at higher lattitudes it will become non-zero.

Applying the geodesic equation, we would say the geodesics are "accelerating towards each other", starting with an initial velocity of zero (at the equator) and picking up a velocity towards each other as we progress towards the north pole, where they meet, then start separating.

Measuring east west distances along a curve of constant lattitude is a common practice, but it's different from the notion of measuring distances along the shortest possible path, which will be a great circle and not a circle of constant lattitude.

Syncrhonus coordinates would be a bit different, if I'm reading Wald correctly a more or less equivalent set of curves would be generated by setting the "origin" of the synchronous coordinates at the north pole. The parameter in our one parameter group would still be the longitude. The other parameter would be the distance along the line of constant longitude (the distance south from the north pole). This would be proportional to lattitude, I think.