Could you use LIGO-technology to measure gravitational redshift?

[Isambard]

Don't know if this is the right forum, but I'll give it a try: If you build a bridge over a gorge that's a couple km deep the middle of the bridge would be as far away from the mountains on both sides as possible (and therefore as far away from the mountains' gravitational field as possible). Under the bridge you attach a long hollow tube which reach the bottom of the gorge, where it is attached.

On one of the mountains you drill a vertical hole, as deep as the gorge. The hole is as narrow as possible, so that light that is sent through it is as close to the mountain's gravitational field as possible.

If you're sending light with the exact same wavelength through both the hollow structure and the narrow hole that goes through the mountain, and they both travel the exact same distance, the light that travels through the mountain will have experienced more redshift than the light that goes through the hollow structure. Not much of course, the difference is so little it is practically impossible to imagine. But there will still be a difference.

Now, if you're sending a laser beam from the bottom of the gorge through a Michelson interferometer, splitting it in two, so that the left beam travel horizontally towards the hollow structure, hits a mirror so it travels vertically through the tube, and the right beam travel horizontally into the mountain, hits a mirror and travels vertically through the narrow hole. On top, both hit a mirror again and travels towards each other. Normally two beams with the same wavelength meeting each other would even each other out, but because one of them has a longer wavelength than the other, would it be possible to register the difference? If so, one could also use the same structure to send a laser from top to bottom, to measure gravitational blueshift.

Of course there would be challenges with a hollow structure exposed to wind, rain, different temperatures and so on. One could probably drill two vertical holes through a mountain, where one is much wider than the other, so that the laser in the center of the widest hole is further away from the surrounding mountain's gravitational field. But it would still be much closer than the laser sent through the hollow structure hanging from the bridge. On the plus side, it would be much easier to control the environment. And the project wouldn't face the same limitations (the depth of the gorge), even if facing the problem with rising temperatures the deeper you go (by drilling from a tall mountain plateau the temperature would rise less than if drilling from sea level).

And if a gravitational wave should pass through the planet from the right angle during one of the tests, it could perhaps be possible to register if gravitational waves have different effect on different wavelengths.

Could it be possible to build a system with this technology sensitive enough to register gravitational redshift or blueshift here on earth by comparing the two laser beams? And could other methods work just as well (even if equally challenging to build)?

 

[Ibix]

If I understand you right you're proposing a square beam path with one emitter and one receiver and light going half way round the path on opposite directions.

Unfortunately, gravitational redshift only depends on the gravitational potential difference between two points. So (a) the beams will have the same frequency both times you can compare them, and (b) you have no way to compare the emitted and received frequencies, which is the only manifestation of gravitational time dilation here.

The Pound-Rebka experiment is an example of a way to measure gravitational redshift. It essentially compares two clocks at different altitudes.

 

[renormalize]

Could it be possible to build a system with this technology sensitive enough to register gravitational redshift or blueshift here on earth by comparing the two laser beams? And could other methods work just as well (even if equally challenging to build)?

Can you explain why you think this optical laser method would be superior to previous gravitational redshift measurements based on gamma or microwave (maser) radiation (see the Pound-Rebka reference in the post by @Ibix)?

 

[Isambard]

If I understand you right you're proposing a square beam path with one emitter and one receiver and light going half way round the path on opposite directions.

Unfortunately, gravitational redshift only depends on the gravitational potential difference between two points. So (a) the beams will have the same frequency both times you can compare them, and (b) you have no way to compare the emitted and received frequencies, which is the only manifestation of gravitational time dilation here.

The Pound-Rebka experiment is an example of a way to measure gravitational redshift. It essentially compares two clocks at different altitudes.

Objects, like mountains and buildings, have their own gravitational field. A beam going through a vertical narrow hole in a massive mountain should therefore experience more gravitational redshift than a beam going through a vertical hollow structure. Time goes faster the further away you are from earth, and it moves a little faster on the top of the roof of a building. But the building has its own gravity, so it would go even faster at the same distance from earth if there was no building there.
And a massive mountain is taller and a stronger gravitational field than a building.

Also, this is not about time dilation, but redshift, where the length of a wavelength changes. Even if time dilation would mean one of the laser beams would need a slightly longer time to travel the same distance.

 

[Isambard]

Can you explain why you think this optical laser method would be superior to previous gravitational redshift measurements based on gamma or microwave (maser) radiation (see the Pound-Rebka reference in the post by @Ibix)?

Well, I asked if it was possible. If it is, we would have two 100% identical laserbeams ending up being not 100% identical anymore, due to different experience to redshift. Which could provide some extra information.

 

[Ibix]

Objects, like mountains and buildings, have their own gravitational field.

I know. The point that seems to escape you is that, if the beams start in the same place and finish in the same place, the gravitational effect on the frequencies of both beams is the same. It depends only on the difference in gravitational potential between start and end points, not the route taken to get from one to the other. So the beams will have the same frequencies when you compare them.

If you want to see a frequency change you have to do something like Pound and Rebka did, comparing the emitted and received beams to some local frequency standard. But then the second beam path is redundant.

Also, this is not about time dilation, but redshift,

They're the same thing in this context - the square root of the ratio of the $g_{tt}$ terms in the weak field metric at the start and end positions.

Even if time dilation would mean one of the laser beams would need a slightly longer time to travel the same distance.

This is actually one of the things you could (in principle) measure with some variant of this setup - the difference in Shapiro delay along the two paths.

 

[Dale]

the light that travels through the mountain will have experienced more redshift than the light that goes through the hollow structure. Not much of course, the difference is so little it is practically impossible to imagine. But there will still be a difference.

I don’t think that this is correct. As @Ibix said, the gravitational redshift in a weak gravitational field depends only on the beginning and ending height. Not on the path.

The reason is that a weak gravitational field has a gravitational potential. And the path independence is a property of the potential. You would need a strong gravitational field to introduce some path dependence.

 

[PeroK]

Well, I asked if it was possible. If it is, we would have two 100% identical laserbeams ending up being not 100% identical anymore, due to different experience to redshift. Which could provide some extra information.

It would be an expensive experiment to carry out.

 

[Ibix]

You would need a strong gravitational field to introduce some path dependence.

And a non-stationary field at that (i.e., what you would loosely call a time varying field). A stationary strong field has no path dependence either.

 

[Dale]

And a non-stationary field at that (i.e., what you would loosely call a time varying field). A stationary strong field has no path dependence either.

I was thinking about that, but I am not sure. I think that maybe non-stationary is too strong a condition. I was thinking that you could probably do it with a non-static metric. But I am not at all confident about that.

 

[Ibix]

I was thinking about that, but I am not sure. I think that maybe non-stationary is too strong a condition. I was thinking that you could probably do it with a non-static metric. But I am not at all confident about that.

I think it's ok in stationary spacetimes too.

Argument: if you have a timelike Killing field $\xi^a$ then $\xi_ap^a$ is constant along a geodesic. Certainly it could be different on each of the sides of the square path under discussion, but then it had to change at the corner. And since $\xi_ap^a$ is what a stationary observer measures as the energy of the light, a stationary observer at the corner would observe that the light changed energy at the corner - i.e. in the interaction with the mirror. You can certainly modify the energy like that, but it's nothing to do with path dependence. Thus $\xi_ap^a$ is constant around the experiment and the frequency is path independent.

 

[Vanadium 50]

There are three problems here.

1. An interferometer, like LIGO, measures a phase change, not a frequency change.
2. As pointed out, the frequency change depends on the difference of heights (technically potentials) between the source and receiver. Bring them to the same point, and the difference iz zero.
3. We can see the frequency change using the earth as a gravitational source. Mountains are much smaller than planets.