# B LIGO and space time changes

#### asca

Summary
How can an interferometer detect Gravitation waves, if the change in space time due to gravity affects all the rulers (and clocks) in that spot?
Sorry if the question has been already answered, but I didn't manage to find it. Let's go back to ligo detection of gravitational waves, my question is the following: if space time changes its texture due to a gravitational effect, all the rulers (and clocks) in that spot will be affected, so also the electromagnetic wave will have its own ruler and clock changing the same way they change for the LIGO arm, and so there should not be any change in the number of wawelenghts and no interference pattern changes, so no detection. What am I missing?

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#### phinds

Gold Member
Summary: How can an interferometer detect Gravitation waves, if the change in space time due to gravity affects all the rulers (and clocks) in that spot?

Sorry if the question has been already answered, but I didn't manage to find it. Let's go back to ligo detection of gravitational waves, my question is the following: if space time changes its texture due to a gravitational effect, all the rulers (and clocks) in that spot will be affected, so also the electromagnetic wave will have its own ruler and clock changing the same way they change for the LIGO arm, and so there should not be any change in the number of wawelenghts and no interference pattern changes, so no detection. What am I missing?
You are missing that the change is directional, oriented towards the source of the gravity waves, and the two arms of LIGO get affected differently and THAT is what the "detection" really is.

#### Michael Price

Summary: How can an interferometer detect Gravitation waves, if the change in space time due to gravity affects all the rulers (and clocks) in that spot?

Sorry if the question has been already answered, but I didn't manage to find it. Let's go back to ligo detection of gravitational waves, my question is the following: if space time changes its texture due to a gravitational effect, all the rulers (and clocks) in that spot will be affected, so also the electromagnetic wave will have its own ruler and clock changing the same way they change for the LIGO arm, and so there should not be any change in the number of wawelenghts and no interference pattern changes, so no detection. What am I missing?
Objects held together only by gravity cannot detect the gravity waves, but objects held together by non-gravitational forces - which is most objects - will sense the purely gravitational (spacetime) distortions.

#### PeterDonis

Mentor
Objects held together only by gravity cannot detect the gravity waves
They can if they exchange light signals. Which is basically what LIGO does. The actual LIGO sensors and mirrors are only "held together by gravity" in the plane of the LIGO apparatus (they do of course have proper acceleration perpendicular to that plane, but that can be ignored when modeling LIGO's workings--and there are plans for a space-based detector that will be entirely in free fall).

#### Orodruin

Staff Emeritus
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Curvature of spacetime essentially manifests itself as tidal gravity, leading to nearby objects not having the same acceleration unless acted upon by a force. This is true also for a gravitational wave. The mirrors in a gravitational wave detector are suspended in a pendulum contraption that in essence means that they move freely in the relevant direction. Since tidal gravity in essence is nearby things moving with different acceleration, the distance between the mirrors does change. The meter is still a meter, leading to a change in travel time for the light and therefore also a changed interference pattern.

Note that the frequency of the gravitational wave is typically lower than the time taken for light to travel back and forth between the mirrors. Thus, there is no relevant stretching of the light wavelength.

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#### PeterDonis

Mentor
if space time changes its texture due to a gravitational effect, all the rulers (and clocks) in that spot will be affected
No, they won't. You are misunderstanding what "changing spacetime geometry" means. It means that the physical distance, for example, between the ends of the arms of a GW detector changes, even though the arms feel no force (because they are only affected by gravity and so they are in free fall). The change in physical distance can be detected because it is a change in how many rulers (or wavelengths of light) will fit in each arm.

#### Ibix

The thing to note about the expansion and contraction is that it is slow compared to the flight time of light in the interferometer. So, although "in flight" light is stretched a bit, new "unstretched" light is being continually injected, so the ratio between wavelength and arm length varies.

#### PeterDonis

Mentor
although "in flight" light is stretched a bit
This is a coordinate-dependent statement, and should probably not be used as a description. The invariant is the variation of the physical distance between the ends of the arms, as shown by interference fringes in a detector like the one in LIGO.

#### pervect

Staff Emeritus
Summary: How can an interferometer detect Gravitation waves, if the change in space time due to gravity affects all the rulers (and clocks) in that spot?

Sorry if the question has been already answered, but I didn't manage to find it. Let's go back to ligo detection of gravitational waves, my question is the following: if space time changes its texture due to a gravitational effect, all the rulers (and clocks) in that spot will be affected, so also the electromagnetic wave will have its own ruler and clock changing the same way they change for the LIGO arm, and so there should not be any change in the number of wawelenghts and no interference pattern changes, so no detection. What am I missing?
While it's possible to view gravity as "rulers and clocks changing", that's not really the most popular analogy to use to describe gravitational waves.

In spite of the fact that it's not popular, let's look at an oversimplified example of how it might work.

Consider Einstein's description of a curved spatial geometry via a heated marble slab, as described in chapter 24 of Einstein's book "Relativity: The Special and General Theory". https://www.bartleby.com/173/24.html.

The basic idea is that you have a heated marble slab, the rods expand with heat and contract with cold, and that you find the geometry of the slab as measured by these rulers isn't Euclidean.

Our heated marble slab represents only spatial geometry, and not space-time geometry. This is oversimplified, but it's the best I can do to remain comprehensible.

One can definitely detect the change in geometry caused by the heat. A simple example illustrates this. Suppose you make a quadrilateral figure with four equal sides, and two equal diagonals, a square.

Then, in an Euclidean geometry, the ratio of the diagonals to the sides will always be $\sqrt{2}$.

Now, lets represent a non-Euclidean geometry, following Einstein's suggestion, by heating the center of the marble slab, but not the outer perimeter. Unheated, the outer rods do not expand, but sections of the diagonal rods do. This causes the ratio of the diagonal length to the outer side length to change from it's Euclidean value.

Then the ratio of the diagonal to the side is not $\sqrt{2}$ anymore, and we see a measurable result from the geometry changing.

Most descriptions of Ligo do not involve the idea of rulers expanding, but it's a possible interpretation of the metric changes, which is what the mathematics actually models.

Ligo does not do the equivalent of taking the ratio of the outside of the square to the diagonal - it's got a different setup. I imagine it might be possible to make a more detailed setup that operates in the same way as the Ligo detector does to detect the changes in geometry, but this short post is not that detailed, it just answers the question of how one could detect the changes induced in the geometry.

#### PeterDonis

Mentor
Then the ratio of the diagonal to the side is not $\sqrt{2}$ anymore, and we see a measurable result from the geometry changing.
Yes, but this measurable result is a result of the geometry changing--from Euclidean to non-Euclidean--not of the rulers changing. If the rulers changed along with the geometry, you would not be able to measure the change in the geometry, because it would change the rulers right along with it. That is basically the misconception that the OP gave.

Most descriptions of Ligo do not involve the idea of rulers expanding, but it's a possible interpretation of the metric changes
No, it isn't. All of the different ways of modeling LIGO (different coordinate charts) agree that the physical lengths of the arms change--more precisely, the physical distances between the sensor and the mirrors changes. This cannot be interpreted as "rulers expanding" for the same reason as given above.

The best way, IMO, to think about a gravitational wave is that it is a wave of changing tidal gravity. Tidal gravity doesn't change the lengths of rulers*. It changes the physical distance between objects following nearby worldlines.

* - Strictly speaking, any real ruler will have a finite tensile strength, so the effects of tidal gravity on it will not be zero, because tidal gravity will induce some finite strain in the material of which the ruler is made. But, first, we can imagine an idealized ruler, the limit of real rulers as the induced strain from tidal gravity goes to zero, and those idealized rulers, strictly speaking, are what I have been talking about. And, second, for a case like LIGO, the induced strain in a real ruler (or in, for example, the substance of the Earth, the tubes enclosing the paths of the laser light in the arms, etc.) is so much smaller than the induced change in arm length between the sensor and the mirrors, that it can be considered zero for analyzing the results.

#### A.T.

if space time changes its texture due to a gravitational effect, all the rulers (and clocks) in that spot will be affected, so also the electromagnetic wave will have its own ruler and clock changing the same way they change for the LIGO arm, and so there should not be any change in the number of wawelenghts and no interference pattern changes, so no detection.g?
This might help:

#### Ibix

This is a coordinate-dependent statement, and should probably not be used as a description. The invariant is the variation of the physical distance between the ends of the arms, as shown by interference fringes in a detector like the one in LIGO.
If the physical distance between the ends of the arms changes then the wavelength of light in flight must also change, surely? To illustrate the point, imagine using light with a wavelength equal to the arm length. If the physical distance to the far mirror changes (on a time scale much less than the flight time of light through the apparatus) then the physical distance to the far end of the wave must also change.

In any case, isn't the invariant that is directly measured the flight time? Or the variation in the difference in flight times for the two arms, more precisely.

#### pervect

Staff Emeritus
Yes, but this measurable result is a result of the geometry changing--from Euclidean to non-Euclidean--not of the rulers changing. If the rulers changed along with the geometry, you would not be able to measure the change in the geometry, because it would change the rulers right along with it. That is basically the misconception that the OP gave.
The whole "shrinking and expanding" rulers idea, to my way of thinking, can be useful as a tool for someone who is reluctant to abandon Euclidean geometry.

One basically imagines that the real, physical, rulers, which we can actually make, are "distorted". Then one can envision a familiar, underlying, Euclidean geometry when one "undistorts" these distorted rulers. But one has to keep in mind that this "undistortion" is basically a mathematical trick - the physical measurments we make are all made with "distorted" rulers.

With this approach, one has to keep track of two sets of rulers then - the actual, physical rulers, and the unphysical, unmeasurable, "undistorted" rulers. One typically winds up with a philosophy that the non-physical , non-measurable rulers are more real than the physical ones.

The philosophical view is not the one I personally prefer. In fact, it's a bit odd, in which one imagines that non-existent, non-observable things are "more real" than the ones we can actually make. But, it's possible to use it to get correct results.

In any case, I do agree that the actual geometry of General relativity (and of gravitational waves), as measured with the physical rulers, the ones we can actually create, is non-Euclidean.

#### Tomas Vencl

Suppose, that I can freeze the gravitation waves while they are passing through the detector. So there will be some static changes in spacetime (or suppose, that frequency of the wave is much smaller than actually is). Can I still see the interference pattern change ?

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#### PeterDonis

Mentor
If the physical distance between the ends of the arms changes then the wavelength of light in flight must also change, surely?
There isn't a single thing called "the wavelength of the light". To construct an invariant related to the wavelength of the light, you would have to specify which observer is observing it and how.

isn't the invariant that is directly measured the flight time? Or the variation in the difference in flight times for the two arms, more precisely
Yes.

#### PeterDonis

Mentor
One basically imagines that the real, physical, rulers, which we can actually make, are "distorted". Then one can envision a familiar, underlying, Euclidean geometry when one "undistorts" these distorted rulers. But one has to keep in mind that this "undistortion" is basically a mathematical trick - the physical measurments we make are all made with "distorted" rulers.
This approach has a good track record of causing confusion--I can think of plenty of past PF threads where that has happened. The problem is that anyone who is capable of really understanding and adhering to your caveat that the "undistorted ruler" thing is a mathematical trick, doesn't need the trick because they have already grasped that the only real geometry of spacetime in the presence of gravity is the non-Euclidean one. The reason people are attracted to the "undistorted ruler" trick is that it allows them to retain their pre-relativistic intuition that those undistorted rulers, or the undistorted Euclidean geometry, are somehow still "real", and that's precisely the intuition they need to let go of.

#### PeterDonis

Mentor
Suppose, that I can freeze the gravitation waves while they are passing through the detector. So there will be some static changes in spacetime (or suppose, that frequency of the wave is much smaller than actually is). Can I still see the interference pattern change?
This doesn't make sense for two reasons.

First, spacetime doesn't "change". It is a 4-dimensional geometry that already contains the entire history of whatever is being modeled. All the "changes" that take place because of the gravitational wave, or anything else, are already contained in that 4-dimensional geometry. So it makes no sense to talk about "changes in spacetime".

Second, "freeze" would mean "look at some particular spacelike slice cut out of the spacetime". But in any such slice, the interference pattern can't change, because the change we see in the pattern is a change from one spacelike slice to another.

#### Ibix

There isn't a single thing called "the wavelength of the light". To construct an invariant related to the wavelength of the light, you would have to specify which observer is observing it and how.
Ah - I see what you are getting at. I need to measure the positions of the endpoints of the wave simultaneously, and in a dynamic spacetime there isn't a unique way to define simultaneity. So I need to specify time in terms of free-floating clocks or whatever.

#### Ibix

Suppose, that I can freeze the gravitation waves while they are passing through the detector. So there will be some static changes in spacetime (or suppose, that frequency of the wave is much smaller than actually is). Can I still see the interference pattern change ?
As Peter says, this doesn't really make sense. However you can definitely consider very low frequency gravitational waves. In principle a LIGO-like detector could detect an arbitrarily low frequency wave. However, waves from cosmological sources will usually have lower powers associated with lower frequency waves and there may well be low-frequency cutoffs in the signal processing, so it would not surprise me if the answer is no in practice.

#### asca

This might help:

I'd like to thank everybody for your attempts at giving an answer. I feel that A.T. provided the most convincing one.
Thank you all again

#### Tomas Vencl

……... However you can definitely consider very low frequency gravitational waves. In principle a LIGO-like detector could detect an arbitrarily low frequency wave. However, waves from cosmological sources will usually have lower powers associated with lower frequency waves and there may well be low-frequency cutoffs in the signal processing, so it would not surprise me if the answer is no in practice.
Yes, this was exactly point of the question. Thank you.
I know, that LIGO has some cutoffs in sensitivity at low and high frequencies, I did not know if this is caused by setting in principle or technical reasons (yes, the boundary is very weak).

#### sandy stone

First, spacetime doesn't "change". It is a 4-dimensional geometry that already contains the entire history of whatever is being modeled. All the "changes" that take place because of the gravitational wave, or anything else, are already contained in that 4-dimensional geometry. So it makes no sense to talk about "changes in spacetime".
So, do researchers treat the emission and detection of gravitational waves as a 4-D solution of the EFE's? Or is there another approach?

(Hopefully this is not too much of a thread hijack.)

#### vanhees71

Gold Member
Well, in theory the gravitational-wave signals of black-hole or neutron-star mergers (which are the ones which are observed by LIGO/VIRGO) are solved by solutions of the EFE's numerically (employing GR hydro or transport for the matter part). For details, see the nice web sites of my colleague Luciano Rezzolla:

https://relastro.uni-frankfurt.de/research/

#### PeterDonis

Mentor
do researchers treat the emission and detection of gravitational waves as a 4-D solution of the EFE's?
Yes.

#### Ibix

do researchers treat the emission and detection of gravitational waves as a 4-D solution of the EFE's?
Yes. At a great distance when the waves are weak enough, you can treat them as a small perturbation to flat spacetime (or other analytically described spacetimes such as FLRW, I presume) and the maths is fairly tractable. Studying the emission is purely numerical work.

"LIGO and space time changes"

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