Light speed and the LIGO experiment

[JohnnyGui]

Hello all,

I was thinking about the speed of light and why it's constant and it brought me to the principle of the LIGO experiment for which I have an assumption that I want to verify. I'm a novice at this so please bear with me.

From what I know, the LIGO experiment splits an emitting light beam into 2 beams that travel along their 2 paths, let's call those paths A and B, which are then reflected to meet each other again at the spot they got split. Furthermore, if I understand correctly, gravitational waves are waves that temporarily stretch the spacetime fabric. If there's no stretch during the experiment (ruling out all other influences of course) the waves of the 2 light beams cancel each other out when they meet each other.

I have 3 questions about why the 2 light beams wouldn't cancel each other out when a gravitational wave passes by.

1. If a gravitational wave passes through and stretches the spacetime fabric of 1 of the two paths, let's say path A, does the light beam that travels along that path "interpret" this stretch as an increase in length that it has to travel along path A?

2. If so, would that increase in length for the light beam also mean an increase in the physical length that we would measure of path A or is it only an increase in length from the view of the light beam, i.e. merely an increase in stretch of the spacetime fabric? In other words, when I myself run along that path instead of the light beam with a constant v, would I also take a bit longer when a stretch in the spacetime fabric passes by that path (even if the increase in length is infinitesimally small)?

3. If it's merely an increase in length from the view of the light beam and we wouldn't measure any difference in the physical length of that path A (not even an infinitesimally small increase), wouldn't that mean that the light beam would take a longer time to travel along path A while we don't measure any difference in the physical path length and thus measure the light having a lower speed than the speed of light?

I have a hunch that my conclusion in point 3 is wrong, but if the first 2 are correct, i.e. there's a stretch in length only with respect to the light beam, then I'd really want to know how the speed of light would still stay the same.

EDIT: Just noticed I posted this in the wrong forum. Sorry about that. Could someone please move this to General Relativity?

 

[Orodruin]

2. If so, would that increase in length for the light beam also mean an increase in the physical length that we would measure of path A or is it only an increase in length from the view of the light beam, i.e. merely an increase in stretch of the spacetime fabric? In other words, when I myself run along that path instead of the light beam with a constant v, would I also take a bit longer when a stretch in the spacetime fabric passes by that path (even if the increase in length is infinitesimally small)?

The things you are describing here are the same thing. An increase in the light-beam travel length is the same thing as an increase in the physical distance. It does not matter if it is you or a light beam traveling.

 

[pervect]

Hello all,

I was thinking about the speed of light and why it's constant and it brought me to the principle of the LIGO experiment for which I have an assumption that I want to verify. I'm a novice at this so please bear with me.

From what I know, the LIGO experiment splits an emitting light beam into 2 beams that travel along their 2 paths, let's call those paths A and B, which are then reflected to meet each other again at the spot they got split. Furthermore, if I understand correctly, gravitational waves are waves that temporarily stretch the spacetime fabric. If there's no stretch during the experiment (ruling out all other influences of course) the waves of the 2 light beams cancel each other out when they meet each other.

I have 3 questions about why the 2 light beams wouldn't cancel each other out when a gravitational wave passes by.

1. If a gravitational wave passes through and stretches the spacetime fabric of 1 of the two paths, let's say path A, does the light beam that travels along that path "interpret" this stretch as an increase in length that it has to travel along path A?

I don't know how a light beam would "interpret" anything - I assume it's a metaphor of some sort, but exactly what the metaphor is is hard for me to imagine. Additionally, lightbeams don't have "points of view". So I'm not sure how to answer this question, or why this is different from point 2 below.

2. If so, would that increase in length for the light beam also mean an increase in the physical length that we would measure of path A or is it only an increase in length from the view of the light beam,''

Well, there are two major things you might mean by "physical distance" One is the current SI definition of the meter, as of 1983. This is

The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.

So by this definition, the answer is obviously yes, the physical distance changes.

Now, especially if you're not familiar with special relativity, you might have a similar understanding of physics as they did back in the late 1800's, when the meter was defined as the length of a prototype platinum bar. Note that the bar itself was too precious to use in routine measurements, various copies of the bar were made to perform actual measurements. You can read about the details in the history of the meter, for instance http://physics.nist.gov/cuu/Units/meter.html. But the fundamental idea was that the a carefully maintained referece bar was the reference standard for all distance measurements.

The answer in this case is also a yes - the distance changes. The modern definition of the meter isn't operationally any different from the 1889 definition in terms of a prototype bar. We can say this because if special relativity is true, the two definitions give equivalent results within experimental error, and we have a high confidence in special relativity.

Now, it's possible that you have some other personal definition of what "physical" distance means that's different from the above two. In this case we can't definitely answer the question unless we know what it is that you mean when you say "physical distance". But if you'd care to give an operational definition of what you mean by distance, we could perhaps give more reassurances, depending on how complex and involved the definition was.

 

[JohnnyGui]

The things you are describing here are the same thing. An increase in the light-beam travel length is the same thing as an increase in the physical distance. It does not matter if it is you or a light beam traveling.

Thanks, this explains to me why light speed stays the same because I wrongly thought that a stretch in spacetime only causes the light to travel a longer distance.

@pervect : Thanks for your answer. What I was trying to ask here is if a stretch in spacetime translates to an increase in length that we can observe. Oroduin confirms it does so I guess that answered my question.

This leads me to the following question though, but now about curved spacetime. Doesn't an area with a curved spacetime also mean that it has its spacetime stretched? If so, and a stretched spacetime means an increase in distance that we observe, doesn't that mean that we should see that an area that lies in a curved spacetime is longer than an area in a flat spacetime?

 

[Orodruin]

This leads me to the following question though, but now about curved spacetime. Doesn't an area with a curved spacetime also mean that it has its spacetime stretched? If so, and a stretched spacetime means an increase in distance that we observe, doesn't that mean that we should see that an area that lies in a curved spacetime is longer than an area in a flat spacetime?

Unfortunately, this question is not clear at all. I do not even dare to guess your intended meaning. Can you perhaps rephrase it?

 

[JohnnyGui]

Unfortunately, this question is not clear at all. I do not even dare to guess your intended meaning. Can you perhaps rephrase it?

Ok, I'll try. Let's take path A that I talked about in the opening post and let's say that path A lies in a flat spacetime. We measure the length of path A to be D. We just said that when a gravitational wave stretches the spacetime in that path, the length of path A would be temporarily longer than D. Now, scratch that wave, and let's curve the spacetime that path A lies in instead. Would that also make the length of path A increase, perhaps because curving spacetime means stretching it as well?

 

[pixel]

Just to tag along here with another question - the LIGO experiment is using a shift in an interference pattern to deduce a small change in a distance. This is based on the assumption that the speed of light is constant. But when the gravitational wave is present, local spacetime is curved. So in addition to changes in distance, is not the speed of light then affected also?

 

[JohnnyGui]

Just to tag along here with another question - the LIGO experiment is using a shift in an interference pattern to deduce a small change in a distance. This is based on the assumption that the speed of light is constant. But when the gravitational wave is present, local spacetime is curved. So in addition to changes in distance, is not the speed of light then affected also?

I think that's indeed what I wanted to know as in my post #6, namely if curved spacetime also translates in an increase in distance or not. If it does, then light speed should stay constant since we'd measure an increase in distance as well. However, if it doesn't, then only light would take longer to travel while we'd still measure the same distance and we'd conclude that light has a lower speed. I could be wrong about what you mean with your question though, sorry if that's the case.

 

[Nugatory]

I think that's indeed what I wanted to know as in my post #6, namely if curved spacetime also translates in an increase in distance or not. If it does, then light speed should stay constant since we'd measure an increase in distance as well. However, if it doesn't, then only light would take longer to travel while we'd still measure the same distance and we'd conclude that light has a lower speed. I could be wrong about what you mean with your question though, sorry if that's the case.

In curved spacetime, different straight-line paths between the same two events may have different lengths and then the time it takes for light to travel the different paths will be different even though the speed of light is the same on both paths. You can see something similar on the curved surface of the earth: If I can get from point A to point B by traveling in a straight line for 5000 kilometers due east, I can also from A to B by traveling 35000 kilometers in a straight line due west. An airplane flying at 1000 km/hr would take 5 hours on one path and 35 hours on the other - but we wouldn't conclude from this that the airplane is faster when it's flying east than when it's flying west.

 

[phyzguy]

Just to tag along here with another question - the LIGO experiment is using a shift in an interference pattern to deduce a small change in a distance. This is based on the assumption that the speed of light is constant. But when the gravitational wave is present, local spacetime is curved. So in addition to changes in distance, is not the speed of light then affected also?

Pixel: I don't understand why you think the speed of light would change. The situation is the following:
(A) The passing gravitational wave increases the distance between the mirrors.
(B) The speed of light stays the same.
(C) The light takes longer to travel the increased distance.
(D) This shifts the interference pattern.

I think that's indeed what I wanted to know as in my post #6, namely if curved spacetime also translates in an increase in distance or not. If it does, then light speed should stay constant since we'd measure an increase in distance as well. However, if it doesn't, then only light would take longer to travel while we'd still measure the same distance and we'd conclude that light has a lower speed. I could be wrong about what you mean with your question though, sorry if that's the case.

JohnnyGUI: This has already been answered. The distance does increase.

 

[pixel]

Pixel: I don't understand why you think the speed of light would change. The situation is the following:
(A) The passing gravitational wave increases the distance between the mirrors.
(B) The speed of light stays the same.
(C) The light takes longer to travel the increased distance.
(D) This shifts the interference pattern.

I thought the constancy of the speed of light only holds for inertial reference frames i.e. flat Minkowski space. In the presence of a gravitational wave, space is no longer flat. Granted it's by a small amount but LIGO is measuring something very small.

A related question: when light bends around a star, is its speed constant i.e. = c for the whole trip as measured in the frame of reference of the star?

 

[Orodruin]

A related question: when light bends around a star, is its speed constant i.e. = c for the whole trip as measured in the frame of reference of the star?

This depends on what you mean by "speed". The star is not in a flat space-time (or the light would not be bent!). It is true that the coordinate speed of light will vary, i.e., how much the spatial coordinates change per temporal coordinate, but this is not a physical speed (it can be made to take any value by a simple change of coordinates). The local speed of light, as measured in a local inertial frame (a coordinate system in a sufficiently small part of space-time) will be the same, c.

 

[PeterDonis]

We just said that when a gravitational wave stretches the spacetime in that path, the length of path A would be temporarily longer than D.

No, that's not what we said. If a gravitational wave is present, spacetime is not flat. So there is no path A. Instead you have a path B, through a particular small region of spacetime, which has a length that is greater than D. And if you pick a small piece of spacetime slightly to the future of path B, you will have a path C, which will have a length that is less than D. And the endpoints of paths B and C will lie on the same worldlines--the worldlines of two ends of an arm in a gravitational wave detector, for example.

In other words, the different path lengths you observe when a gravitational wave is present are not because spacetime is being "stretched" from flat. It is because the non-flat spacetime has a geometry in which the path lengths between two inertial worldlines are not constant, the way they would be if spacetime were flat. Spacetime doesn't "change"; it is a 4-dimensional geometry that already contains in it all the things we usually think of as "change", because we're not used to thinking of time as one of the dimensions of spacetime. "Changing with time" just means that if you look at parts of the geometry that are separated along the time dimension, they can contain curves with different path lengths that connect the same pair of inertial worldlines.

 

[Orodruin]

So there is no path A. Instead you have a path B, through a particular small region of spacetime, which has a length that is greater than D. And if you pick a small piece of spacetime slightly to the future of path B, you will have a path C, which will have a length that is less than D. And the endpoints of paths B and C will lie on the same worldlines--the worldlines of two ends of an arm in a gravitational wave detector, for example.

Are you referring to the world lines of the light signals now? Because if you do this is confusing. The light signal world lines are always null and not be related to any distance D. If not you will run into problems with defining what "path" means as it will depend on the particular separation of space-time into space and time. I would assume that you want to pick pieces of space-time which contain much less than a full period of the gravitational wave and always contains the full experiment. Such a region might be considered approximately static and it starts making sense to talk about "paths", but that also requires a simultaneity convention in that region.

 

[Nugatory]

I thought the constancy of the speed of light only holds for inertial reference frames i.e. flat Minkowski space. In the presence of a gravitational wave, space is no longer flat.

You are right that the constant speed of light only necessarily holds for inertial frames. However, you can have inertial frames even in non-flat spacetimes. An easy example would be the free-fall frame of someone in orbit around the earth; the inside of the ship is pretty clearly locally inertial even though gravity is sending it on a curved path around the earth. What you cannot have in a curved spacetime is a global inertial frame.

 

[PeterDonis]

Are you referring to the world lines of the light signals now?

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.

If not you will run into problems with defining what "path" means as it will depend on the particular separation of space-time into space and time.

That's true; the particular split I was thinking of was roughly the one that would correspond to the center of mass frame of the apparatus as a whole, with spatial axes oriented so that two of them are transverse to the incoming GW wave vector. As I understand it, this is basically the split that is used by the LIGO team in analyzing the experiment.

 

[pixel]

...It is true that the coordinate speed of light will vary, i.e., how much the spatial coordinates change per temporal coordinate, but this is not a physical speed (it can be made to take any value by a simple change of coordinates).

Trying to understand this so please indulge me - you can transform to another set of coordinates, but these coordinates would still have physical units and it should be possible to calculate the path length of light in those coordinates. As an example, we can go from x,y to r,θ. If the speed of light is constant, then the new spatial coordinates would have to change in such a way that the speed of light along the path taken would still come out to c meters/sec.

Wouldn't the idea of "coordinate speed" also be present in special relativity.? If I have a reference frame with coordinate x such that the speed of light is c in the x-direction, dx/dt = c, and define a new coordinate x'=2x, then dx'/dt = 2c.

Maybe you can provide a lucid reference that discusses the speed of light in GR.

 

[Orodruin]

Wouldn't the idea of "coordinate speed" also be present in special relativity.? If I have a reference frame with coordinate x such that the speed of light is c in the x-direction, dx/dt = c, and define a new coordinate x'=2x, then dx'/dt = 2c.

Yes, but the usual thing in SR is to stick to Minkowski coordinates. So you will rarely encounter this issue.

but these coordinates would still have physical units and it should be possible to calculate the path length of light in those coordinates

It does not matter if your units have physical dimension or not. Coordinate speed remains unphysical. In order to extract something physical you will need to use the metric.

 

[JohnnyGui]

No, that's not what we said. If a gravitational wave is present, spacetime is not flat

I didn't mean to say that if a gravitational wave is present, spacetime would stay flat. I meant to ask what a gravitational wave would turn this spacetime into. You kind of answered my question with this since I understand from this that gravitational waves make spacetime curved and that curving spacetime makes a length change.

"Changing with time" just means that if you look at parts of the geometry that are separated along the time dimension, they can contain curves with different path lengths that connect the same pair of inertial worldlines

That's a bit hard to comprehend for me, but do you mean with this that a passing gravitational wave is just a part on the time dimension which is more curved than the rest?

 

[PeterDonis]

I meant to ask what a gravitational wave would turn this spacetime into.

The wave doesn't "turn the spacetime into" anything. Once again: spacetime doesn't "change" from one geometry to another. It is a 4-dimensional geometry that already contains all the information about "change"--"change" is just "things are different in one place in the geometry compared to another".

do you mean with this that a passing gravitational wave is just a part on the time dimension which is more curved than the rest?

No. Have you seen a spacetime diagram? It has time as one axis (usually vertical) and space (one dimension of space) as another axis (usually horizontal). In the flat spacetime of SR, the diagram can be drawn at true scale on a flat sheet of paper, and can be ruled with grid lines that are just like the ones on an ordinary sheet of graph paper. The "time" grid lines are exactly parallel and the same distance apart everywhere, and so are the "space" grid lines, and the two sets are perpendicular to each other everywhere.

Now imagine trying to draw the same sort of diagram on a surface that has wavelike ripples in it. Those ripples are gravitational waves. (More precisely, they are part of them--you can't completely describe gravitational waves using only one dimension of space, but I'm trying to give a heuristic description.) On such a rippled surface, if you try to draw grid lines (in one direction for the time dimension, in another direction for the space dimension), you will find that they can't be drawn with the same properties as they have on a flat sheet of paper. For example, two nearby "time" grid lines won't always be parallel and won't always be the same distance apart. That kind of variation is what LIGO is detecting: the two "time" grid lines are like the worldlines of two endpoints of an arm in the LIGO detector (in other words, each gridline describes the history of one endpoint over time, and each point on the gridline corresponds to one particular instant in the history of that endpoint).

 

[Dr_Zinj]

I'm going to chime in and hopefully stimulate some thought and not confuse the issue.
Gravitational wave increases the distance the light has to travel.
Increase in space will decrease the density of matter in that space.
Should that not result in an infinitesimal increase in the speed of light since it is a more rarified vacuum?

 

[Orodruin]

I'm going to chime in and hopefully stimulate some thought and not confuse the issue.
Gravitational wave increases the distance the light has to travel.
Increase in space will decrease the density of matter in that space.
Should that not result in an infinitesimal increase in the speed of light since it is a more rarified vacuum?

No. It is essentially a perfect vacuum already.

 

[JohnnyGui]

No. Have you seen a spacetime diagram? It has time as one axis (usually vertical) and space (one dimension of space) as another axis (usually horizontal). In the flat spacetime of SR, the diagram can be drawn at true scale on a flat sheet of paper, and can be ruled with grid lines that are just like the ones on an ordinary sheet of graph paper. The "time" grid lines are exactly parallel and the same distance apart everywhere, and so are the "space" grid lines, and the two sets are perpendicular to each other everywhere.

Now imagine trying to draw the same sort of diagram on a surface that has wavelike ripples in it. Those ripples are gravitational waves. (More precisely, they are part of them--you can't completely describe gravitational waves using only one dimension of space, but I'm trying to give a heuristic description.) On such a rippled surface, if you try to draw grid lines (in one direction for the time dimension, in another direction for the space dimension), you will find that they can't be drawn with the same properties as they have on a flat sheet of paper. For example, two nearby "time" grid lines won't always be parallel and won't always be the same distance apart. That kind of variation is what LIGO is detecting: the two "time" grid lines are like the worldlines of two endpoints of an arm in the LIGO detector (in other words, each gridline describes the history of one endpoint over time, and each point on the gridline corresponds to one particular instant in the history of that endpoint).

Picturing this in my head now, I've got it. It is space which is getting stretched over time so that a photon would take a longer time to make the distance between the 2 endpoints (worldlines) of an arm in the LIGO detector.

I might be overthinking this, but I'm trying to extrapolate this explanation to why time is getting slowed down around massive objects. If spacetime is curved around a massive object, wouldn't that mean that a unit of length in that curved spacetime is longer than a unit length in a less curved one (i.e. far away from that massive object) just as an area, in which a gravitational wave passes by, has a (albeit temporary) length increase because of its curved spacetime?

If so, why would we interpret the spacetime curve in the LIGO experiment as an increase in distance while we interpret the spacetime curve around a massive object as a slowdown of time? It is as if I say, it took me longer than normal to run the distance D of a LIGO arm at v because the distance has temporarily increased by a gravitational wave, but in case of when I'm on a massive object, it took me longer than normal to run that same distance D at v because time has slowed down for me with respect to someone in a less curved spacetime. Aren't both spacetime curves practically the same thing so that the cause of me taking longer than normal to travel the same distance should be the same cause i.e. in both cases either an increase in distance or a slowdown of time?

 

[PeterDonis]

It is space which is getting stretched over time so that a photon would take a longer time to make the distance between the 2 endpoints (worldlines) of an arm in the LIGO detector.

That is one interpretation, but not the only possible one. Whenever you talk about "space" or "time" instead of "spacetime", that should be an indication to you that you are making an interpretation that depends on a particular choice of coordinates, and therefore will not necessarily apply if you make a different coordinate choice. In other words, you're not building your interpretation on invariants.

I'm trying to extrapolate this explanation to why time is getting slowed down around massive objects.

That's not a good idea. See below.

If spacetime is curved around a massive object, wouldn't that mean that a unit of length in that curved spacetime is longer than a unit length in a less curved one (i.e. far away from that massive object) just as an area, in which a gravitational wave passes by, has a (albeit temporary) length increase because of its curved spacetime?

This interpretation is sometimes adopted, but it has serious limitations. To take just the most obvious: a person who is close to the massive object won't notice any change in the "unit of length"; his meter sticks will work the same as your meter sticks (assuming you are far away from the massive object). And if you travel down to meet him, carrying your meter stick with you, your meter stick will be the same length as his.

(This is in addition to the fact that this "change in the unit of length due to curved space" is coordinate-dependent, as I said above. You can pick coordinates on spacetime around the massive object that make "space" flat, i.e., Euclidean. No interpretation based on "changing unit of length" will work in such coordinates.)

why would we interpret the spacetime curve in the LIGO experiment as an increase in distance while we interpret the spacetime curve around a massive object as a slowdown of time?

Because these are two different choices of coordinates in two different spacetime geometries. "Spacetime curvature" is not just one thing. There are different geometries of spacetime just as there are different shapes of objects in ordinary geometry. You can't expect an interpretation that works for one shape to work for another--much less that an interpretation that works for one particular choice of coordinates will work for another.

Aren't both spacetime curves practically the same thing

Most emphatically not. See above. You are both overthinking and oversimplifying. "Spacetime curvature" is not as simple as you are trying to make it, and you can't shoehorn all possible spacetime geometries into a single interpretation. It won't work.

 

[JohnnyGui]

Thanks for the extensive reply.

(This is in addition to the fact that this "change in the unit of length due to curved space" is coordinate-dependent, as I said above. You can pick coordinates on spacetime around the massive object that make "space" flat, i.e., Euclidean. No interpretation based on "changing unit of length" will work in such coordinates.)

Do you mean with this that the meter stick will not change in length, not just because you're moving from a flat space into a curved one? Such that, even after compensating for the curved space around the massive object (making space flat around the massive object as you stated) you'd still get the same length of the meter sticks between the one on the massive object and the one far away from it? Also, is this the meaning of the meter stick being invariant to coordinates?

Because these are two different choices of coordinates in two different spacetime geometries. "Spacetime curvature" is not just one thing. There are different geometries of spacetime just as there are different shapes of objects in ordinary geometry. You can't expect an interpretation that works for one shape to work for another--much less that an interpretation that works for one particular choice of coordinates will work for another.

I'm trying to fully understand the meaning of "two different coordinates". Do you mean with this that the amount/shape of curvature is different among the two scenarios (curved space at a gravitational wave and curved space at a massive object)? If so, what if we consider that the amount of curvature/shape is the same among the two coordinates. Is it then still not possible to dumb this down to asking why we interpret an increase in distance in one and a slowdown of time in the other?

 

[PeterDonis]

Do you mean with this that the meter stick will not change in length, not just because you're moving from a flat space into a curved one?

It won't change in length as measured by someone traveling with the meter stick, no.

Such that, even after compensating for the curved space around the massive object (making space flat around the massive object as you stated)

You can't "make" space flat. You can only choose coordinates (possibly) such that "space" in those coordinates is flat. That doesn't change anything about the spacetime geometry. It just changes the coordinates you use to describe it.

you'd still get the same length of the meter sticks between the one on the massive object and the one far away from it?

This is not a local measurement, at least if I'm understanding you correctly. You are now talking about putting meter sticks end to end between the observer close to the massive object and the one far away from it. You can do this, but what will you compare the result to to see if it has "changed"? Changed compared to what?

is this the meaning of the meter stick being invariant to coordinates?

I'm not sure what you mean by a meter stick being "invariant to coordinates".

I'm trying to fully understand the meaning of "two different coordinates".

It's like polar coordinates vs. Cartesian coordinates on a flat piece of paper. The piece of paper is the same; but a particular point on the piece of paper will have different coordinates (sets of numbers) assigned to it if you use polar coordinates, vs. if you use Cartesian coordinates.

Do you mean with this that the amount/shape of curvature is different among the two scenarios (curved space at a gravitational wave and curved space at a massive object)?

The geometry of spacetime--how it is curved, what "shape" it has--is independent of your choice of coordinates. The two scenarios have different geometries of spacetime--different shapes. But the fact that, for each of those shapes, you can choose different coordinates to describe it, is an additional fact, over and above the fact that the two shapes are different.

It's like the surface of a ball vs. a flat sheet of paper. Both are 2-dimensional manifolds, so coordinates on both will be mappings of pairs of numbers to points on the manifold. But they have different shapes. And, in addition to that, there are multiple coordinate charts you could use to describe each shape: polar vs. Cartesian (and others) for the flat sheet of paper; latitude/longitude, Mercator, stereographic, etc. for the surface of the ball.

Is it then still not possible to dumb this down to asking why we interpret an increase in distance in one and a slowdown of time in the other?

I don't think so. I think you would be better served by taking the time to learn about how spacetime geometry and spacetime curvature works in more detail, and to learn how different the two cases are that you are trying to simplistically compare.

What you are doing is something like taking a sheet of paper that has some wrinkles in it, and the surface of a ball, and asking me to give you a single description that applies to both shapes. It won't work. You need to take the time to understand each shape separately.

 

[JohnnyGui]

snip

You're right about me taking time to learn about spacetime geometry and curvature. I should dive into this a bit more.

However, judging by your reply, I have a feeling my question isn't clear enough. I didn't mean to choose coordinates in a curved spacetime such that those coordinates lies in a more or less flat part of that curved spacetime. What I want to know is, if person A is holding a metre stick in a flat spacetime and person B is holding a similar metre stick while standing on a massive object such that the coordinates of that metre stick lies in a curved spacetime (not a flat part of the curved spacetime, I know it's a bit of a stretch since a metre is a very small unit compared to the curved spacetime it lies in), will the metre stick of person B look longer with respect to person A?

 

[PeterDonis]

if person A is holding a metre stick in a flat spacetime and person B is holding a similar metre stick while standing on a massive object such that the coordinates of that metre stick lies in a curved spacetime (not a flat part of the curved spacetime, I know it's a bit of a stretch since a metre is a very small unit compared to the curved spacetime it lies in), will the metre stick of person B look longer with respect to person A?

What does "look longer with respect to person A" mean?

The issue you are missing here is that coordinates have no physical meaning. It is true that the meter stick B is holding occupies a smaller increment of the radial $r$ coordinate (assuming we're using standard Schwarzschild coordinates) than the meter stick A is holding does. But that has no physical meaning in itself. So if you want an answer to your question that has physical meaning, you have to first specify the question. What measurement are you imagining A to make that tells him the length of B's meter stick "with respect to him"?

 

[JohnnyGui]

What does "look longer with respect to person A" mean?

The issue you are missing here is that coordinates have no physical meaning. It is true that the meter stick B is holding occupies a smaller increment of the radial $r$ coordinate (assuming we're using standard Schwarzschild coordinates) than the meter stick A is holding does. But that has no physical meaning in itself. So if you want an answer to your question that has physical meaning, you have to first specify the question. What measurement are you imagining A to make that tells him the length of B's meter stick "with respect to him"?

Ah, ok. If A knows his distance to B and measures the angle between both ends of the metre stick that B is holding perpendicular to A's viewing angle, wouldn't A be able to calculate the length of the stick B is holding?

 

[PeterDonis]

If A knows his distance to B and measures the angle between both ends of the metre stick that B is holding perpendicular to A's viewing angle, wouldn't A be able to calculate the length of the stick B is holding?

He would also have to allow for the effect of spacetime curvature on the paths of light rays. If he corrected for that, he would find that B's stick is the same length as his own.

 

[JohnnyGui]

He would also have to allow for the effect of spacetime curvature on the paths of light rays. If he corrected for that, he would find that B's stick is the same length as his own.

So if he didn't correct for that, he would measure a different length of the meter stick?

 

[PeterDonis]

So if he didn't correct for that, he would measure a different length of the meter stick?

No; if he didn't correct for that, he would not be making a "measurement", he would be making a physically meaningless calculation.

 

[JohnnyGui]

No; if he didn't correct for that, he would not be making a "measurement", he would be making a physically meaningless calculation.

Ok, so not correcting for a distance that lies in a curved space would be meaningless. Now, when it's about a gravitational wave "stretching/curving" the space around one of the LIGO arms making that arm increase in length, and you correct for that "stretch/curving", shouldn't the length of that LIGO arm be the same as its initial length (i.e. without the presence of a gravitational wave)?

 

[PeterDonis]

not correcting for a distance that lies in a curved space would be meaningless

You're stating this wrong. You observe light rays from each end of the object coming into your detector separated by a certain angle. You calculate from that angle and the known distance of the object the "size" of the object. But in doing that calculation, you have to use the correct paths of the light rays, in the actual curved spacetime they are traveling through. The usual formula for "size" in terms of distance and angle subtended in your field of view does not do that; it assumes that spacetime is flat. So you have to "correct" that formula to take into account the actual paths of the light rays in the actual curved spacetime. That's not "correcting for a distance"; it's using the correct formula instead of an incorrect one.

when it's about a gravitational wave "stretching/curving" the space around one of the LIGO arms making that arm increase in length, and you correct for that "stretch/curving",

No, you don't "correct for the stretch/curving". You calculate the correct path of the light rays, in the curved spacetime. Whether the geometry of that curved spacetime can be interpreted as "stretching/curving space" depends on your choice of coordinates. But the paths of light rays in the curved spacetime geometry do not; they are what they are regardless of what coordinates you choose. And in LIGO, the paths of the light rays in the curved spacetime geometry are what determine the interference pattern at the detector. You don't have to "correct" for anything.

 

[JohnnyGui]

No, you don't "correct for the stretch/curving". You calculate the correct path of the light rays, in the curved spacetime. Whether the geometry of that curved spacetime can be interpreted as "stretching/curving space" depends on your choice of coordinates. But the paths of light rays in the curved spacetime geometry do not; they are what they are regardless of what coordinates you choose. And in LIGO, the paths of the light rays in the curved spacetime geometry are what determine the interference pattern at the detector. You don't have to "correct" for anything.

I understand that it depends on the choice of coordinates and that you don't need to correct for anything to measure the interference pattern. I'm only focusing on physical length of the LIGO arm as well as when you choose such a particular set of coordinates in that length that can be interpreted as stretching/curving the space around that LIGO arm. In that case, if you correct the formula for calculating the length of the arm and take into account the actual paths of the light rays in the curved space caused by the gravitational wave, would you calculate the same length as the initial length of the LIGO arm in the absence of a gravitaitonal wave? Just like you'd calculate the same length of the meter stick previously mentioned if you correct for the effect of spacetime curvature on the paths of the lightrays?