Expanding Space: How Do We View It?

[wolram]

With this idea that space is expanding every where, i assume that every unit of expanding space has a shape, and must have some limit as to how far it can expand, the shape must be very uniform for all, unless fuzzy edges are allowed, how do people view expanding space?

 

[Wallace]

Space, in the way you are speaking of it, does not exists. The phrase 'the expansion of space' is not a physical theory, it is a loose metaphor or analogy. When you try and push the analogy you come up with non-sensical question such as those that you are asking.

The 'idea that space is expanding everywhere' is not an idea that comes from science, but from science journalism, so unfortunately your questions have no meaningful answer.

If it means anything at all, 'the expansion of space' simply means that there is more distance between any two objects. If you lift your coffee cup off the table you could say 'the space between the table and my cup expanded' and this is just as valid a use of the expression as it is in cosmology. The reason that the coffee cup left the table was not the increase in distance, the increase in distance occurred because you moved the cup. In the same way galaxies do not move apart because the space between them expanded, they move apart because they are moving apart already (i.e. they have a relative velocity and hence momentum, and this momentum is conserved). The increase in distance occurs because of this momentum. If you choose to call this 'the expansion of space' then that is fine, but there is clearly no need to turn this into a physical theory, and then make up properties that some none existent medium must have.

 

[FunkyDwarf]

I disagree entirely with the above statement. GR clearly gives a mathematical framework for distortion of the underlying metric. How you want to interpret this i suppose is up for debate but i don't see anything wrong with the stretching space idea. The distance between objects in space gets larger not because of their momentum (although this is sort of the case in cosmological expansion but not in generality) but rather because some energy or momentum (not neccisarily that of the objects mentioned) has distorted space in that region.

In terms of cosmological expansion of space this IS caused by mass flying apart and thus distorting space as it goes, but if you take that distortion as read then the metric certainly isn't euclidean and if you throw particles around in that distorted space you will find it acts as if the space has indeed been stretched. Its not a perfect analogy ill grant you but its certainly not as flawed as you suggest.

As to the OP

i assume that every unit of expanding space has a shape,

Not sure what you mean by this. By and large the discretised parts of the metric (the underlying space stuff thing) do not change locally due to disturbances but globally they do, as to their shape changing, its quite possible to create a situation where shapes are distorted, but again I am not sure what you mean by the units of space changing shape.

 

[Wallace]

I disagree entirely with the above statement. GR clearly gives a mathematical framework for distortion of the underlying metric. How you want to interpret this i suppose is up for debate but i don't see anything wrong with the stretching space idea.

The idea is only wrong when you use it wrongly. You can think of an FRW metric in terms of expanding space, but it is fraught with danger. I common problem is that it encourages mixing first and second derivatives of motion. So people ask 'why don't galaxies expand with space'. If someone asks such a question then they have been duped by this expanding space concept into making a simple concept much harder (i.e. when you realize that the expansion of space is simply a description of one non-unique co-ordinate system, you see that there is obviously no expansion occurring in a galaxy and hence no reason why any would start. Too often people interpret the expansion of space as 'causing' things to move apart, and that is the real problem).

The distance between objects in space gets larger not because of their momentum (although this is sort of the case in cosmological expansion but not in generality) but rather because some energy or momentum (not neccisarily that of the objects mentioned) has distorted space in that region.

Really? I think the Bianchi Identities will have something to say about that! You still have conservation laws in GR that need to be satisfied, and these reduce to the familiar conservation of momentum in Newtonian physics. Do you really want to explain than when you throw a ball in the air the reason it gets further from your hand is not really because of it's momentum, but because of the distortion of space between the ball and your hand??

It is not 'sort of' the case that conservation laws apply to the motion of cosmological bodies. They most definitely apply. Conservation laws tell you why expansion continiues, the presence of matter and energy in the space tell you how that mometum changes (i.e. since the energy content of the universe determines a(t) in the FRW metric).

In terms of cosmological expansion of space this IS caused by mass flying apart and thus distorting space as it goes, but if you take that distortion as read then the metric certainly isn't euclidean and if you throw particles around in that distorted space you will find it acts as if the space has indeed been stretched. Its not a perfect analogy ill grant you but its certainly not as flawed as you suggest.

The fundamental problem is that you fail to define space, then insist that it (whatever 'it' is) expands, distorts or whatever and then throw terms like 'non-euclidean' around as if they were relavant in this context. 'Space' in the FRW metric can in fact be perfectly flat (if we take k=0, and this is observationally favoured). In this way we see that in fact only 'time' is curved, but this is admitadly a very confusing way of putting things. The bottom line is that Newtonian physics acutally gives the correct answer, even for 'non-flat' FRW models out to surprisingly great distances. Eventually of course relativistic effects kick in such that purely Newtonian cosmology doesn't get the right answer. BUT, you can understand the expansion of the universe qualitatively without recourse to fuzzily defined quasi-relativistic concepts like the expansion of space. Newtons first and second law can get you there with much more clarity (and this all happens in a 'Euclidean' space).

If you disagree please answer this question, what experiment (however fanciful) could ever be performed that demonstrated that space is expanding?

 

[sylas]

The idea is only wrong when you use it wrongly. You can think of an FRW metric in terms of expanding space, but it is fraught with danger. I common problem is that it encourages mixing first and second derivatives of motion. So people ask 'why don't galaxies expand with space'. If someone asks such a question then they have been duped by this expanding space concept into making a simple concept much harder (i.e. when you realize that the expansion of space is simply a description of one non-unique co-ordinate system, you see that there is obviously no expansion occurring in a galaxy and hence no reason why any would start. Too often people interpret the expansion of space as 'causing' things to move apart, and that is the real problem).

There are several issues I have with this.

Primarily, the reason a galaxy does not pull apart is simply that it is held together with gravity. Space is expanding (or whatever phrase you perfer, if you want some other phrase) and that includes everything within the galaxy as well. The thing is simply that objects move through space, and a galaxy holds together much too strongly to fall apart with expanding space.

In principle, with a continuous increase in the rate of expansion of space ... or whatever you want to call the behaviour described with the FRW solutions ... you can indeed get galaxies torn apart. Even solar systems or (in the ultimate extreme) even atoms. This albeit fanciful proposal was considered theoretically, as a solution for the conventional equations involving a kind of super acceleration, or super-inflation. It's sort of like a dark energy term that keeps getting strong. It's called the "Big Rip".

This is not really a credible model for our universe; it involves a value for the cosmological parameter ω that is less than -1. The conventional cosmological constant, or "inflation" has ω = -1, and our present universe is apparently "accelerating", which means ω < -1/3. The point is that the notion of what is usually called "expansion of space" can, in principle, tear apart a galaxy. It's simply that expansion rates as they stand are much too small, and so the galaxy continually falls together under it's own gravity, even as space "expands".At least, that is my understanding of the matter.As for experiments that demonstrate space is expanding: the observations of luminosity vs z relations in distant supernovae qualify for me. I expect you to say it's not "really" space expanding; but that's just words. It's what I call expansion of space, and I think this is quite conventional.

Another good experiment, in principle, though not practical in reality, would be to take two particles at a very large separation distance, and hold them steady at that separation. Then release. It's a thought experiment described by Davis and Lineweaver in their paper on the nature of the expansion of space, and the results are surprising at first. However, IMO it becomes intuitive once you grasp that it's all about space expanding, and not about any "substance" that expands or drags on a galaxy. The distant tethered galaxy would be, at point of release, moving towards us through space at just the velocity to equal the expansion of space. See Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects (Davis and Lineweaver, arXiv:astro-ph/0104349, 2001).

In a "cold dark matter" model (no dark energy term) the galaxy continues to move towards us, and in fact approaches us as the expansion of space "slows", until eventually it passes right by us and (eventually) joins in with the Hubble flow on the other side of the sky!

In an accelerating expansion (as we apparently have at present), the galaxy will move towards us, but the expansion of space accelerates which leads to an increasing distance between us and the initially "tethered" galaxy.

This behaviour is a prediction of the FRW models, and it really only makes sense in the light of expansion of space itself, IMO. Which is why that is how it is conventionally described.

Cheers -- Sylas

 

[Wallace]

There are several issues I have with this.

Primarily, the reason a galaxy does not pull apart is simply that it is held together with gravity. Space is expanding (or whatever phrase you perfer, if you want some other phrase) and that includes everything within the galaxy as well. The thing is simply that objects move through space, and a galaxy holds together much too strongly to fall apart with expanding space.

No! This is simply wrong. There is no other way to put it, on this level it is not a question of personal taste and interpretation. To suggest that the 'expansion of space' is trying (but failing) pull a galaxy apart violates the equivalence principle. Really it's a sorry state of affairs. The concept of the expansion of space relates well to situations in which co-moving co-ordinates are an approrpriate co-ordinate system. But clearly that is not the case in a galaxy.

Think about the physics, you are suggesting that the expansion of space is a force it is this force that 'causes' galaxies to move apart. This is non-sense, if galaxies have relative motion inertia says they will continue that motion. Forces act on the second derivative of motion, not the first. What is the physical origin of your alleged expansion of space that 'acts everywhere'?

The fallacy is that expansion is a universal force. This is untrue. It is a universal motion (on large enough scales) but on small scales, if things aren't moving apart then there is not mystical process dubbed the expansion of space that tries to get them to start moving apart.

You are trying to elevate analogies to physical theories.

In principle, with a continuous increase in the rate of expansion of space ... or whatever you want to call the behaviour described with the FRW solutions ... you can indeed get galaxies torn apart. Even solar systems or (in the ultimate extreme) even atoms. This albeit fanciful proposal was considered theoretically, as a solution for the conventional equations involving a kind of super acceleration, or super-inflation. It's sort of like a dark energy term that keeps getting strong. It's called the "Big Rip".

This is not really a credible model for our universe; it involves a value for the cosmological parameter ω that is less than -1. The conventional cosmological constant, or "inflation" has ω = -1, and our present universe is apparently "accelerating", which means ω < -1/3. The point is that the notion of what is usually called "expansion of space" can, in principle, tear apart a galaxy. It's simply that expansion rates as they stand are much too small, and so the galaxy continually falls together under it's own gravity, even as space "expands".

You are once again mixing first and second derivatives of motion. Dark energy is something that imparts a force. But dark energy is most definitely NOT what we 'normally call the expansion of space'. Dark energy changes the rate at which expansion occurs but it does not cause expansion to occur, this would happen without any dark energy.

Galaxies do not 'continually fall together under their own gravity'. Rather, they are bodies which have virialised, their kinetic and gravitational potential energies are balanced in accordance with the virial theorm. They do not expand or contract, they just sit there. In the case where dark energy has strange behaviour (such as phantom DE with w<-1) then eventually this balance changes and a galaxy can be ripped apart, but this is due to the decrease in potential energy due to the changed balance of forces within the galaxy. So in this case the force due to the DE causes the galaxy to 'de-virialise' but to equate this acceleration due to dark energy with the motion of galaxies moving apart (the expansion of space) is clearly non-physical.

At least, that is my understanding of the matter.

Please take the time to comprehend the above, I hope your understanding will be much improved. There are some very pervasive myths relating to the expansion of the Universe, and they are so often repeated it can be hard to unlearn them. For what it's worth I am a professional cosmologist. Not that I'm trying to beat you down with the letters after my name but I'm not just some ranting hack, unfortunately what you are arguing above flies in the face of some very basic concepts. I know if feels like you have a good grip on things, because some much bad pop-sci agrees with you, but this doesn't make it right!

As for experiments that demonstrate space is expanding: the observations of luminosity vs z relations in distant supernovae qualify for me. I expect you to say it's not "really" space expanding; but that's just words. It's what I call expansion of space, and I think this is quite conventional.

Again, conventional in what circles? The relationships you refer to come straight from GR, so any interpretation must be consistent with GR, and the 'universal expansion of space' violates the equivalence principle, so can't be compatible with GR.

Another good experiment, in principle, though not practical in reality, would be to take two particles at a very large separation distance, and hold them steady at that separation. Then release. It's a thought experiment described by Davis and Lineweaver in their paper on the nature of the expansion of space, and the results are surprising at first. However, IMO it becomes intuitive once you grasp that it's all about space expanding, and not about any "substance" that expands or drags on a galaxy. The distant tethered galaxy would be, at point of release, moving towards us through space at just the velocity to equal the expansion of space. See Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects (Davis and Lineweaver, arXiv:astro-ph/0104349, 2001).

In a "cold dark matter" model (no dark energy term) the galaxy continues to move towards us, and in fact approaches us as the expansion of space "slows", until eventually it passes right by us and (eventually) joins in with the Hubble flow on the other side of the sky!

In an accelerating expansion (as we apparently have at present), the galaxy will move towards us, but the expansion of space accelerates which leads to an increasing distance between us and the initially "tethered" galaxy.

Right, but this doesn't demonstrate that space expands. Try this. In an FRW universe with a(t)=t (equivalent to an 'empty' universe) work out the motion of the particle in the tethered galaxy problem. In co-moving co-ordinates it does a complicated dance in which it moves slowly towards the origin, however if you transform to a Minkowski metric you find that the particle is in fact statinary the whole time. So 'space expands' in one co-ordinate system but not in another. Clearly then, this is not a physical phenomenon (which would be co-ordinate independant).

In this completely empty universe you can make up all kinds of sophostry about how the expansion of space is acting on the particle when in fact it is doing nothing at all. The danger the expansion of space interpretation is that you are holding up the 'motion' of particles with respect to one non-unique co-ordinate system (co-moving co-ordinates) as being in some way special. You are elevating the behaviour of co-ordinates to the level of physics and this is non-sense.

This behaviour is a prediction of the FRW models, and it really only makes sense in the light of expansion of space itself, IMO. Which is why that is how it is conventionally described.

It's also the prediction of Newtonian physics. Does Newtonian physics need 'the expansion of space' to make sense?

Take a matter only model. You setup the comoving fluid in exactly the same way and set a particle at rest with respect to the origin at distance R. What does it do? It moves towards the origin of course, because Gauss's law says that all the matter in a sphere centered on the origin and of radius R acts as if it was placed at the origin. The motion of the particle is understood perfectly well by Newtonian physics. You don't need complex ideas like ' the particle is moving towards us through space at just the velocity to equal the expansion of space', this is silly. We said the particle was at rest, so let it just be at rest. Once we let it move it picks up a velocity due to the effects of gravity. We can do this problem in co-moving co-ordinates, in which case you need to have a peculiar velocity that matches the co-moving one, but you only need to do that because of the particular co-ordinates. If you are interested in the test particle it is easier to do the problem in a more natural co-ordinate system in which it is simply at rest.

 

[Ich]

This behaviour is a prediction of the FRW models, and it really only makes sense in the light of expansion of space itself, IMO.

True, but that's simply because the setup of the thought experiment is set up and analyzed in a specific coordinate system. This coordinate system is not the standard one for defining velocities, yet is is supposed to be by the authors. That's why counterintuitive results follow that only make (limited) sense in the "expanding space" picture.
Try calculating the "x"-position of a really tethered galaxy in empty spacetime. Again you will find that there is much counterintuitive behaviour of the coordinates, not of physics, and that Davis and Lineweaver fail to account for that.

 

[sylas]

True, but that's simply because the setup of the thought experiment is set up and analyzed in a specific coordinate system. This coordinate system is not the standard one for defining velocities, yet is is supposed to be by the authors. That's why counterintuitive results follow that only make (limited) sense in the "expanding space" picture.
Try calculating the "x"-position of a really tethered galaxy in empty spacetime. Again you will find that there is much counterintuitive behaviour of the coordinates, not of physics, and that Davis and Lineweaver fail to account for that.

How do you define "really tethered galaxy"?

(For other readers: is a serious question. Ich is my new hero for helping me fix up my errors, and to understand an error made in a different Davis and Lineweaver paper concerning SR models of ballistic expansion in a static space. (Ref: [post=2144968]msg #71[/post] of Why "expanding space"?) I hope that here he can do it again, or else that I can return the favour. I don't know which it will be as yet.)

Obviously, galaxies are not really tethered, and so to be precise, we would have to define what we mean by giving world lines in some co-ordinate system. The definition used in Davis and Lineweaver is explicit. They mean that "proper velocity" is zero, which means the rate of change of "proper distance" with "proper time".

Furthermore, the question is not about what happens as the tether remains fixed, but what happens when a particle is released from the hypothetical tether, and follows an unaccelerated world line thereafter. Davis and Lineweaver are explicit that the problem is about a galaxy that has been "let go" from the initial condition defined by the hypothetical tether.

Given this definition, the rest follows independently of co-ordinates. For example, in a critical density model, they state that an unaccelerated particle with a proper velocity initially equal to zero will gradually approach the observer, and pass through to the other side of the sky, where it will eventually (in the limit) come to rest against the Hubble flow.

Here's a review of the relevant co-ordinate stuff, as I see it.

(A)Proper time and proper distance co-ordinates

I find it easiest to use the co-ordinates of proper time and proper distance. This is commonly used in cosmology. For example, these are the co-ordinates used in Ned's formulae for distances in expanding space.

In this co-ordinate system, there is an essentially arbitrary choice of a world line for a non-accelerating reference particle to define distance zero, and any other spacetime point is defined by a radial distance r, by two polar co-ordinates for direction, and by a proper time co-ordinate. With these co-ordinates, the scale factor is simply a function of proper time. Different models for the universe correspond to different functions for the scale factor.

There is also a notion of "co-moving distance", which is defined as the proper distance divided by the scale factor. Co-moving observers are ones that maintain a fixed co-moving distance. In our universe, presuming a uniform cosmological background radiation, all observers that are (locally) at rest with respect to the background radiation (as determined by redshift of the radiation being equal from all directions) are co-moving observers with each other. Our Sun is currently moving at a velocity of about 600 m/s locally wrt to the cosmological background, which is a minor complication. For cosmology, it is convenient to use cosmological background as a reference point for zero proper distance, rather than the Earth itself.

The proper distance has a simple intuitive consequence. Suppose that all of space is filled with co-moving observers, each of which holds a ruler of a fixed length, defined by a light clock. That is, the ruler measures distance by seeing how far light moves, locally, in a small fixed increment of proper time. At a previously agreed common instant of proper time, all the co-moving observers between two galaxies get a reading for their rulers. The distance measured as the sum of all the rulers at a common instant of proper time is the proper distance co-ordinate, at that point in proper time. The rulers have to be co-moving, which means that the number of fixed length rulers between two co-moving objects in an expanding space is always increasing.

(B) Recession velocities

You can get different notions of velocity when you have different notions of distance. Distance, and velocity, are numbers that depend on a co-ordinate system. I'll use a "proper velocity" as the rate of change of "proper distance" with "proper time". This is a pretty conventional definition.

You can easily give other definitions. I've deleted some remarks here about other notions as a side track. But I can give them and work with them if needed.

For a co-moving particles, the proper velocity is directly proportional to rate of change of scale factor.

(C) Defining "tether"

My understanding of "tethered galaxy" is one for which the "proper velocity" is zero. It has a world line for which the proper distance remains constant with proper time. This is explicitly what Davis and Lineweaver use.

In my opinion, they are right to do so. The "proper velocity" is a standard for talking about recession velocities in cosmology, and people who have trouble understanding the notion of expanding space might just possibly benefit from thinking about what will happen to an unaccelerated particle at some great distance if it starts from a point in time with zero "proper velocity".

If they won't benefit from this thought experiment, then the Davis and Lineweaver paper is no good to them. I, however, believe I have gained useful insights into the FRW solutions by considering the tethered galaxy problem as defined and solved by Davis and Lineweaver.

If there are actual errors in their analysis, given their explicit definitions, I'll be surprised. But I've been surprised before, with respect to some plain errors in another paper concerning an incorrectly derived SR formula.

(D) Homework

You have proposed for me a homework exercise. This worked well for me last time, and I'll try it again. It may take me a bit of time, as I am busy with other projects also.

As I understand it, your homework problem can be defined formally as finding the world line of an unaccelerated particle that initially has a proper distance "r" at a time "t", and initial proper velocity 0, in an empty universe where the scale factor is simply a(t) = H₀ t. Is that right?

For reference, I am going to assume that the local momentum of an unaccelerated particle in motion wrt to the co-moving background is inversely proportional to scale factor. This is not a violation of conservation of momentum, for the same reason cosmological redshift of photons is not a violation of conservation of energy. Indeed, the locally measured momentum of photons has precisely this relation to scale factor. I believe it holds also for any particle. I mentioned this some three years ago on the BAUT forums, where I worked on similar problems. (linky) The reference given for explaining this decay of locally measured momentum is Misner, Thorne and Wheeler. I read MTW in small doses, because it goes well beyond my own level of expertise.

Cheers -- Sylas

 

[Chalnoth]

One way to think about why expanding space isn't an accurate description of what's happening within a galaxy is this:

A galaxy is a gravitationally-bound object. It is held together by the mutual gravitation of all of the stars, gas, and dark matter within it (mostly the dark matter). A galaxy has a relatively stable shape because this is a relatively stable gravitational system. In fact, if you neglect friction and radiative cooling, and only have normal + dark matter, a galaxy would be perfectly stable.

In a similar way, if we look at the universe as a whole, the expansion of the universe is driven by gravity. Gravity is the only force that is in operation on such large scales, and thus the way gravity interacts with the matter/energy that makes up the universe determines how that expansion behaves.

Combine these two and it should become clear: if gravity is the exact same thing that is keeping a galaxy stable as that which drives expansion, then the expansion cannot pull apart a galaxy, because the expansion is driven by gravity, while a galaxy is held together by that same force. We've already taken into account the expansion when we've figured out how gravity holds the galaxy together!

 

[sylas]

Thanks Chalnoth! And Wallace, I understand that you are a professional cosmologist. That is great. I have long been interested in this, and I have no pretensions of being an original theorist. I am aiming to understand (and explain sometimes to others, to the best of my ability) some of the details of conventional modern cosmology.

To this end, I will in all seriousness be very grateful indeed to any professional who is willing to take the time to help an amateur.

Now I mean no offense, but there's another side to the coin. I don't know you at all; so I'm not just going to take everything at face value all at once. You don't seem to give any details of your real life work, which is fair enough. Your posts stand on their own merits, and they look very helpful. But I still have no choice but to evaluate your posts critically, to the best of my limited ability. So I'll work through things and take my own time at it. Fair enough?

---

I was half way through this reply, including the worked example you proposed, but given that Chalnoth just above gives a useful comment, and because this is relevant to what I've written already, I'll post this much now, to acknowledge this point exactly. I think.

I've still got a heap to learn, but I'm not a complete novice, so here we go.

It seems to me that you dislike the notion of "expanding space", as a misleading metaphor. I can agree with that; except that I don't think there is any better metaphor around. In the end, the real deal is the formal mathematical account, and when explaining this for a legitimately interested popular audience, there's no choice but to use some metaphors of some kind. I completely understand that the metaphor is not the reality, or the formal description of reality. So I will continue to speak of "expanding space", until something better comes along, while also trying to explain how things work on the basis of the underlying physics – again, to the best of my own limited abilities.

Primarily, the reason a galaxy does not pull apart is simply that it is held together with gravity. Space is expanding (or whatever phrase you perfer, if you want some other phrase) and that includes everything within the galaxy as well. The thing is simply that objects move through space, and a galaxy holds together much too strongly to fall apart with expanding space.

No! This is simply wrong. There is no other way to put it, on this level it is not a question of personal taste and interpretation. To suggest that the 'expansion of space' is trying (but failing) pull a galaxy apart violates the equivalence principle. Really it's a sorry state of affairs. The concept of the expansion of space relates well to situations in which co-moving co-ordinates are an approrpriate co-ordinate system. But clearly that is not the case in a galaxy.

Well, on a minor point, I don't say that expansion of space is trying to pull a galaxy apart. I say that gravity is holding a galaxy together, which still seems legitimate. I also am not thinking at all of the expansion of space as some kind of force that pulls on particles. I also agree that the expansion of space is (almost by definition) only really useful on scales where co-moving co-ordinates become a useful notion (that is, where they are distinct from proper distance co-ordinates).

I do take your point about the equivalence principle, however. The simple FRW solutions I am used to presume a homogenous universe, but that's a convenient simplification. In reality, mass is not evenly distributed, and a galaxy represents a region of high mass density. Gravity is thought of as a force in classical physics, but in GR it is treated in geometric terms. Now I am not expert in GR at all. But I should be wary of suggesting that gravity and expansion of space are different things. And in particular it's incorrect to say that the expansion of space is uniform through space. To say a galaxy is held together by gravity might (I guess?) be another way of saying there's no increasing scale factor within a galaxy.

More coming, when I get the equations worked out for some of the problems you proposed for me.

Cheers -- Sylas

 

[Ich]

How do you define "really tethered galaxy"?

Close to what a tether would really do. You pull at your galaxy such that the tether is tight. In an empty universe you need almost no force to do so, so we neglect elongation of the tether. Alternatively, you mount the galaxy on the tip of a long pole. If you release the galaxy, it will -empty universe again- stay at the tip of the pole / end of the tether.

Now to D&L's definition:

Given this definition, the rest follows independently of co-ordinates. For example, in a critical density model, they state that an unaccelerated particle with a proper velocity initially equal to zero will gradually approach the observer, and pass through to the other side of the sky, where it will eventually (in the limit) come to rest against the Hubble flow.

Yes, that is what follows from their setup. I do not say that their calculations are wrong. I just say that their galaxy was not "really" tethered, and that's the reason why it does not behave as one would expect from a tethered galaxy.Therefore I disagree strongly with

The fact that approaching galaxies can be redshifted
and receding galaxies can be blueshifted is an interesting
illustration of the fact that cosmological redshifts are not
Doppler shifts.

To the contrary, it shows that the used coordinates are not minkowsi coordinates and therefore their values and time derivatives should not be expected to behave like normal distances and velocities.
Instead of pointing out that the weird, unexpected behaviour shows that the terms "proper distance" and "recession velocity" should be taken with a grain of salt, they take them at face value and instead claim that physics is much more complicated (in some undefined mystical way) when cosmology is involved.

I also have some comments to your definitions

I find it easiest to use the co-ordinates of proper time and proper distance. [...]
In this co-ordinate system, there is an essentially arbitrary choice of a world line for a non-accelerating reference particle to define distance zero, and any other spacetime point is defined by a radial distance r, by two polar co-ordinates for direction, and by a proper time co-ordinate. With these co-ordinates, the scale factor is simply a function of proper time.

I'm not sure whether you write what you mean here. Comoving coordinates have the simple metric ds²=dt²-a²dr². The scalefactor comes rather unintuitively into the metric of proper time and distances.

...The rulers have to be co-moving, which means that the number of fixed length rulers between two co-moving objects in an expanding space is always increasing.

There is another consequence which is the important one in this context: Those rulers measure some entity that is not comoving, but at rest wrt some point (e.g. a tether) length contracted. As time goes by, the relevant rulers get slower and slower, and their measurement approaches the rest length of the tether. This "decontracting" is part of the so-called "recession velocity" and causes the counterintuitive behaviour.

As I understand it, your homework problem can be defined formally as finding the world line of an unaccelerated particle that initially has a proper distance "r" at a time "t", and initial proper velocity 0, in an empty universe where the scale factor is simply a(t) = H0 t. Is that right?

That's the inverse problem: take D&L's definition and show that this galaxy cannot possibly be at the end of a tight tether and should consequently not be called "tethered". Transform to minkowski coordinates. It's the same transformation as to and from Rindler coordinates, but with time and space swapped.

 

[sylas]

Close to what a tether would really do. You pull at your galaxy such that the tether is tight. In an empty universe you need almost no force to do so, so we neglect elongation of the tether. Alternatively, you mount the galaxy on the tip of a long pole. If you release the galaxy, it will -empty universe again- stay at the tip of the pole / end of the tether.

Then forget the tether. It's not about tethers at all. The tether is just a metaphor and this is explained quite clearly in the paper. They are also clear that you could get the same effect by giving a particle a local velocity wrt a background of co-moving observers. The whole idea is to consider the subsequent motions of any unaccelerated particle initially having zero rate of change of proper distance, wrt to proper time. That is all there is to it.

If you don't like the particular metaphor they've chosen for the concept of zero rate of change in proper distance, then fine. I can understand that, but I don't think it matters. It's the same as arguments over analogies with balloon skins or rising dough. In either case the underlying reality is the maths, and the rest is just attempts to give some convenient picture. That's just a subjective choice and I don't think it matters.

It's not about tethers at all. It's exclusively about having a zero rate of change of proper distance: the distance denoted D_now in Ned Wright's definitions of different distance notions, which I cited earlier.

I find it easiest to use the co-ordinates of proper time and proper distance. [...]
In this co-ordinate system, there is an essentially arbitrary choice of a world line for a non-accelerating reference particle to define distance zero, and any other spacetime point is defined by a radial distance r, by two polar co-ordinates for direction, and by a proper time co-ordinate. With these co-ordinates, the scale factor is simply a function of proper time.

I'm not sure whether you write what you mean here. Comoving coordinates have the simple metric ds²=dt²-a²dr². The scalefactor comes rather unintuitively into the metric of proper time and distances.

Shrug. I don't find it unintuitive at all; not any more. The scale factor is just the ratio between proper distance and co-moving distance co-ordinates.

Co-moving co-ordinates are certainly very nice for a lot of the basic calculations, I'll grant that. It makes things really simple, in lots of ways. The problem is that it doesn't match up with the conventional notions of distance we use locally. People want to know how far away a galaxy is. The answer depends, of course, on what you mean by "how far", and it is possible to explain some of the different notions of distance you could use. The notion I find most convenient is the proper distance co-ordinate. But I can convert to others as needed.

There is another consequence which is the important one in this context: Those rulers measure some entity that is not comoving, but at rest wrt some point (e.g. a tether) length contracted. As time goes by, the relevant rulers get slower and slower, and their measurement approaches the rest length of the tether. This "decontracting" is part of the so-called "recession velocity" and causes the counterintuitive behaviour.

You are absolutely correct about rulers measuring something that is not co-moving. But I was explicit (and this is important!) that the rulers themselves must all be co-moving. They cannot, therefore, correspond to a "tether", and they are certainly not at rest with respect to a hypothetical tether. They can only measure a tether, if one is used. Co-moving rulers will themselves all be moving apart from each other with the Hubble flow. You should think of a potential ruler being located at every point in space, and all of them are co-moving. That's what measures proper distance.

I think you should forget tethers altogether. It's not about tethers. It's about a particle with (initially) zero rate of change in proper distance.

That's the inverse problem: take D&L's definition and show that this galaxy cannot possibly be at the end of a tight tether and should consequently not be called "tethered". Transform to minkowski coordinates. It's the same transformation as to and from Rindler coordinates, but with time and space swapped.

Of course. There are galaxies we can see right now that are, and always have been, receding from us in proper distance co-ordinates with velocities greater than the speed of light. It's not physically possible to have a galaxy at that distance with zero rate of change of proper distance. Again, this is not about tethers. You can't physically have a particle moving that fast, not with a tether, not with ANYTHING.

And that's worth knowing. It's the converse of the common confusion people have about seeing a galaxy that is "receding" fast than lightspeed. We have no problem seeing such galaxies, and the recession velocity is not a motion through space, but a statement about the rate of change of proper distance with time.

-----

By the way, I have completed the first part of the exercise on unaccelerated motions. For a non-relativistic particle moving through space where for simplicity the only component of motion is radial to us (the observer), I get the following relation:

y = a ( y_0 + v_0 \int_{t_0}^{t} a^{-2} dt )​

In this equation, a is the scale factor, defined to be 1 at time t₀. y is a proper distance for a moving particle. t is the proper time. y₀ is the proper distance at time t₀. v₀ is the "peculiar velocity" at time t₀, which means the local velocity wrt to a co-moving local background.

This is a general equation, applying for any development of the scale factor, and any initial velocity wrt background. It requires that the velocity v₀ be non-relativistic, because it is based on momentum being inversely proportional to velocity.

A galaxy or particle with zero rate of change of proper distance at time t₀ must have v_0 = - y_0 \dot{a}(t_0). (\dot{a} is a proper time derivative. The dot is hard to see in the inline LaTeX.) In the special case of an empty universe, where the scale factor is simply the linear function a = t/t_0, this whole equation reduces to y = y_0. The particle remains forever at the same proper distance.

I'm still working on some of the other cases, but I think this was the case you asked for.

Thinking about it, this empty universe example is probably a good illustration of the equivalence principle that people have been mentioning. So thanks. It's a helpful illustration!

Cheers -- Sylas

 

[Ich]

Then forget the tether. It's not about tethers at all. The tether is just a metaphor and this is explained quite clearly in the paper. They are also clear that you could get the same effect by giving a particle a local velocity wrt a background of co-moving observers. The whole idea is to consider the subsequent motions of any unaccelerated particle initially having zero rate of change of proper distance, wrt to proper time. That is all there is to it.

If that were all there is to it, I would not complain. As I said, I don't disagree with the maths. This paper could have been a good essay about the properties of cosmological coordinates, if they had analyzed and clearly stated what deviations from the usual meaning of "distance" and "velocity" there are and where they come from. A real tether would have been the ideal comparison to make the distinction clear.
Instead, and that's what I critizise, there is not a single word about these interesting relations, but a totally unfounded and misleading conclusion about the nature of redshift.
I quoted it before, but "The fact that approaching galaxies can be redshifted" is not a proof of mystic physics in cosmology, it is a prime example of what happens when one uses words like "approaching" if one really means "negative time derivation of cosmological proper distance".
The redshift they're talking about stems solely from the fact the the respective galaxy is not approaching at all, but receding. It is still a doppler shift. That's a missed opportunity to explain some coordinate aspects, and provokes even more confusion as there already is.

Shrug. I don't find it unintuitive at all; not any more. The scale factor is just the ratio between proper distance and co-moving distance co-ordinates.

True, but the metric I was talking about becomes
ds^2=(1-H^2 x^2)dt^2 + 2 H x\, dx dt - dx^2
That are standard coordinates to first order, but the higher order deviations are neither trivial nor accounted for when people speak about measurements in these coordinates.

And that's worth knowing. It's the converse of the common confusion people have about seeing a galaxy that is "receding" fast than lightspeed. We have no problem seeing such galaxies, and the recession velocity is not a motion through space, but a statement about the rate of change of proper distance with time.

Yes, it's worth knowing that "recession velocity" is never the same as a velocity, even in the empty universe. It's a coordinate velocity, and one should know how to relate it to observables.

I'm still working on some of the other cases, but I think this was the case you asked for.

Yes in principle, but the interesting deviations are of higher order in the empty case.

I will say some more about momentum decay in another post. That's an easy route to get the correct equations of motion in "private space".

 

[FunkyDwarf]

If you disagree please answer this question, what experiment (however fanciful) could ever be performed that demonstrated that space is expanding?

As far as i know this has already been performed, ie showing that galaxies are moving apparently faster than the speed of light yet incurring no relativistic penalites, this is, as far as I am concerned, evidence for the expansion of space.

 

[sylas]

As far as i know this has already been performed, ie showing that galaxies are moving apparently faster than the speed of light yet incurring no relativistic penalites, this is, as far as I am concerned, evidence for the expansion of space.

Except, of course, that they aren't "moving" faster than the speed of light at all. Light moves past and through those distant galaxies just like it does here at the Milky Way.

What you really mean is that the distance between us and the distance galaxies is increasing. That's the difference between objects moving, and space expanding. It's not the same thing. And that, indeed, is a part of your point. You're arguing (as I am inclined to do) that cosmological redshift is better explained in terms of increasing separation distance than in terms of a proper motion.

Except -- and this is the kicker -- what do you mean by "distance"? There are quite a number of useful notions of distance in cosmology. For most of them, the distance to those galaxies (with high cosmological redshift, like z=3, for example) is indeed increasing at a rate greater than c.

Or there's the notion of "co-moving" distance, which Ich has recommended to me in the thread, because (as he points out) it simplifies calculations a lot. In that co-ordinate system they are not moving at all.

And finally, there's a notion of distance based on presumptions of a constant light speed. Conceptually, you think of a photon traveling from us to the galaxy, and then back to us again -- which can happen, because we are seeing the galaxies after all. Then impute a distance by taking the light travel time we measure locally, divide by 2 to account for the return trip, and multiply by c. And lo and behold, with that distance measure, the distant galaxies are receding at subluminal velocities.

The question that you, and I, have been asked, is whether or not we can give a physical experiment to demonstrate that the galaxies are not actually moving in this way, and that the effect is "really" expansion of space.

Just saying that high redshift galaxies have superluminal recession velocities is, in a sense, presuming the answer. How do you show that with an experiment? You can't just use a theory imputing superluminal recession, as there really are different imputed velocities for different distance conventions.

I'm trying to think this through myself. It's not a simple question quite so easily answered.

Cheers -- Sylas