Motion in an expanding space

Lots of people have been talking about expanding space in the enormous thread Message 315. This thread is to explore a simple puzzle about expanding spaces. The solution may help folks understand what is going on a bit better.(Percy, you are going to love this one!)As background, be aware that astronomers observe that all galaxies are moving away from us, and the further they are away the faster they move.Modern physics explains this as expanding space. For every MegaParsec of separation distance between object in deep space, we get 70 new kilometres of space every second. (Actually it is 71, but let’s use 70 to keep calculations easy.) Thus galaxies that are 30 MegaParsecs away from each other are receding from each other, on average, at 2100 km/sec. A MegaParsec is about 3*10¹⁹ kilometres.On top of that, galaxies can be moving locally through space. A galaxy moving at 300 km/sec with respect to nearby galaxies will have this velocity added on to the recession velocity due to expanding space.People sometimes think that space pulls things along with it, somehow. Let’s see.Take two particles, and hold them at exactly the same (large) separation, while space is expanding. For simplicity, let particle "A" be at rest in the frame of background radiation, and let particle "B" be one MegaParsec away. Assume that these two particles are out in deep space, in one of the great voids. That is, there are no other galaxies anywhere near these particles to push or pull them with local gravitational attractions.Space between these two particles is expanding at 70 new kilometers every second. To remain at the same separation distance, therefore, particle "B" must have a local motion of 70 km/sec towards particle "A". That is, the particles have a local velocity towards each other that exactly matches and cancels out the rate at which space is stretching between them.Now release the particles, so that they are in free fall. Will the distance between them increase, or decrease, or stay the same?I’ll give the solution, but I want to give people the chance to think about it first. This is a good way to get a feel for whether your intuitions about expanding spaces are correct, or not.Cheers — SylasThis message has been edited by Sylas, 01-28-2005 18:31 AM

Very good! You have got the key point, which is that expanding space does not actually pull on the particles to drag them apart.There is a bit more to say than this. In fact, I did not quite give enough information to solve the puzzle! The description you have given is correct for inflationary expansion.The rate of expansion 70 km/sec/MPsec is called the "Hubble constant". The term is misleading, because the Hubble constant is not in fact a constant at all! It is a constant value through all space, but it changes with time.That is, anywhere in the universe (as far as we can tell) which is 13.7 billion years from the singularity, like we are, will see space expanding at 70 km/sec (sticking to this rather than 71 for helping any maths when we get around to it).But what happens as time passes? In a simple zero density case, a galaxy at 1 MegaParsec distance, which is receding with the Hubble flow at 70 km/sec, will continue to recede at that same rate, even as it moves further away. So, taking a MegaParsec as 3*10¹⁹ km, then after 4.3*¹⁷ seconds, the galaxy will be 2 MegaParsecs away. This is about 14 billion years.(Does than number sound familiar? It should! It is about the age of the universe, since the time when galaxies are at zero separation distance.)At that time, the Hubble flow rate will be only 35 km/sec/MPsec. In other words, the flow rate is inversely proportional to age.Now there are a few more quibbles. A "cosmological constant" tends to maintain the Hubble flow at a fixed rate. As galaxies move further away, therefore, the rate of recession increases. In the extreme case this is what happens in so-called "inflation". If we have an inflationary expansion, so that the Hubble constant remains at 70 km/sec/MPsec indefinitely, then your answer is correct. (I think! I've only been learning about this myself quite recently.)On the other hand, gravity from mass in the universe tends to retard expansion.At present, there seems to be a small cosmological constant, but not enough for inflation, and also a small mass density which acts to retard expansion rates. So to a first approximation, we can just use the zero density model. Assume that the expansion of space is such that objects moving with the flow of expansion continue to recede at the same velocity as they move further away.Cheers -- SylasThis message has been edited by Sylas, 01-28-2005 19:39 AM

Recommendations and references coming soon. First, you have to make a stab at the question. The previous posts almost give the answer, but see if you can push it a bit further.Imagine a sequence of massless particles, A, B, C, D, E, F, G, ...Right now, imagine them all in a long line, at 1 MegaParsec separation. They are all at rest in the local Hubble flow; and so there is 70 km/sec recession between A and B, B and C, C and D, and so on, due to expansion of space, but with no other local movements.Imagine that the Hubble flow is such that the separation rate of these particles remains 70 km/sec, even as they recede from each other.But we have another particle "d", which is initially at the same location as "D", and moving back towards "C" at 70/km/sec to compensate for expansion between "C" and "D". That is, (initially at least) the "C" and "d" remain at a steady 1 MegaParsec separation, and "d" is moving past "D" at 70 km/sec.What happens to "d" as lots more time passes?Cheers -- Sylas the evil instructor

There are indeed problems with the notion of distance in general relativity; but we don’t need to go into the matter. There is a concept called proper distance in cosmology, which I am using without bothering to define it in detail. This does make sense of having masses at a certain distance, so yes, it is sensible to speak of masses at a certain distance.The expansion of the universe means that objects at a given distance will recede from each other at a rate proportional to distance. Gravity can slow the universal expansion over time, and with a very dense universe it can even eventually reverse expansion and collapse the universe (the Big Crunch). All indications are that this model is wrong; there is not enough mass to stop expansion, and in fact there seems to be another effect at work that is accelerating expansion. So no, gravity in our universe does not seem to be strong enough to overcome expansion of the whole universe.What this thread is about is the effects of additional local motions. All of the above is referring to objects that are at rest (in a sense I have not fully defined) with respect to the rest of the universe. Objects at rest in this sense see all the rest of the universe expanding away from them. Two objects at rest in this sense will recede apart from each other due to the expansion of space between them.If massive objects are close together, then gravitational attraction will make them move; they end up in an orbit. One effect of this orbit is that the objects remains close together, so gravity does overcome the expansion of space just for these two objects, by keeping them moving in space to remain in a close orbit while all the rest of the universe that is not close enough to be gravitationally bound continues to recede.This means, for example, that the solar system remains the same size even despite the expansion of space.

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PS. For other participants in this thread. Kudos to ohnhai for a good explanation of centers (Message 10), and to buz for an excellent question (Message 9) and for comprehending the answer with such rapidity (Message 11). Well done folks. Remember, you don't have to agree with all aspects of a model to understand it! Anyone who starts to understand the model is doing very well, regardless of whether you accept that it corresponds to reality.Tom, I’m sorry, but your comments in Message 13 just make no sense at all. Light does not "return in a curve". The galaxies don't get closer; they get further apart. Light does not refract as a curve in space. Galaxy size is not inflated (or at least, not by any detectable amount) because gravity holds them together even as space expands within and around them. You need to learn a lot more very basic stuff. This thread is too advanced for you at present.Cheers -- SylasPPS. I'm still learning all this as well. I have made a couple of minor errors in the thread already. Specifically, I was wrong in Message 4 with the comment about Inflation. The actual conditions for "A" and "B" in the initial post to remain at a fixed separation are apparently the Ω=0 case. The reason why is still confusing me.This message has been edited by Sylas, 01-29-2005 04:29 AM

Good questions, Ben. I'll make an attempt to answer, and I'll warn you when I am likely to be a bit unreliable. It turns out that at any point in space, there is one unique reference frame that fits easily with what we see of the universe.The universe is filled with background radiation, coming from every corner of the sky. This is one of the major lines of evidence for the "Big Bang", and study of this radiation has been very useful in cosmology. It has the spectrum of a perfect blackbody radiator, with a supercold temperature of 2.725 degrees above absolute zero.This radiation defines the rest frame. If you move through the radiation, it looks a bit hotter in the direction of motion, and a bit cooler looking back. In fact, the Sun is moving through this radiation at about 369 km/sec, and so from Earth, one half of the sky is just a bit hotter than the other half.The velocity of 369 km/sec is measured with respect to the local rest frame of the background radiation. At any point in the universe, you can do the same thing; place yourself at rest with respect to the background radiation that fills the universe.A set of observers spread through space, all of whom are at rest in this sense, are called the co-moving observers. They all recede from each other at a rate determined by the Hubble constant. That is, two co-moving observers at a separation of 1 MegaParsec will at the present instant be receding from each other at 71 km/sec.A local motion means a motion with respect to the co-moving observer at your present location. Caution, I am approaching the limits of my ability again.An inertial reference frame is a very important concept in special relativity, but it does not carry over to general relativity. In general relativity, the best we can manage is a local inertial frame. That is, for any point in space time, we can have different inertial rest frames centered on that point (but differing in velocity) that are a very good approximation to nearby spacetime, but become progressively less reliable as you move away.Put another way, there is no inertial reference frame containing two particles that are 1 MegaParsec apart.The notion of distance is dubious in general relativity. However, in cosmology there is a useful notion of proper distance. Basically, it is a distance between co-moving observers of the same age, all in a straight line and all adding up touching rulers. This distance expands over time. Hubble flow refers to the rate of increase of proper distance between co-moving observers at a point in time.If you have a local motion, then you are effectively moving past a line of adjacent co-moving observers. Your velocity to another very distant point is in two parts. The local velocity is the rate at which you are passing by adjacent co-moving observers. The recession velocity is the rate at which proper distance is expanding between the observer you are next to right now and the distant observer. You add these to get the rate of change in proper distance to a distant observer.Again, my standard reference for much of this stuff is Ned Wright’s Cosmology Tutorial. The notion of a comoving observer is discussed in the first section.Cheers -- Sylas

You are not agreeing with Einstein. You are disagreeing with Einstein, and don't know enough to even recognize this. Your links are perfectly consistent with the information in this thread. They are trying to explain a basic point about relativity, without using heavy maths. Your comments about dimensions are incorrect. Galaxies and light are all moving in a a spacetime of three spatial dimensions and one time dimension. That spacetime is expanding. The association of curvature and expansion is much more complex than you represent; but in any case, whether curvature is positive, negative or flat (the three cases mentioned in your links) you still have the 71 km/sec/MPsec expansion. Curvature impacts on how expansion develops over time. It does not replace expansion as an alternative for the observations.

Sorry about the delay in posting this solution. The delay is due to my having some incorrect assumptions of my own!The solution (which is the opposite to what I thought when I first posted!) is that the two particles do actually start to move apart from each other; at least if the current most favoured model for the universe is correct. The previously favoured model, before the discovery of accelerated expansion, was for the particles to approach each other! The puzzle I set for Charles was actually too difficult. Sorry Charles. There is a complication I neglected to consider in the first instance!The reference I have been using, which has some undergraduate level maths, is Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects, by Tamara M. Davis, Charles H. Lineweaver, and John K. Webbc, at the Department of Astrophysics, University of New South Wales, Australia. I recommend it; it has excellent credentials, and is intended to help in teaching some physical consequences of expansion which are not always well understood.People sometimes think that space pulls things along with it, somehow. It doesn't. Space just expands.The problem I posed concerns two particles, which are held at the same fixed separation distance, while space expands. I assume that the particles are 1 MegaParsec apart, and that space expands at 70 km/sec/MPsec. This value is called H₀, the Hubble constant.The expansion of space means that there are 70 new kilometres of space between the particles every second. To remain at the same separation, the particles must have a local movement towards each other, of 70 km/sec, just enough to cancel the effect of expansion on separation distance.If the particles are allowed to drift free of any forces, then to a first approximation they remain at a fixed separation. There is no force to accelerate the particles, so they keep moving locally at 70 km/sec. Space does not act to drag them in any direction. In the meantime, space continues to expand at 70 km/sec, so nothing changes, at least in the short term.1. The Hubble constant is not constant. The Hubble constant H₀, of 70 about km/sec/Mpsec, is not actually constant. It is a constant through all space, but it changes over time. How it changes depends on various properties of the universe. The most important properties are the density of the universe, and the cosmological constant. But in the absence of these two effects, the recession velocity of two galaxies, like a normal velocity in Newtonian physics with no forces, remains constant. This has the effect of reducing H. In fact, in this model H is simply the inverse of the age of the universe! We use H₀ for H at the current instant.Let’s consider how this changes things.Let a million years pass by; which is about 3*10¹³ seconds. The galaxies that were previously one MegaParsec away from A are now on average an additional 2.1*10¹⁵ kilometers (70*3*10¹³) away, due to the expansion of space. A MegaParsec is about 3*10¹⁹ kilometers, so those galaxies are an additional 7*10^-5 MegaParsecs away. But the rate at which space is increasing is still 70 km/sec between those galaxies and A. This means that the rate of expansion, after a million years, is now 70 km/sec for each 1.00007 MegaParsecs, which is 69.995 km/sec/MPsec.Assuming that the local velocity of A and B remains 70 km/sec, they now will be approaching each other at about 5 meters a second.Originally, I had hoped this would be enough to give a feel for what goes on. But it isn’t2. Momentum decays. It turns out that just like the expansion of space gives photons a redshift, it also causes the momentum of any moving particles to decay. If the universe is empty of mass, so that galaxies maintain a fixed recession velocity as they move further away, it turns out that the decay of momentum matches the slowing of the Hubble constant, and the test particles in my problem statement remain at a fixed separation.This means I was wrong in Message 4. I said that separation remains fixed in inflationary space. In fact, I should have said it remains fixed in zero-density case.3. Mass retards expansion. It turns out that mass in the universe slows the expansion of the universe; a bit like pulling it in on itself. If we take into account the effects of mass, the expansion rate slows more rapidly than the local momentum, and so the test particles actually move closer together. This was the most favoured model for the universe up until several years ago.4. The cosmological constant pushes expansion. Einstein’s equations allow for a so-called cosmological constant that accelerates expansion. Evidence presently suggests that there is small cosmological constant, at work in the universe. In this case, the expansion rate while momentum decays, and the particles will start to move away from each other as time passes. If current models are correct, then the cosmological constant is the more powerful effect; and the particles will indeed move apart from each other after release.This is a pity; I was hoping for the opposite answer, to underline the fact that space does not pull on particles. Unfortunately, the current most favoured model for the parameters of the universe is such that the released particles will begin to move apart.From hear on I get a little more technical, just to go into the answers in a little move depth. Proceeding is optional.

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5. Scale factor, comoving observers. I’ve explained (Message 17) how background radiation defines a kind of natural rest frame anywhere in the universe. As particles move through space, we can calculate their momentum at any point with respect to their local rest frame at that point. This value for momentum is what decays. But note; the background radiation does not define a single inertial rest frame for the universe. General relativity does not have such a concept. The background defines many different inertial frames that work over small scales at each point in spacetime. These frames are called the co-moving observers.There is a value called the scale factor of the universe, written a, that changes with time. This number represents how much space has expanded. It is defined to be one at the present instant, and at all other times is represents the factor by which distances between co-moving observers has changed.Putting this in simple mathematical terms let R₀ be the distance between two given comoving galaxies at the present instant. The scale factor after a time t is a(t), and the distance between the galaxies at time t can be given as R₀*a(t).The rate of separation for these two galaxies is thus R₀da/dt, and the Hubble constant is always equal to (da/dt) / a.For the simple case given above, the scale factor simple keeps increasing at a fixed rate, and so da/dt is a constant. Hence the value of the Hubble constant at any time is inversely proportional to the scale factor. H(t) = H₀ / a.6. Momentum inversely proportional to scale factor. This was my error; I initially neglected to consider this aspect of the situation.Think first about background radiation. It fills the universe, from every direction. As space expands, the wavelength of the photons gets stretched as well. This has the effect of a redshift over time.Photons have a momentum, inversely proportional to wavelength. Hence, as space expands, photons in space actually lose momentum. Their wavelength expands proportional to the scale factor, and so their momentum is initial momentum divided by a.What about conservation of momentum, I hear you cry! What about conservation of energy! I’m not completely sure. One answer is that this may actually be compensated for by momentum and energy effects arising from the expansion itself. Another is that (hold on to your hat) conservation laws are only approximations anyway. The conservation laws are only defined locally, and in general relativity we don’t have a clear notion of a frame for the whole universe in which these laws can be easily expressed. Truth to tell, I don’t fully understand this aspect of general relativity yet, and don’t trust myself to give a good explanation. So for the moment, take it on trust, and we’ll see some consequences. I encourage interested readers to explore this further; and anyone who wants to take a stab at explaining conservation laws in the light of general relativity, fire ahead.In any case, what I neglected to consider is that the same thing applies for particles moving at sub-relativistic velocities. Particles at motion gradually lose momentum with respect to their local rest frame, proportional to 1/a, just like photons.Here is how I think about this.... as a moving particle moves into a frame for a new portion of space, there is an associated co-ordinate transformation, and with this transformation there is a loss of momentum just like we calculate different momentums depending on the different velocities of observers in normal Newtonian physics. That basically is what we have with moving particles. I am not really sure if this is a useful way to think about it, it seems to give the numbers that match theory.7. The effect of momentum decay on particle separation. If we consider again the simple case in which receding galaxies retain the same recession velocity as space expands, then the rate of separation is proportional to 1/a, and so is the local recession velocity. Let t₀ be the present time (the age of the universe) we have the relation a = t/t₀ Thus both local velocity and recession velocity decay at the same rate, and the particles remain at the same separation.8. The effects of mass. Einstein’s equations applied to the spacetime of the entire universe indicate that mass in the universe has the effect of slowing expansion. There is a certain critical value for mass density which is just enough to reduce the expansion rate to nothing in the limit, without quite being enough to halt expansion and let the universe re-collapse. Up until fairly recently it was thought that our universe had critical mass density, even though we were not sure where all the mass was. This has the effect of slowing the Hubble expansionIf the mass of the universe has critical density, it turns out that a(t) = (t/t₀)^2/3, and the released particles eventually approach each other.9. The effects of a cosmological constant. This would need to be solved by numeric methods, as there is no simple analytic solution for scale factor. Basically, however, released particles do, eventually, start to recede from each other.It is getting too late for me to run some sample numbers, so I’ll leave it at that for tonight.Cheers -- Sylas

I'm quite dissatisfied with this thread. I do not think I have explained the matter well, and the explanation is more complex than really necessary. Sometime I might know enough to explain it better.... I'm not sure I understand the question.I think the answer may be "no". Einstein's equations admit all kinds of solutions for the universe, depending on various parameters. With enough mass, expansion is slowed and eventually reverses to become a compression. With a cosmological constant, expansion accelerates and proceeds forever. With a careful balance of these features (which is what Einstein proposed) the universe is static, neither expanding nor contracting. Unfortunately, Einsteins model was unstable, and hence cannot explain an infinite static universe as he tended to assume.The point is that in all of these models, on local scales mass continues to experience accelerations.Expansion is not a separate thing that can ehlp explain local gravitational interactions. Rather, our current best theory of gravity represents gravity as acting upon spacetime to alter its geometry. Falling objects move along geodesics in spacetime. And on cogmological scales, the spacetime has global features like expansion, and a geometry (flat, open, closed).General relativity is the model for both local and global aspects. The global expansion is not a separate thing that could stand as inducing the local interactions. No. This is a common reaction, but in fact local forces, which move things through spacetime, are more than enough to hold things together while space is expanding.This was been considered in considerable detail in a couple of posts in other threads. Have a look at Message 296. There is also a link to a good paper which calculates the effects of expansion on the gravitationally bound solar system. There is an effect, but it is far too small to detect. Gravity works to keep the orbit looking much the same, whether space is expanding, contracting, or remaining the same.Cheers -- SylasThis message has been edited by Sylas, 02-05-2005 16:41 AM

Right. The paper I have cited and tried to use does employ numerical solutions, and I could do the same myself from the formulae provided.What I was trying to do... and failed... was give a simpler intuition for a potentially surprising result, with as little maths as I could get away with.Part of the difficulty is that the (Ω_M, Ω_Λ) = (0.27,0.73) solution that is now widely cited as the best model consistent with current observations, will actually have tethered particles receding on release. Models from a few years ago with no Ω_Λ component have tethered particles approaching on release.By the way, welcome back. I've been writing a lot on cosmology recently, and I have badly needed your expertise! Watch your posting identity. You can select whether you are Admin or not in a post. You want to be back with the proles in this thread, I think.Cheers -- Sylas

Yes! I alluded to this problem myself (or at least, I think I did, indirectly!) in one of my posts that I can't now find.The paper I have cited several times (eg, in Message 286 of another thread) that calculates perturbations in the solar system arising from cosmological expansion has Cooperstock as one of the authors. It is The abstract reads as follows: The paper seems to take a simplified approach, and simply assumes expansion is universal (as opposed to varying locally) and calculates the consequent pertubations on the Solar system and galaxy.Is this the same paper you were thinking of? It does not seem to be quite the same thing, and I would very much like to see a reference that considers in detail the problems you mention.I had a quick look, and just as a minor detail I found a wonderful page of Cooperstock quotations, which were apparently collected by admiring students in his relativity lectures. Cooperstock quotes.Sounds like quite a guy!Cheers -- Sylas

Hit the "peek" button in the bottom right hand corner of a post to see how text was actually entered. I used HTML special character codes.

Yes, I am still very active there. But I don't have much to say about cosmology in that forum; it would tend to be a bit over the head of nearly all contributors. The issues there are a bit more basic. I like TWEB, and am generally appreciated by the management. I get a bit more explicitly theological; and when discussing science topics I tend to focus on the evolution related matters.

The statement you quote is cited in first message of my the new thread. See Message 1. The major issue they have is funding. Frankly, many of the signers don't deserve a penny. But let's consider them on a case by case basis in the new thread. Funding must be decided case by case, not by open slather to anyone with a novel criticism. But not in this thread please; it is not relevant here.Cheers -- Sylas