Why doesn't the universe expand on small scales?

[Fredrik]

I'm posting this to confess that I have never understood how the universe can be expanding on large scales and at the same time not be expanding on small scales. I'm hoping someone can explain that to me.

The only argument I've been able to come up with, is that the solutions that describe an expanding universe are what we get from Einstein's equation when we decide to look for solutions that describe a homogeneous and isotropic universe. Since our universe is both homogeneous and isotropic on large scales, we can expect one of those solutions to be a reasonably accurate model of the large-scale behavior of the universe, but we can't expect it to be a good model of the small-scale behavior. So even though those solutions predict that e.g. a meter stick should expand (right?), which would make the expansion undetectable, it still shouldn't come as a huge surprise that meter sticks, solar systems and galaxies don't expand.

I haven't heard any better arguments from anyone else. Someone said that the reason why meter sticks don't expand is that EM forces are much stronger than gravitational forces. That sounds like an explanation that someone just pulled out of their you-know-what, but I don't know how to respond to that since I don't know that the real answer is.

I understand that on large scales space-time should look like a FRW space-time (is that the standard name for it?), and near a star it should look like a Schwarzschild space-time, but I can't see how those pieces fit together. Is it possible to draw some kind of picture that makes this easy to understand?

I'm looking forward to hearing your answers and learning something new.

 

[Count Iblis]

If you make a meter stick by placing two test masses that do not interact at all one meter apart in deep space, then this will expand. In general, you have on top of this expansion all the other forces that act on the two test masses. These will completely samp the expansion effect.

The expansion of the universe tries to pull away the test masses. In principle this force also exists in a meter stick on Earth, it is just too small to be measured. Compare this with gravitational wave detectors like LIGO. Here on let's mirrors float. A gravitational wave, which is a perturbation in the space-time metric will cause the distance betwen the mirrors to vary, which an be measured using lases. There are plans to build such a gravitational wave detctor in space.

Also there are detectors that are just metal bars. The changes in the metric will give rise to an effective force which can cause the bar to vibrate.

There are cosmological models in which the expansion rate of the universe will become infinitely large within a finite time, the so-called Big Rip. In this scenario the expansion rate will become large enough to dominate over the local forces. So, the solar system will become unbound, the Earth will explode, atoms will be ionized, etc.

 

[CJames]

Think of two objects in free fall. If I drop one ball, and then another, they will start out near each other. But the more time given, the greater the distance between the two. They will accelerate away from each other. Now, if I do the same experiment, but with one of the balls a greater distance from the other, they will accelerate away from each other faster.

Just apply this principle to the whole universe, and you have your answer.

 

[Fredrik]

Just apply this principle to the whole universe, and you have your answer.

I assume your point is that the velocity is proportional to the distance, and that this implies that the velocities will be small on small scales. I don't think that solves the problem. I think that is the problem.

It implies that the distance between the ends of a meter stick will double in exactly the same time it takes the size of the universe to double, so the distance to faraway galaxies will always be the same number of meters.

If you make a meter stick by placing two test masses that do not interact at all one meter apart in deep space, then this will expand.

OK, that sounds reasonable. After all, test masses (by definition) don't cause any curvature, so the geometry of that region of space-time must be shaped by the matter in other parts of the universe. This will make the metric in that region some kind of intermediate form between Minkowski and FRW, because a) the region is empty, and b) the large-scale distribution of matter is homogeneous and isotropic.

The world lines of the test masses will be isometries of that metric, so the test masses will move apart, but since the metric isn't quite FRW, they won't move apart quite as fast as Hubble's law predicts. This is cool, I never realized this before.

My follow-up question was going to be this:

Now replace one of the test masses with a neutron star and put the other test mass in a circular orbit around the neutron star. Will the cosmological expansion increase the radius of that circle?

I realize now that the same line of reasoning will answer this question. The geometry in that region will be a little bit like FRW and a lot like Schwarzschild. So there will be an expansion, but it will be much slower than the cosmological expansion.

In principle this force also exists in a meter stick on Earth, it is just too small to be measured.

I don't think that explains why the stick doesn't expand. Why would interactions in the stick (mainly electromagnetic) pull its component atoms away from geodesic motion in some space-times (FRW) but not in others (Minkowski).

 

[Hurkyl]

I don't think that explains why the stick doesn't expand. Why would interactions in the stick (mainly electromagnetic) pull its component atoms away from geodesic motion in some space-times (FRW) but not in others (Minkowski).

Even in classical mechanics, the electromagnetic forces are pulling the component atoms away from geodesic motion -- that's why you have a meter stick consisting of many atoms vibrating in place relative to each other (i.e. a solid) rather than a bunch of meter stick atoms diffusing into space (i.e. an ideal gas).

 

[Fredrik]

Even in classical mechanics, the electromagnetic forces are pulling the component atoms away from geodesic motion -- that's why you have a meter stick consisting of many atoms vibrating in place relative to each other (i.e. a solid) rather than a bunch of meter stick atoms diffusing (i.e. an ideal gas).

That's a little nitpicky, but OK, you have a point. If an atom is vibrating back and forth around an average position, the atom isn't doing geodesic motion. But the average position is. So my question becomes, why do the interactions in a solid pull the average positions away from geodesic motion in some space-times (FRW) but not in others (Minkowski)?

We can also imagine cooling the stick to a temperature that's extremely close to absolute zero in order to (almost) stop the vibrations.

I have to get some sleep, but I'll be back in 9 hours or so.

 

[yuiop]

Hi,

If the average orbital radius of the Earth from the Sun (or the orbital radius of the Sun from the centre of our Galaxy) is constant over time while the universe is expanding then that implies that without the expansion the normal behavior of orbiting bodies is to gradually spiral inwards and that spiral motion is canceled out by the expansion on small scales.

Measurements of two neutron stars orbiting each other, show they do in fact spiral inwards towards each other as they lose energy in the form of gravitational waves. Presumably the Earth and the Sun are also losing energy, but at a much slower rate and therefore should be spiralling towards each other but this effect is too small to measure.

If the expansion of the universe is accelerating then, in the very distant future the expansion will be noticeable at the solar scale and the Earth will start moving away from the Sun.

[EDIT] Perhaps I should add that dark matter is thought to be present inside most galaxies and this effectively increases the gravitaional force experienced by orbiting bodies so perhaps we should add to the question in the Op, why is that gravitational effect of dark matter is not detected in the solar system so that solar system bodies have to orbit faster than Newtonian gravity predicts?

 

[Hurkyl]

So my question becomes, why do the interactions in a solid pull the average positions away from geodesic motion in some space-times (FRW) but not in others (Minkowski)?

Quick question: does a spinning ball (classical or special relativistic) count as having the 'average position' of its particles following geodesic motion?

The intermolecular forces simply seek to try and maintain relative position: as particles separate, the force increases, and as particles come close together, the force decreases; they don't care one whit about geodesic motion.

Taking a guess at one thing you might mean... the whole point of expanding/contracting space and tidal forces is that an ideal cloud of noninteracting, comoving dust will tend to expand/contract/shear. So if you have a cloud interacting dust particles, the interparticular forces will act to resist such deformation, thus giving you nongeodesic motion.

 

[Fredrik]

Quick question: does a spinning ball (classical or special relativistic) count as having the 'average position' of its particles following geodesic motion?

Of course not, but if it's not spinning...

I don't mind that you're pointing out that I could have been more careful about the details of my question, but you haven't exactly answered anything.

The intermolecular forces simply seek to try and maintain relative position: as particles separate, the force increases, and as particles come close together, the force decreases; they don't care one whit about geodesic motion.

And yet they preserve geodesic motion in the case of a solid at zero temperature, at rest in an inertial frame in Minkowski space. But if you imagine the same solid in a FRW space-time, in a weird initial state such that every atom is at rest in the cosmological rest frame, those intermolecular forces would immediately pull every atom away from geodesic motion. (Yes, I know we wouldn't actually be able to prepare that initial state in the real world, but this is a thought experiment).

I'm not saying I have a good reason to expect that forces in a solid should preserve geodesic motion in particular. I just want to know what they do preserve, and mentioning geodesics was just a way to make the question more specific.

If it's not clear what I mean by a cosmological rest frame, I mean a coordinate system such that points in space that move apart with the cosmological expansion stay at the same spatial coordinates.

Taking a guess at one thing you might mean... the whole point of expanding/contracting space and tidal forces is that an ideal cloud of noninteracting, comoving dust will tend to expand/contract/shear. So if you have a cloud interacting dust particles, the interparticular forces will act to resist such deformation, thus giving you nongeodesic motion.

What I might mean? Haven't I made the point of this thread clear? Why don't meter sticks expand so that cosmological expansion is undetectable? Why doesn't the distance between two adjacent atoms in a solid grow with the cosmological expansion? Why don't the individual atoms grow with the cosmological expansion? Nothing I have seen in this thread suggests an answer. (Except maybe the reasoning I used myself in #4, but that leads to some conclusions that I expect to be false, so for the moment I don't think that what I said in #4 is the whole answer).

 

[Fredrik]

Measurements of two neutron stars orbiting each other, show they do in fact spiral inwards towards each other as they lose energy in the form of gravitational waves.

I'm aware of this. That's why I talked about a neutron star and a test mass in #4 instead of two neutron stars. (I also think the neutron stars will move apart as a result of the energy loss, not move closer together).

perhaps we should add to the question in the Op, why is that gravitational effect of dark matter is not detected in the solar system so that solar system bodies have to orbit faster than Newtonian gravity predicts?

That question is very different from my question in the OP, but maybe you should start another thread about that. (I don't know the answer).

 

[Count Iblis]

You can generalize the Schwarzschild solution by including a cosmological constant term:

http://arxiv.org/abs/gr-qc/0602002

 

[Hurkyl]

Why doesn't the distance between two adjacent atoms in a solid grow with the cosmological expansion?

I presume you understand how intermolecular forces in a solid act to maintain the shape of that solid. I really and truly do not know why you are conceptually losing that understanding when you pass from classical to the general relativistic case.

 

[Fredrik]

I presume you understand how intermolecular forces in a solid act to maintain the shape of that solid. I really and truly do not know why you are conceptually losing that understanding when you pass from classical to the general relativistic case.

Your answers aren't helpful. You just keep implying that the explanation is trivial. If it is, then why haven't you just told me what it is? This isn't the homework forum.

I think I get it now though, at least roughly. What those forces are preserving are proper lengths of certain space-like curves. I haven't figured out (yet) exactly which curves this applies to, but it seems to be enough that it holds for one particular class of space-like curves to solve this problem:

Define "space" at time t to be the space-like hypersurface where the cosmological time coordinate is t. The proper distance in space, between the points where the world lines of the endpoints of a meter stick intersects space at time t, must be independent of t. This means that difference between their spatial coordinates in the cosmological rest frame must decrease with increasing t.

Is it really trivial that forces in a solid preserve those proper distances? Wouldn't we have to examine the details of the theory of electromagnetism in curved space-times to know that?

 

[yuiop]

I presume you understand how intermolecular forces in a solid act to maintain the shape of that solid. I really and truly do not know why you are conceptually losing that understanding when you pass from classical to the general relativistic case.

You have a good point Hurkyl, but I have a couple of additional questions.

Does gravity act to maintain the size of an orbiting system in the same way?

How do the intermolecular forces act in a meter stick that is moving relative to us (length contracted) or a meter stick placed vertically deep in a gravitational well? As far as I know there are no stresses in a length contracted meter stick with relative motion so the intermolecular forces are not acting against the length contraction. It could be argued that the intermolecular forces adjust to universal expansion in the same way that they adjust to gravitational or kinetic length contraction.

 

[yuiop]

...
That question is very different from my question in the OP, but maybe you should start another thread about that. (I don't know the answer).

I agree the question of dark matter is very different, but it is sort of relevant because it acts in the opposite direction to expansion. If for example dark matter in our galaxy (and in our solar system) could be working to reduce the orbital size of the Solar system while the the expansion is working to increase the orbital radius of the Solar system. Without knowing the relative magnitude of the two effects how can you be certain they do not cancel each other out and so we would not expect to see any effect at the Solar scale? I am also pretty sure the loss of energy due to radiating gravity waves causes an inward spiral rather than an outward spiral so that will also be acting to cancel out any observation of the expansion at the local scale in a gravitationally bound system. All I am saying is that you need to know the magnitude (and direction) of all these different effects to know what to expect to observe on the local scale.

 

[Fredrik]

I agree the question of dark matter is very different, but it is sort of relevant because it acts in the opposite direction to expansion. If for example dark matter in our galaxy (and in our solar system) could be working to reduce the orbital size of the Solar system while the the expansion is working to increase the orbital radius of the Solar system. Without knowing the relative magnitude of the two effects how can you be certain they do not cancel each other out and so we would not expect to see any effect at the Solar scale?

I'm not, but that's not the point. Dark matter can't explain it on several different length scales: atoms, meter sticks, solar systems, galaxies. Also, it would be quite a coincidence if those two cancel each other out, even in the case of a solar system.

I am also pretty sure the loss of energy due to radiating gravity waves causes an inward spiral

I'm thinking that lower energy means lower orbital speeds, which means higher orbits. I also remember reading that the moon is moving away from Earth because of the energy loss due to work done by tidal forces. Why would neutron stars behave differently? (I'm not saying that they definitely don't. I'm just saying that if I'm wrong, I would like to know what my mistake was).

All I am saying is that you need to know the magnitude (and direction) of all these different effects to know what to expect to observe on the local scale.

Yes, but I don't need to know exactly what to expect. I'm just trying to understand why it would be possible in principle to measure the expansion in the absence of all those other effects. I believe that I do understand it reasonably well now. What I said in #4 explains it for gravitational systems, and what I said in #13 explains it for non-gravitational systems. I wouldn't mind seeing more rigorous arguments though, or more rigorous versions of my arguments.

 

[DaveC426913]

Hi,

If the average orbital radius of the Earth from the Sun (or the orbital radius of the Sun from the centre of our Galaxy) is constant over time while the universe is expanding then that implies that without the expansion the normal behavior of orbiting bodies is to gradually spiral inwards and that spiral motion is canceled out by the expansion on small scales.

No it does not. The logic is flawed. See below.

You just keep implying that the explanation is trivial. If it is, then why haven't you just told me what it is?
At the risk of patronizing I'm going to resort to the basic-yet-venerable balloon analogy.

Glue a bunch of pennies to a balloon. Inflate the balloon. Ask why the pennies don't expand. Simple: the expanding balloon's pull on the penny's structure is far inadequate to overcome the penny's solidity.

Likewise, the expansion of space is far inadequate to sub-galatic-scale objects to expand (the forces between stars/stars and stars planets are just way too strong).

Likewise, a lack of expansion does NOT mean the orbits would decay, just like the lack of inflating the balloon does NOT mean the pennies would starting shrinking.

 

[rahuldandekar]

I think that the relevant fact is that the geometry of the universe is not FRW on small scale. It is Schwarzschild in, for example, the solar system. Hence, the metric on these scales is not expanding. Only on very large scales, is the universe's metric approximately FRW, hence expanding. You can visualise the universe as expanding on large scales, with smaller "pockets" remaining intact. That, atleast for me, is the most plausible-seeming explanation.

The meter scales, then, on the earth, are not in the FRW metric, but rather in a Schwarzschild one. There is no reason for them to expand.

Now what if a meter stick was placed in the intergalactic space? One can imagine a very long meter stick, about 10 Mpc in length, placed between two comoving observers that distance apart. If it is composed of small masses, it'll probably expand with the universe, hence the two observers won't see any expansion. If it is composed to masses attracted to each other by gravity, and the attraction being appreciable, we can treat the stick as a perturbation in the FRW metric, and it'll collapse slowly. Instead, if it is a "rigid" body like our terrestrial meter sticks, it'll may stay the same length as before, that is, it'll move away from both the observers. That is, if the expansion of the universe is not fast enough to overcome the EM forces.

I may be wrong, because I'm only a relative newbie to GR.

 

[Chronos]

The short answer is gravity. The universe expands on large scales because gravity is too weak to bind large scale structures. At 'short' distances gravity, as well as other local forces overwhelm the relatively weak dark energy force.

 

[yuiop]

I'm not, but that's not the point. Dark matter can't explain it on several different length scales: atoms, meter sticks, solar systems, galaxies. Also, it would be quite a coincidence if those two cancel each other out, even in the case of a solar system.

Sure, but dark matter is thought to be a sphere of weakly interacting matter with uniform distribution throughout the galaxy. Until they find they find vast quantities of brown dwarfs or minature black holes to account for the missing mass (and they hardly count as weakly interacting) then it is possible that dark matter is an all pervasive material throughout the galaxy and the Solar system and possibly even at the meter scale. The only scale that dark matter would not be active over would be the inter galactic scale and that kind of fits neatly with why expansion is seen on large scales but not the sub galactic scale. Agreed it would be a huge coincidence if the two effects canceled each other out at the Solar scale but we are talking about measuring one tiny effect and an opposing tiny effect which means the overall effect at the Solar scale may be be even smaller, even if they do not exactly cancel.

I'm thinking that lower energy means lower orbital speeds, which means higher orbits. I also remember reading that the moon is moving away from Earth because of the energy loss due to work done by tidal forces. Why would neutron stars behave differently? (I'm not saying that they definitely don't. I'm just saying that if I'm wrong, I would like to know what my mistake was).

It is true that the moon is moving away from the Earth, but the reason it is, is a little more complex. The Eart completes a full rotation about its spin axis in about one day while the moon goes around the Earth in about a month. The friction drag is slowing down the spin speed of the Earth and increasing the angular velocity of the moon. The loss of energy of the system is in fact causing the moons kinetic energy and angular momentum to increase. The slow down of the Earth's angular momentum is compensated by the moon's increase in angular momentum as the orbital radius of the moon increases. For example let's say when the Earth-moon system eventually becomes orbitally locked, the Earth day becomes 48 hours, then the orbital period of the moon will also be 48 hours which is a lot faster than it is now.

The case for binary neutron stars is also not so straight forward. It would seem from the angular momentum equation L=mvr or L=mwr^2 where w=v/r is the angular velocity, that a reduction in v or w would result in an increase in radius to conserve angular momentum. A search on the internet shows that the words like "coalescing", "merging", "inspiral" are nearly always associated with the description of binary neutron stars indicating the opposite is the case.

For example:

wikpedia article http://en.wikipedia.org/wiki/PSR_1913+16

"The rate of decrease of orbital period is 0.0000765 seconds per year, the rate of decrease of semimajor axis is 3.5 meters per year, and the calculated lifetime to final inspiral is 300,000,000 years.[2]"

wikipedia article http://en.wikipedia.org/wiki/PSR_J0737-3039

"As a result of energy loss due to gravitational waves, the common orbit shrinks by 7 mm per day. The two components will coalesce in about 85 million years."

A paper by Joseph H. Taylor who won a Nobel prize for his studies of binary neutron stars http://arxiv.org/PS_cache/astro-ph/pdf/0407/0407149v1.pdf

"The loss of orbital energy results in shrinkage of the orbit, which is most easily observed as a decrease in orbital period."

http://www.astrophysicsspectator.com/topics/generalrelativity/TestBinaryPulsar.html

"When a binary pulsar emits gravitational radiation, it loses orbital energy and angular momentum, which causes the orbit to shrink and the period to decrease."

The clue to the cause of the non intuitive behavior is in the last quote. The gravitational waves are are not only carrying energy away, but are also carrying angular momentum away.

I can not find any references that suggest the orbital radius of a binary netron star pair increases over time.

Yes, but I don't need to know exactly what to expect. I'm just trying to understand why it would be possible in principle to measure the expansion in the absence of all those other effects. I believe that I do understand it reasonably well now. What I said in #4 explains it for gravitational systems, and what I said in #13 explains it for non-gravitational systems. I wouldn't mind seeing more rigorous arguments though, or more rigorous versions of my arguments.

As for what happens with a meter stick and the electromagetic forces within it, I am not sure. It seems to me that is the scale of the universe doubles over a given period due to expansion, then if meter sticks doubled in length over the same period we would either notice the speed of light has halved over the same period or the speed of light would have actually double over the same period so that we do not measure any change in the speed of light over time. I think it is generally accepted that the speed of light is constant over time so that sort of rules out the expansion of meter sticks. It can also be noted that if meter sticks increased over time and the speed of light increased over time at the same rate then we would not be able to measure any expansion of the universe, so that also rules out the expansion of meter sticks, (because we do measure the universe as expanding.)

 

[Dale]

And yet they preserve geodesic motion in the case of a solid at zero temperature, at rest in an inertial frame in Minkowski space. But if you imagine the same solid in a FRW space-time, in a weird initial state such that every atom is at rest in the cosmological rest frame, those intermolecular forces would immediately pull every atom away from geodesic motion. (Yes, I know we wouldn't actually be able to prepare that initial state in the real world, but this is a thought experiment).

The forces in a solid lattice work to preserve the distance between the lattice points. This implies parallel worldlines for the lattice points of a material at rest. In Minkowski spacetime if one such world line is a geodesic then all of the parallel worldlines are also geodesics. In other spacetimess worldlines that are parallel to a geodesic are not generally also geodesics.

 

[Fredrik]

I think that the relevant fact is that the geometry of the universe is not FRW on small scale.

Yes, this is my conclusion too. (See #4).

The meter scales, then, on the earth, are not in the FRW metric, but rather in a Schwarzschild one. There is no reason for them to expand.

They are in a metric that's a lot like Schwarzschild and a little bit like FRW, so there should be an expansion, but it should be very slow compared to the cosmological expansion. Two test masses falling in a vacuum on Earth would move away from each other very slowly, but a meter stick isn't going to expand at all, because of how solids behave. See my comments in #13 and DaleSpam's comments in #21.

Now what if a meter stick was placed in the intergalactic space?

It was by considering this exact question that I realized that what I said in #4 can't be the whole story.

if it is a "rigid" body like our terrestrial meter sticks, it'll may stay the same length as before

I'm pretty sure that it will, and that the reason is what I said in #13 (or something very similar).

 

[Fredrik]

The forces in a solid lattice work to preserve the distance between the lattice points. This implies parallel worldlines for the lattice points of a material at rest. In Minkowski spacetime if one such world line is a geodesic then all of the parallel worldlines are also geodesics. In other spacetimess worldlines that are parallel to a geodesic are not generally also geodesics.

Hm, this sounded right at first, but now I don't think it is. I'm thinking that the forces in a solid can't keep the world lines "parallel" because of how that concept is defined. Given two curves and a point on each of them, we can't even say if the curves are parallel at those points, because the answer depends on what path we choose when we parallel transport a tangent vector from one of the points to the other.

I've been thinking about this, and now I think that what the forces in a solid work to maintain, are the distances between nearby parts of the object in co-moving local Lorentz frames.

I should explain what I mean by that. Pick any particle P in the object and consider the tangent to its world line at some event A on it. The tangent is defined as the geodesic that has the same tangent vector at A as the world line of P. Choose proper time as the parameter of the tangent. We can use the tangent to define a coordinate system in a small neighborhood of A. Let the tangent be the time axis and choose the spatial coordinates so that the speed of light in these coordinates is 1. (What we're doing here is just using the fact that GR only deals with metrics that look like the Minkowski metric on small scales). Now pick a nearby particle Q and consider its world line. What I'm saying is that the forces in a solid works to maintain the distance between P and Q in these frames.

This could be wrong, but I think it looks correct, and it's definitely the best I've been able to come up with so far.

 

[Fredrik]

The short answer is gravity. The universe expands on large scales because gravity is too weak to bind large scale structures. At 'short' distances gravity, as well as other local forces overwhelm the relatively weak dark energy force.

The universe would be expanding even without dark energy. Dark energy is only needed to explain why the rate of expansion is increasing.

At the risk of patronizing I'm going to resort to the basic-yet-venerable balloon analogy.
Glue a bunch of pennies to a balloon. Inflate the balloon. Ask why the pennies don't expand.

But that's not what I'm asking. I'm asking why we should think of galaxies, solar systems and meter sticks as pennies, and not as drawings on the ballon's surface. (I think I have a pretty good answer now).

Simple: the expanding balloon's pull on the penny's structure is far inadequate to overcome the penny's solidity.

Likewise, the expansion of space is far inadequate to sub-galatic-scale objects to expand (the forces between stars/stars and stars planets are just way too strong).

These things don't explain anything. What we need to know is why those forces would even "try" to work against the expansion. (And as I said, I think I have an answer now).

 

[Hurkyl]

Given two curves and a point on each of them, we can't even say if the curves are parallel at those points, because the answer depends on what path we choose when we parallel transport a tangent vector from one of the points to the other.

On sufficiently small scales, we can (approximately) do that -- that's the content of the requirement that spacetime be 'locally Minkowski'.

 

[Dale]

What I'm saying is that the forces in a solid works to maintain the distance between P and Q in these frames.

I don't think that you are saying anything different than what I am saying. I think the forces in a solid work to maintain the proper distances of the lattice, which I believe is exactly what you were saying (except more concise).

My geometric point was that, two geodesics that are parallel to each other at one event in a flat spacetime will be parallel to each other at all events, so the proper distance between any two particles on those geodesics will remain constant. Such is not the case for curved spacetimes where, as you mentioned, even the concept of "parallel" is non-trivial.

In any case, it seems like you have answered the question in your own mind and are satisfied that there is nothing particularly troubling about the fact that the motion of particles an inertially moving extended solid body is not geodesic in curved spacetime.

 

[yuiop]

...

In any case, it seems like you have answered the question in your own mind and are satisfied that there is nothing particularly troubling about the fact that the motion of particles an inertially moving extended solid body is not geodesic in curved spacetime.

Is it not precisely because "the motion of particles an inertially moving extended solid body is not geodesic in curved spacetime" that natural orbiting bodies are torn apart at the Roche limit? I think the rings of Saturn are thought to have originated from moons that were torn apart in that manner.

 

[Dale]

I don't know too much of the details of the Roche limit, but I think that is correct.

 

[yuiop]

For what is worth, I have worked out that Pluto should be receding at a rate of about 7 meters per year due to expansion assuming a Hubble constant of 100 (on the high side) and an orbital radius of 5.913*10^9 kms. The values quoted for the inspiral of binary neutron stars is of a similar order of magnitude but it is fair to assume the inspiral of Pluto due to radiating gravity waves would be many orders of magnitude less because of the significantly less extreme gravitational conditions within the Solar system. Question is, do we have instruments capable of measuring a change of 7 meters per year at a range of six thousand million Kilometers? Quite possibly we do.

 

[Fredrik]

I don't think that you are saying anything different than what I am saying. I think the forces in a solid work to maintain the proper distances of the lattice, which I believe is exactly what you were saying (except more concise).

OK, we agree then, but I think we have to mention the "local Minkowski stuff" since there's (in general) no way to define which slice of space-time is "space" and hence no way to even specify which set of events is the lattice.

Is it not precisely because "the motion of particles an inertially moving extended solid body is not geodesic in curved spacetime" that natural orbiting bodies are torn apart at the Roche limit? I think the rings of Saturn are thought to have originated from moons that were torn apart in that manner.

I had never heard of the Roche limit before, but a quick look at the Wikipedia page tells me that those objects are torn apart at the Roche limit because gravity is the only force holding them together in the first place. So they are not solid objects in the same sense as a meter stick.

For what is worth, I have worked out that Pluto should be receding at a rate of about 7 meters per year due to expansion assuming a Hubble constant...

Didn't you like the argument about the metric in situations like this being "a little bit like FRW and a lot like Schwarzschild", which would imply that space is expanding locally, but at a much slower rate than the cosmological expansion?

 

[yuiop]

...
Didn't you like the argument about the metric in situations like this being "a little bit like FRW and a lot like Schwarzschild", which would imply that space is expanding locally, but at a much slower rate than the cosmological expansion?

Well, I did not see it so much as an argument, but more as a statement. What is the explanation for why the FRW metric is valid cosmologically valid but not locally valid? Where is the demarcation between the FRW metric and Schwarzschild metric? Is it defined by the limit we loosely refer to as a gravitationally bound system? In our neighbourhood of the universe we have the Solar System, our galaxy, the local group and then the local super cluster (which is collapsing), so presumably the FRW metric only applies outside the local super cluster. The "big rip" theory which seems popular recently, has it that the expansion of the universe will eventually tear apart the solar system and even atoms eventually, but I am not sure how serious a theory that is.

Basically, I just not clear if orbiting bodies are self regulating in radius in the same way a meter stick is. Does the gravititional force within the Solar system simply overwhelm the expansion or just totally ignore it? GR has it that the curvature of spacetime around a massive body defines the geodesics of particles orbiting around it. Now the picture I have of the expanding universe is that spacetime on the universal scale is stretching and carrying along galaxies that are effectively embedded in it. I picture the large scale stretching of spacetime as slightly flattening the curvature around massive bodies. Maybe I do not have the correct "picture" in my head :P It certainly does not seem to coincide with the picture of spacetime as being staic as described by MeJennifer.

 

[Dale]

What is the explanation for why the FRW metric is valid cosmologically valid but not locally valid? Where is the demarcation between the FRW metric and Schwarzschild metric?

The FRW metric is for a homogenous and isotropic distribution of matter. The Swarzschild metric is for a spherical mass surrounded by empty space, which is most definitely not homogenous. If you can point to a round central mass that isn't spinning too fast use Swarzschild, if you have a rather homogenous scattering of mass everywhere use FRW. I don't think there is too much danger of overlap, between the solar-system scale and the universe scale there are the galactic and cluster scales that would not be well-described by either Swarzschild or FRW.

 

[Fredrik]

What is the explanation for why the FRW metric is valid cosmologically valid but not locally valid?

It should be possible to calculate this explicitly, but the fact that everything I drop falls to the ground is proof that the metric is more like Schwarzschild.

Where is the demarcation between the FRW metric and Schwarzschild metric?

That's a good question. I'd say that in any system that can be described reasonably well by Newtonian gravity, the "Schwarzschildness" of the metric dominates over its "FRWness". I think that means that everything up to the scale of clusters of galaxies won't expand much.

The "big rip" theory which seems popular recently, has it that the expansion of the universe will eventually tear apart the solar system and even atoms eventually, but I am not sure how serious a theory that is.

I've been thinking about that too. I don't know how those models work. I can imagine that there will be a big rip if the density of dark energy is increasing, but I don't see how it can happen if we're just talking about a large value of the cosmological constant. I'm going to have to think about that some more (unless someone explains it to me).

Basically, I just not clear if orbiting bodies are self regulating in radius in the same way a meter stick is.

I don't know that either (yet), but it shouldn't be too hard to figure out. The Schwarzschild solution predicts the existence of circular orbits. A test mass in such an orbit stays at a constant r coordinate. Does "constant r" imply "constant proper distance" from the star? Hm, is that even the right question to ask? I'm too lazy to think about that right now. Maybe tomorrow.

I picture the large scale stretching of spacetime as slightly flattening the curvature around massive bodies.

That's sounds reasonable to me. The "slightly flattening" effect is caused by the "FRWness" of the metric that's "a bit like FRW and a lot like Schwarzschild", and it can only be a small effect, otherwise I'd be taking off from this chair right now.

 

[MeJennifer]

I picture the large scale stretching of spacetime as slightly flattening the curvature around massive bodies. Maybe I do not have the correct "picture" in my head :P It certainly does not seem to coincide with the picture of spacetime as being staic as described by MeJennifer.

Do not confuse spacetime with the dynamics of consecutive foliations of spacetime. There are many ways to slice spacetime, each way of slicing will generally give you a different kind of dynamics.

 

[rahuldandekar]

What is the explanation for why the FRW metric is valid cosmologically valid but not locally valid? Where is the demarcation between the FRW metric and Schwarzschild metric?

As has been said previously, that's a very good question. (Although I don't know how accurate it is to say that the metric is partly FRW and partly Schwarzschild.) The FRW metric is valid only if the matter distribution is homogeneous and isotropic. Thus, it only applies on the >~100 Mpc scale.

Thus, the metric of the universe is almost FRW at distances of 10⁸ pc.

Let's consider some scales:

Distance of Andromeda from Milky way ~ 1 Mpc = 10⁶ pc
Radius of Milky way ~ 100 kpc = 10⁵ pc
Radius of Solar system < 1 pc = 10⁰ pc
Distance of Earth from sun ~ 10^-5 pc

To get the "validity" of Schwarzschild-like metrics, we can consider the ratio of a typical distance in that system to it's Schwarzschild radius.

(I do this because, remember, it's R_S/r that counts in the S metric. The numbers below are order of magnitude, rough estimates.)

Large scales ~ approx infinity, since mass is evenly distributed, so S radius = 0.
Galaxy ~ distance of sun from center of milky way/ (GM/c²) ~ 10¹⁹ meters / 10^-17 meters = 10³⁶
Solar system ~ distance of Earth from sun/ (GM/c²) = 10⁹ meters /10³ meters = 10¹²

So the solar system metric is probably very Schwarzschild-like, and negligibly FRW-like.

I picture the large scale stretching of spacetime as slightly flattening the curvature around massive bodies.

I think, given the numbers above, it's very, very slight. As a result, Hubble's-law-like statements about expansion may not noticeably hold.