# Effect of expansion on a tether

1. Sep 18, 2012

### Chiclayo guy

If a string could be tied to two galaxies that were not gravitationally bound (i.e., the distance between them is increasing), would expansion cause the string to break, or would expansion or some other force act on the string such that tension on the string is not increased?

My thanks for any replies.

2. Sep 18, 2012

### marcus

How far apart do you imagine the galaxies to be to start with? How are you defining the rate that distance is changing?

You are asking about Hubble law expansion. This is defined in terms of PROPER DISTANCE between pairs of observers who are both at rest relative to Background.
Defining proper distance requires the concept of universe time---time as measured by stationary observers (to whom the Background has the same temperature in all directions). And the Hubble law concerns the RATE that proper distances are changing at a given moment of universe time, the rate being measured in terms of universe time.

The short answer is that if the distance is only, say, 139 million lightyears and it is a strong string then it wouldn't necessarily break. In fact it might go slack.
It all depends on the distance you start with and the initial individual motions (relative to Background) which the galaxies have to start with.

EDIT: This is a pretty brief condensed answer. You might want to ask more questions, like how is proper distance defined...or look up some stuff in the FAQ pages.

Last edited: Sep 18, 2012
3. Sep 18, 2012

### marcus

Assuming you understand the ideas of proper distance and universe time, the Hubble law is pretty easy to state. The current Hubble time is estimated to be 13.9 billion years. (Let's abbreviate that 13.9 Gy to save trouble writing.)
You just take any distance between stationary observers, say 139 million lightyears, and divide by Hubble time and that gives the rate that distance is expanding.

In this example, 139 Mly/ 13.9 Gy = 0.01c = a hundredth of the speed of light.

That's only 3000 km/second. So if galaxy A happens to be moving towards galaxy B (which let us say is at universe rest) and happens to be moving just that fast relative to Background, then the two galaxies will stay the same distance apart for as long as the Hubbletime remains the stated 13.9 Gy.

Eventually the string will go slack, though. Because the Hubbletime is slated to increase slowly towards a limiting value of 16.3 billion years. It is on a course to plateau there.
So the denominator of that fraction is increasing and therefore the rate that the given distance is expanding will decrease. Then the speed A is going (relative to Background) will be more than enough to cancel expansion, and A will tend to creep towards B.

========
The answer to your question depends a lot on the size of the distance. If the distance is too big then it will be expanding too fast for the local individual motions of the galaxies to cancel it.

For example take a distance of 27.8 billion lightyears. What is 27.8 Gly divided by 13.9 Gy?

Most of the galaxies that we can see with a telescope are currently more than 27.8 billion lightyears from us and therefore the distances to them (measured by stationary observers) are growing at more than twice the speed of light. There is no way their local individual motions could cancel that out!

Last edited: Sep 18, 2012
4. Sep 19, 2012

### Chiclayo guy

My question was actually prompted by thinking about the balloon analogy. After inflating the balloon a line drawn between the dots lengthens, and I wondered what would happen hypothetically if the line were a string. Thank you Marcus for your response.

5. Sep 19, 2012

### marcus

A friend just pointed out that I made a mistake: the line will not go slack! This has to do with there being positive cosmological constant (accelerated expansion.) He referred me to a very interesting article about just this problem---the tethered galaxy problem---by some top people. I will get the link so you can have a look too, if you wish to.

http://arxiv.org/abs/astro-ph/0104349
Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects
Tamara M. Davis, Charles H. Lineweaver, John K. Webb (The University of New South Wales)
(Submitted on 21 Apr 2001 (v1), last revised 1 May 2003 (this version, v3))
We use the dynamics of a galaxy, set up initially at a constant proper distance from an observer, to derive and illustrate two counter-intuitive general relativistic results. Although the galaxy does gradually join the expansion of the universe (Hubble flow), it does not necessarily recede from us. In particular, in the currently favored cosmological model, which includes a cosmological constant, the galaxy recedes from the observer as it joins the Hubble flow, but in the previously favored cold dark matter model, the galaxy approaches, passes through the observer, and joins the Hubble flow on the opposite side of the sky. We show that this behavior is consistent with the general relativistic idea that space is expanding and is determined by the acceleration of the expansion of the universe -- not a force or drag associated with the expansion itself. We also show that objects at a constant proper distance will have a nonzero redshift; receding galaxies can be blueshifted and approaching galaxies can be redshifted.
8 pages including 6 figures, to appear in Am. J. Phys., 2003.

Last edited: Sep 19, 2012
6. Sep 19, 2012

### Lino

Chiclayo, (I thought that there was an FAQ on this - what is effected by expansion, but I'm obviously wrong) The short answer is that in a simple example - galaxies that are not bound gravitationally to each other or any other objecst and at rest with respect to the CMB - the bonds of the individual atoms / molecules would easily overcome the local expansion of the space between the atoms / molecules , causing the tether to tighten and then snap (if the tether is not elastic). In relation to the ballon analogy, think of the tether as a string resting on the surface of the balloon rather than a line drawn on the balloon.

Regards,

Noel.

7. Sep 19, 2012

### marcus

Hi Lino, I think that is just the point. Tethered galaxies are NOT both at rest with respect to the CMB.
In my example the tether is strong and the galaxies are 139 Mly apart. Observers at CMB rest initially separated by that distance would find the distance between them increasing at 1% of speed of light.

Tension in the tether would pull the galaxies towards each other so as to keep the distance constant. If you imagine one much more massive so that it stays essentially at CMB rest then the other must acquire an individual local motion of 3000 km/s towards its partner, to compensate.

That's all stable and intuitive, galaxies are allowed to have individual local velocities. That is the situation which the "tethered galaxy problem" ASSUMES. Check out the paper by Lineweaver et al (http://arxiv.org/abs/astro-ph/0104349) for more discussion.

The T. G. problem is then to say what happens AFTER this situation is established, when the tethered galaxy is RELEASED from tether. As per the Lineweaver paper, the galaxies have been tethered for some period of time, kept at a fixed distance from each other, before they are released.

8. Sep 20, 2012

### Lino

Understood Marcus. Thanks for that.

Regards,

Noel.

9. Sep 20, 2012

### Jorrie

I have once attempted to quantify the tension in such a tether in terms of the instantaneous "cosmological tidal acceleration" that would result if the tether is cut at age T. This means how fast would the two galaxies start to accelerate towards or away from each other, from a proper distance perspective. One must keep in mind that there are forces on the galaxies before the tether is cut, but not thereafter, when they become inertial.

The effort resulted in this graph (done for a flat, $\Omega_\Lambda = 0.74$ universe, and slightly mis-titled as "Tidal Force").

The acceleration crosses from negative to positive around 7 Gy, when the accelerated expansion started. It scales linearly with proper distance D between the 'tethered galaxies'. I have used this equation:

$$\frac{d^2D}{d^2T} =DH^2_0(\Omega_\Lambda - \frac{\Omega_m}{2a^3})$$

which is essentially just the cosmic deceleration parameter scaled with D.

References:

Gregory S. Adkins et. al (2006): Cosmological perturbations on local systems

Matteo Carrera and Domenico Giulini (2006):On the influence of the global cosmological expansion on the local dynamics in the Solar System

Last edited: Sep 20, 2012
10. Sep 20, 2012

### marcus

That's neat! I'll try to make it intuitive with a thought experiment. You point out that after around year 7 billion growth of distances (between stationary observers) accelerates, instead of slowing down. Let's imagine two galaxies one much more massive at CMB rest (stationary) and the other, a smaller "test" galaxy, tethered at distance 139 Mly. A stationary observer whose location at that moment coincides with it, sees the test galaxy moving 3000 km/s towards the massive one. This is the motion that has been imparted by the pull of the cable.

If expansion were not accelerating then the observer would see the test galaxy continue to move at a constant speed of 3000 km/s away from him and towards the main galaxy. He would deduce that there was no tension in the cable, since no acceleration. But this is not fully realistic, since distance growth accelerates between stationary objects.

Now suppose the cable is cut. The test galaxy by its own inertia (as seen by the observer) continues to move away from him at 3000 km/s

However since the distance between stationary galaxy and stationary observer is growing at accelerating pace, it will soon be increasing at 3001 km/s. The test galaxy motion cancels 3000 km/s of this. So now the distance between the galaxies is increasing at 1 km/s.

As time goes on this rate will increase. As long as the stationary observer's reference frame is approximately valid we can say that it increases approximately in step with the observer's own distance from the main galaxy. When his distance is increasing at 3002 km/s, the test galaxy's distance will be increasing at nearly 2 km/s.

The upshot is that after the cable is cut the test galaxy drifts farther way. So while it WAS tethered there must always have been tension in the cable. Even though the test galaxy was staying at a constant distance from main.

I don't know if others would find it helpful, but for me it makes this more intuitive to imagine it from the standpoint of an observer who is at CMB rest and whose location momentarily coincides with that of the tethered test galaxy.

Last edited: Sep 20, 2012
11. Sep 20, 2012

### Jorrie

I found it more comfortable to put the massive galaxy and the observer at the origin and let them stay there, creating an inertial frame at rest with the CMB.The test galaxy then is first tethered at proper distance D, with zero proper velocity in the reference frame, but with proper velocity -HD against its local Hubble flow.

If the tether is cut before matter-lambda equality (~7 Gy), the test galaxy will fall through the origin and later join the Hubble flow at the other side. If it is cut at matter-lambda equality, it would hover for a while and then very slowly gather positive recession speed, to join the Hubble flow at the side where it started.

Here is a graphical illustration of the case, showing proper distance and all the relevant coordinate velocities.

The velocities shown may be a bit confusing, so here is an attempt to clarify the diagram.

At cosmic time t = 0.2 Gy after the BB, the tethered galaxy was chosen to be at D = –0.1 Gly. At that stage, the expansion rate was decreasing under the dominant (99.96%) matter density influence of the time. The Hubble velocity at D=-0.1 Gly was a whopping –0.335c and hence the peculiar velocity of the tethered galaxy at that time was 0.335c towards the origin. But, due to the decreasing expansion rate, the Hubble velocity quickly diminished for that proper distance; hence when untethered, the galaxy started to 'fall' rapidly towards the origin. It 'fell through' the origin at some 1.5 billion years and then continued to move in the positive D direction.

The proper velocity (red) started at zero (because the galaxy was tethered). When untethered, proper velocity increased rapidly until the galaxy passed through the origin and then the proper velocity started to decrease, marginally, only for as long as matter density dominated and the expansion rate still slowed down. At around 7 Gy, the accelerating effect of the cosmological constant more or less balanced out the deceleration of expansion caused by the (decreasing) matter density. For a while the expansion rate remained more or less constant and the proper velocity of the galaxy also remained constant.

After 8 Gy age the cosmological constant started to win and the expansion rate started to increase (and so did the proper velocity of the galaxy). During all this, the peculiar velocity (green) continuously decayed and will keep on doing so, approaching zero. The proper velocity eventually joins the Hubble flow.

Last edited: Sep 21, 2012