A look at tethered and untethered galaxies
|For a more recent version of this paper see Tethered galaxies and the expanding space paradigm|
AbstractThe calculations in the paper 'Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects.'  lead to the counterintuitive result that a stationary or receding galaxy can be blueshifted, even in an empty universe. However, this is due to the definition of distance (comoving distance), and here we demonstrate that 'radar distance' is a more reasonable definition for which the effect disappears in an empty universe. However, in a non-empty universe the results still hold for radar distance.
The tethered galaxy in an empty universeIn  Davis, Lineweaver and Webb examine the behaviour of a 'tethered galaxy', that is a hypothetical object at a constant distance from us, within a cosmological context. According to special relativity such an object should not show any blueshift or redshift. However, the calculations in  show that such a galaxy would exhibit a blueshift, even in the case of the universe without gravitating matter or a cosmological constant - the (0,0) universe - which one would expect to be equivalent to the universe of special relativity. This gives strong support to the claim that the universe cannot be thought of in special relativistic terms. However a more accurate calculation shows that the object cannot in fact be considered stationary, and that there is no blueshift for a truly stationary object.
A homogeneous, isotropic universe is considered using the Friedman-Robertson-Walker metric, which can be given comoving coordinates (only radial distances are considered):
ds2 = - dt2 +a(t) 2 dχ2,
Here a(t) depends on the parameters of the model being considered, in particular on the density of gravitating matter and the cosmological constant. Here we consider the (0,0) case in which a(t)=t/t0. (Note: all calculations use c=1)
The galaxy is tethered at a constant distance until proper time t0 at which time its position is given by χ=χ0. We take χ0>0, and the galaxy is taken to be within the Hubble sphere so that its motion does not exceed the local speed of light. (The coordinates are scaled so that a(t0)=1). In  constant distance is taken to mean constant a(t)χ. However, suppose we wanted to measure the distance to such an object. We would send a light signal and see how long it took to receive a return signal. Suppose we (at χ=0) send a light signal at time t1 which meets the object at coordinate (χ,t) and it is reflected back to be received at time t2 . Then a textbook on general relativity (e.g. ) says that
The untethered galaxyWhile the object is tethered it might be argued that choice of what one regards as constant distance is just a matter of preference. Clavering  has argued why radar distance is the correct definition (and this paper duplicates his work to some degree), but comoving distance (aχ) also seems reasonable. The deciding point is what happens when the tether is removed, in particular in the (0,0) case. In the absence of forces one would expect the object to stay where it is. A calculation in  shows that (in the (0,0) case) tχ remains constant after the tether is released. Hence constant a(t)χ seems to be the correct definition of constant distance. However this calculation uses a non-relativistic approximation and so is not precisely correct. Although this would seem unimportant, it does in fact lead to the non-physical result for large t . So the question is, what does the correct calculation predict for an object with the given initial conditions? It is possible to repeat the calculation in  including relativistic effects, but here here we proceed via the geodesic equation: (Note that a dot indicates a derivative with respect to cosmological time t)
Clearly tχ constant is not a solution. The solutions are of the form t.χ=t0 tanh(w-χ/t0) where w is a constant, leading to t sinh(w-χ/t0)=constant
Taking the initial condition a0.χ0 = -.a0χ0, that is .χ0 = -χ0/t0 in the (0,0) case, we get
As t→∞ , χ→ w t0=χ0-t0tanh-1(χ0/t0)<0
This initial condition does not represent a galaxy at constant distance. It represents a galaxy which starts at χ>0 and moves towards us, passing us and joining the Hubble flow on the other side of the sky. This explains why such a galaxy is seen to be blueshifted.
Note that if the tether is taken to represent constant radar distance then we have t sinh χ =constant, which is the solution of the geodesic equation with w=0, and so the object will continue on the same geodesic when untethered. (Note that w is simply the speed in 'special relativistic' coordinates) This shows that radar distance is the more reasonable definition of distance in this context.
The tethered galaxy in a non-empty universeThe above discussion indicates that 'radar distance' is strongly preferable when considering a tethered galaxy, and eliminates the blueshift in the Milne universe. The question then arises, what happens in a non-empty universe? Consider the matter dominated, critical density universe given by a=t2/3. Then the calculation of radar distance proceeds as follows:
|u2/3 du||= 3(t1/3-t11/3)|
= t - t2/3χ + t1/3χ2/3 - χ3/27
Hence radar time and distance are given by
Tr= t+t1/3χ2/3 and Xr=t2/3χ+χ3/27
Now consider the case when the object has constant radar distance, and emits a signal from point (χ,t), which is received by us at time T.
|u-2/3 du||= 3(T1/3-t1/3)|
For constant Xr we have
Let β=t-2/3χ2/9 . Then:
.T=1+(1/3)t-2/3χ2/3 + (2/3)t1/3.χχ
=1+β-4β/(1+β) = (1-β)2/(1+β)
Note that t is cosmological time. The object is moving with respect to the comoving background, and so its proper time τ will differ from this, but can be obtained directly from the metric.
dτ2=dt2-a(t)2dχ2 .τ2 =1-t4/3.χ2 =1-(4/9)t-2/3χ2/(1+β)2
Considering the frequency of the emitted signal, we see that the redshift is the rate of change of the time of observation with respect to the proper time of emission.
Hence z+1=.T/.τ=1-β <1 so z<0,
This means that the object is blueshifted, agreeing with the findings in  for comoving distance.
This isn't necessarily counterintuitive. In the matter dominated universe, each comoving observer may think of themselves at the bottom of a gravitational potential well, and so would expect a signal from an object at constant distance to be blueshifted.
Similar calculations can be done for other metrics
|Description of universe||Metric||Redshift of tethered galaxy?||Notes|
|Empty||a=t||Zero, (as indicated above).|
|Radiation dominated, critical density||a=t½||Blueshift (as with matter).||Comoving distance and radar distance coincide.|
|Cosmological constant dominated||a= eHt||Redshift||Radar distance is a function of comoving distance, so the definitions of constant distance coincide.|
DiscussionGeneral relativity allows considerable freedom in choosing coordinates to represent the metric of a given spacetime. In cosmology the proper time of comoving objects is usually taken as the time coordinate, giving coordinates for the (0,0) case of ds2 = - dt2 +(t/t0)2 dχ2 . However there is the also 'Special relativistic' alternative, with a different time coordinate, ds2=-dt2+dx2 ,. Note that these represent the same metric and so should always lead to the same result in any calculation.
In comoving coordinates there are objects moving apart faster than light. This is usually explained by saying that it is the expansion of space between the objects rather than the motion of the objects themselves. Unfortunately this has lead to many misunderstandings. There is a belief that if an object is moving away from us faster than light then we cannot see it - it is beyond a horizon. Also it is implied that space has physical properties, and so it is assumed that it influences the matter within it, causing it to join in with the Hubble flow. The belief that expanding space is needed to explain why the galaxies are receding may then encourage the pre-Newtonian idea that a moving object requires a cause to stay in motion.
Adding to the confusion is the the recent discovery that the universe may have a positive cosmological constant due to vacuum energy, is which case it is more reasonable to think of space as having physical properties and objects merging into the Hubble flow. In academic circles it is always clear whether the model which is being discussed involves a cosmological constant or not. Unfortunately in popular accounts this is often not the case, and the distinction between different models is not made. This is a particular problem in discussions of why local objects do not join in with the expansion.
A large part of this confusion has been addressed by Lineweaver and Davis in . However they argue that thinking in 'Special Relativistic' terms is part of the source of the confusion. I would argue that it is the other way round, and that it is the insistence on sticking to comoving coordinates which has lead to the attribution of physical causes for what is actually an effect of coordinate choice . In this paper I have shown that one argument against special relativistic intuition, that a stationary object in an empty universe can be blueshifted, is false. I hope that the reader is encouraged to a greater belief in such intuition.
- T. M. Davis, C. H. Lineweaver, J. K. Webb, "Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects." Am.J.Phys. 71 (2003) 358-364 (astro-ph/0104349)
- R. Wald. General Relativity University Of Chicago Press (1984)
- W. J. Clavering "Comments on the tethered galaxy problem" 2005 (astro-ph/0511709)
- T. M. Davis, C. H. Lineweaver, "Expanding confusion: common misconceptions of cosmological horizons and the superluminal expansion of the universe", 2003 (astro-ph/0310808)
- D. N. Page, "No superluminal expansion of the universe" 1993 (gr-qc/9303008)
- S. Lee "Cosmology, Special Relativity and the Milne Universe" 2004 (http://www.chronon.org/articles/milne_cosmology.html)
- S. Lee "Stretchy Space?" 2004 (http://www.chronon.org/articles/stretchyspace.html)