Cosmic Tidal Force: Measuring CTF With Strain Gauge

[Jorrie]

TL;DR Summary

I want to investigate/discuss/quantify the strength of the cosmological tidal force caused by accelerating cosmic expansion

In order to make it as simple and non-controversial as possible, I propose a setup where inhomogeneity and change of reference frames play as little role as possible, i.e. in free space and with its center of mass comoving. I'll call that point the origin and set up my lab there. Rockets deliver two identical masses a distance D/2 in opposite directions, attach a cable from each to the lab and allow the situation to stabilize as something like this, with the cables taking some cosmological stress due to accelerating expansion.

We can measure the cosmological 'tidal' force (CTF) on the cable in the lab by inserting a strain gauge. But before that we should calculate the OOM, so that we can insert a properly scaled strain gauge. Using standard cosmological equations, I get the CTF:

$$CTF = D H_0^2 \left(\Omega_{\Lambda} - \Omega_M/(2a^3)\right)$$
(From Peebles 1993, eq. 13.3, p 312)
At present, with a=1 and the Omegas 0.7 and 0.3, with Ho roughly 0.072/Gyr², it works out to:^[1]

CTF ~ 2.9x10^-11 D Newton/kg, with D in Glyr, or in easier terms:

29 nano-Newton per kg mass per Glyr separation.

Unless I have made some serious blunders in conversion, this is a small force, really small, unless we make the two masses very large...

Note [1] Since I often play around with Gyr, meaning acceleration is geometrically in the units Gyr^-1, I find it very convenient that
1 Gyr^-1 is very close to 1 nano-g (Precisely 0.970735785686955 ng).

​ ​

 

[anuttarasammyak]

I am afraid that the wire is loosened by the approach of the two massive bodies to each other due to universal gravitation force of
$$-G\frac{M^2}{D^2}$$.

 

[PeroK]

Summary: I want to investigate/discuss/quantify the strength of the cosmological tidal force caused by accelerating cosmic expansion

In order to make it as simple and non-controversial as possible, I propose a setup where inhomogeneity and change of reference frames play as little role as possible, i.e. in free space and with its center of mass comoving. I'll call that point the origin and set up my lab there. Rockets deliver two identical masses a distance D/2 in opposite directions, attach a cable from each to the lab and allow the situation to stabilize as something like this, with the cables taking some cosmological stress due to accelerating expansion.

View attachment 313372

We can measure the cosmological 'tidal' force (CTF) on the cable in the lab by inserting a strain gauge.

​ ​

Why would space be expanding in a lab?

 

[PAllen]

Why would space be expanding in a lab?

This is an excellent point, with a key caveat. In general, FLRW metric is an emergent, large scale average of local geometry that would not be expected to governed by FLRW metric at all. The exception is the cosmological constant. IF the cosmological constant correctly models physical reality, then it does apply at arbitrarily small scales (though with undetectable effects for realistic parameter choices).

 

[Bandersnatch]

I am afraid that the wire is loosened by the approach of the two massive bodies to each other due to universal gravitation force

Not here. The gravitational interactions between (all the) masses (in the universe) are already taken into account in the evolution of the scale factor, together with the influence of the cosmological constant. As you can see from the equation, with sufficiently diluted mass density the contribution from the cosmological constant takes over and keeps the string taut.

Why would space be expanding in a lab?

It's not the point of the setup that the space should or shouldn't expand in the lab. The setup has the two masses at cosmological distances attached to strings, and the ends of those strings are attached to a strain gauge in the lab so that the force can be measured. The tension in the string is then due to the dynamics of the expansion, and the lab is there just to measure it unambiguously.

 

[PeroK]

If there is a cable of cosmological scale under strain, then does that represent a non negligible stress energy within the experimental setup?

 

[Jorrie]

I apologize to all who have misunderstood the scenario that I have presented. I should have made it more clear that I was talking about cosmological distances and I should have done a better job of the schematic.

That said, I should also have stressed that the equation that I've used is a weak field approximation and not a full relativistic one. This implies that the two masses should not be excessively large and the distances large, but not something near a Hubble radius apart.

The equation should however hold for galaxy (or even cluster) sizes and masses and distances into the hundreds of millions of light years, where we can assume weak gravitational fields and Hubble rates well below the speed of light.

This is where the effect of the accelerating expansion is simple enough to be very closely approximated by the (Peebles) equation given in the OP. If anyone knows a reference for a more general relativistic equation, I would be grateful.

 

[Jorrie]

If there is a cable of cosmological scale under strain, then does that represent a non negligible stress energy within the experimental setup?

The idea was that the cables are only attached to a strain gauge in the relatively small lab; so the lab itself is under negligible strain.

 

[PeroK]

The idea was that the cables are only attached to a strain gauge in the relatively small lab; so the lab itself is under negligible strain.

My point was that you no longer have a vacuum. Even if you neglect the mass of the cable, there is still the strain, which cannot necessarily be neglected. But ought to be calculated.

 

[Jorrie]

My point was that you no longer have a vacuum. Even if you neglect the mass of the cable, there is still the strain, which cannot necessarily be neglected. But ought to be calculated.

Correct, but I was thinking that the very symmetrical situation would enable a fairly reliable OOM determination. As always, the "devil" might well be in the details...

 

[PeterDonis]

If there is a cable of cosmological scale under strain, then does that represent a non negligible stress energy within the experimental setup?

This is an interesting question. The key thing is energy density, and what we need to compare is the energy density of dark energy, since that is what is causing the strain, and the energy density of the cable itself. (Since the OP has specified the weak field case, the strain in the cable should be small compared to the cable's energy density, so the latter is what we need to estimate.)

Even if you neglect the mass of the cable, there is still the strain, which cannot necessarily be neglected.

If we are in the weak field regime, then, as above, the strain in the cable must be small compared to its energy density, so you can't neglect the mass of the cable.

 

[anuttarasammyak]

Not here. The gravitational interactions between (all the) masses (in the universe) are already taken into account in the evolution of the scale factor, together with the influence of the cosmological constant. As you can see from the equation, with sufficiently diluted mass density the contribution from the cosmological constant takes over and keeps the string taut.

Thanks. I think of a similar thought experiment which we do not have to introduce mass.

Let us prepare a ideally light, long retractable measure. At initial time the end of the tape is fixed at a point A and the holder is fixed at point B. Here "fixed" means fixed to points where CMG is observed homogeneous. When distance AB is our daily scale nothing happens. As distance AB becomes longer, we start noticing the tape is pulled out from the holder. Longer the distance AB, faster the speed of pull out and it would become beyond c, when physical mechanism of the measure cannot follow.

 

[PAllen]

Thanks. I think of a similar thought experiment which we do not have to introduce mass.

Let us prepare a ideally light, long retractable measure. At initial time the end of the tape is fixed at a point A and the holder is fixed at point B. Here "fixed" means fixed to points where CMG is observed homogeneous. When distance AB is our daily scale nothing happens. As distance AB becomes longer, we start noticing the tape is pulled out from the holder. Longer the distance AB, faster the speed of pull out and it would become beyond c, when physical mechanism of the measure cannot follow.

This is different because if A and B are comoving, we don't have a tethered problem at all. When AB distance is large and they are comoving, they are already separating by initial conditions and this has nothing to do with tidal forces - the subject of this thread. Instead, B can be replaced with a free fall body whose initial state is such that the initial motion of the measure end is zero on both sides and there is no tension. This is possible for very large distances even in a universe with accelerated expansion. Then (in a universe with accelerated expansion), the tape measure will start being pulled (put under tension) despite the initial set up. But you can start with conditions where the initial tension is zero.

 

[PAllen]

We can dodge any discussion of material of the string or measure by using mathematical constructs and proper acceleration rather than force. Just consider some comoving body A. Extend from it a spacelike geodesic 4-orthogonal to it. Now, consider a timelike geodesic, B, that is initially 4-orthogonal to the other end of this "mathematical string". This, by construction, guarantees that initial rate of change of distance between A and B per this distance definition, is zero. If there is accelerating expansion, a spacelike geodesic from a later moment along A, that is 4-orthogonal to it and intersects the B geodesic will:

1) be longer than it was at the starting moment.
2) will no longer be 4-orthogonal to the B geodesic.

If, instead, we replace the B geodesic with world line that preserves distance measured as above, it will have a proper acceleration in the direction from B to A. This corresponds to tension in the string without any debates about string properties.

 

[PeterDonis]

When AB distance is large and they are comoving, they are already separating by initial conditions and this has nothing to do with tidal forces - the subject of this thread.

The "tidal forces" the OP is referring to are the changes in the rate of separation of comoving objects, in the particular case of a dark energy dominated universe: i.e., the relative acceleration away from each other of comoving geodesics. In a matter dominated universe in which the expansion was decelerating, the relative acceleration of comoving geodesics would be towards each other, but it would still be relative acceleration. Both of these are manifestations of spacetime curvature, and in GR "tidal gravity" is synonymous with spacetime curvature, so both of these could be termed "tidal forces" (or "tidal gravity", which is a better term since comoving geodesic objects feel no force).

The actual forces in the OP's scenario, the tension in the cables, is due to the fact that the cables prevent the masses from following geodesic paths, so the masses feel nonzero force (proper acceleration). Such forces are commonly referred to as "tidal forces", but this is really a misnomer.

 

[PeterDonis]

We can dodge any discussion of material of the string or measure by using mathematical constructs and proper acceleration rather than force.

The expression the OP gives from Peebles actually is for proper acceleration, since it is in units of Newtons per kilogram, i.e., force per unit mass, i.e., acceleration.

I agree that it is generally simpler and easier to look at proper acceleration rather than force. However, one of the questions raised in this thread has been whether the cables themselves will have non-negligible stress-energy and will therefore affect the observations. This question cannot be answered just by looking at proper accelerations.

 

[PAllen]

The "tidal forces" the OP is referring to are the changes in the rate of separation of comoving objects, in the particular case of a dark energy dominated universe: i.e., the relative acceleration away from each other of comoving geodesics. In a matter dominated universe in which the expansion was decelerating, the relative acceleration of comoving geodesics would be towards each other, but it would still be relative acceleration. Both of these are manifestations of spacetime curvature, and in GR "tidal gravity" is synonymous with spacetime curvature, so both of these could be termed "tidal forces" (or "tidal gravity", which is a better term since comoving geodesic objects feel no force).

The actual forces in the OP's scenario, the tension in the cables, is due to the fact that the cables prevent the masses from following geodesic paths, so the masses feel nonzero force (proper acceleration). Such forces are commonly referred to as "tidal forces", but this is really a misnomer.

But why have both bodies be comoving? That would cause stretching of a connector between them that has nothing to do with tidal forces. It would be present in the Milne cosmology. My understanding of tethered galaxy problems is to have at least one of them not be comoving, so the there is initially no stretching of a tether.

 

[PAllen]

The expression the OP gives from Peebles actually is for proper acceleration, since it is in units of Newtons per kilogram, i.e., force per unit mass, i.e., acceleration.

I agree that it is generally simpler and easier to look at proper acceleration rather than force. However, one of the questions raised in this thread has been whether the cables themselves will have non-negligible stress-energy and will therefore affect the observations. This question cannot be answered just by looking at proper accelerations.

I don't see that in the OP at all. That was introduced later, without fully answering the quantitative questions in the idealized scenario of the OP.

 

[PAllen]

The actual forces in the OP's scenario, the tension in the cables, is due to the fact that the cables prevent the masses from following geodesic paths, so the masses feel nonzero force (proper acceleration). Such forces are commonly referred to as "tidal forces", but this is really a misnomer.

I agree, and that is what I described, but you definitely do not want to start with both bodies being comoving. In fact, if you fix the length of the connecting geodesic 4- orthogonal to one that is comoving, the other end will never be comoving, as I'm sure you know. (And I don't have any issue with calling such things tidal forces - basically you measure deviation between two geodesics that are initially parallel by some construction, or proper acceleration of the non-geodesic world line that prevents the deviation).

 

[anuttarasammyak]

This is different because if A and B are comoving, we don't have a tethered problem at all. When AB distance is large and they are comoving, they are already separating by initial conditions and this has nothing to do with tidal forces - the subject of this thread. Instead, B can be replaced with a free fall body whose initial state is such that the initial motion of the measure end is zero on both sides and there is no tension. This is possible for very large distances even in a universe with accelerated expansion. Then (in a universe with accelerated expansion), the tape measure will start being pulled (put under tension) despite the initial set up. But you can start with conditions where the initial tension is zero.

Thanks. Now I understand the essence of the setting of OP that only Lab, which is located at the middle of the wire, is comoving and the two ends are free so that the ends are observed moving and under acceleration at their local comoving frames of reference. The tension of wire is measured at the lab whose division by the end mass would show tidal force ( per unit mass ).

I find an analogy in rotation. The center lab is rotation center and the ends undertake centrifugal force per ends mass of $D/2\ \omega^2$. The center observe tension / mass of $D \omega^2$. $D \omega < 2c$ is required to keep the mechanism of the experiment.

I feel mechanism of transmitting tension should be investigated further in OP setting as well as my previous setting of retractable tape measure. The speed of tension cannot go beyond c. I am not sure for such a long D the distance between the ends are kept constant by wire tension.

 

[Jorrie]

If we are in the weak field regime, then, as above, the strain in the cable must be small compared to its energy density, so you can't neglect the mass of the cable.

It is easy to calculate the mass of the cable and also how it is distributed. What is not so easy is to calculate is the force contributed by the distributed mass. The cable mass will contribute zero at the lab and more and more for the more distant parts being forced off their geodesics.

H. Nikolic's Relativistic contraction of an accelerated rod analyzes a different problem, but his method described in section 5 pages 7 to 9, applicably adapted, might be useful here, e.g.

 

[PAllen]

Thanks. Now I understand the essence of the setting of OP that only Lab, which is located at the middle of the wire, is comoving and the two ends are free so that the ends are observed moving and under acceleration at their local comoving frames of reference. The tension of wire is measured at the lab whose division by the end mass would show tidal force ( per unit mass ).

I find an analogy in rotation. The center lab is rotation center and the ends undertake centrifugal force per ends mass of $D/2\ \omega^2$. The center observe tension / mass of $D \omega^2$. $D \omega < 2c$ is required to keep the mechanism of the experiment.

I feel mechanism of transmitting tension should be investigated further in OP setting as well as my previous setting of retractable tape measure. The speed of tension cannot go beyond c. I am not sure for such a long D the distance between the ends are kept constant by wire tension.

I think the limiting factor in size of tethered experiment for accelerated expansion is better viewed in terms of infinite proper acceleration rather the superluminal speed or tension. An SR analogy is that if the right end of a rod has proper acceleration to the right of 1 g, then the maximum size of constant length rod is 1 ly. As this limit is reached, the left end of the rod would need unbounded proper acceleration. I believe that with accelerated expansion there is some similar limit, where the proper acceleration of of a body maintaining constant 4-orthogonal distance from a comoving body would approach infinity as the length approaches some maximum.

 

[Jorrie]

I believe that with accelerated expansion there is some similar limit, where the proper acceleration of of a body maintaining constant 4-orthogonal distance from a comoving body would approach infinity as the length approaches some maximum.

Yes, I think that limit would be the Hubble radius, relative to the position of the lab. But I also think that the low speed/weak field equations will break down long before that, as the peculiar radial velocity grows to a significant fraction of the speed of light.

This is all rather academic, but is a tasty challenge to amateur-armchair relativists-cosmologists like myself.

 

[PeterDonis]

why have both bodies be comoving?

The OP's proposal did not. But depending on what the OP actually wants to measure, having both bodies be comoving could be useful.

That would cause stretching of a connector between them that has nothing to do with tidal forces.

I disagree; the stretching is due to the divergence of comoving geodesics in a dark energy dominated universe, which is tidal gravity (spacetime curvature). (It is not "tidal forces" because comoving objects feel no force, and the "connector" would have to have zero structural strength and would therefore register zero strain, which of course is not possible with any real material.)

It would be present in the Milne cosmology.

Yes, and in that particular edge case, spacetime curvature is zero so there is no tidal gravity. But that does not generalize to a dark energy dominated universe (accelerating expansion), which is what the OP was considering.

 

[PeterDonis]

I think that limit would be the Hubble radius

More precisely, the cosmological horizon distance. The cosmological horizon is the point at which the proper acceleration diverges.

 

[PeterDonis]

I also think that the low speed/weak field equations will break down long before that

I agree.

 

[PeterDonis]

I am not sure for such a long D the distance between the ends are kept constant by wire tension.

They will be eventually, but it will take at least as much time as the light travel time from one end to the other for the situation to reach equilibrium. Until equilibrium is reached, the overall motion of the cables will indeed not be Born rigid--the various parts of the cable will be in motion relative to each other. The strain measurement the OP wants to make cannot be made until equilibrium is reached.

 

[Jorrie]

They will be eventually, but it will take at least as much time as the light travel time from one end to the other for the situation to reach equilibrium.

Sure, but the two spaceships that hook up the cables with the lab will take much longer than that. The cables could presumably be rolled out from the two spaceships with just the right tension to keep the two masses at the required constant proper distance from the lab.

It will be an interesting challenge to program the spaceship computers to achieve that accurately. Both the thrust and the brakes on the cable drums will need to have an increasing profile of just the right shape. Cool homework problem?

 

[PAllen]

I disagree; the stretching is due to the divergence of comoving geodesics in a dark energy dominated universe, which is tidal gravity (spacetime curvature). (It is not "tidal forces" because comoving objects feel no force, and the "connector" would have to have zero structural strength and would therefore register zero strain, which of course is not possible with any real material.)

And I disagree. To measure geodesic deviation or focusing, you start with initially parallel geodesics and see whether they diverge or converge after this. The tidal force arises because, with accelerated expansion, a geodesic initially 4-orthogonal to the spacelike geodesic extended 4-orthogonally to a reference comoving observer will diverge - increase in distance defined the same way. Having comoving observers on both ends does not measure curvature efffects at all - it is totally dominated by recession from initial conditions.

Yes, and in that particular edge case, spacetime curvature is zero so there is no tidal gravity. But that does not generalize to a dark energy dominated universe (accelerating expansion), which is what the OP was considering.

No, there is a perfect analogy. Two distant comoving observers in Milne model have initial high relative velocity, and this has nothing to do with presence or absence of curvature. Instead, in the Milne model, corresponding to some reference comoving observer, there is a distant time like geodesic which remains at constant distance to the reference, measured along 4-othogonal spacelike geodesics. This is the feature that requires flatness. The corresponding case in realistic cosmology is exactly as described in my post #14, and it is properties (1) and (2) there described that indicate tidal forces = curvature.

 

[PAllen]

The "tidal forces" the OP is referring to are the changes in the rate of separation of comoving objects, in the particular case of a dark energy dominated universe: i.e., the relative acceleration away from each other of comoving geodesics. In a matter dominated universe in which the expansion was decelerating, the relative acceleration of comoving geodesics would be towards each other, but it would still be relative acceleration. Both of these are manifestations of spacetime curvature, and in GR "tidal gravity" is synonymous with spacetime curvature, so both of these could be termed "tidal forces" (or "tidal gravity", which is a better term since comoving geodesic objects feel no force).

As stated, what you say is correct, but, IMO, not a useful way to get at the essence of tidal gravity. The key to understanding force on a string that maintains, e.g. Born rigidity, is to compare its path to a geodesic initially tangent to the string end world line (which would be one 4-orthogonal to the spacelike geodesic representing the string at some chosen point). These diverge when there is curvature, while they don't diverge when there is no curvature. Comparing the string end to a colocated comoving observer is irrelevant because the dominant effect here is that this geodesic has enormous relative velocity to the colocated geodesic I construct above.

The actual forces in the OP's scenario, the tension in the cables, is due to the fact that the cables prevent the masses from following geodesic paths, so the masses feel nonzero force (proper acceleration). Such forces are commonly referred to as "tidal forces", but this is really a misnomer.

I agree with this, and I don't see any problem calling this tidal acceleration.

 

[PeterDonis]

To measure geodesic deviation or focusing, you start with initially parallel geodesics and see whether they diverge or converge after this.

That's a common way of doing it, and the way it is usually explained in textbooks, but it is not the only way of detecting that spacetime curvature is present. Anything that shows a nonzero Riemann tensor is sufficient.

The tidal force arises because, with accelerated expansion, a geodesic initially 4-orthogonal to the spacelike geodesic extended 4-orthogonally to a reference comoving observer will diverge

And with decelerated expansion, a geodesic initially 4-orthogonal to the spacelike geodesic extended 4-orthogonally to a reference comoving observer will converge (another version of a tethered galaxy scenario that was discussed in a recent thread). So either way we have nonzero tidal force--it's just divergence in one case and convergence in the other.

Some physicists would object to calling this "tidal force" on the grounds that that term should only be used for Weyl curvature, which is zero in both cases--all of the spacetime curvature is Ricci curvature. I don't personally agree with that objection.

Having comoving observers on both ends does not measure curvature efffects at all - it is totally dominated by recession from initial conditions.

I disagree: "recession from initial conditions" cannot explain either acceleration or deceleration. The only case where "recession from initial conditions" explains the entire motion is the Milne universe, since that is just flat Minkowski spacetime in disguise and there is zero spacetime curvature.

No, there is a perfect analogy.

I don't think "perfect analogy" is a good term to use, because I disagree with your claim that the relative motion of comoving observers does not measure curvature effects; see above.

I do understand the general scenario you want to set up: you have a reference object which is comoving, you take a spacelike geodesic orthogonal to its worldline and extend it for some fixed proper distance, and you put a test object at the other end. You are comparing two possible choices for what state of motion to put the test object in: "comoving" vs. "orthogonal". So consider the following:

The spacelike geodesic in question does not lie in a single spacelike hypersurface of constant comoving time (i.e., a surface of constant FRW coordinate time). In fact, the FRW coordinate time at the test object end of the spacelike geodesic is earlier than the FRW coordinate time at the reference object end.

This means that, if we consider the relative velocity of the two possible test objects at the other end of the spacelike geodesic, the "comoving" and "orthogonal" ones, it will not necessarily be the same as the peculiar velocity of a test object at rest relative to the reference object at the same proper distance in a spacelike hypersurface of constant FRW coordinate time. (Btw, I think this means that the formula given in the OP from Peebles is not exactly correct for the cases being considered, since I think that formula assumes that everything is evaluated at the same FRW coordinate time.) In fact, if we call the two velocities $v_r$ (relative velocity of test object at the end of the orthogonal spacelike geodesic with proper length $D$) and $v_p$ (peculiar velocity of test object at relative rest at distance $D$ in constant FRW coordinate time surface), we can distinguish [Edit: crossed out part that is not correct; see post #34 below] three cases:

(1) $v_r < v_p$: this indicates accelerating expansion.

(2) $v_r = v_p$: this indicates the Milne universe.

(3) $v_r > v_p$: this indicates decelerating expansion.

In other words, (2) is what you would expect if "recession from initial conditions" were the only effect present. So if you have (1) or (3) instead, you know that spacetime curvature is present.

These three cases also correspond to the three possible effects on a cable (or rod) between the reference object and the test object whose proper length is stipulated to be constant:

(1) Cable is under tension (this is the case described in the OP of this thread);

(2) Cable/rod has zero stress/strain;

(3) Rod (a better term than "cable" for this case) is under compression.

As I understand it, you are using these last three effects to distinguish the cases.

 

[PAllen]

I disagree: "recession from initial conditions" cannot explain either acceleration or deceleration. The only case where "recession from initial conditions" explains the entire motion is the Milne universe, since that is just flat Minkowski spacetime in disguise and there is zero spacetime curvature.

What I mean here is that in the proposal of using an extensible tape measure, the initial motion of the tape measure end using comoving observers has nothing to do with curvature. I agree that evolution over time does give curvature information. However, it seems much cleaner to me to eliminate the initial conditions issue, and start with zero motion for the tape measure connected to an appropriately selected inertial test object. Then, accelerated expansion implies the tape measure will start extending, decelerating expansion implies it will 'retract' (even if the expansion itself never ceases - this is a possible case), while the Milne edge case will have the tape measure neither extend nor retract.

I do understand the general scenario you want to set up: you have a reference object which is comoving, you take a spacelike geodesic orthogonal to its worldline and extend it for some fixed proper distance, and you put a test object at the other end. You are comparing two possible choices for what state of motion to put the test object in: "comoving" vs. "orthogonal". So consider the following:

The spacelike geodesic in question does not lie in a single spacelike hypersurface of constant comoving time (i.e., a surface of constant FRW coordinate time). In fact, the FRW coordinate time at the test object end of the spacelike geodesic is earlier than the FRW coordinate time at the reference object end.

I'm well aware of this. In fact, I was considering giving another view of why the tether scenario becomes impossible for sufficiently large distances in an accelerating expansion case. The issue is that there exist events such that all 4-orthogonal geodesics from the distant comoving world line intersect the initial 'big bang surface' without reaching such an event, even out to infinite proper time along the comoving world line. In other terms, there are regions in the causal future of the entire Fermi-Normal coordinate patch constructed from a comoving world line - when you have accelerated expansion (and only then). [For decelerated expansion there is no problem with global coverage by FN coordinates, and also obviously not for the Milne case].

This means that, if we consider the relative velocity of the two possible test objects at the other end of the spacelike geodesic, the "comoving" and "orthogonal" ones, it will not necessarily be the same as the peculiar velocity of a test object at rest relative to the reference object at the same proper distance in a spacelike hypersurface of constant FRW coordinate time. (Btw, I think this means that the formula given in the OP from Peebles is not exactly correct for the cases being considered, since I think that formula assumes that everything is evaluated at the same FRW coordinate time.)

I agree and fully understood this. However, given my focus on the tether scenario, I was interested in pointing out the huge relative velocity between a free fall test object that initially does not 'extend or retract a tape measure' and a colocated comoving observer. I would certainly call this peculiar velocity even though it is at an earlier cosmological time than other end of the tether.

In fact, if we call the two velocities $v_r$ (relative velocity of test object at the end of the orthogonal spacelike geodesic with proper length $D$) and $v_p$ (peculiar velocity of test object at relative rest at distance $D$ in constant FRW coordinate time surface), we can distinguish three cases:

(1) $v_r < v_p$: this indicates accelerating expansion.

(2) $v_r = v_p$: this indicates the Milne universe.

(3) $v_r > v_p$: this indicates decelerating expansion.

I think it may be more complicated than this. Consider the Milne case. Consider the reference comoving observer to be the t axis of a standard Minkowski coordinate system (by symmetry, there is no loss of generality in this assumption). Pick some event on the t axis. If I understand your $v_r$, you would take a horizontal line in these coodinates some distance D from this event, and compute the relative velocity between a vertical world line at the end, and a world line from the origin to this same end event.

Meanwhile, for $v_p$, you would extent a family of hyperbolas from the t axis, and consider the relative velocity of the the locus of constant D measured along these versus a world line from the end directly to the origin - with this comparison being made at the event D along a hyperbola from the same t axis event as the prior paragraph.

A priori, I see no reason for your claim of equality between these. I haven't calculated it, but I certainly don't see any reason for your (2) above to be true for this case.

These three cases also correspond to the three possible effects on a cable (or rod) between the reference object and the test object whose proper length is stipulated to be constant:

(1) Cable is under tension (this is the case described in the OP of this thread);

(2) Cable/rod has zero stress/strain;

(3) Rod (a better term than "cable" for this case) is under compression.

As I understand it, you are using these last three effects to distinguish the cases.

Yes, this last is the way I think is most useful for understanding the tethered galaxy problem

 

[PAllen]

Since @PeterDonis has brought the following issue up, it is worth discussion:

Why use a spacelike geodesic 4-orthogonal to a world line as a model for rigid rod rather than a geodesic of the submanifold of a surface of constant cosmological time (with metric induced from the overall manifold)? Note these are not the same, even in the Milne special case. I don't know of any FLRW case where they are the same.

The reason is that it is a known theorem that any Born-rigid congruence is also a Ferm-Normal congruence (the converse is not generally true). Meanwhile, the second option above violates local rigidity.

 

[PeterDonis]

I mean here is that in the proposal of using an extensible tape measure, the initial motion of the tape measure end using comoving observers has nothing to do with curvature.

Actually, it does, because if you define the initial motion as "the motion of a comoving observer at the far end of the spacelike geodesic", then exactly what that motion is, if we hold the proper distance along the spacelike geodesic fixed, will depend on which universe you are in. More precisely, the spacetime angle between the spacelike geodesic and the initial 4-velocity of the test object, if the test object is comoving, will be different for the three cases I described.

it seems much cleaner to me to eliminate the initial conditions issue, and start with zero motion for the tape measure connected to an appropriately selected inertial test object.

I don't necessarily disagree with this, and it does correspond better with the original scenario in the OP.

there exist events such that all 4-orthogonal geodesics from the distant comoving world line intersect the initial 'big bang surface' without reaching such an event, even out to infinite proper time along the comoving world line.

For the accelerated case, yes, agreed.

If I understand your $v_r$, you would take a horizontal line in these coodinates some distance D from this event, and compute the relative velocity between a vertical world line at the end, and a world line from the origin to this same end event.

Yes.

for $v_p$, you would extent a family of hyperbolas from the t axis, and consider the relative velocity of the the locus of constant D measured along these versus a world line from the end directly to the origin

I'm not sure I understand, so let me describe $v_p$ from scratch in my own words. Take the hyperbola that meets the reference object's worldline at the same reference event where the spacelike geodesic (horizontal line) meets it. Find the event on that hyperbola at arc length $D$ from the reference event. $v_p$ is then the relative velocity of a worldline from the origin to that $D$ event, as compared to a vertical worldline at the same event.

I see now, though, that your suspicion that we might not have $v_r = v_p$ for the Milne case is correct (if $T$ is the comoving proper time of the reference event, then we will have $v_r = D / T$ but $v_p = \tanh D / T$), so that particular trichotomy that I gave won't work. So we would need to use the tension/zero stress/compression trichotomy instead.

 

[Jorrie]

This is an interesting question. The key thing is energy density, and what we need to compare is the energy density of dark energy, since that is what is causing the strain, and the energy density of the cable itself. (Since the OP has specified the weak field case, the strain in the cable should be small compared to the cable's energy density, so the latter is what we need to estimate.)If we are in the weak field regime, then, as above, the strain in the cable must be small compared to its energy density, so you can't neglect the mass of the cable.

After this reply, the thread went into a "highly A" level, so I thought about a way to simplify it somewhat. Is it possible to do away with the cable and all its complexities? Perhaps.

If we (one day) have the technology to put two labs, each of non-negligible mass M, somewhere in the intergalactic void, separated by a proper distance D, is there a 'critical distance' D_critical, call it D_c for brevity, where the gravitational acceleration and the cosmological acceleration cancel each other?

In the weak field limit (at least), the answer is yes. They will each create an attractive acceleration of (geometric) magnitude $\frac{M}{(0.5D)^2}$ in the frame of their common center of gravity, for a total of $\frac{0.5M}{D^2}$ when added together. This is to be compatible with the measurement of the OP's strain gauge at the midpoint.

So we can find D_c by equating this acceleration to the OP's CTF, i.e.
$$\frac{0.5M}{D^2} = D H_0^2 \left(\Omega_{\Lambda} - \Omega_M/2\right)$$
and I found
$$D_c = \left(\frac{M}{H_0^2} (\Omega_{\Lambda}-\Omega_M/2)\right)^{1/3}$$

If this is correct for the present time, does it give us a potential way to measure the Cosmic Tidal Force (CTF)? That is, by precisely measuring that they stay at the constant distance D_c for long enough?