Do Cosmological Tidal Forces Prove the Expansion of Space?

[jonmtkisco]

The team of Francis, Barnes, James & Lewis have published several very helpful papers in recent years about the expansion of space (with their names in various orders.) I suggested in a recent thread about their Radar Ranging article that they are trending away from explaining particle behaviors as resulting from "the expansion of space," toward explaining such behaviors as being motivated solely by gravity. They're still not admitting that they're all the way there, but I expect they will eventually, at a pace sufficiently measured to prevent their cosmology bretheren and sisteren from rejecting them out of hand as radicals.

In any event, in light of the evolution in their papers, it's not surprising that their earlier papers may lean a little too far towards rationalizing why "space itself" is expanding. Still, I am disappointed by one section in their 7/07 http://arxiv.org/abs/0707.0380" "Expanding space: root of all evil?" in which they described a particular particle behavior as "cosmological tidal forces". I believe that their expanation and terminology on this subject (sec. 2.6.2) are misleading and confusing, perhaps in an effort to bend over backwards to justify that space itself is expanding. Here are pertinent quotes:

Consider an object of many particles with no internal forces. It is shot away from the origin $$( \chi = 0)$$ with speed $$v_{0}$$, the first particle leaving at time $$t_{0}$$ and the last at $$t_{0} + \Delta t_{0}$$. The length of the object is $$l_{0} = v_{0} \Delta t_{0}$$. The object travels to an observer in the Hubble flow at $$\chi$$, who measures its speed relative to him $$(v_{f})$$ and the time of arrival of the first $$(t_{f})$$ and last particle $$(t_{f} + \Delta t_{f})$$ in order to measure its length $$( l_{f} = v_{f} \Delta t_{f})$$.

... (equations) ...

Then, following the method of Barnes et al. (2006) to calculate $$v_{f} = \dot{\chi} (t_{f}) R (t_{f})$$ and substituting for C we have that

$$\frac{l_{f}}{l_{0}} = \frac{v_{f} \Delta t_{f}}{v_{0} \Delta t_{0}} = \frac{R(t_{f}) }{R(t_{0}) }$$

Hence, the length of the object has increased in proportion with the scale factor. This result answers the question: what if an object had no internal forces, leaving it at the mercy of expanding space? This is a rather strange object — it would very quickly be disrupted by the forces of everyday life. Nevertheless, it is a useful thought experiment. The above result shows that the object, being subject only to expanding space, has been stretched in proportion with the scale factor. These are essentially cosmological tidal forces.

We therefore have clear, unambiguous conditions that determine whether an object will be stretched by the expansion of space. Objects will not expand with the universe when there are sufficient internal forces to maintain the dimensions of the object.

Clearly the only reason the particle string "expands" lengthwise is that it is being measured by an observer who is comoving with the Hubble flow. The comoving observer is moving away from the coordinate origin in terms of proper distance; so naturally he observers an increasing time interval between the passage of each particle and the subsequent particle in the stream.

As far as I can deduce, the proper distance between the particles in the stream does not change at all as a function of time, regardless of the expansion of the universe. How could it? This then is simply another example of a coordinate system-dependent calculation which does not reflect an underlying coordinate-independent physical reality. I strongly suspect that if the authors could do a "take-back", they would now retract this example, or at least significantly change their description of what it means

I think it is important to re-explain this sort of example, because I think the correct answer helps to illustrate two important points: (1) there is no need to consider space itself to be expanding in order to explain peculiar particle motions, and (2) peculiar particle motion in a flat, homogeneous dust-filled universe (per Gauss' law) demonstrates clearly that "spacetime curvature" exists even in the absence of any tidal forces. By definition, there are no differential gradients in the matter distribution, and therefore in the gravitational force, in such a universe. Tidal forces may = spacetime curvature, but a more complete statement is that spacetime curvature = tidal forces + nontidal gravitational forces.

Jon

 

[Chronos]

What observer is not comoving with the Hubble flow?

 

[jonmtkisco]

Hi Chronos,

What observer is not comoving with the Hubble flow?

Your point is valid as far as it goes... but the recent papers on this subject make it clear that many misconceptions about the expansion of space have arisen primarily because of the overuse of a single coordinate system, comoving coordinates. Examining non-Hubble flow particle motions in terms of proper motion or conformally flat coordinates helps to clarify which conclusions are coordinate-dependent and which are invariant.

Jon

 

[jonmtkisco]

As far as I can deduce, the proper distance between the particles in the stream does not change at all as a function of time, regardless of the expansion of the universe.

Well, my point about the Francis & Barnes explanation is valid, but I made a misstatement. The particle string they define is just another variant on the Tethered Galaxy problem. As such, the question of whether the proper length of the particle string will expand or contract depends on the cosmological model. If the universe has a cosmological constant, the proper length of the particle string will expand and accelerate, and will asymptotically approach but never reach the Hubble expansion rate. On the other hand if Lambda=0 and the universe is at critical density, the particle string will contract.

If we consider a ball of equally spaced particles rather than a string, in the Lambda=0 universe the ball will collapse to a single point (located at the point on the ball's original surface which is closest to the coordinate origin) and then if the particles narrowly avoid colliding, they will pass through the convergence point and move away in the opposite direction at a proper velocity that declines slowly over time - thus creating an ever-expanding ball.

I suppose it's reasonable to refer to this latter behavior as "cosmic tidal forces". On further consideration, I think that ANY scenario involving two or more separated particles will experience tidal forces, whether their origin is cosmic or due to a discrete massive body.

Consider another example, a ball of equally spaced particles placed in circular orbit around a non-rotating planet. If each particle is initially given the correct angular momentum for a stable orbit, the angular momenta of the particles will vary depending on their individual radial distance from the planet. As a result, the ball will experience rotational shear: the particles located further from the planet will lag behind the particles closer to the planet, in terms of orbital motion, resulting at a smearing-out of the ball. This is of course because the orbital period lengthens as the radius of the orbit increases. When there are multiple particles at the same orbital radius, they will not contract together in the direction of orbital motion, but I think they will collapse together in the direction transverse to the orbital motion and parallel to the planet's surface. This behavior again is an example of tidal forces.

So in it may be accurate to say that spacetime curvature is always characterized by some tidal force; it is absolutely unavoidable if there are two or more separated particles. This is because a uniform gravitational field is physically impossible. If there were such a thing as a uniform gravitational field, it would not have tidal forces. Even a homogeneous, expanding dust cloud has gravitational gradients as calculated by Gauss' Law or the Newtonian Shell Thereom, even though no directional matter gradient whatsoever occurs in the dust cloud.

But in any event I do not think that tidal force can meaningfully be considered to be the essence of spacetime curvature. When I stand outside and throw a ball straight up in the air, it will experience dramatic spacetime curvature effects (it will return to me) but at this scale the tidal force makes no significant contribution to the ball's motion.

Jon

 

[chronon]

I think that you're right to be disappointed with this section of their paper. If you think of the particles as being at the wave crests of an electromagnetic wave then they show that the wave will be redshifted according to another observer - they've discovered the Doppler effect!

 

[Wallace]

If you read that section in context you will see that this in essence the point being made. The question that people ask is why does the expansion of space stretch photons but not atoms (for instance). This re-derivation of the cosmological doppler shift demonstrates the reasons for this by seeing what it would take to get an effective cosmological doppler shift in a stream of massive particles. Here and elsewhere in that paper it is shown that Hubble law initial conditions are required in order to observe 'the expansion of space', demonstrating how this is not a physical law, just a phenomenological description.

That paper made a very specific definition of expanding space and all references to the term should consider that definition, not whatever idea of this the reader has to start with. I'm not how this can be described as 'bending over backwards' to defend expanding space. The definition of expanding space outlined in that paper makes it clear that it is not a physical law being discussed.

 

[jonmtkisco]

Hi Wallace,

Here and elsewhere in that paper it is shown that Hubble law initial conditions are required in order to observe 'the expansion of space', demonstrating how this is not a physical law, just a phenomenological description.

I think I've made it clear that I think this paper, along with the other papers by the same team, make very insightful and important contributions to understanding particle motions and the expansion of space. In fact I think they've made a huge contribution to the cosmological community. There's no reason for anyone to get defensive about a critique of one aspect of one section of one paper.

There is an "apparent" inconsistency between this section and a statement in Section 2.2 of the paper which says that John Peacock's bedroom would expand locally along with the expansion of space only if a number of conditions are met, including the condition that "the particles making up the wall were at rest with the cosmological fluid which, importantly requires that they not be initially at rest with respect to one another..." I agree with that condition.

Contrast that with the example of the particle string shot away from the origin in Section 2.6.2, where by all indications the particles are indeed at proper rest with respect to one another immediately after emission. According to the requirements of Section 2.2, the particle string therefore should not expand locally, yet as I quoted, Section 2.6.2 says:

The above result shows that the object, being subject only to expanding space, has been stretched in proportion with the scale factor. ... We therefore have clear, unambiguous conditions that determine whether an object will be stretched by the expansion of space. Objects will not expand with the universe when there are sufficient internal forces to maintain the dimensions of the object.

I think it's fair to say that leaving the apparent conflict with Section 2.2 unresolved makes this section confusing and could potentially lead a reader astray.

Jon

 

[Wallace]

If the conflict you suggest was real then that section would have to be in error mathematically. It was demonstrated that the initial conditions for this strange 'object' result in the redshift like behavior. These initial conditions are quite different than those for the walls of John Peacock's bedroom, were its constituent material to suddenly lose all internal forces.

The pair of traveling particles do not measure 'local' expansion, the comparison is made between times in the emitted and observed frame over cosmological distances. This is a different question to asking what happens in a frame co-moving with one particle or the other.

 

[jonmtkisco]

Hi Wallace,
When you explain what the authors had in mind, the point of that section becomes a little clearer. Thank you for the insight.

The term "tidal forces" normally applies to gravitational effects. At the very least, it is confusing and potentially misleading to apply that terminology to describe merely that the observed frame is drifting with the Hubble flow, away from the emitted frame.

Jon

 

[jonmtkisco]

Of course one advantage of the two-particle scenario modeled in the Francis, Barnes paper (compared to a conventional scenario involving recessionary redshift of light) is that radar ranging could be used (with appropriate interpretational adjustments) by the comoving observer to measure that the proper distance between the two particles does NOT expand with the Hubble Scale factor. The observer could then conclude that the Doppler Effect is the sole cause of the observed time lag in the arrival of the second particle. This would seemingly allow the observer to unambiguously rule out any possibility that the expansion of "empty space itself" causes the proper distance between particles detached from the Hubble flow to increase with the scale factor.

This statement excludes any effects correlating to the deceleration or acceleration of the background expansion rate of the cosmic fluid, which as I noted earlier would in fact be associated with a contraction or expansion, respectively, of the proper distance between the two particles. As shown by the Tethered Galaxy problem.

Jon