Much has been written in the debate about whether the space between galaxies is "really" expanding, or whether on the other hand the motion of galaxies is a kinematic recessional motion through space. The equivalence principle strongly suggests that it is impossible to detect any observational difference between the two models, indicating that we have no reasonable alternative but to treat both as being equally valid and mathematically interchangeable. This point was made in a very helpful 2007 http://arxiv.org/abs/0707.0380v1" [Broken] "Expanding Space: the Root of all Evil?" by Francis, Barnes, James & Lewis. Unfortunately though, the conclusion of the paper is murky and tepid: "This description of the cosmic expansion should be considered a teaching and conceptual aid, rather than a physical theory with an attendant clutch of physical predictions." The paper warns us not to be deceived by the simplistic balloon analogy, even though the analogy of an ant marching on the surface of an inflating balloon works well. And the paper includes one glaring erroneous conclusion. The tethered galaxy problem The tethered galaxy thought problem is most frequently used to explore particle behavior in an expanding universe. A (massless) test galaxy is held at rest (tethered) with respect to the coordinate origin at a cosmological distance, meaning that its proper velocity toward the origin is zero. Then the tether is released, and we calculate whether the galaxy moves toward the origin or away from it. In a matter-only (Lambda=0) homogeneous dust universe at critical density ([tex]\Omega=1[/tex]), the galaxy accelerates rapidly toward the origin, passes through it, and asymptotically regains its initial comoving radius on the opposite side of the sky (all as measured in proper distance). After the galaxy passes the origin, its proper velocity decelerates very slightly over time, and its peculiar velocity (velocity relative to the local Hubble rate) decays over time at the well known rate of 1/a, where a is the dimensionless scale factor of the universe. In an influential http://arxiv.org/abs/0809.4573v1" [Broken] on this subject first published in 2001, Prof. Peacock describes this result as a "paradox." He says that the 'expanding space' idea would suggest that the test particle should recede rather than approach. "The acceleration is negative, so the particle moves inwards, in complete apparent contradiction to our 'expanding space conclusion that the particle would tend with time to pick up the Hubble expansion.... In no sense, therefore can 'expanding space' be said to have operated.... The behavior can be understood quantitatively using only Newtonian dynamics.... This analysis demonstrates that there is no local effect on particle dynamics from the global expansion of the universe: the tendency to separate is a kinematic initial condition, and once this is removed, all memory of the expansion is lost." Peacock's hand waving conclusion that the untethered galaxy's motion cannot be explained by the 'expanding space' model is flat out wrong. A 2001 paper by Lineweaver & Davis explains that the paradox can be sorted out by considering the motion from the galaxy's rest frame. The "Root of all Evil" paper builds on the Lineweaver & Davis explanation: "In this frame, we see the original observer moving at vred,o and the particle shot out of the local Hubble frame at vpec,o, so that the scenario resembles a race. Since their velocities are initially equal, the winner of the race is decided by how these velocities change with time. In a decelerating universe, the recession velocity of the original observer decreases, handing victory to the test particle, which catches up with the observer. ... The original observer should view the initial conditions of the test particle, not as neutral, but as a battle between motion through space and the expansion of space. The expansion of space has been momentarily nullified by the initial conditions, so we must ask how the expansion of space changes with time. We contend that this explanation successfully incorporates test particle motion into the concept of expanding space. In particular, it shows why it is wrong to expect, on the basis of the balloon analogy, that expanding space would carry the particle away." If the authors intend to dismiss the inflating balloon as an analogy for the 'expanding space' model, they are unjustified in doing so. The balloon model provides an excellent analogy for 'expanding space' if it is interpreted correctly. I will expand a bit on the analysis later in this post. It is easiest to sort out the recessionary motion of galaxies by considering it to result from the combination of 3 independent vectors. The original expansion resulting from the Big Bang (or inflation) is a residual velocity vector (the first derivative of proper distance). A velocity vector retains kinetic energy (momentum) from the initial conditions, but it is not a "force". A velocity vector can be exactly offset by a matching peculiar velocity vector in the opposite direction. On the other hand, the deceleration of the expansion, caused by the self-gravitation of the mass-energy contents of the universe, is an acceleration vector (the second derivative of proper distance). It acts as a "force" and decelerates the relative recession velocities of particles (in a kinematic model) or alternatively decelerates the expansion velocity vector (in an 'expanding space' model). An acceleration vector obviously cannot be exactly offset by a velocity vector (except potentially asymptotically in the infinite future), because the velocity imparted by the acceleration vector progressively changes. The third vector is Lambda (dark energy, etc.) which is also an acceleration vector. I will set Lambda aside for this discussion, leaving us to deal with one velocity vector and one acceleration vector. Now let's visualize how the untethered galaxy moves toward the origin. The inward peculiar velocity imparted to the galaxy by the tether is exactly equal to, and in the opposite direction of, the local Hubble flow relative to the origin. It is not meaningful to say that the local "Hubble expansion" is "nullified" or "forgotten" because of the peculiar motion. Rather, the galaxy's initial peculiar momentum has tossed it "upstream against the Hubble flow" at a rate that keeps it exactly stationary relative to the origin. Then, as the expansion rate decreases over time due to the gravitational deceleration vector, the Hubble flow falters and begins "pushing" the galaxy away "downstream" at a lower velocity than the conserved momentum of the galaxy's initial peculiar velocity continues to carry it "upstream." One could say that the galaxy accelerates toward the origin. Or one could just as well say that the origin accelerates toward the galaxy. It is accurate to say that everything accelerates toward everything else, in all directions. It may be helpful to think of a set of comoving "gravity coordinates" where particles remain at rest in the inward moving gravitational flow even as their proper distance coordinates converge. We can analogize the galaxy to a motorized baggage cart which initially is winched onto the "downstream" end of an airport moving sidewalk, facing "the wrong direction", ie facing against the sidewalk's motion. The moving sidewalk's velocity is 1 m/s, and upon untethering the baggage cart's motor tries to move the cart "upstream" against the moving sidewalk's motion also at 1 m/s. Obviously the cart will remain motionless on the moving sidewalk, with its wheels turning. Now if we progressively turn down the speed of the moving sidewalk, the cart will begin to move forward, at increasing speed, and eventually exit the upstream end of the moving sidewalk at a velocity approaching 1 m/s. Did the cart accelerate toward the upstream end of the moving sidewalk (let's ignore the force required for the cart to overcome friction), or did the sidewalk decelerate its own motion? In this case we know the answer, but in cosmology we have no stationary reference frame to help us judge. [Edit:] You may have had the same experience on a treadmill. If you walk at a constant velocity, then if you decrease the treadmill's speed, you will move forward (upstream). If you increase the treadmill's speed, you will move backward (downstream.) The expanding balloon or rubber sheet analogy In another http://arxiv.org/abs/astro-ph/0609271v1" [Broken] Barnes, Frances, James & Lewis introduce another analogy: "A popular way of visualizing expanding space is a balloon or a large rubber sheet. Imagine yourself and a friend at rest on a large rubber sheet. We cannot directly observe spacetime, so we will do this thought experiment in the dark. Suppose you both observe a glowing ball moving away from you. 'The rubber sheet is being stretched,' you say. 'No it’s not,' replies your friend, 'the sheet is still and the ball is rolling away.' Together, you come up with an ingenious way of finding out who is right. You take another glowing ball, and drop it onto the sheet a certain distance away. If the sheet is expanding, then we expect it to carry the ball away; if the sheet is still then the recession of the first ball was due to a kinematical initial condition. Once this is removed, so is the recession." We are now in a position to see clearly why the dichotomy put forward by the authors is false. Even if the 2nd glowing ball is dropped on a stretching rubber sheet, it will not be carried away by the expansion. Instead, we will see the ball remain stationary in the observers' (presumably stationary) ground frame (of course we assume zero friction). If transverse stripes are painted on the rubber surface, we will notice that the ball will continue rolling "backward" across the stripes in the "upstream" direction, although the ball's position doesn't change in the observers' frame. This happens because, in the rubber sheet's frame, the rubber sheet is stationary and the ball lands on it with a relative or peculiar motion in the "upstream" direction. If the rubber sheet's rate of stretching is progressively decelerated, then when viewed from the ground frame the ball will be seen to begin "accelerating" toward the observers. We know that this is really a fictional "pseudo-acceleration" because we know that it is the stretching that is slowing. Let's examine the analogy of an ant marching at constant local velocity across the surface of an inflating balloon. First we mark an arbitrary origin point on the balloon. Then we place the ant at a distance away from (and facing) the origin, such that the ant's constant marching speed (say 1 mm/s) is exactly equal to the rate at which the ant's location on the balloon surface is comoving away from the origin. As in the other analogies, while the balloon continues inflating at a constant rate, the ant's proper distance from the origin will not change. But then if we begin slowing the rate at which the balloon is inflated, the ant's proper distance from the origin will begin to decrease. Eventually the ant will march right over the origin and speed away in the opposite direction, at a peculiar velocity approaching the full 1 mm/s. So we see that the ant on the balloon is a very good 2-dimensional analogy for particle motion in 3-dimensional 'expanding space.' Now we can also recognize the error in the "Root of all Evil" paper (which Ich pointed out). In section 2.6.2 the authors describe two particles shot away from the origin at the same speed, in expanding space, with a time interval between their firing. The authors claim to prove mathematically that the distance between the two particles will increase over time in proportion with the expansion of the universe's scale factor. They refer to this as "essentially cosmological tidal forces." But in an [tex]\Omega=1[/tex] universe, if we consider the first particle to be the coordinate origin (ie at rest in its own frame), then upon firing the second particle will have zero proper velocity toward the first particle, and the ant and balloon analogy helpfully instructs us that, as the Hubble rate decelerates thereafter, the second particle will approach the first particle, eventually pass it, and move progressively further and further ahead of the first particle. So the two-particle string will initially shrink rather than stretch, turn itself inside out, and then stretch indefinitely. It's nice that the lowly ant and balloon analogy works correctly in a situation where the math of professional cosmologists fails! (However, please don't take away a message that math is dispensable. Not at all.) We can also address Peacock's claim that once a tethered galaxy is "removed from the Hubble flow" it will "forget" the Hubble flow. The fact that a galaxy has a peculiar velocity does not mean it has been removed from the Hubble flow. (If it did, then the Hubble flow would be a useless concept, since virtually every galaxy has some peculiar motion.) Imagine that once the tethered galaxy is untethered, we use rockets to immediately re-accelerate the galaxy radially away from the origin. We turn the rockets off at the point where the galaxy's proper velocity away from the origin is exactly equal to the local Hubble rate relative to the origin. Thus we have restored the galaxy to being "at rest" in the local Hubble flow (albeit at a different location than it began). The galaxy has now been "retaught" the Hubble flow it had forgotten, and will henceforth behave exactly like any galaxy that has remained undisturbed in the Hubble flow. This illustrates the natural equivalence of kinematic motion with the expansion of the universe. Why isn't Brooklyn expanding? [EDIT: On further consideration I have changed this section to align it more closely with the Barnes Lewis analysis.] Often the question is asked, why doesn't 'expanding space' make local structures (like Brooklyn) expand. There is debate in the literature about whether gravity and the electromagnetic internal forces of matter dynamically compete with the underlying expansion and effectively put it on hold. Peacock says yes, and the Barnes Lewis team says no. The latter says that local structures don't expand because the static Schwarzschild metric applies locally instead of the expanding FRW metric. That answer is a bit cryptic. Isn't the important question why a static metric applies in a region of expanding space, regardless of whether the density of that region happens to be homogeneous with the cosmic density? I believe the Barnes Lewis approach is fundamentally correct however. We consider that at the global level the gravitational acceleration vector progressively causes enduring change to the expansionary velocity vector. Simply put, in an [tex]\Omega=1[/tex] universe the global velocity vector decreases as a function of time. Consistent with that dynamic, it makes sense to say that a gravitational acceleration vector which varies from location to location around the universe (due to local overdensity or underdensity) also causes enduring differential changes to the local expansion velocity vector. Thus at a point in time the local expansion velocity vector may have decreased less than the global average (e.g. for a void), or more than the average (e.g. for a supercluster). Galaxies can be thought of as locations where the local expansion velocity vector has stabilized at zero. Operationally this local variance creates the same dynamic within a galaxy as if the local expansion velocity vector remained positive and the particles comprising it had attained a zero inward proper velocity. Even if we think of the latter picture as being valid, the ant and ballon analogy instructs us that Brooklyn would not stretch. Again, this result would occur because, when the proper velocity of the local Hubble expansion rate has become exactly matched by an inward peculiar velocity of each particle in the structure, the positive expansion velocity vector must be disregarded. So either way, Brooklyn isn't expanding. On the other hand, our comoving gravitational coordinate system tells us that when particles have zero proper velocity relative to each other, gravity causes the comoving gravitational coordinate of every particle to converge toward (and eventually through) every other particle. Brooklyn's particles begin with relative proper velocities of zero, so why don't they all collapse toward each other in such a universe, turning Brooklyn inside out? [A comedian will say that already happened!] Of course the answer is that the internal electromagnetic forces of Brooklyn's matter are sufficiently strong to entirely prevent gravitational collapse (in a kinematic model using proper distance coordinates), or to impart an outward peculiar acceleration (in the 'expanding space' model using comoving gravitational coordinates). We should think the same way about the orbits of massive bodies. Why doesn't gravity cause an orbiting planet's coordinate location to eventually converge with the sun's coordinate location? Because the orbiting planet retains sufficient angular momentum to overcome the gravity of the orbital system. (In other words, the planet-sun system is fully virialized.) Or because there is sufficient angular momentum to push the planet outward (in comoving gravitational coordinates.) Remaining questions [I have edited this section] I'll conclude by noting that while the equivalence principle suggests that 'expanding space' and 'kinetic recession' are merely different descriptions of the same thing, that doesn't mean we have sorted out all of the complexities. In particular, no complete and demonstrably valid explanation has been provided as to how the cosmological redshift and the Superluminal recession of distant galaxies can be explained by a purely kinetic model. However, the equivalence principle encourages optimism that eventually the kinematic model will be fleshed out enough to provide a fully kinematic alternative explanation of those phenomena. On the other hand, we may eventually conclude that a fully kinematic explanation would result in measurable observational differences. That remains a real possibility, despite the faith we put in the equivalence principle. In which case, we will eventually be able to know for a fact whether 'expanding space' causes particles to separate, or whether the particles are moving kinematically through space.