Does Space Expand?

There seems to be something in the air these days that is making people speak out against the idea that space is expanding. For evidence, check out these recent papers:

The kinematic origin of the cosmological redshift
Emory F. Bunn, David W. Hogg

A diatribe on expanding space
J.A. Peacock

Expanding Space: the Root of all Evil?
Matthew J. Francis, Luke A. Barnes, J. Berian James, Geraint F. Lewis

Admittedly, my first sentence is unfair. The correct thing way to paraphrase the underlying argument here is to say that “space is expanding” is not the right way to think about certain observable properties of particles in general-relativistic cosmologies. These aren’t crackpots arguing against the Big Bang; these are real scientists attacking the Does the Earth move around the Sun? problem. I.e., they are asking whether these are the right words to be attaching to certain indisputable features of a particular theory.

Respectable scientific theories are phrased as formal systems, usually in terms of equations. But most of us don’t think in equations, we think in words and/or pictures. This is true not only for non-specialists interested in science, but for scientists themselves; we’re not happy to just write down the equations, we want sensible ways to think about them. Inevitably, we “translate” the equations into natural-language words. But these translations aren’t the original theory; they are more like an analogy. And analogies tend to break under pressure.

So the respectable cosmologists above are calling into question the invocation of expanding space in certain situations. Bunn and Hogg want to argue against a favorite cosmological talking point, that the cosmological redshift is not an old-fashioned Doppler shift, but a novel feature of general relativity due to the expansion of space. Peacock argues against the notion of expanding space more generally, admitting that while it is occasionally well-defined, it often can be exchanged for ordinary Newtonian kinematics by an appropriate choice of coordinates.

They each have a point. And there are equally valid points for the other side. But it’s not anything to get worked up about. These are not arguments about the theory — everyone agrees on what GR predicts for observables in cosmology. These are only arguments about an analogy, i.e. the translation into English words. For example, the motivation of B&H is to do away with confusions in students caused by the “rubber sheet” analogy for expanding space. Taken too seriously, thinking of space as an expanding rubber sheet convinces students that the galaxy should be expanding, or that Brooklyn should be expanding — and that’s not a prediction of GR, it’s just wrong. In fact, they argue, it is perfectly possible to think of the cosmological redshift as a Doppler shift, and that’s what we should do.

Well, maybe. On the other hand, there is another pernicious mistake that people tend to make: the tendency, quite understandable in Newtonian mechanics, to talk about the relative speed between two far-away objects. Subtracting vectors at distinct points, if you like. In general relativity, you just can’t do that. And realizing that you just can’t do that helps avoid confusions along the lines of “Don’t sufficiently distant galaxies travel faster than light?” And reifying a distinction between the Doppler shift and the cosmological redshift is a good first step toward appreciating that you can’t compare the velocities of two objects that are far away from each other.

The point is, arguments about analogies (and, by extension, the proper words in which to translate some well-accepted scientific phenomenon) are not “right” or “wrong.” The analogies are simply “useful” or “useless,” “helpful” or “misleading.” And which of these categories they fall into may depend on the context. Personally, I think “expanding space” is an extremely useful concept. My universe will keep expanding.

181 Comments

181 thoughts on “Does Space Expand?”

  1. Geraint, lets mess up things even a little bit more… 😀

    And talk about Dark Energy, Expanding Universe and Empty Space.

    How does your paper deal with DE? I can see the reasoning about; “the energy density will not increase with time and bound structures will remain bound and stable”

    But what about the argument on Empty Space, is it really empty? If 73% of the universe is made of DE, it must be somewhere …?

  2. Hi Speedy,

    “A will catch the ball after 5s, when B has been running another 15m

    B will get confused since he was able to run another 15m before A catches the ball, and he should only have been able to run another 6m??”

    You must state clearly whether A and B are aware of who is standing still and who is moving. If both parties have perfect knowledge, then nobody is confused about why B runs another 15m in 5s at 3m/s. And if A thinks he is standing still and B is running away, his calculations will be correct. On the other hand, if B thinks he is standing still, he won’t perceive himself to have moved at all, let alone 15m. He will perceive that A has run a total of 25m, which is the correct answer in his frame of reference.

    “E smiles secretly because he knows that the clocks for A & B are not synchronized…”

    We should not refer to a non-relativistic Doppler Shift as a clock synchronization problem. A’s and B’s clocks are entirely in synch in this example. The classical Doppler Effect is a “pseudo-time dilation”, because it looks like one but it’s not.

    Jon

  3. Hi Speedy,

    “Ehhh, how can the photon “surf” and on what?”

    Please understand that I’m using informal terminology to try to make the explanation as intuitive as I can. In that spirit, imagine a two-dimensional infinite plane expanding universe. As a photon moves radially away from the emitter in its frame of reference, it MUST accelerate to catch up to successively more distant galaxies, because the Hubble rate increases in direct proportional to distance. The photon must pass each successive galaxy at a local speed of exactly c. (Think of the receding galaxies as the waves being surfed, ok maybe it’s a weak analogy.) If you think of the distribution of galaxies being extremely granular, then the acceleration (relative to the emitter) is smooth and continuous. After a moderate amount of cosmic time has passed (e.g., well before the scale factor doubles), the photon will start decelerating (relative to the emitter) because the cosmic gravitational deceleration starts to exceed the rate of increase in the Hubble rate (relative to the emitter).

    “And doesn’t the “Hubble flow” have a “slowing down effect” all the way from start up to my nose??”

    Yes, but there are two separate effects at work here: (1) the Hubble rate which increases with distance, requiring the photon to accelerate, and (2) the gravitational contraction, which slows the Hubble rate as a function of time.

    I agree with your implication that there is an interesting dilemma here. Surely all changes in the momentum and speed of the photon should result directly from gravitational force. So how is it that the FLRW metric causes the photon first to accelerate rapidly, and then to decelerate more slowly (in the emitter’s frame)? At this point I can’t give you a crisp answer.

    Gauss’s Law seems to give a gravitational “free ride” to the photon when calculated from the observer’s perspective. That is, the collapse action of the cosmic gravitation imparts a blueshift (like an accelerating moving sidewalk) to the photon, which exactly offsets and compensates for its classical Doppler redshift. The photon does not need to “expend” any energy/momentum of its own in order to accelerate. On the other hand, there is no apparent gravitational source to “boost” the photon up to the increasing recession velocity of the Hubble flow, as a function of distance from the emitter.

    I suppose a theoretical answer is that, if the photon is “required” to accelerate to keep pace with the Hubble flow, then it must surrender a corresponding amount of energy/momentum, relative to both the emitter and the observer. That loss of energy/momentum then is manifested as the cosmological redshift. While this thought seems somewhat tidy, I am quite bothered by the concept that the photon can spontaneously accelerate its speed, without any gravitational boost, merely because the higher speed is “required” in order to pass each successive local galaxy at exactly c. And I’m not ready to accept the concept that there is some GR “frame dragging” effect at work here, with the receding motion of the galaxies itself applying a force directly onto the photon. And as I’ve said, I don’t think SR time dilation provides a useable answer.

    Well, it’s all food for more thought.

    Jon

  4. >How does your paper deal with DE?

    In GR, you have fluids that describe the energy density – and dark energy is added by putting in a fluid of a particular equation of state.

    NOWHERE in relativity does it say dark energy is a property of space – that is only achieved with some quantum-mechanically hand waving.

    Jon – I’ll stick with E=-p.u

  5. Hi Geraint,

    E=-p.u is “merely” a mathematical equation, not a physical description of what’s happening. This equation is indispensible, but alone I don’t see how it gives us a robust understanding of the complexities of cosmological redshift.

    I want to understand how a long series of infintesimal SR time dilations can accumulate as a photon travels the cosmological distances (z>1) between Galaxy A and Galaxy B, without violating the FLRW constraint that all clocks in the Hubble flow remain synchronized.

    I want to understand how cosmic gravitational time dilation can explain cosmological redshift if the emitting and receiving galaxy clocks remain in synch.

    I want to understand why cosmological redshift is PRECISELY equal to the ratio between the distance of a galaxy at the time of emission and at the time of reception of a photon. Surely a convoluted integration of SR and gravitational time dilation in a universe with or without expansive dark energy thrown into the mix would not be expected to serendipitously generate such a profoundly simple relationship.

    I want to understand why we calculate a photon to reposition itself exactly (z+1) times as far away when viewed from the emitter’s frame as when viwed from the observer’s frame, yet the parties in both frames calculate the same elapsed travel time.

    I want to understand how cosmic gravitation can explain how a photon emitted from Earth will accelerate (in the sense of repositioning itself a relatively longer distance away in a short time interval) and then decelerate (reposition itself a a relatively shorter distance away in a longer time interval), as viewed from our frame of reference.

    And, as something of a diversion, I want to understand your comment in an earlier post that photons are only exchanged between particles, not shot blindly into empty space. When the CMB surface of last scattering set some photons on a path roughly toward where our solar system would someday form, how did the source particle “anticipate” that the photon would eventually hit our orbiting satellite which we wouldn’t decided to launch until gigayears after the photon began its journey? If the Australian space agency cuts the budget before the photons arive, are the photons instantly recalled to home base, or are they permitted to retroactively decide to never have begun the journey at all? (No, because the CMB source particle may subsequently no longer exist or may have relocated.) Are you suggesting instantaneous action or quantum entanglement across cosmological distances? Does the source particle’s energy level change when it first emits the photon, or only later when the photon becomes assured of striking an identified target? If a tree falls in the forest and no one is there, did it make a sound?

    Jon

  6. Lawrence B. Crowell

    These calculations of the comoving velocity for z =~ 1100 by Jon strike me as off the mark. Given the Hubble rule (FRW cosmology etc) that

    1 + z = exp(v/c)

    and for z = 1100 indicates that v/c = ln(1100) = 7, or an apparent velocity of 7c. Things are not moving nearly as fast as is being presumed here.

    Lawrence B. Crowell

  7. Jon Corthell, Thank you very very *MUCH*!!

    In my opinion this comment is the most interesting and intelligent in this post so far. To cover the feeling in my heart in one line:

    “The important thing is not to stop questioning. Curiosity has its own reason for existing.” — Einstein

    Jon, I’m particularly happy/impressed by the line: “E=-p.u is ‘merely’ a mathematical equation, not a physical description of what’s happening.”

    We have a lot of truly intelligent guys here who clearly grasp the math beyond the public’s imagination. But, if you can’t communicate the “product” to the public in plain language, it must be something wrong, or at least a reason to rethink.

    Sometimes I get the feeling that it is more or less “taboo” for physicists to say: “Well guys, we don’t know everything yet, there some really weird things going on in the universe, but we are working on it.”

    It’s obviously safer to say: “Well guys, everything is perfectly clear, we have run the equations and it works, there is nothing to discuss about the fundamental behavior of the universe. It’s a problem though, that you don’t understand the math.”

    Michael S. Turner who coined the term dark energy, elaborate his thoughts in this video (mov 18.4 MB) about where science stands today (June 22, 2003) in the understanding of the universe. In an open and sincere way he comes to the conclusion that “we are a kind of six blind cosmologists and the universe” as an allegory to the Six Blind Men and An Elephant.

    I’m confused…

    Some really heavy Nobel Laureates like Steven Weinberg says: “…how is it possible for space, which is utterly empty, to expand? How can nothing expand? The answer is: space does not expand. Cosmologists sometimes talk about expanding space, but they should know better.” (Quoted in – Expanding Space: the Root of all Evil?)

    Some says that there must be an expanding space to explain the stretching of light wave frequencies from very distant objects in the universe, sending photons in our direction.

    Some says: “There is no speed limit on the universe.”

    Some says: “There is no such thing as expanding faster than the speed of light.”

    Some says: “For the Hubble law this gives a v ~ 6c for the velocity of the material we are observing.”

    Some make logical acrobatics and says: “…there’s no such thing as expanding faster than the speed of light, then it doesn’t make sense for there to be a speed limit on it, does there?”

    Some says: “…clocks tick at a uniform rate for all comoving galaxies in the FLRW metric.”

    Some says: “Comparing velocities at different spatial locations is simply not a valid operation in General Relativity.”

    Michael S. Turner says: “Only knowing everything there is to know about the Universe would be worse than knowing all the questions to ask about it. Without doubt, as our understanding deepens, new questions and new surprises will spring forth.”

    I’m confused…

    (and maybe I’m not the only one)

    And Jon, finally a question to you: “…if the photon is “required” to accelerate to keep pace with the Hubble flow…”

    What framework does the photon use to “calculate” and “adjust” its speed? How does the photon know who is watching (thrower/catcher)? It’s getting even worse, since there is no way of comparing velocities at different spatial locations? Is that little bastard the almighty God??

  8. Lawrence B. Crowell

    Photons don’t accelerate. It is that the null geodesics exist in arcs. In fact if you project the spatial surface into Fermat coordinates, you get a Poincare disk (3-ball) where null geodesics are great arcs on the disk. A look at an Escher print of tessilations on a disk, fish or devils and angels. gives an illustration of this.

    Looking at this according to null geodesics is preferrable to language about expanding spatial surfaces.

    Lawrence B. Crowell

  9. E=-p.u is not *merely* an equation – it is *the* equation that you are trying to find a picture for – but the point is that the E is the observable, a coordinate independent quantity, but the other side, the p, the u and even the . are coordinate dependent. So, the picture you want to paint depends of the coordinates you choose. *But*, and again this is the point, there is no choice of coordinates that are “more correct” than another. Some might make calculations easier, or make a picture somewhat more pretty, but none are the “correct” choice.

    If I choose FRW, then there is no spatial component to u for comoving observers, and so we say “oooooh – they aren’t moving and so the redshift is due to “space expanding in between”.

    But if I transfer FRW into its conformal representation, which describes exactly the same geometry, then there is a spatial component to u, but the redshift is exactly the same, then we go “ooooooh – now it’s moving and so some of the redshift must be Doppler”

    When we perform a series of transformations into orthonormal frames at a series of points along the path, and the redshift is still exactly the same, and we go “ooooh look, now we have a series of Doppler shifts along the path”.

    Or I could play coordinate games between now and the end of (conformal*) time, and I could continually reinterpret the same redshift in a myriad of ways.
    But none of them is *the* correct description – every description is coordinate dependent.

    – Geraint

    * conformal time is bounded into our future

  10. Hi Geraint,

    Thanks for the reply, but if I am interpreting your response correctly, we are talking past each other. I am pretty familiar with the characteristics of different coordinate systems such as FLRW, conformally flat Minkowski, Milne, etc. I read your Coordinate Confusion paper as well as all the relevant papers by your team, Davis & Lineweaver, Whiting, Chodorowski, Abrahmson, etc.

    My comments and questions relate to Proper Distance and Proper Speed, not Comoving Distance and Speed. I understand that comoving galaxies are stationary in comoving coordinates, that’s why it’s so good at obscuring interesting things that are going on. It doesn’t make it wrong, just limits its usefulness. I know that there is more than one way to characterize redshift.

    I would appreciate if you could respond specifically to as many of my redshift questions as you feel comfortable with. I have some very specific concerns and I hope you can help me with them.

    My recollection is that the papers you and your team wrote did not directly address the question of whether cosmological redshift can be formulated in terms of infintesimal SR and/or gravitational time dilations (in appropriate coordinate systems). You ducked the question in your papers it seems to me. If you believe that time dilation is the answer, please let me know whether you’ve run calculation using those equations that yield cosmological redshift solutions that match the traditional (observed) results at both low and high z values. I.e., DOES IT ACTUALLY WORK? I have yet to see anyone run the results that way and generate the correct answer with observational data.

    Thanks, Jon

    p.s., what about my question about photon exchanges?

  11. YES! FINALLY AN EXPLANATION IN PLAIN LANGUAGE!

    Universal Expansion
    The expansion of the universe causes distant galaxies to recede from us faster than the speed of light, if comoving distance and cosmological time are used to calculate the speeds of these galaxies. However, in general relativity, velocity is a local notion, so velocity calculated using comoving coordinates does not have any simple relation to velocity calculated locally. Rules that apply to relative velocities in special relativity, such as the rule that relative velocities cannot increase past the speed of light, do not apply to relative velocities in comoving coordinates, which are often described in terms of the “expansion of space” between galaxies. This expansion rate is thought to have been at its peak during the inflationary epoch thought to have occurred in a tiny fraction of the second after the Big Bang (models suggest the period would have been from around 10^-36 seconds after the Big Bang to around 10^-33 seconds), when the universe may have rapidly expanded by a factor of around 10^20 – 10^30.

    Astronomical Observations
    Apparent superluminal motion is observed in many radio galaxies, blazars, quasars and recently also in microquasars. The effect was predicted before it was observed by Martin Rees and can be explained as an optical illusion caused by the object partly moving in the direction of the observer, when the speed calculations assume it does not. The phenomenon does not contradict the theory of special relativity. Interestingly, corrected calculations show these objects have velocities close to the speed of light (relative to our reference frame). They are the first examples of large amounts of mass moving at close to the speed of light. Earth-bound laboratories have only been able to accelerate small numbers of elementary particles to such speeds.

    From Wikipedia

    This of course doesn’t deal with redshift, as this webpage do.

  12. > how did the source particle “anticipate” that the photon would eventually hit our orbiting satellite which we wouldn’t decided to launch until gigayears after the photon began its journey?

    Have a read of the Wheeler-Feyman absorber theory (or transaction theory).

    Speedy – that’s really no different to what we’ve been saying.

  13. “that’s really no different to what we’ve been saying.”

    Geraint, thanks and of course you are absolutely right. Those last posts from me only prove what a complete nutcase I am in physics! 😀

    I have to explain something though (to save my face 😉 ). English in not my native language and I’m not a physicist, so if everyday talks runs pretty okay, I’m completely lost in space when pros start talking about “null geodesics” and “gauge condition” etc, and I have to run thru half the Internet before I get the picture. And then add “hieroglyphic” equations to that and you got the picture of a totally confused amateur very late at night! 😀

    Lawrence B. Crowell is of course also absolutely right when saying; “People should maybe exercise the discipline to use that marvelous multi-media machine we have had from our evolutionary roots.”

    Law No. 1: There is only one way to learn – use your brain.

    Sorry guys, I’ve been rambling on too much about “it must be something wrong”, and I take it back.

  14. Hi Geraint,
    OK so I read about the Wheeler-Feynman absorber theory, and it’s about as wacky a theory as one could imagine. From a web article that describes it as the most like cause of mass having inertia:

    “This theory says that when you push on something, it creates a disturbance in the gravitational field that propagates outward into the future. Out there in the distant future the disturbance interacts with chiefly the distant matter in the universe. It wiggles. When it wiggles it sends a gravitational disturbance backward in time (a so-called “advanced” wave). The effect of all of these “advanced” disturbances propagating backward in time is to create the inertial reaction force you experience at the instant you start to push (and cancel the advanced wave that would otherwise be created by you pushing on the object).”

    This travel back and forward in time is right up there with string theory and colliding branes on the wacky scale. I understand that a lot of brilliant people contributed to these theories, and the math really does work. But there is no guarantee that a theory supported by valid math is itself physically valid. I like many people think we need to apply more common sense to evaluating these far out theories.

    Jon

  15. Hi Lawrence,

    >”These calculations of the comoving velocity for z =~ 1100 by Jon strike me as off the mark. Given the Hubble rule (FRW cosmology etc) that

    1 + z = exp(v/c)

    and for z = 1100 indicates that v/c = ln(1100) = 7, or an apparent velocity of 7c. Things are not moving nearly as fast as is being presumed here.”

    I believe you are looking at the instantaneous recession velocity of the EMITTER as measured in the OBSERVER’S (Earth’s) frame of reference. According to my spreadsheet and the Wright and Morgan online cosmic calculators, an emission source at z=1089 had a recession velocity of 56.6c then and 3.3c now, as measured in the observer’s frame of reference. Not sure why your equation calculates a different number.

    The figure I gave of 1048c is measuring something quite different: the average TRAVEL SPEED of a PHOTON during the SEGMENT of its trip between z=1023 and z=511, as measured in the EMITTER’S frame of reference.

    Jon

  16. I have an idea how cosmic gravity might first cause the redshifted photon to accelerate and then later to decelerate, thereby enabling it to pass each succeeding galaxy in the Hubble flow at exactly 1c.

    I submit that, rather than applying Gauss’ Law to a cosmic sphere centered on EITHER the observer or emitter, it should be applied to BOTH such cosmic spheres, at every point along its path, and then netted, to yield a combined “forward and backward” figure for gravitational influence at each point. This set of effects can then be integrated to calculate the geodesic across the entire path.

    Intuitively, early in the photon’s travel the gravity of the sphere centered on the observer will dominate, because the radius of that sphere is enormously larger, and during this historical epoch the cosmic mass/energy density is also relatively very high. So the initial acceleration toward the observer (which causes blueshift) is very high. Late in the photon’s travel the sphere centered on the emitter will now dominate, because the radius of that sphere is enormously larger. So the net effect of cosmic gravity will be to decelerate the photon. However, this deceleration will be milder than the early acceleration, even though the sphere’s radius is larger, because the cosmic density has decreased by much more proportionally than the propoprtion by which the late observer sphere’s radius exceeds the early emitter sphere’s radius.

    The math remains to be done, but I am encouraged by the logic. I believe that the cosmic gravity field MUST provide the sole explanation for the photon’s early acceleration and late deceleration. In the absence of accumulated SR time dilation (which seems to be ruled out by the FLRW metric), the photon can’t self-accelerate just because it is “required to” in order to pass every galaxy at exactly 1c.

    Jon

  17. Lawrence B. Crowell

    Jon Corthell on Oct 19th, 2008 at 4:49 pm
    I have an idea how cosmic gravity might first cause the redshifted photon to accelerate and then later to decelerate, thereby enabling it to pass each succeeding galaxy in the Hubble flow at exactly 1c.

    ————————

    Photons don’t accelerate! Null geodesics might be curved and the photon red or blue shifted, but there is no acceleration of a photon.

    Lawrence B. Crowell

  18. Lawrence B. Crowell

    Jon Corthell: According to my spreadsheet and the Wright and Morgan online cosmic calculators, an emission source at z=1089 had a recession velocity of 56.6c then and 3.3c now, as measured in the observer’s frame of reference.

    ——————

    Again something is amiss. SNI data tells us that the recessional velocities of galaxies in increasing, not decreasing.

    Lawrence B. Crowell

  19. >This travel back and forward in time is right up there with string theory and colliding branes on the wacky scale. I understand that a lot of brilliant people contributed to these theories, and the math really does work. But there is no guarantee that a theory supported by valid math is itself physically valid. I like many people think we need to apply more common sense to evaluating these far out theories.

    Why do you think your “common sense” in anyway is a good predictor of the way the universe works?

  20. Hi Geraint,

    Well I can’t say the absorber theory is definitely wrong, but I think it is a fundamental principle of the scientific method that all possible approaches should be exhausted that rely on close-to-conventional physics before radically exotic “new” physics is accepted. Time travel is just inherently too radical, and is not used in any other mainstream physics theory. And the concept of time travel is known to have many theoretical impediments. For example, both the emitter and receiver will move substantial distances during the wave travel period. I don’t see how there’s anything in the theory to accomodate that spatial relocation angles between the past, present and future.

    In my humble opinion the closer a cosmology theory is to purely kinematic (with GR gravity of course) the more likely it is to be correct.

    Jon

  21. Hi Lawrence,

    > “Again something is amiss. SNI data tells us that the recessional velocities of galaxies in increasing, not decreasing.”

    Dark energy is causing recessional velocities to increase in late times (e.g., since about 7Gy), but it will be many Gy before the are as fast as they were shortly after the inflation era ended. Gravity caused recession speeds to slow dramatically during the first 7 Gy before they turned around and started increasing again.

    > “Photons don’t accelerate! Null geodesics might be curved and the photon red or blue shifted, but there is no acceleration of a photon.”

    I agree that photons don’t accelerate LOCALLY. But at cosmological distances, their travel speed relative to both the observer and emitter MUST change. Otherwise, how could distant galaxies have been receding at superluminal velocities at the time of emission, yet the photons approach us at exactly c today?

    GR terminology such as “spacetime curvature” is all well and good, but we should not slavishly constrain ourselves to use terminology which obscures the fact that something real and tangible is occuring out there. Terminology is here to help us communicate, not to get in the way. When I talk about photons accelerating, I am simply referring to the distance they relocate over some elapsed period of time, as measured from a particular frame of reference. Let’s focus on the substance rather than the terminology.

    Jon

  22. > Terminology is here to help us communicate, not to get in the way. When I talk about photons accelerating, I am simply referring to the distance they relocate over some elapsed period of time, as measured from a particular frame of reference. Let’s focus on the substance rather than the terminology.

    Sigh – this is a coordinate dependent quantity and so there is no single “correct” answer.

  23. Lawrence B. Crowell

    General relativity removes the whole notion of force and acceleration from gravitation. Photons exist on curved geodesics in cosmology which loop back (so to speak) so distant objects comoving at apparent v > c can be observed. But this really does not involve accelerations as such.

    Lawrence B. Crowell

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