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Kinematic GR models of expansion

  1. May 14, 2008 #1
    M. Chodorowski argues in his 27 Mar 07 paper that the concept of "expansion of space" is not necessary to explain the expansion of the universe, and therefore Occam's Razor tells us that we should consider discarding the concept entirely. He says that expansion is sufficiently described by a kinematic model of the cosmic dust particles moving apart because they've always moved apart; coupled with GR to define the effects of the dust field's self-gravity. Applying GR is also necessary to explain why the universe can't be described by a single global Minkowski inertial frame, and therefore why SR is valid only in local inertial frames.

    Chodorowski cites J. Peacock who asserts that the expansion of space is fictitious locally but is valid at the global level. An interesting dichotomy that Chodorowski disagrees with. It is noteworthy that Tamara Davis (of Davis & Lineweaver fame) is the referee for Chodorowski's paper. That doesn't mean she agrees with his conclusions, but he does ascribe some semi-supportive statements to her in the form of "private conversations."

    Francis & Barnes et al argue in their 3 July 07 paper that the concept of the expansion of space should be preserved because it is a useful teaching and conceptual aid. They worry that "throwing the baby of an intuitive framework out with the bathwater of misconceptions leaves us only with bare mathematics, which in the case of general relativity is particularly daunting for the uninitiated, and useless as a conceptual device." But then they define their idea of the expansion of space to be "neither more nor less than the increase over time of the distance between observers at rest with respect to the cosmic fluid." They suggest that newly-created space "wells up" between particles as a result (and definitely not as the cause) of those particles moving apart in the Hubble flow. They also say that any dynamic effects attributed to the expansion of space actually are dependent on the choice of coordinates and therefore that it is fruitless to ask whether they are physically "real" phenomena.

    The ongoing dialogue I briefly summarized above touches on some fundamental concepts of big bang cosmology. In that context, I thought it would be useful to start a discussion about the idea that space itself might not be expanding, and to explore what the implications might be of extending that idea to its ultimate logical conclusions. For lack of a name I'll call this the "pure kinematic-GR framework".

    1. Space is nothing. In this framework, space should be viewed as nothing, simply the empty interval between quantum particles. As such, space has no structure, no geometry, no curvature, no movement, and no ether; it simply doesn't exist. What does exist are matter and radiation. Particles, pictured as a homogeneous dust field, are moving away from each other. Does their movement apart "create" new space, or are they moving through pre-existing space?

    2. Expansion "of" or "through" space. Chodorowski says "Galaxies do not move through space or in space. In a Machian view, they move instead with space: they simply enable space to exist." This sounds like waffling to me, especially from an advocate of the idea that space is not expanding. If it's not expanding now, then wasn't it already there? Why not simply start with the presumption that matter is expanding through pre-existing space; that is, through pre-existing physical coordinate regions?

    3. Origin and Edge of the universe. Perhaps the most fundamental ground rule of big bang cosmology is the Copernican Principle, which says that no point or direction in space is any more important than any other. At very large scales, the distribution of matter, the Hubble flow, and the anisotropies of the CMB are all observed to be highly isotropic and homogeneous. Hubble's law accurately predicts (at very large scales) that recession velocity in all directions is proportional to distance, and this is quite reasonably believed to be true for all observers regardless of where they may be located in the universe.

    If particles are moving apart because they were previously moving apart, then they must have been much closer together at some earlier era. They may have been tightly packed together at t=0, or even formed a singularity. This is the generally accepted picture, with variations on the theme. However, the accepted description is that the expansion did not begin at a single point or origin; rather, it began "everywhere" in the universe, in the sense that the initial size of the universe was limited to the region where the particles were (which may have been finite or infinite), and the movement of the particles away from each other "created" new space between them, together resulting in an expanding universe. The accepted explanation may be true, but I know of no reason why it is could not also be possible that the universe had its origin at a single coordinate point in space, subsequently expanding through pre-existing empty space.

    Specifically, consider the scenario in which all of the particles in the universe departed the origin coordinate simultaneously, at various speeds ranging evenly from 0 to c. As Chodorowski says, this is like the Milne model, except with gravity. As these particles travel isotropically and homogeneously in every direction away from the origin, observers on each particle (or eventually on each galaxy) would observe all of the other particles moving away from them, with speed proportional to distance. Exactly in accordance with Hubble's law.

    Consider the analogy of the NY Marathon. Imagine 100,000 runners starting approximately simultaneously, but each running at a different speed. After a couple hours, the runners would be spread out across much of the course. A middling runner would observe the faster runners ahead moving away, at "comoving" speeds that vary with distance. The same runner would observe runners behind falling yet further behind, again at "comoving" speeds that vary with distance. Consider a scenario where half of the runners start the race at the same time but run in the opposite direction. One of them doesn't run at all and remains at the origin. If the speeds of the runners in both directions were perfectly distributed by speed, the middling runner in one direction looking backwards would observe that the "Hubble flow" of runners behind was continuous all the way through the origin and then through the runners going in the opposite direction of the observer. If the observer did not know that the Marathon started on the Verazzano Bridge, he'd have no way of distinguishing the origin from any other point in the "Hubble flow" of runners travelling in both directions.

    Thus, even if there were a physical origin point in the universe for the expansion of the particle flow, a middling observer in the comoving Hubble flow would have no clue whatsoever to enable her to distinguish the origin from any other point in the universe. She could not even discern which direction was "towards" the origin, vs. "away" or "transverse".

    If the the universe is infinite in extent, then once the Hubble flow begins, the concept of an origin becomes meaningless. Its unique identity is immediately lost forever. On the other hand, in a finite universe it would theoretically be possible to reconstruct the location of the origin if one could observe the expanding outer surface (edge) of the dust field. Clearly however, we are not in a position to make such an observation. The fact that the CMB is homogeneous in all directions from our location indicates that we are too far from the edge (if one existed) to ever detect its presence. The CMB radiation coming at us from the "edge" direction would have been travelling ever since the recombination era, and our observations currently are obscured beyond that distance. (In the event there is an "edge", please don't ask me what lies beyond it. I have nothing to say about that subject.)

    By measuring the Hubble parameter we could calculate how fast the dust field (at various observable locations) is moving with respect to the origin, and how long ago it must have departed the origin. But we would still have no clue in which direction the origin lies.

    4. Proper velocity measurements. The tethered galaxy exercise involves bringing a distant test particle to zero proper velocity with respect to an arbitrarily selected origin point. After the test particle is untethered, its comoving and proper motion with respect to the origin are observed. An important thing to keep in mind with respect to the pure kinematic-GR framework is that any so-called origin point defined for such a problem isn't actually a fixed location relative to the actual (but undetectable) origin coordinate of the dust field's expansion. The "exercise origin" is itself comoving with respect to the "universal" origin (if it exists). This explains why a sphere of massless test particles shot away from the exercise origin will "expand with the Hubble flow", as described in section 2.6.2 of the Francis & Barnes paper. They describe this effect (rather confusingly) as "cosmological tidal forces", and in my view incorrectly attribute it to expanding space. Tsk, tsk.

    5. Superluminal recession. As both Chodorowski and Francis & Barnes explain, the observation that distant galaxies are receding from us faster than the speed of light is a phenomenon which is entirely dependent on choice of coordinates in a gravitating dust field, and it has no bearing on whether space is or is not expanding. The observed redshifts can equally well be explained as a combination of SR doppler shift and gravitational redshift.

    I'll stop here and hope to generate a robust discussion.

    Jon
     
  2. jcsd
  3. May 14, 2008 #2
    Well in my opinion when we are talking about GR it is best to talk about spacetime instead of thinking about space and time separately.

    Sometimes we get idiotic descriptions for instance terminology like "metric expansion of space". This is simply confusing terminology as space by itself has no metric that changes under GR.

    Many people, especially cosmologists, like to talk about the effects due to the selection of a particular coordinate chart as if they are physical properties of a particular spacetime.

    Seriously if a metric of spacetime were to expand, GR would be in serious trouble, since the metric is fixed under GR solutions.
     
  4. May 14, 2008 #3

    Wallace

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    I agree with MeJennifer, apart from the attack on cosmologists. There are various co-ordinate systems used by cosmologists (co-moving co-ordinates, super co-moving co-ordinates are just two examples) to describe the Universe and there are certainly various idioms (expanding of space etc) that are used to describe processes in these co-ordinates, however for the most part cosmologists are fully aware that these are convenient shorthands and don't have genuine physical significance.

    But the key point that was well made is that the metric is fixed in a GR solution and certainly doesn't expand or anything else.
     
  5. May 14, 2008 #4
    Hi MeJennifer and Wallace.

    I agree that the metric is fixed in the GR solution. GR is a wonderful set of abstract mathematical equations that can be applied validly in a universe with multiple sets of moving parts, without regard to the choice of coordinate system and without regard to whether there is such a thing as "absolute" coordinates or velocities. GR is sort of like a specialized computer: You feed data in, and it generates mathematical results. Sadly however, the "GR computer" is limited to calculating mathematics; it does not supply or mandate any particular physical description of particles or space.

    Indeed, the GR computer will generate results which accurately map to observations, regardless of whether "new space" is being created by the ongoing separation of the dust particles, or whether those particles are simply moving apart through "pre-existing" space. One must conclude that we simply lack the data we would need to use the GR computer to calculate mathematically which of these two models better describes the underlying physical reality.

    Therefore, my suggestion is that cosmologists should stop instructing people that (a) the expansion is creating new space between the dust particles, and (b) that the universe did not have a single origin point and, if it is finite, does not have an edge. Instead, we should say "we don't know." We can say that it is extremely difficult to determine which description is better, and we don't have enough facts to offer more than a range of reasonable alternatives. That way we avoid tying cosmology to philosophical preferences.

    Jon
     
    Last edited: May 14, 2008
  6. May 18, 2008 #5
    The most straightforward physical description of a kinematic-GR origin of the expansion is that all of the mass-energy particles were packed tightly together in a homogeneous, isotropic, spherical mass at t=0 and then they began accelerating apart from each other all at once. To be precise, each particle would have accelerated away from each other particle with the same acceleration rate. This suggests an anti-gravity phase of acceleration, which might possibly have been very quick (analagous to the timescale of inflation theories) or alternatively more protracted.

    The origin point in space, at the original physical center of the packed particle body, need not have any special significance. It just would have happened to be the centerpoint of the packed particle body at the time the anti-gravity phase began. As I suggested, the particles would have been accelerating away from each other, not away from the origin coordinate itself.

    The cumulative effect of the particles uniformly accelerating away from each other would have been that, by the end of the anti-gravity phase, each particle would have been imparted a proper expansion velocity proportional to how many particles were interposed roughly on the radial line between it and the coordinate origin. Particles toward the center of the original packed particle body ended up with low proper expansion velocities; particles further outward ended up with higher proper expansion velocities (which is simply the result of vector addition). Theoretically, the total number of particles in the original packed body could have been either finite or infinite.

    Each particle subsequently would have continued coasting away from each other particle at a relative velocity which is a geometric function of the two particles' (1) difference in radial proper speed away from the origin coordinate, (2) difference in angle of departure from the origin coordinate, and (3) elapsed time since t=0.

    Each particle's proper velocity away from the origin coordinate (and therefore away from each other) would also decrease as a function of time, due to the mutual gravitational attraction of all of the particles (assuming Lambda=0).

    A fundamental observer on any one of the particles (provided there is no outer "edge" to the particle expansion, or the observer is too far away from the edge to observe its effects) might perceive the nearby Hubble expansion rate to be decelerating much faster than the proper velocity of the nearby particles away from the coordinate origin was actually decelerating gravitationally. (Of course the proper velocity away from the origin could not be directly measured.) This is because as more time passes, the particles whose proper speed away from the origin, as well as particles with large differences in angle of departure from the origin, would have moved relatively much farther away from the observing particle. The more time passes, the the more the nearby particles (at any given distance) would be those particles which originally departed the origin with ever smaller differences in proper velocity and angle relative to the observing particle.

    In fact, even if the gravitational deceleration were exactly offset by Lambda with (w = 1/3), the observing particle would perceive a continuing deceleration of the nearby Hubble flow, due to this effect. As Wallace points out, that's just another way of saying that in expanding universes, the Hubble rate at any given proper distance declines over time (except to the extent that a large Lambda rate would counteract it.)

    Again, I see nothing in the kinematic-GR framework that requires that space be thought of as "expanding" as a result of the particle motions. Particles could be moving in pre-existing space, or creating new space as they coast away.

    Jon
     
    Last edited: May 18, 2008
  7. May 19, 2008 #6
    I think if you want to understand this you need to start thinking in terms of fluid and pressure, not in terms of dust.
     
  8. May 20, 2008 #7
    Hi MeJennifer,

    Well, I am thinking about dust in terms of it being a pressureless fluid with an equation of state w=0.

    I assume you are referring to what I described as the "anti-gravity phase". From a GR perspective, this could refer to some condition causing negative pressure, e.g. an equation of state w= -1. As I understand it, both inflation and the cosmological constant are characterized by that equation of state. However, in both of those examples the negative pressure is an attribute of some unidentified quantum particle that "resides" in otherwise empty space (e.g. inflatons, dark energy, etc.), and therefore is not an inherent attribute of the conventional matter (e.g. dust) and radiation particles themselves. Is that the point you're getting at?

    It seems cleaner to me (and more consistent with the kinematic-GR concept that space is NOT expanding) if the anti-gravity effect were an inherent attribute of conventional matter and/or radiation (albeit one that switches on and off under certain unknown conditions), rather than requiring the invention yet another quantum particle that "resides" in otherwise empty space. But if the latter is what is needed, then so be it. It may well be that a cosmological constant (of empty space) is needed to explain the late-times acceleration of the expansion rate.

    I would appreciate if you could expand on your comment.

    Jon
     
    Last edited: May 20, 2008
  9. May 20, 2008 #8
    I'd like to reinforce the point that the kinematic-GR physical framework I summarized (as inspired by Chodorowski) apparently results in a fully homogeneous (unlike Milne) and isotropic distribution of dust particles over time, and complies with Hubble's law.

    I suggest that even though such a model has a coordinate origin "point" (and possibly an outer edge), it does not violate the cosmological principle because the underlying physics are no different at the origin coordinate than at any other point in the universe.

    Another aspect that is particularly clear about (although not necessarily a distinguishing feature of) a kinematic-GR model is that the universe does not mysteriously lose enormous amounts of energy as a result of the redshift of free radiation in expanding space. Since space does not expand, no energy is permanently lost. Instead, distant observers detect a reduced amount of radiation (relative to the original emitter) (doppler shift) only because the observers are in motion away from the source, i.e. they are in a receding reference frame.

    And, most importantly, my understanding is that the concept that "there was no single origin point, rather the universe began expanding everywhere" requires a 3-dimensional manifold embedded in 4 spatial dimensions. Since there has been no scientific demonstration whatsoever that a 4th spatial dimension is a physical reality (as opposed to a mathematical construct of GR), it makes good sense to me that perhaps more scientific effort should be expended on exploring models based on a vanilla 3 spatial dimensions rather than an exotic 4.

    Jon
     
    Last edited: May 20, 2008
  10. May 22, 2008 #9
    I don't know how to contribute more to what you write, I fear further comments will be viewed as unhelpful. I think you are on the wrong track, adopting a kinematic model of GR is like drawing Euclid's shapes in quicksand.

    With respect to spacetime, all we can say is that it works, if it is physical or not is mostly a matter of philosophy in my view.
     
  11. May 22, 2008 #10
    Hi MeJennifer,

    Your comments are always helpful to me, and I'm sure to others. Please don't be reluctant to make specific comments. I am not philosophically tied to the kinematic-GR model; I just started exploring it recently after studying Chodorowski's and Whiting's papers. I am trying to explore it in some depth to understand why it is or is not a useful way of thinking about the expansion of the universe.

    I'd like to understand why you think this model is wrong per se, as opposed to just being a different physical interpretation of the same set of observations and using the same GR mathematics. Isn't it true that GR defines mathematical relationships but does not dictate a specific physical model of whether or not the empty space between particles (disregarding Lambda for the moment) is expanding or pre-existing? As I mentioned, John Peacock asserts that the expansion of space is valid globally but invalid locally, without explaining why that isn't self-contradictory. Barnes & Francis on the other hand assert that it is meaningless to ask whether or not the space between the particles is expanding, which implies to me that they don't ascribe much importance to possibility that the expansion of space is not a valid physical concept.

    Here's another way to look at it: In the tethered galaxy scenario, after it is released, the observer at the origin sees the test particle being passed by nearby galaxies moving away from the origin in the Hubble flow. So this observer can legitimately ask, wasn't the local space around the test particle "pre-existing", in the sense that this local space was already there before a galaxy passed by the test particle in the Hubble flow? Thus the observer may conceptually conclude that the passing-by of a galaxy did not "create new space" in the local region around the test particle. This question of course could be expanded indefinitely by defining a very large number of such untethered test particles, each with its own pre-existing local space.

    The question becomes abstract to the point of meaninglessness: Did our mere act of inserting the massless test particle really "create some new space" around it? Or perhaps that local space is "fictitious" until some real massive object passes by, which then annoints it as "real" space?

    For that matter, how in a physically real sense can the mere passage of a massive object (whether in the Hubble flow or peculiar to it) with or through coordinate space act to create new space around it? This question creates a serious issue if the "new space" is not entirely featureless, for example if it contains inherent energy such as a cosmological constant. Surely the coasting movement of a massive object with or through a coordinate system cannot physically cause the universe to gain energy in any form. The causation seems more logical if we instead say that a dust field expanding with the Hubble flow is incorporating pre-existing energy (cosmological constant) into the coordinate space the dust field occupies, rather than actually causing the creation of that energy.

    It just seems to me that the idiosyncracies of a particular coordinate system (comoving coordinates) perhaps are being assigned too much physical meaning on this topic.

    Jon
     
    Last edited: May 22, 2008
  12. May 23, 2008 #11
    It's a myth that Milne rejected homogeneity, indeed he saw it as a central principle. e.g.
    You might like to look at what I have written on this subject (which also relates to your tethered galaxy thread) http://www.chronon.org/papers/current/untethered.pdf
     
    Last edited: May 23, 2008
  13. May 23, 2008 #12
    Hi Chronon,

    I had remembered that the Milne model was not homogeneous, but apparently that is incorrect. My mistake. It is homogeneous, isotropic, and obeys Hubble's law. Here is a paper describing the Milne model in more detail.

    Thanks for the link to your paper on the tethered galaxy problem, it was interesting. Are you going to update it to address AB Whiting and Francis & Barnes?

    Jon
     
  14. Jun 1, 2008 #13
    Here's another thought about fluid and pressure.

    Standard cosmology models assume that the very early universe (well before the CMB was released) was radiation-dominated, meaning that there were many orders of magnitude more of mass-energy in the form of free radiation particles than in the form of matter particles (aka "dust"). The standard FLRW model says that the free radiation fluid has the equation of state attribute of positive pressure. GR says that positive pressure itself causes extra gravitation, such that free radiation has a higher gravitational potential per unit of mass-energy density than does matter. The standard FLRW model says that because the universe is deemed to have no outer edge, there is no "outer surface" with a pressure gradient, and therefore the positive pressure of the radiation fluid does not exert any physical expansion force on the universe.

    On the other hand, let's consider as a mind experiment the possibility of a finite, expanding kinematic-GR universe model that does have an outer edge. If space outside the edge were empty, then the positive pressure of the free radiation fluid would cause a pressure gradient to exist at the outer surface. In the very early universe, could this positive pressure gradient be a possible source of the original expansion? If initially the dominant quantity of free radiation particles were smoothly interspersed among (or attached to) the tightly packed matter particles, it seems to me that the positive pressure gradient could have motivated the original expansion which I referred to as the "antigravity phase". At the same time, the extra gravitational potential of the positive pressure causes rapid deceleration. Presumably both the expansionary acceleration created by the positive pressure gradient and the extra gravitational deceleration it caused would both become insignificant around the time that the universe became matter dominated due to the redshift of the free radiation. Thereafter such a universe (with Lambda = 0) would follow a standard matter-dominated Einstein-de Sitter deceleration.

    Jon
     
    Last edited: Jun 1, 2008
  15. Jun 7, 2008 #14
    The outcome of the tethered galaxy problem seems to be quite different in a model universe that follows a kinematic-GR framework, than in a conventional "expanding space" framework with Lambda = 0.

    Francis & Barnes mention in their paper "Expanding Space is the Root of all Evil" (7/07) that "the kinematical view sees no difference between recession and peculiar velocities...." In that view, all galaxies are in peculiar motion in space, moving away in proper distance from a single original expansion coordinate (let's call it the "expansion origin".) So the "origin" point defined arbitrarily in the thethered galaxy problem (let's call it the "problem origin") apparently is actually comoving away from the expansion origin at a fixed proper velocity.

    When the tethered galaxy is brought to zero proper velocity compared to the problem origin and released, both of them continue to move away from the expansion origin at the same proper speed, and both of them will experience the same absolute deceleration rate from the background dust gravity. Thus the untethered galaxy will not move relative to the problem origin and the proper distance between them will remain constant indefinitely. (Compare this to the non-kinematic model, where the untethered galaxy accelerates towards the problem origin, passes through it, and speeds away in the opposite direction.)

    Imagine the NYC marathon again, with the runners who left the start line at the same instant but at different speeds and spread smoothly away from each other over time. If one of these runners is arbitrarily designated as the "problem origin" runner, we can then "tether" another runner to him by speeding the latter up if she's behind the problem origin runner, or slowing her down if she's ahead of him. Thereafter, if we assume that all of the runners decelerate smoothly and equally (in terms of proper absolute velocity, not relative velocity) as a function of time, the proper distance between the untethered runner and the problem origin runner will remain constant until the end of the race.

    It is unsettling that the tethered galaxy problem has such a fundamentally different outcome in a kinematic-GR framework than in a conventional "expanding space" framework.

    Jon
     
  16. Jun 7, 2008 #15
  17. Jun 7, 2008 #16
    Hi shalayka,

    I don't have a problem with the analysis in this Lewis, Francis, Barnes & James paper, and I am not aware of anyone referring to it as idiotic, including Chodorowski. On the simplest level, the paper says that Chodorowski's math was wrong (or unjustified) when he purported to demonstrate that the recession velocity of distant objects cannot be superluminal (faster than the speed of light) in conformal coordinates. The math in the present paper shows that superluminal velocities do occur in those coordinates. Since Chodorowski's mathematical conclusion is said to be wrong, his use of that conclusion to bolster the argument that space is not expanding is also misplaced.

    Of course, this doesn't mean that other forms of analysis can't be used to probe whether the expansion of space is a physical reality or not. Trying out different coordinate systems can often shed new light on the dynamics of the universe. Perhaps Lewis et al disagree with that perspective; they say: "This work concludes by emphasizing that the expansion of space is perfectly valid in the general relativistic framework, however, asking the question of whether space really expands is a futile exercise." I think it's way too early in the history of cosmology to declare that this is a futile exercise. We need to keep plugging away, looking for new perspectives.

    Did I address your question, or are you trying to get to another point?

    Jon
     
    Last edited: Jun 7, 2008
  18. Jun 8, 2008 #17
    Hi Jon, you answered my question far beyond what I had expected actually. :) Thank you for that.

    My half-joking comment re: "idiotic" was pointed at a particular someone (not you) who either does not keep up with the field, or has simply succumbed to prejudice. Pretty sad either way, considering that as an amateur I obviously trump them in both respects.

    On the other hand, I think that your comment is the very definition of a truly scientific mind:

    "I think it's way too early in the history of cosmology to declare that this is a futile exercise. We need to keep plugging away, looking for new perspectives."
     
    Last edited: Jun 8, 2008
  19. Jun 8, 2008 #18
    Thanks for the encouragement shalayka. A particularly challenging part of this effort is that one often must rediscover for oneself ideas that were originally thought of long ago, but aren't mentioned too much in current textbooks and technical papers. One of my pet peeves is that too many cosmology textbooks (I've read 3 in the last 9 months) devote most of their effort showing complete proofs of how the many important equations are derived, and way too little effort describing what the many curious little details mean (and don't mean) in practical terms, or why alternative theories are ruled out.

    On the other hand, I just finished "Exploring Black Holes" by Edwin Taylor and the recently departed John Archibald Wheeler. I highly recommend it to anyone looking for a thorough and accessible explanation of both the basic math and detailed intuitive concepts. It focuses on black holes, but it teaches comprehensively about gravity. It doesn't get much into FLRW cosmology. Anyway, it's head and shoulders above any other textbook I've read on gravity, for people at my level. Unfortunately like most textbooks it's >$50.

    Jon
     
  20. Jun 9, 2008 #19
    CMB and Kinematic GR

    The isotropy of the CMB poses a challenge for the Kinematic Gr model if the universe is finite and has an outer edge. The CMB looks very nearly the same to us today no matter in which direction we view it. But the universe was much smaller at the time the CMB was emitted, in fact its radius was about 1050 times smaller. If our location today has moved beyond the surface where the outer edge (if there was one) of the universe was at CMB emission, then we would see CMB photons approaching us today only from our cosmic hemisphere which faces towards the expansion origin, and no CMB photons approaching us from the opposite direction.

    The only solution I can think of is that our location today needs to be physically inside the envelope of the universe as it was at CMB emission time. Not only that, but it needs to be a full Hubble radius inside where the outer edge was at CMB emission time, so that the edge (if there is one) does not affect the isotropy. Depending on how far proportionately we are from the expansion center to the edge, the full universe at CMB emission time must have had a radius of no less than 1050 - 2100 times the radius of our observable universe at that time. (The same ratio would hold today).

    This issue doesn't exist if the universe is infinite, of course.

    Jon
     
    Last edited: Jun 9, 2008
  21. Jun 10, 2008 #20
    I think I misstated this point about the tethered galaxy problem in a kinematic-GR framework. The Newtonian Shell Theorem says that runners further from the start line will "feel" more gravitational deceleration than runners closer to the start line. In fact, contrary to what I said, in order to keep the proper distances between the runners constant in absolute terms as they decelerate, I think that each runner must decelerate at a rate proportional to its own proper speed. That seems to be consistent with what the Shell Theorem would produce; more deceleration for the fastest runners who have run the farthest.

    Jon
     
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