A Dark, Misleading Force

Certain subsectors of the scientifically-oriented blogosphere are abuzz — abuzz, I say! — about this new presentation on Dark Energy at the Hubblesite. It’s slickly done, and worth checking out, although be warned that a deep voice redolent with mystery will commence speaking as soon as you open the page.

But Ryan Michney at Topography of Ignorance puts his finger on the important thing here, the opening teaser text:

Scientists have found an unexplained force that is changing our universe,
forcing galazies farther and farther apart,
stretching the very fabric of space faster and faster.
If unchecked, this mystery force could be the death of the universe,
tearing even its atoms apart.

We call this force dark energy.

Scary! Also, wrong. Not the part about “tearing even its atoms apart,” an allusion to the Big Rip. That’s annoying, because a Big Rip is an extremely unlikely future for a universe even if it is dominated by dark energy, yet people can’t stop putting the idea front and center because it’s provocative. Annoying, but not wrong.

The wrong part is referring to dark energy as a “force,” which it’s not. At least since Isaac Newton, we’ve had a pretty clear idea about the distinction between “stuff” and the forces that act on that stuff. The usual story in physics is that our ideas become increasingly general and sophisticated, and distinctions that were once clear-cut might end up being altered or completely irrelevant. However, the stuff/force distinction has continued to be useful, even as relativity has broadened our definition of “stuff” to include all forms of matter and energy. Indeed, quantum field theory implies that the ingredients of a four-dimensional universe are divided neatly into two types: fermions, which cannot pile on top of each other due to the exclusion principle, and bosons, which can. That’s extremely close to the stuff/force distinction, and indeed we tend to associate the known bosonic fields — gravity, electromagnetism, gluons, and weak vector bosons — with the “forces of nature.” Personally I like to count the Higgs boson as a fifth force rather than a new matter particle, but that’s just because I’m especially fastidious. The well-defined fermion/boson distinction is not precisely equivalent to the more casual stuff/force distinction, because relativity teaches us that the bosonic “force fields” are also sources for the forces themselves. But we think we know the difference between a force and the stuff that is acting as its source.

Anyway, that last paragraph got a bit out of control, but the point remains: you have stuff, and you have forces. And dark energy is definitely “stuff.” It’s not a new force. (There might be a force associated with it, if the dark energy is a light scalar field, but that force is so weak that it’s not been detected, and certainly isn’t responsible for the acceleration of the universe.) In fact, the relevant force is a pretty old one — gravity! Cosmologists consider all kinds of crazy ideas in their efforts to account for dark energy, but in all the sensible theories I’ve heard of, it’s gravity that is the operative force. The dark energy is causing a gravitational field, and an interesting kind of field that causes distant objects to appear to accelerate away from us rather than toward us, but it’s definitely gravity that is doing the forcing here.

Is this a distinction worth making, or just something to kvetch about while we pat ourselves on the back for being smart scientists, misunderstood once again by those hacks in the PR department? I think it is worth making. One of the big obstacles to successfully explaining modern physics to a broad audience is that the English language wasn’t made with physics in mind. How could it have been, when many of the physical concepts weren’t yet invented? Sometimes we invent brand new words to describe new ideas in science, but often we re-purpose existing words to describe concepts for which they originally weren’t intended. It’s understandably confusing, and it’s the least we can do to be careful about how we use the words. One person says “there are four forces of nature…” and another says “we’ve discovered a new force, dark energy…”, and you could hardly blame someone who is paying attention for turning around and asking “Does that mean we have five forces now?” And you’d have to explain “No, we didn’t mean that…” Why not just get it right the first time?

Sometimes the re-purposed meanings are so deeply embedded that we forget they could mean anything different. Anyone who has spoken about “energy” or “dimensions” to a non-specialist audience has come across this language barrier. Just recently it was finally beaten into me how bad “dark” is for describing “dark matter” and “dark energy.” What we mean by “dark” in these cases is “completely transparent to light.” To your average non-physicist, it turns out, “dark” might mean “completely absorbs light.” Which is the opposite! Who knew? That’s why I prefer calling it “smooth tension,” which sounds more Barry White than Public Enemy.

What I would really like to get rid of is any discussion of “negative pressure.” The important thing about dark energy is that it’s persistent — the density (energy per cubic centimeter) remains roughly constant, even as the universe expands. Therefore, according to general relativity, it imparts a perpetual impulse to the expansion of the universe, not one that gradually dilutes away. A constant density leads to a constant expansion rate, which means that the time it takes the universe to double in size is a constant. But if the universe doubles in size every ten billion years or so, what we see is distant galaxies acceleratating away — first they are X parsecs away, then they are 2X parsecs away, then 4X parsecs away, then 8X, etc. The distance grows faster and faster, which we observe as acceleration.

That all makes a sort of sense, and never once did we mention “negative pressure.” But it’s nevertheless true that, in general relativity, there is a relationship between the pressure of a substance and the rate at which its density dilutes away as the universe expands: the more (positive) pressure, the faster it dilutes away. To indulge in a bit of equationry, imagine that the energy density dilutes away as a function of the scale factor as R-n. So for matter, whose density just goes down as the volume goes up, n=3. For a cosmological constant, which doesn’t dilute away at all, n=0. Now let’s call the ratio of the pressure to the density w, so that matter (which has no pressure) has w=0 and the cosmological constant (with pressure equal and opposite to its density) has w=-1. In fact, there is a perfectly lockstep relation between the two quantities:

n = 3(w + 1).

Measuring, or putting limits on, one quantity is precisely equivalent to the other; it’s just a matter of your own preferences how you might want to cast your results.

To me, the parameter n describing how the density evolves is easy to understand and has a straightforward relationship to how the universe expands, which is what we are actually measuring. The parameter w describing the relationship of pressure to energy density is a bit abstract. Certainly, if you haven’t studied general relativity, it’s not at all clear why the pressure should have anything to do with how the universe expands. (Although it does, of course; we’re not debating right and wrong, just how to most clearly translate the physics into English.) But talking about negative pressure is a quick and dirty way to convey the illusion of understanding. The usual legerdemain goes like this: “Gravity feels both energy density and pressure. So negative pressure is kind of like anti-gravity, pushing things apart rather than pulling them together.” Which is completely true, as far as it goes. But if you think about it just a little bit, you start asking what the effect of a “negative pressure” should really be. Doesn’t ordinary positive pressure, after all, tend to push things apart? So shouldn’t negative pressure pull them together? Then you have to apologize and explain that the actual force of this negative pressure can’t be felt at all, since it’s equal in magnitude in every direction, and it’s only the indirect gravitational effect of the negative pressure that is being measured. All true, but not nearly as enlightening as leaving the concept behind altogether.

But I fear we are stuck with it. Cosmologists talk about negative pressure and w all the time, even though it’s confusing and ultimately not what we are measuring anyway. Once I put into motion my nefarious scheme to overthrow the scientific establishment and have myself crowned Emperor of Cosmology, rest assured that instituting a sensible system of nomenclature will be one of my very first acts as sovereign.

326 Comments

326 thoughts on “A Dark, Misleading Force”

  1. Lawrence Crowell

    Right, its like doing a survey. One can platt out the land in any way possible, but how you do this does nothing to the hill, valley or dale. The land is what it is. Now given a certain lay of the land there might be more convenient ways to do the survey. In the case of cosmology the FRW coordinate system is convenient, though we have found departures and the universe appears to be asymptotically approaching a deSitter spacetime.

    Ah yes, to the chosen few, that band of brothers who know general relativity! Even if it can be an infernal pain in the ass at times.

    Lawrence B. Crowell

  2. Lawrence,

    It is back to square one, but that just means I have to go around it again to explain why I see it as a cycle, not a line.

    I understand that spacetime is defined locally, by its contents. That’s why I originally asked you if expanding space meant the speed of light is affected. I know it can be a matter of relative effects, but the simple point here isn’t complicated. It’s like 2+2=4. If we measure space as a function of lightspeed and two points are moving apart, as measured by the time it takes light to pass inbetween them, then there is more space/distance between them, not that the space is stretched, if the space is still being measured by lightspeed. Otherwise we need to say that lightspeed is not the basis of measuring space.

    As you point out, the speed of light is apparently slowed emerging from the vicinity of a black hole, thus we say space is contracted, because the space is curved/contracted, but we don’t say the black hole is shrinking and will vanish in the near future. Why not?

    BUT we do say the opposite; That since light crossing space between these gravitational wells is redshifted, then they must actually be flying away from each other and will eventually vanish from the view of each other.

    As you say,

    There is no preexisting space, nor is there a background space or spacetime upon which general relativistic spacetime exists.

    And Jason further qualifies,

    The geometry of space-time is intimately connected to the matter that inhabits it. The Einstein equations describe the relationship between space-time and matter, and if we have matter that is moving apart, then space is necessarily expanding.

    The reason gravity wells don’t vanish is because they are constantly drawing in fresh energy and matter that defines space. (Being gravity, mostly matter.)

    Now the radiation escaping from all these wells goes back out to fill all that nearly empty space. Is it a surprise then that it appears to expand?

    So gravity doesn’t cause the space around it to vanish, because there is always more material being drawn in. Space between these galaxies is expanding, as it is continually filled with more radiation. but the distance between these galaxies doesn’t actually grow, because the material defining this space eventually falls into these wells.

    Now I know you say the physics of how energy turns into matter doesn’t fit this model, but is our knowledge of this complete, or are we drawing conclusions from what we can measure in our particular environment, when this would have to be something happening at nearly the level of absolute zero, over very large volumes of space?

    The alternative involves saying that space expands, but our most basic measure of space is stable. As well as superluminal expansion and a number of other highly speculative features.

    Jason,

    There is no ruler out in space. None. The numbers we place on space-time are entirely arbitrary, human constructions.

    Yes, but we can’t start contradicting what we said previously, or it really does get Kafkaesque. If we say that lightspeed is our fundamental ruler of space, then if we wish to say that space expands, it must be because this measure is curved/expanded, not that this measure is the stable ruler we use to show that space is expanding. That isn’t even a self-referential argument. That is a self-contradictory argument.

  3. Lawrence,

    Those last five paragraphs prior to the reply to Jason were not supposed to be in quotations.

  4. John,

    The expansion of space has nothing whatsoever to do with the gravity wells that inhabit space.

  5. Jason,

    No, they are the contraction of space. Einstein was originally concerned that gravity would cause the entire universe to collapse. So he added a cosmological constant to maintain a stable universe.

    Redshift was evidence the space between these gravity wells is expanding.

    If Omega=1, or appears even close to it, then presumably gravity and expansion are in inverse proportion, or nearly so.

    Could it be that redshift is actually evidence of a cosmological constant? Measurements of the Dark Energy effect indicate the expansion is comparable to a c.c.

  6. Lawrence Crowell

    Gravitating bodies in the universe tend to counter the expansion. A popular model of the FRW cosmology was a recollapse model, which is similar to ballistic flight. It appears this is ruled out, and as the universe expands in an eternal inflationary acceleration it will asymptotically approach a deSitter spacetime. There matter in the cosmology will dilute into an ever smaller density and asymptotically approach being a vacuum spacetime.

    The Omega = 1 appears to be the case. Yet this factor has three parts

    Omega = Omega_m + Omega_{dm} + Omega_{de}

    form m = matter, e = energy and d = dark. As this blog-o-page is based around the dark misleading force, we can look at dark energy in a misleading way as due to a “vacuum field” with the equation of state w = -1 so the pressure is negative. So this “flip” in how gravity operates means that this cosmological gravitational field is repulsive. Dark energy constitutes about 70% of the Omega, dark matter about 27% and ordinary matter only 3%. These are the percentages I recall at the moment without looking them up.

    This contributes to the cosmological constant, in the dark misleading form, if we consider the vacuum as a momentum-energy tensor and trace over that to obtain the cosmological constant /.

    Lawrence B. Crowell

  7. If Omega=1, or appears even close to it, then presumably gravity and expansion are in inverse proportion, or nearly so.

    No. Firstly, “gravity and expansion” is a false dichotomy. Gravity drives the expansion, and, depending upon the matter content, can either slow it down or speed it up.

    Secondly, it wholly depends upon the sort matter content with Omega=1 what’s happening with the universe. Omega=1 does not remotely indicate a balance between expansion and contraction. What it does mean is that space is, on average, flat. If there is only normal/dark matter, then any such universe, were it to start in an expanding state, would end up with eternal expansion asymptotically approaching no expansion. If there was a negative cosmological constant, then the universe would recollapse in on itself. If there is a positive cosmological constant, then it expands (accelerating in the future). There is simply no balance to be had in expansion/contraction.

    And yes, it is certainly possible that the dark energy is a cosmological constant. We can’t yet say it is, because we haven’t ruled out other possibilities. But, suffice it to say, we haven’t yet found any better idea than it’s just a cosmological constant: as bad as the cosmological constant idea is theoretically, the other ideas are even more ad-hoc and contrived.

  8. Lawrence,

    Minuscule quibble, but I’d like to mention that there’s also photons and neutrinos that make up the fractional energy density of the universe. Of course, their density is much smaller, and diluting more rapidly, than matter, so they don’t matter a whole lot. But I thought it might be a point of interest.

  9. These are factored into the Omega_m. The PMN matrix for neutrino oscillations predicted masses for neutrinos, which were found by the super Kamiokande detector in 1997 or so, with m = 10-100ev. People had thought that neutrinos would close the universe up. But the masses are too small and dark energy came on the scene, which overwhelms any gravitational influence from neutrinos. There is also the axion particle issue, putative small mass particles which “protect” QCD from CP violations. So that is an open question at this time, but they are expected to have very small masses.

    The original cosmological constant was put into the Einstein field equation

    R_{ab} + 1/2Rg_{ab} + /g_{ab} = -8piG/c^4 T_{ab}

    to counter the gravtational collapse of the universe. Without the momentum-energy tensor the / in the field equations predicts that the universe is an Einstein space where the Ricci curvature is proportional to its metric. If we just look at / this way then dark energy is a pure gravitational effect, or due to the structure of spacetime in the absence of any source. On the other hand if we say that dark energy is due to the ZPE of the vacuum (I have a lot to say about this and how much of this is frankly B.S.) then it is a source that defines a T_{ab}, which when traced over gives the /. So there is a dichotomy of sorts in how to interpret this dark energy, which is a part of the issue of how this is a misleading force.

    Lawrence B. Crowell

  10. These are factored into the Omega_m. The PMN matrix for neutrino oscillations predicted masses for neutrinos, which were found by the super Kamiokande detector in 1997 or so, with m = 10-100ev.

    Actually, the detectors so far have only detected the differences in the masses between neutrino flavors, but the mass differences are in meV, not eV. We actually don’t know their absolute masses at all, though from experience we might suspect that it’s of the same order of magnitude as the mass differences.

    But I thought I’d like to point out something that you didn’t make clear:

    It only makes sense to factor in the neutrino mass density with Omega_m, and then only after the neutrinos have sufficiently cooled that they are no longer relativistic. Radiation will always have to be handled as a separate sort of matter density.

    The problem, however, is that due to the uncertainty in the absolute neutrino masses, we can’t actually say that we can adequately factor them into Omega_m. But, fortunately it’s really a non-issue as the energy density is so small that it’s pretty irrelevant anyway. I just thought it was of mild interest.

  11. Lawrence Crowell

    The neutrino mass differences are

    Delta_{12} =~ 0.009

    Delta_{13} ~ Delta_{23} =~ 0.05

    For 1, 2, 3 = e, mu, tau. When I wrote that I got the tenths and hundredths mixed with 10’s and 100’s. The delta mass squared enter into the neutrino phase

    e^{-i Delta m_{ij}L/2E},

    for L the oscillation length.

    The usual neutrino has ~1-10 MeV energy, so that is a bigger factor in the end for an y contribution to Omega

    In this latter matter dominated phase the contribution of radiation is pretty small. I can’t quote right off what it contributes to Omega_m, but it is a small percentate of it and Omega_m ~ .03*Omega. That is not much to worry about.

    Lawrence B. Crowell

  12. Er, Omega_m is ~0.23. Omega_r is around 10^-5 currently (this is measured from the thermal spectrum of the CMB).

  13. Lawrence, Jason,

    Gravitating bodies in the universe tend to counter the expansion.

    This is the basic relationship. Observations suggest expansion exceeds contraction, though. I could ask whether the incredible amounts of energy bound up in mass might be a mitigating factor, but that might be too technical a question for me to follow the math.(Given how much of it is “dark,” are we positive there are no more surprises out there that might balance the two columns?)

    Right, its like doing a survey. One can platt out the land in any way possible, but how you do this does nothing to the hill, valley or dale. The land is what it is.

    Omega=1 does not remotely indicate a balance between expansion and contraction. What it does mean is that space is, on average, flat.

    Given the analogies involved, is it any wonder I might assume “flat” means all the hills and valleys cancel out?

    This still doesn’t explain which of the two metrics of space based on different quantities of light, a stable one described by lightspeed, or an expanding one based on redshift of distant sources, is the real measure of space.

  14. To Jason:

    Omega_m = .23 if you include dark matter, though I though it was a bit higher than this, but I’d have to look it up. I was thinking of luminous matter or “ordinary stuff.” That is only about omega = .05. Agreed the radiation stuff is pretty small, it is way down on the list.

    As for John,

    The surveyor analogue can’t be taken quite the way you think. Maybe navigating on the Earth is a better way to think about it. You are sailing in some ocean and you have your local chart or map. Somebody else is sailing elsewhere far away, and they each have their respective maps. Yet globally we know that one captain is upside down, or nearly so, relative to the other. Yet both captains are using maps which has them standing horizontally. This is what confused people in centuries past about a spherical Earth’s surface. In effect, whether you know it or not, your insistence on using your “local map” and imposing it on another coordinate region globally removed is a GR fallacy similar to what had people confused about living on a round Earth.

    The speed of light is measured on a local frame that is small enough to be considered flat spacetime, just as some local region you see looks like a flat Earth. But globally your chart is different from that of someone elses, which in their local frame is also flat. If you come up with a curved geometric system that patches these maps together you find a more general rule for geometry. This is loosely how general relativity works.

    Lawrence B. Crowell

  15. This is the basic relationship. Observations suggest expansion exceeds contraction, though. I could ask whether the incredible amounts of energy bound up in mass might be a mitigating factor, but that might be too technical a question for me to follow the math.(Given how much of it is “dark,” are we positive there are no more surprises out there that might balance the two columns?)

    No. The energy bound up in mass is what makes it gravitate in the first place. It’s the relationship between the matter and pressure that determines how it gravitates, and normal matter has zero pressure on large scales. The dark energy, whatever it is, has pressure less than -1/3 its energy density. This negative pressure property is what causes its gravitational effect to be an acceleration of the expansion rate (note: this is assuming the “dark energy” isn’t actually a misunderstanding of gravity, that GR is actually accurate on very large scales…but that’s another whole can of worms).

    Given the analogies involved, is it any wonder I might assume “flat” means all the hills and valleys cancel out?

    No. The universe could still be quite spatially curved on a whole and still have all the “hills and valleys” cancel out (by some measure). It’s more of a question of initial conditions or our region of the universe.

    And no, space isn’t “stable” as described by light. Light is redshifted in passage, and can take a very long time to reach what was originally a rather short distance. That’s not an indication of stable space at all. For example, the light that we see in the CMB was emitted about 13.7 billion years ago, a few hundred thousand years after the classical big bang singularity. The surface that emitted those photons was then only a few hundred thousand light years from our present location. The only explanation or why it’s taken so long to reach us is that as space expanded, it had more and more distance to travel.

  16. Slight clarification:

    It’s not the relationship between matter and pressure, but rather between matter’s energy and matter’s pressure that determines how it gravitates. This relationship determines how it collapses, whether it collapses, and how it affects the expansion of the universe as a whole.

  17. Lawrence,

    Even though curvature may be imperceptible at the local level, it still must exist for it be manifest at larger scales. Truly flat space would be a plane held against the surface of the ball.

    The earth is quite literally curved because it is way over on one side of the equation, ie. it is shaped by the contraction effect of gravity. By “flat” space, I take it to mean that the sum total of all universal contraction and expansion effects cancel out. If you want to argue that actually the universe is expanding more the it is contracting and that eventually everything will disappear from sight, no problem, but that means the universe is curved out, not flat.

    I’m not arguing that the speed of light represents flat spacetime, or not. I’m simply saying that we define distances in space using lightspeed as the metric, so if two locations are further apart then they were, as measured by lightspeed, that would be increased distance, if you are still using lightspeed as the metric, not expanded space. So far as we know, the speed of light could be variable, as measured against some unknown dimension, but we don’t know that because at these distances, lightspeed seems the most stable parameter. For example, the speed of light is slowed passing through some mediums, but we don’t say the medium is expanded because it takes light longer to pass through it. We say the medium is stable and lightspeed is slowed.

    Jason,

    And no, space isn’t “stable” as described by light. Light is redshifted in passage, and can take a very long time to reach what was originally a rather short distance. That’s not an indication of stable space at all. For example, the light that we see in the CMB was emitted about 13.7 billion years ago, a few hundred thousand years after the classical big bang singularity. The surface that emitted those photons was then only a few hundred thousand light years from our present location. The only explanation or why it’s taken so long to reach us is that as space expanded, it had more and more distance to travel.

    So you are saying space, as measured by the emitting objects, is expanding faster then the speed of light? Of course, that is classic Inflation theory. Or maybe lightspeed was much slower then and it has been increasing and finally caught up with us.

    It’s the same problem I pointed out to Lawrence; Curvature may be imperceptible at the local level, but it still must exist in order to manifest at larger scales and if spacetime is curved enough at the local level to become apparent at intergalactic scales, then it should be carrying light along with it, as well as the emitting objects, such that while objects might appear to be flying apart, the measured distance doesn’t change over time. A topological analogy might be having two towns on either side of a lake and you don’t have a boat, so while they look only as far apart as the diameter of the lake, to get there means you have to walk around the lake and it takes longer, so you think they must be moving apart, but that’s not how it works.

  18. So you are saying space, as measured by the emitting objects, is expanding faster then the speed of light? Of course, that is classic Inflation theory.

    No. The units are different. The expansion rate of space has units of inverse time. The speed of light has units of distance over time. It’s like saying that a car is faster than a stopwatch: the claim is nonsensical.

    Or maybe lightspeed was much slower then and it has been increasing and finally caught up with us.

    Impossible. The speed of light is intimately tied to the properties of the electromagnetic force. As a result, a changing speed of light results in a change in electromagnetic interactions, which we can be pretty confident were at least nearly identical to the way they are today all the way back to the emission of the surface of last scattering.

    Interestingly enough, some work has been done to attempt to discover if a faster speed of light in the distant past might actually be an explanation for inflation (i.e. what we view as “inflation” was actually a faster speed of light at that time, instead of an accelerated expansion). But it looks like this work just wasn’t promising, and it appears that the scientists working on it have largely abandoned it.

    It’s the same problem I pointed out to Lawrence; Curvature may be imperceptible at the local level, but it still must exist in order to manifest at larger scales and if spacetime is curved enough at the local level to become apparent at intergalactic scales, then it should be carrying light along with it, as well as the emitting objects, such that while objects might appear to be flying apart, the measured distance doesn’t change over time. A topological analogy might be having two towns on either side of a lake and you don’t have a boat, so while they look only as far apart as the diameter of the lake, to get there means you have to walk around the lake and it takes longer, so you think they must be moving apart, but that’s not how it works.

    Okay. But if the universe was static then the CMB wouldn’t exist.

  19. Lawrence Crowell

    John Merryman on Jan 11th, 2008 at 12:33 pm
    Lawrence,

    Even though curvature may be imperceptible at the local level, it still must exist for it be manifest at larger scales. Truly flat space would be a plane held against the surface of the ball.

    —————–

    We might be making progress. A flat plane held against the ball, or its spherical boundary is a tangent plane. Now think of putting postage stamp sized flat planar elements at many points, or every point, on the sphere. How these little planes deviate from each other from point to point tells you how a vector field on the sphere will change. Each local plane has its (x,y) coordinate system, but they are unique to each of these little planar segments. So if you have a vector V at a point with coordinates (x,y) and another vector V’ with coordinates (x’,y’) nearby then there difference between these two vectors V’ – V is

    (V’ – V)^i = nabla_jV^i (x’^j – x^j)

    where x^1 = vector in the x direction and y^2 = y directed vector, and “i” is an index over the coordinate labels. nabla_j is a derivative the vector in the jth direction. In some books this is called a directional derivative. The stuff on the right hand side is telling us what has to change to get V to match up to V’.

    BTW, I hope you have had calculus and know what a derivative is.

    Now this idea gets generalized some, for there are rotational symmetries for these coordinate systems. In the case of an (x,y) coordinate system this is given by a matrix with cosines and sines (easily looked up) of an angle. This structure is then written according to these local transformations (unique on each “postage stamp” tangent to the sphere. This assembly of tangent planes is then given by a tangent bundle, where we think of there being a unique set of local symmetries at every point (or small coordinate chart) on the sphere.

    This gets generalized to higher dimensions with larger local groups. In the case of spacetime this is the Lorentz group (three ordinary rotations plus three hyperbolic transformations called boosts. This can also be extended to the Poincare boosts for translations included as well. The Lorentz group is the structure of special relativity, which is also applied to any small local flat frame. The hyperbolic transformations or boosts have asymptotes, which correspond to the fundamental invariant of special relativity — the speed of light. Each local flat frame has its own Lorentz group, and how these “mesh together” in a tangent bundle describes the curvature of spacetime.

    This is a 101-introduction to the topic of Riemannian geometry, but this is basically how general relativity is mathematically structured.

    The space sometimes called the Hubble frame is either a very large sphere, or it could be flat. Yet the spacetime it is embedded in is not flat, which is why it is being stretched out in this eternal inflation. It is worth pointing out that how one puts this spatial frame into the spacetime is a matter of one;s coordinate choice. It is analogous to a gauge in electromagnetism. We observe galaxies not on this frame, but from light rays on a past light cone. This is a projective space in spacetime — with lots of interesting properties. But I’d better not delve in that direction right now.

    Lawrence B. Crowell

  20. Jason,

    The expansion rate of space has units of inverse time.

    This means the further we look, the faster it recedes?

    The speed of light is intimately tied to the properties of the electromagnetic force. As a result, a changing speed of light results in a change in electromagnetic interactions, which we can be pretty confident were at least nearly identical to the way they are today all the way back to the emission of the surface of last scattering.

    So you agree that lightspeed provides the most stable metric of space and if a source of light were actually receding, in terms of lightyears, this would be increasing distance in stable space, not expanding space?

    Okay. But if the universe was static then the CMB wouldn’t exist.

    Static and stable are not the same. If you have an infinite universe of collapsing gravity wells, balanced by expanding radiation, then there is a definite curvature of space and the further light manages to travel, the more it is curved, so that eventually it is redshifted to the point that the source appears to be receding faster then the speed of light, which creates a horizon line of visibility, but not necessarily of radiant energy, so the spectrum of this energy begins where visible light ends, out about 13.7 billion lightyears. Since it is lower then red, it is black body.

    Lawrence,

    BTW, I hope you have had calculus and know what a derivative is.

    Are you joking? I ran off to join the circus at 16. Actually I changed my mind. Some drifter told me Florida cops were mean, so I ended up working in a polo club in Santa Barbara CA. I did eventually graduate from school, but mostly it’s just self motivated curiosity and my math abilities ran out around trig.

    My appreciation for relativity is due to it helping make sense of my world, as opposed to all the litttle postage stamp sized absolutes.

  21. Well don’t laugh too much at the postage stamp idea. Think of the glitterball in the karioke or dico dance floor that is made of may little flat reflecting elements. It is a sphere approximated by lots of little flat areas pasted together. In calculus there is the limit idea, and if you take the limit that each of these elements becomes very small or infinitesimal and the number of them increase to cover the sphere you recover the sphere. Each of these little flat areas are similiar to a local inertial frame in spacetime.

    Lawrence B. Crowell

  22. This means the further we look, the faster it recedes?

    Effectively, yes.

    So you agree that lightspeed provides the most stable metric of space and if a source of light were actually receding, in terms of lightyears, this would be increasing distance in stable space, not expanding space?

    I’ve already explained with this is nonsense.

    Static and stable are not the same. If you have an infinite universe of collapsing gravity wells, balanced by expanding radiation, then there is a definite curvature of space and the further light manages to travel, the more it is curved, so that eventually it is redshifted to the point that the source appears to be receding faster then the speed of light, which creates a horizon line of visibility, but not necessarily of radiant energy, so the spectrum of this energy begins where visible light ends, out about 13.7 billion lightyears. Since it is lower then red, it is black body.

    Then this wouldn’t explain the temperature of the CMB, it wouldn’t explain the anisotropies we see in the CMB or how those anisotropies are correlated with the distribution of matter in the nearby universe, it wouldn’t explain why we see significant evolution of galaxies (that is, galaxies that are at a redshift of 2 are very different from galaxies at a redshift of 1), it wouldn’t explain the primordial (before stellar processing) abundances of hydrogen and helium, and finally it describes a universe where the energy in every region of space is decreasing with time, a situation that is not stable at all (redshift is a decrease in energy).

  23. Lawrence,

    I wasn’t making fun of it. It is similar to my observation that three dimensions are simply a coordinate system, not space itself and that any number of such frames can be used to define the same space, just like any number of planes can be used to define the surface of this planet. The point about postage stamp absolutes is because people tend to view others from their own set of references and try to fit everything into the same set of coordinates. This goes a long way towards explaining political/relgious conflict. You might say the Arabs and the Israelis used diffferent coordinates to describe the same space.

    In fact, Jason and I have been over this before. His position is that since three dimensions are the most efficient description of space, that they are the ideal form of space and presumably all the messy irregularities are just clutter. My position is the opposite, that three dimensions are just a reductionist model and that the reality is far more multifaceted, so three dimensions are just a mapping device, not the basis of the reality.

    Jason,

    Effectively, yes.

    Could that be due to space being curved, so that redshift is compounded by crossing ever more space, rather then that the universe was simply expanding faster, earlier in time?

    I’ve already explained with this is nonsense.

    Just because you keep pushing your mental reset button hasn’t given me sufficient reason to erase it off my hard drive.

    Then this wouldn’t explain the temperature of the CMB, it wouldn’t explain the anisotropies we see in the CMB or how those anisotropies are correlated with the distribution of matter in the nearby universe, it wouldn’t explain why we see significant evolution of galaxies (that is, galaxies that are at a redshift of 2 are very different from galaxies at a redshift of 1), it wouldn’t explain the primordial (before stellar processing) abundances of hydrogen and helium, and finally it describes a universe where the energy in every region of space is decreasing with time, a situation that is not stable at all (redshift is a decrease in energy).

    It would explain large scale structure as far as we can detect, since it removes any age limit on it. It would explain why redshift is isotropic in all directions without having to describe space as expanding, when electromagnetic properties would seem to maintain a fairly static dimension of space, as measured by lightspeed. It wouldn’t need Inflation Theory and all the issues involved, to explain why space appears flat. It wouldn’t need dark energy to explain why redshift appears as a curvature/cosmological constant, rather then the cooling remnant of a singularity. I certainly don’t have all the answers, but I suspect that you are young enough not to really appreciate how concepts that seem so all encompassing can suddenly pop and everyone(still alive) just walks away. It happens in politics over the course of decades. It can happen in religion over the course of centuries. It can happen alot of ways, even in science. Solid ground is always temporary, but it’s all we have to stand on.

  24. It would explain large scale structure as far as we can detect, since it removes any age limit on it. It would explain why redshift is isotropic in all directions without having to describe space as expanding, when electromagnetic properties would seem to maintain a fairly static dimension of space, as measured by lightspeed. It wouldn’t need Inflation Theory and all the issues involved, to explain why space appears flat. It wouldn’t need dark energy to explain why redshift appears as a curvature/cosmological constant, rather then the cooling remnant of a singularity. I certainly don’t have all the answers, but I suspect that you are young enough not to really appreciate how concepts that seem so all encompassing can suddenly pop and everyone(still alive) just walks away. It happens in politics over the course of decades. It can happen in religion over the course of centuries. It can happen alot of ways, even in science. Solid ground is always temporary, but it’s all we have to stand on.

    Oh, I’m sorry. By “wouldn’t explain” I meant “is completely incompatible with our observations of”. Oh, and by the way, you don’t get redshift from purely spatial curvature. It doesn’t work that way.

    And, by the way, I have changed my mind a great number of times, occasionally resulting in a complete transformation of the way I view the world.

  25. Jason,

    I think I’ve presented my thoughts to the best of my ability and it’s certainly reasonable for you to disagree with what isn’t the standard model. You have done an admirable job of arguing for the established position and I thank you for the time. That said, I have certainly changed my view of things before, but you haven’t closed the case for the current model being on the right track, in my mind, so I suppose we will have to wait for more evidence to come in, one way or another.

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