Tuesday, November 19, 2013

Why positive pressure causes deceleration

...explaining a puzzling sign to Mr Joe Public...

Sean Carroll shouted a completely wrong claim about basic general relativity, namely that the "accelerated expansion has nothing to do with the negative pressure", and I clarified his misguided remarks yesterday.

But it may be useful to mention that Carroll was provoked to write his text by Peter Coles' challenge in the article:
A Dark Energy Mission.
Coles points out that we need a negative pressure (e.g. one coming from the cosmological constant) to achieve a repulsive gravity producing the accelerated expansion in the FRW cosmology.

That may look strange because we think that it's the positive pressure, like the positive pressure of some gas in a bottle, that is trying to make the bottle explode and expand (at an accelerating rate). So why is the sign in front of pressure in the second Friedmann equation opposite to what we expect based on the experience with gases in the PET bottles?

Coles asked his readers to produce an explanation for Mr Joe Public. I don't know this man – except for knowing that he has 2.3 children and didn't manage to sign up for Obamacare. ;-) He has also recorded the song arguing that "you've got to live and learn before your bridges burn" along with his clones and namesakes (see above). But because all the answers at Coles' blog are completely wrong (for example, Phil Gibbs is building on a non-existent conserved total energy in the FRW cosmology), let me try to clarify some of the reasons.

Well, recall that the second Friedmann equation is\[

\frac{\ddot a}{a} =-\frac{4\pi G}{3} \left(\rho + \frac{3p}{c^2}\right)

\] Here, \(\rho\) is the mass density and I added the factor of \(1/c^2\) to convert the pressure (energy density) to the units of mass density. Also, \(a\) is a scale factor that may literally be understood as the distance between two particular galaxies (which is a function of time \(t\)). It's helpful to return to the \(c\neq 1\) units because we may immediately recognize that the pressure is a (special) relativistic correction of a sort.

Before we begin, I want to say that in locally Minkowskian coordinates, \(T_{00}=T^{00}=\rho\) is non-negative everywhere in Nature while the doubly spatial components \(T_{ii}=T^{ii}=p\) are the pressure if we talk about isotropic environments. Note that we got no minus signs from raising or lowering the indices because two indices of the same kind were raised or lowered simultaneously. Only \(T^{0i}\) and \(T_{0i}\) would have opposite signs. Of course, the components \(T^0_0\) or \(T^i_i\) with one upper or one lower index would have the opposite signs than the doubly upper or doubly lower components.

Fine. Let me return to the mission.

First, let us answer the following question: What's wrong with the argument that a positive pressure, like the pressure of gas in the PET bottle, wants to repel the walls of the bottle and cause an accelerated expansion of the bottle (and the Universe)?

Well, there's nothing wrong with the claim about the bottle. But this simple effect of pressure is a completely different effect than the effect of the pressure in Friedmann's equations. There are two main differences. In the case of the bottle,
  1. the force actually depends on the gradient of the pressure rather than the pressure itself
  2. the force isn't proportional to Newton's constant \(G\) at all because it is not a gravitational force at all
  3. it's actually the force we calculate quickly; in the case of the Universe, it's the acceleration and we have to multiply it by the mass of the "probe" as well to get the force
To explain the first point, let me mention that you may lower a cube of volume \(V\) to a vessel with water. The hydrostatic pressure \(p=h\rho_{\rm water} g=-z\rho_{\rm water} g\) increases with the depth and the density of force acting on the cube is simply the gradient of the pressure (with the minus sign):\[

\frac{d\vec F}{dV} = -\nabla p

\] If you realize that \(-\nabla p\) is simply \(\rho_{\rm water} g\) pointing in the positive \(z\) direction, you may easily integrate this force density and get the total upward force \(V\rho_{\rm water} g\) acting on the cube. In other words, we have just derived Archimedes' principle. Note that the upward force only exists because the pressure depends on the depth. If it didn't depend on the depth (or location), all the forces would cancel each other.

To understand the second point, let me mention that the water has a nonzero mass and this mass therefore induces a gravitational field as well. But we actually never measure the gravitational force caused by this water. This force is proportional to the tiny constant \(G\) and decreases as \(1/r^2\) with the distance from the molecules of the water. This has clearly nothing to do with the reason why the cube is being pushed upward. The upward force is proportional to \(\nabla p\), just the gradient of the pressure, rather than \(Gp\), the pressure multiplied by Newton's constant.

Well, the hydrostatic pressure depends on the depth of the water because of gravity but it's the gravity caused by the Earth. If you remove the Earth, the water will lose the ability to push the cube upward (there won't be any preferred upward direction, after all).

Fine, so this was a negative explanation, an explanation why the term that depends on the pressure in the Friedmann equation is a completely different term than the usual pressure that we know from gases and liquids.

But why is there the term \(+3p\) in the Friedmann equation for the acceleration?

As I have already indicated by the restored factor \(1/c^2\), it is a relativistic correction, something that only matters if you appreciate all the (special) relativistic phenomena that occur when the velocities are no longer negligible relatively to the speed of light \(c\).

We may talk about the mass density and pressure and say that the non-relativistic limit is only applicable if \(p/c^2\) is negligible relatively to the mass density \(\rho\). For everyday materials, this is the case because \(p,\rho\) are numbers of order one in the SI units while \(1/c^2\) is a tiny number. But the corrections scaling like \(1/c^2\) are there, anyway. If you want to be accurate enough (for example because you are able to perform very accurate measurements), you can't omit them. And for all forms of matter except for dust, \(p/c^2\) is actually "comparable" to \(\rho\).

Now, I must emphasize that the only "truly controllable" way to derive the Friedmann equation is the full calculation in GR. You start with Einstein's equations and simply insert the Ansatz for the metric tensor – the Ansatz that defines the FRW cosmology (a uniform, isotropic universe whose geometric properties – well, the overall size dictated by the linear scale factor \(a\) – only depends on time). With this Ansatz, Einstein's equations reduce to the simpler Friedmann equations.

The problem is that Mr Joe Public didn't understand the previous paragraph because he had enough trouble with the Obamacare website. Can we please give a physical justification of the \(\rho+3p/c^2\) pattern while avoiding words like Einstein's equations and Ansatz? The answer is the same as the answer that turned out to be wrong in the case of Obamacare: Yes, we can. ;-)

I don't want to perform the full nonlinear, exact relativistic calculation – only to estimate the leading correction to \(\rho\) in a particular environment with some particular (positive) pressure, namely some gas.

What we have is a grid of galaxies that may be arranged to cubic boxes. Look at one edge of a box. The vertices host two galaxies whose relative distance is \(a(t)\). That's the proper length of a "spatially" straight line through the \(t={\rm const}\) slice. (It is not a geodesic in the whole spacetime because a geodesic would prefer to visit other values of \(t\) as well.) Why is the expansion of the Universe accelerating or decelerating in the first place?

We often hear (and say) that the Hubble expansion is something miraculous, intrinsically general relativistic, that cannot be visualized as a simple explosion that shot galaxies in different directions. But the truth is that when we talk to Mr Joe Public, it is actually totally legitimate to visualize the Hubble expansion as a simple explosion. The reason is that a sufficiently thin hypercylindrical "strip" of the curved four-dimensional spacetime (in our case, one of the small boxes in the cosmological grid, taken at all times \(t\)) may always be flattened i.e. embedded into the Minkowski space. And once the galaxies are embedded in this auxiliary Minkowski space, their receding motion simply has to have the usual special relativistic interpretation – after all, it's just some relative motion.

(This special relativistic visualization of the expansion of the Universe has one undesirable feature: it invites us to think that one of the galaxies simply has to "at rest", in the center of the primordial explosion, while others are not at the center. But this conclusion is actually unnecessary even in special relativity – all motion is relative even in special relativity. So it's not such an insurmountable problem. The real virtue of a proper general relativistic description is that it allows us to consistently avoid the subtle questions how to "cancel" the infinite force from the galaxies on the left side with the infinite force from the galaxies on the right side and similar issues.)

Why is the expansion of the Universe decelerating for \(p=0\), \(\rho\gt 0\)? It's simple: it's because of the gravitational field created by the matter inside the box of volume \(V=a^3\). This box has mass \(m=\rho V=\rho a^3\). Because of the inverse square law, we may see that the acceleration at the distance of order \(a\) from the center of the cube, e.g. at one of the vertices, is of order \[

a_{\rm acc} = \frac{Gm}{a^2}=\frac{G\rho a^3}{a^2} =G\rho a.

\] So this box filled with some mass of density \(\rho\) will simply attract the galaxies at the vertices with the acceleration comparable to \(g\rho a\). To derive the numerical factor, one would either have to deal with messy questions about the gravitational fields of cubes or do the general relativistic calculation. But the acceleration \(a_{\rm acc}\) is nothing else than the \(-d^2 a / dt^2\). I included a minus sign because a positive mass inside the box will tend to shrink \(a\) in the far future (deceleration i.e. attractive gravity) while \(a_{\rm acc}\) was defined to be positive by the displayed equation above.

(Note that you don't have to worry about the attractive forces exerted by the other boxes on the same galaxy. They will be responsible for the evolving size of the other cubes – which will of course evolve in the same way as our box. But the assumption of a uniform Universe allows us to "localize" the whole problem and study the effect of a single box only.)

Up to the numerical constant \(4\pi/3\) and assuming \(p=0\), I have just derived the equation\[

\frac{d^2 a}{dt^2} = - \frac{4\pi}{3} G\rho a.

\] We usually multiply it by \(1/a\) so that \(a\) disappears from the right hand side while the left hand side becomes \(\ddot a / a\). Great. So a part of the Friedmann equation could now be clear to Mr Joe Public. It is nothing else than the motion of probes – the galaxies – in the gravitational field produced by some distribution of mass – and it's only the mass in the single cube that mattered for our problem. Again, let me stress that this explanation involving the usual "gravitational attraction" you know from the basic school isn't just an analogy: general relativity is still a theory of gravity so even when we discuss the expansion of the Universe, the physics still has to agree with some basic mechanisms of gravity (up to some nonlinear corrections for strong fields and fast motion etc.).

Finally, we would like to see why \(\rho\) should be replaced by the relativistically corrected \(\rho+3p/c^2\). The previous section already boasted a headline that promised such an explanation but we're finally getting there.

Imagine that the cubic box in between the galaxies (the edge has length \(a\)) is filled with gas of mass density \(\rho\) whose molecules have the velocity \(v\) and mass \(m\). What is the pressure? Well, start with a molecule bouncing from the left to the right and back. It takes the time \(t=a/v\) for the molecule to get to the other side and during each collision, the molecule changes its velocity by \(2v\) so it deposits the momentum \(2mv\) to the wall. The momentum deposited to the walls per unit time is \(2mv/(a/v) = 2mv^2/a\) and that's nothing else than the definition of the (average) force. Here, \(v\) was really \(v_x\) but we must add the motion (and momentum transfer) to the remaining \(6-2=4\) faces of the cube which is equivalent to replacing \(v_x^2\) by simply \(v^2\), the total one. Finally, the total force \(2mv^2/a\) must be divided by \(6a^2\), the surface of the cube, to get the pressure.

We see that the pressure contributed by one molecule is \(mv^2/3a^3\). Multiply it by the number of molecules \(N\) to get the total pressure \[

p= \frac{Nmv^2}{3a^3}=\frac{M_{\rm total}v^2}{3a^3}=\frac{\rho v^2}{3}.

\] We explicitly see that \((p/c^2)/\rho\) goes like \(v^2/3c^2\). It approaches a number of order one exactly when the speed of the molecules (or whatever the gas is composed of) approaches a speed comparable to the speed of light. Note that \(3p\) is nothing else than \(\rho v^2\) and\[

\frac{\rho+ \frac{3p}{c^2}}{\rho} = 1+\frac{v^2}{c^2}.

\] We're assuming \(v\ll c\) but we want to derive the first subleading correction. Why is the force going like \(1+v^2/c^2\) and not just \(1\)?

The answer may be found by a careful counting of the Lorentz factors\[

\gamma = \frac{1}{\sqrt{1-v^2/c^2}} \approx 1+\frac{v^2}{2c^2}+\dots

\] How does the counting go? It's relatively simple:

The rest mass of the molecule is actually smaller than the total mass, by a factor of \(\gamma\) – that's the usual relativistic increase of the total mass with the velocity. So if we switch to the rest frame of the molecule where the gravitational field caused by the molecule is calculable by the usual methods, we could think that a higher speed (and nonzero pressure) makes the total force that is decelerating the Universe smaller.

However, there are actually 3 factors of \(\gamma\) that go in the opposite direction, so that the total deceleration is \(\gamma^3/\gamma\approx 1+v^2/c^2\) times the non-relativistic expectation. Why are there three "positive" factors of \(\gamma\)? All of them are related to the Lorentz contraction.

One of them appears because we want to switch to the rest frame of the moving molecules (those are the source of the gravitational field and we only know how to describe this gravitational field in the sources' rest frame) but in this frame, the boxes are Lorentz contracted and we may therefore squeeze \(\gamma\) times more molecules into the same cube in the coordinate space.

Two additional powers of \(\gamma\) emerge because the gravitational acceleration goes like \(1/r^2\) and this \(r\) is Lorentz-contracted (reduced) which means that \(1/r^2\), the acceleration, is increased by the factor \(\gamma^2\approx 1+v^2/c^2\). Just to be sure, only the component of the force that goes in the same direction as the separation of the two galaxies – separation whose magnitude we call \(a\) – will be nonzero.

At any rate, when you combine the factors together, the enhancement from the nonzero velocities is by \[

\gamma^2\approx 1+\frac{v^2}{c^2}\approx \zav{\rho+\frac{3p}{c^2}}\frac{1}{\rho}

\] as we wanted to prove. Once Mr Joe Public learns something about general relativity, he will notice that the stress-energy tensor enters Einstein's equations linearly so the pressure must enter the second Friedmann equation linearly, too. That's why the form of the acceleration involving \(\rho+3p/c^2\) is actually exact even for \(v/c\sim 1\).

The environment with negative pressure cannot be visualized as a gas of molecules (we would need imaginary velocities to obtain a negative pressure) but due to the linearity in \(p\), it's clear that the environment with a negative pressure will contribute to the accelerated expansion of the Universe.


  1. Hi Lubos,

    This is somewhat off topic but I don't know where to ask this. I am an undergraduate student and want to learn quantum mechanics. I have read your article http://motls.blogspot.com/2011/05/copenhagen-interpretation-of-quantum.html and an article of Englert you gave a link of a couple weeks ago. These articles made many things clear for me, but for a systematic study I couldn't found a book which touches these points. Today I went to library and looked to 7 books, some of them are pretty common for undergraduate and graduate courses, but none of them talks about, for example subjectivity of wave function. I have seen one that says quantum mechanics is nonlocal. So can you name a book, which is good about foundations of quantum mechanics. It doesn't need to contain sophisticated math or applications, I can learn them after I have understood physical ideas behind quantum mechanics completely. Thanks.

  2. I don't really understand what Sean is getting at. A positive energy density leads to a negative pressure which causes expansion. By saying that dark matter doesn't get diluted by the expansion has nothing to do with a positive energy density---it is still positive.
    Because he doesn't intuitively understand what the equations mean doesn't mean that others don't.

  3. john,
    What you want to understand is not easy to accomplish because, like everyone else, you can only try to form a mental picture in terms of what you already know, classical mechanics. This really does not work as evidenced by countless posts on TRF and elsewhere. Lubos has given many beautiful explanations of the basics and I highly recommend all of them but if you don’t get it, which is likely, I can think of no better introduction than the first chapters of Dirac’s wonderful textbook on the subject.
    Still, you will have to deeply ponder situations where classical mechanics utterly fails, such as the double slit experiment. When the double slit experiment makes perfect sense to you then, and only then, will you grasp what Lubos is getting at.
    Just don’t think it will be easy and please don’t fall, into the trap of thinking prematurely that you do understand. Above all, keep asking questions.

  4. Leonard Susskind has a complete course online and there are courses at EdX etc.
    The lecture by Sidney Coleman, "Quantum Mechanics in Your Face" is a good place to start...not that I am any expert on QM.

  5. Ok, read it. Wow! An amazing description.

  6. Thanks, Gordon, except that this thread is about something else. ;-)

  7. Thanks for your excitement, Dimension10!

  8. Thanks , I will look to them.

  9. Hi gene,
    Thanks for your reply. I didn't have a quantum mechanics course yet so my knowledge comes from various places where I looked occasionally. I don't know many things for example I don't know what is a density matrix. To understand completely what lubos says one should have a fair knowledge of quantum mechanics. I couldn't begin a through study because most of the books are very weak on foundations. For example they give the interpretation that collapsion of a wave function is a physical process. Or for solution of EPR paradox they say one can't send instantaneous message. They don't say wave function is about subjective knowledge of observer about the system (this is what I mean by subjectivity of wave function) different observers can have different wave functions and there is no other meaning of wave functions. After I learned this many of my confusions disappeared. But I don't know what measurement really is or why x and p becomes a classical observables while other states dont. I will look to Dirac's book but I don't think it has answers for some questions I have for example decoherence, which I think is necessary to completely understand quantum mechanics even though Copenhagen interpretation is correct.

  10. This is an easy starter, but it should rather be considered as a help to better understand more comprehensive textbooks / courses


    Some people consider the not negligible amount of typos as a big nuisance, but they are easy enough to spot such that they do not hamper the good introduction and explanation of the most important concepts in QM.

    Typos and bugs in the theoretical physics demystified books can be reported here BTW


  11. I think there is some misunderstanding. He is not saying what you imply that he does. He is concerned only with pedagogy. He doesn't say that this or that is wrong, he says that it would be the wrong way to explain it that way to your kids because they will not get it. And he suggest a different explanation.

  12. It doesn't matter. His different explanation is just *wrong* and it is a problem.

    The accelerated character of the expansion is *undoubtedly* due to the negative pressure. One may understand it or one may misunderstand it or one may deny it but that lists the possibilities he can do about this fact.

    The beginner who plans to master the equation *must* learn the right reason and neither "pressure" nor "negative" are really tough words.

    The layman who never wants to learn GR at any quantitative level may have a problem with anything here - that the pressure may be negative; that the pressure affects gravity; that the cosmological constant carries a negative pressure, and other things - but that's just fine if he is confused by these things because these things arguably may be confusing for the layman. It is right to think that the negative pressure is "bizarre" because the pressure we encounter in the environments is positive. It is right to be confused that not just matter density but also pressure influences the curvature of the spacetime - because in Newton's gravity, only matter density matters. And so on.

    But these "new facts" are critical for the accelerated character of the expansion and it's just wrong, wrong, wrong to mask this fact. It's wrong, wrong, wrong to replace the right statements by wrong ones because they might be "more acceptable".

    It is just wrong, wrong, wrong for superficial and marginally dishonest folks to sell an increasing number of distortions and untruths as physics.

  13. Why should I learn all this complicated math in order to understand General Relativity? Its very premise tells me not to bother to spending the time. Einstein never addressed the basic problem with Newton's theory. Newton had serious concerns about how some, innate ability of mass had to attract other mass. Einstein just gives mass an even more fantastic ability than to just attract other mass. He gives it the ability to warp space. And we have been living in Alice in Wonderland ever since. Now for GR to workk we have to assume that the universe is filled with material 95% of which we cannot understand. The recent, negative LUX results just makes the highly trained GR mystery lovers all the more confident in their excessively complex theory. If we assume as Kepler did before Newton got us believing in the mysterious powers of mass that it is the light of the sun attracts that attracts the surrounding mass, then the posted graph shows some well established facts
    can be used to explain cosmic acceleration.

  14. You are lucky that I am @work and therefore can not log in, so my downvote has to wait till I'm @home ... ;-)

    But maybe others can help out with clicking the button ...

  15. There are already several geniuses on this forum (some of them have been silent recently but Uncle Al has been making up for that) and I am not sure if our host believes that there is a vacancy for another one. Moreover, you will be the first one to come out of the closet as a math-illiterate and I doubt that this is going to count in your favour.

  16. Thanks for reply. Although its name is a distraction (because of past experince with books with similar name) I will look to it if I can find it.

  17. No worries! I just keep on waiting those Juno's Earth flyby results. Meanwhile, I have other duties to fulfill, like daytime job and family :-)

  18. Perhaps laymen can get a mental image of negative pressure if you tell them to think about tension, as in a solid (or even in a liquid if they happen to know how tall trees work).

  19. You should stop dwelling on the idea that I am either a math-illiterate or genius and focus on the confirmed observational finding that the "dimming of the universe" slightly precedes the onset of "cosmic acceleration". What on earth would cause cosmic acceleration? How many people believe its the cosmo-illogical constant?
    If experiments show that light is gravitationally attractive, an idea that should have been thoroughly tested
    years ago , then the coincident of these two rare "global" events--cosmic acceleration and dimming of the universe--will provide a "close to experience" explanation of why the universe is unexpectedly accelerating. Not only my self finance experiments show that light is "attractive" but now also A. Dmitriev''s experiments do as
    well as PE Shaw's 1916
    experiments .http://arxiv.org/abs/1201.4461

  20. Happy to oblige :)

  21. Have been wondering the same thing (why the boss tolerates Mr. Fred [not Mr. Ed, who is a horse]).

    My best guess: TRF's radar defenses are tuned to only one crackpot frequency. Just as there are concentric circles in Dante's Hell, there are different wavelengths of crackpottery. The boss is concerned only with the longest and least egregious, people who accept all of classical physics and some of quantum mechanics but are seriously confused about parts of QM.

    But as one descends into the madness, shorter and more energetic wavelengths appear. Perhaps the most popular is the Einstine [sic] Is Wrong faction. The present specimen eschews not only Einstein but throws overboard the 19th, 18th and 17th century as well: Bohr Was Wrong, Einstein Was Wrong, Maxwell Was Wrong, Newton Was Wrong.

    But why stop there? That Archimedes guy never seemed too trustworthy, either. Inductive reasoning is the source of all our woes. It all began when the Monolith arrived and taught us the use of hand tools. We must subject humanity to a cleansing bath of gamma rays and go back to the state of nature before then.

  22. Indeed, I think I already suggested more or less the same thing to another "specimen". Actually, it seems to me that the "Einstein is wrong" faction largely overlaps with the "I am the new Einstein" faction and if you suffer to read through the ramblings of almost any member of either of these, you will arrive at a reference to "my theory, which has not received the financing that it deserves" or (see below) "my self finance experiments".
    "Self finance" - aren't you touched by the bitterness of the phrase?

  23. Thanks Eugene :-)

    I was now finally able do insert downvotes at appropriate places too ...

  24. You're welcome :-)

    The Demystified books are no Dummy books ...

    They introduce the most important ideas and concepts of the repective topic at a serious technical level in really fine grained steps.

  25. Okay. Have you ever wondered why all the crackp^H^H^H unsung geniuses seem to cheerfully coexist. They all attack "mainstream science" but not one another. Shouldn't their waves cancel one another out via destructive interference (in a trough) or merge into something else (in a peak). Well, if their respective frequencies differ only slightly then the lowest common denominator of the respective wavelengths could be really high and interference rare.

    But what I think is more important is that they see themselves and each of their "colleagues" as participants in a lottery. I'm sure you've seen many quotations from famous scientists to the effect that for a breakthrough discovery, you need talent, education, hard work... and luck.

    Our "friends" harbour no doubts concerning their talent, so that's taken care of. As for education and hard work... why bother with that when they don't guarantee a breakthrough because you still need luck? Luck could even be the essential ingredient!

    They see themselves as playing a lottery. Any string of characters could be the winner, although natural-language words are preferred because easier to handle. The winner will be chosen, at random, by nature, not some ivory-tower resident bent on protecting his turf.

  26. Well, I am not quite sure of that. The reason for my doubt is that all of them believe that they are victims of great injustice because people ignore their "work" and even refuse to read it. But, they resolutely refuse to read each other's work or even comment on the posts of the other "geniuses". I think a better explanation is the one that I learned about from Gordon:


  27. Dear Eugene, I don't read this crank's comments, just sometimes downvote them if the button is nearby.

    The reason why my tolerance for this nutcase is higher is that he is less aggressive *and* it is more self-evident to almost everyone - I believe - that he is a lunatic. So I suppose that by allowing him to post him here sometimes, I am helping his health.

  28. Only recognition by "mainstream science" would provide the ego gratification that they crave. Hence their complaint is not about "people ignoring their work" but about bona fide scientists not doing so. Interesting story about the "Three Christs of Ypsilanti", hadn't heard about it.

    I'll stop pontificating about crackpots now. There is a danger in overdoing it, one may become smugly complacent and lazy by feeling superior to them, which is too easy to do.

  29. I think that more of our revered scientists who view this blog should be worried about Christopher Stubbs reaction to the fact of cosmic acceleration. It is interested that in his colloquium he has picture of a ostrich sticking its head in the sand. He mentions the fact that cosmic acceleration brings to light a 120 orders magnitude discrepancy between theory and observation. He explains "Why dark energy constitutes a crisis in fundamental physics" . My paradigm shifting radiation-based gravity theory was developed way before the observation of cosmic acceleration. i hope you will agree that light output in the universe can diminish over comic time while the amount of mass in the universe cannot as easily decrease over cosmic time. But is light attractive? I have experiments showing that it is, These results are confirmed by A. Dmtriev' and P.E. Shaw 1916 work. All three experiments bring into question the idea that if its mass is given an amount of heat given by E its mass will increase by and amount m =E/c^2.
    C Stubbs' talk:

  30. Dear Lobos, what about this idea on the origin of the cosmological constant ? http://xxx.lanl.gov/abs/0711.4897 Thanks

  31. The minimum speed is interesting - well, the minimum acceleration is a theme that I consider most promising *assuming* that we will have to develop some MOND theory - but due to the Lorentz-breaking in that paper, I think it's wrong to call the resulting effect "cosmological constant". By definition, the cosmological constant is a coefficient of a Lorentz-invariant term in a Lorentz-invariant theory.

    Once we break it, a can of worms is opened and almost everything potentially ceases to work. Sorry, I won't read that paper - or any superficially similar paper on DSR etc.

  32. Under what transformation are the Xi scalars? I'm confused about the information they contain. They are supposed to represent the coordinates of all the particles (as well as interactions) or do the particles come about after some dynamics?