Monday, November 18, 2013

The expansion is accelerating due to negative enough pressure strings attached...

In recent years, I got used to the fact that Sean Carroll is confused about some very basic physics – the postulates of quantum mechanics as well as thermodynamics (and the very basic insight due to Boltzmann and others that its laws are microscopically explained by statistical physics and not, for example, by cosmology). And I won't even threaten your stomach by memories of the Boltzmann brains, doomsdays, and similar delusions.

But I thought he could rationally think at least about classical general relativity. His book was pretty good, I thought, although I have never read the whole volume. However, I don't think so anymore after I finished reading Carroll's insane tirade called
Why Does Dark Energy Make the Universe Accelerate?
in which he tries to assault an elementary fact that the reason behind the acceleration of the expansion of the Universe is the negative pressure (caused by dark energy).

His crusade is made even more paradoxical given the apparent fact that he knows the equation and other key pieces needed to understand why it's accurate to say that the acceleration is caused by negative pressure. But like a schoolkid who has just mindlessly memorized an equation but can't understand what it means, he just can't sort out what the basic implications of the equations are. So he wants to "ban" the fact that the negative pressure is the reason for the acceleration from expositions of cosmology. You may imagine that a progressive (i.e. Stalinist) like himself thinks that such a ban would be "a great step forward". Bans aren't a good step forward, especially not bans of key scientific insights.

(Brian Greene would be among those whose books would be banned; he wrote a crisp explanation of these matters in The Hidden Reality. Tony Zee's GR book would be on the black list, too. Zee mentions beginners' i.e. Carroll's confusion of the velocity and acceleration in 2nd paragraph on page 500 – and more generally, between pages 499 and 507.)

Since the late 1990s, we've known that the Universe was not only expanding but the rate of the expansion was increasing. It was a surprise for many because most people were expecting that the rate was slowing down. The substance driving this expansion is "dark energy" – the cosmological constant with \(p=-\rho\) is the simplest and most natural "subtype" or "more detailed explanation" of dark energy that is so far compatible with all statistically significant experimental results.

The property that allows dark matter or cosmological constant to accelerate the expansion is its negative pressure \(p\lt 0\). Why is that? Well, it is because of the so-called second Friedmann equation\[

\frac{\ddot a}{a} =-\frac{4\pi G}{3} (\rho + 3p)

\] The numerator on the left hand side contains the second derivative of the scale factor \(a\). You may literally imagine that in some units, \(a\) is nothing else than a distance between two particular galaxies (well, the proper length of a line that connects them through the \(t={\rm const}\) slice which is, let's admit, not a geodesic, but it is some coordinate distance, anyway).

The second derivative of this distance is fully analogous to the acceleration \(a_{\rm acc}=-\ddot h\) of a ball that you threw somewhere. Note that Earth's gravity implies \(a_{\rm acc}=-\ddot h=g\) which means that the ball will ultimately fall down (unless its speed exceeds the escape velocity: we would have to modify the equation if the ball could reach substantial distances from the surface) towards the Earth.

Note that we use the convention in which a positive acceleration \(a_{\rm acc}\gt 0\) means that the ball is attracted to the Earth i.e. the second derivative of its height is negative. That's why we had to insert the minus sign.

The second Friedmann equation is completely analogous. It's not just some vague popular analogy; it is a mathematical isomorphism. The distance between two galaxies is fully analogous to the distance between the ball and the Earth's surface. In both cases, they are attracted by the gravitational force (of a sort). In the second Friedmann case, the gravity follows somewhat more accurate laws imposed by the general theory of relativity – the Friedmann equations are what Einstein's equations of GR boil down to if we assume a uniform, isotropic Universe.

You may see that the role of the Earth's gravitational acceleration \(g\) is being played by\[

\frac{4\pi G}{3} (\rho + 3p)

\] The minus sign in front of the right hand side is there for the same reason as in the case of the ball: attraction (deceleration of the outward speed) is identified with a negative second derivative of "the" quantity (the height of the ball or the distance between two galaxies).

So the total force is "attractive" if\[

\rho + 3p \gt 0.

\] For example, if the Universe were filled with the dust only, and the dust has \(p=0\), this expression would surely be positive and we would get an attraction i.e. decelerated expansion. A positive energy density implies attraction for the same reason why the Earth's positive energy (and energy density) is able to attract the ball. Ordinary gravity is simply attractive. If the Universe were filled with radiation and nothing else, \(p=+\rho/3\) (with the plus sign) and the two terms would actually have the same sign and double: an even clearer deceleration.

The type of matter that has \(p=-\rho/3\) is actually "cosmic strings". If the Universe were filled with cosmic strings only (in chaotic directions), they would contribute nothing to the acceleration. Cosmic domain walls (membranes of a sort) would have \(p=-2\rho/3\) and the expression would already be negative. The domain walls would make the expansion accelerate.

Similarly, the cosmological constant – the most motivated type of dark energy – has \(p=-\rho\) so \(\rho+3p\) is negative. In general, you see that you get an accelerated expansion if \(p\) is not only negative but smaller than \(-\rho/3\),\[

p \lt -\frac{\rho}{3}.

\] This is the only refinement of the claim that "a negative pressure is the cause of the acceleration". In fact, we need a "sufficiently negative pressure", one obeying the inequality above. But otherwise the statement is 100% accurate – and not just at the level of popular presentations. It's also completely accurate to say that the total gravitational force operating in between the galaxies becomes repulsive – you may even call it "antigravity" – when the pressure in between is sufficiently negative.

Carroll tries to claim that there is something wrong with the proposition that "the acceleration is caused by a [sufficiently] negative pressure" but his argumentation seems utterly irrational. Well, the core of his would-be argument is probably the following:
But, while that’s a perfectly good equation — the “second Friedmann equation” — it’s not the one anyone actually uses to solve for the evolution of the universe. It’s much nicer to use the first Friedmann equation, which involves the first derivative of the scale factor rather than its second derivative (spatial curvature set to zero for convenience):\[

H^2 \equiv \zav{ \frac{\dot a}{a} }^2 = \frac{8\pi G}{3} \rho

So Carroll told us that we should switch to this equation because "people use it more often" and it is "nicer". The problem with this would-be justification is that it is no justification at all. If an equation is used more often or looks "nicer" to someone (for other irrational reasons), it does not imply that this equation is the right equation to explain a pattern or to answer a question.

In this case, we want to explain why the acceleration is negative and the acceleration is simply related to the second derivative of the height or the second derivative of the scale factor \(a\). The last displayed equation above, the first Friedmann equation, doesn't include the second derivative \(\ddot a\) at all, so it can't possibly be the right equation that tells us whether the acceleration is positive or negative!

I am stunned that Carroll isn't capable of figuring this simple point out.

So the mathematical formalization of the reason why the expansion is accelerating is the second Friedmann equation and it doesn't matter a single bit whether this equation is used more often or less often to calculate other things or answer other questions.

What is the alternative proposition that Carroll proposed instead of the correct one? It isn't quite clear but it seems that it's the bold face sentence below:
Second, a constant energy density straightforwardly implies a constant expansion rate \(H\). So no problem at all: a persistent source of energy causes the universe to accelerate.
But this sentence is just incorrect. The energy density carried by dust or anything else is "persistent" in the sense that it remains nonzero forever but the dust implies a decelerating expansion (much like most other known types of energy density). If the word "persistent" were interpreted as "constant", the sentence above would be marginally correct but it would completely obscure the reason why the energy density is able to stay constant in an expanding Universe. The reason for this is the negative pressure, too; Sean has only offered a sleight-of-hand to mask the actual reason, the negative pressure. Even a German blogger knows that.

The relevant quantity for the sign of the acceleration is \(\rho+3 p\) and not just \(\rho\) (or by \(\dot\rho\)) as Carroll incorrectly suggests. This influence of the pressure on the curvature of the spacetime (in this particular case, the acceleration of its expansion) is one of the "refinements" that general relativity brought us relatively to Newton's gravity where only the total energy density mattered for all gravitational fields. In GR, the whole stress-energy tensor (not just the energy density but also the pressure and the density of momentum etc.) matters for various aspects of the spacetime curvature.

Carroll correctly states that the first Friedmann equation and the second Friedmann equation are consistent with one another because one may be derived from the other using the general relativistic form of the energy conservation law (which does depend on the pressure as well). All this stuff is OK but it changes nothing about the fact that he gave a completely incorrect answer to the key question which of the equations is the right one to calculate the sign of the acceleration of the expansion of the Universe.

These sentences of mine are no "popular presentations" and surely not "misleading popular presentations" and whoever understands them really understands what drives the acceleration etc. – it is not just an illusion of the understanding – while Carroll apparently does not understand these basic facts. He does not understand that the pressure has become relevant for some questions about the spacetime curvature – in particular, for the question whether the expansion is accelerating.

The relevant equation is unquestionably the second Friedmann equation, whether a pervert finds it nicer or not, and I urge all writers to keep on writing the absolutely valid claim that the negative pressure is the reason and notice that Sean Carroll is just [being?] an idiot.

And that's the memo.


  1. ... no strings attached ... :-D !?

    Then I can read it (looking forward to reconsider some cosmology) ... :-P ;-)

  2. This is such a nice and clear explanation why the second Friedmann equation and therefore \rho + 3p is the right quantity to look at in the context of questions about the accelerated expansion, that really every dimwit should get it .. :-)

    Explanations about the derivation of the equations of state for cosmic strings and domain walls I would appreciate and welcome here :-P

    Hope they dont shoot me down ...

  3. kashyap vasavadaNov 18, 2013, 3:43:00 PM

    Hi Lubos:

    Possible typo."negative expansion is the reason for the acceleration". You mean negative pressure.

  4. Dear Lubos, what is the relation between p and pho for gluons and also for quarks? What kind of (exotic) matter would imply p = - pho? Thanks

  5. Dear NumCracker, all matter that is made of "isolated slow particles" has p=0 at the end because it behaves just like gas and the usual derivation pV=nRT for the gas applies. If the molecules of the gas, regardless of their composition, are much slower than the speed of light, their pressure on the walls is negligible relatively to the mass density times c^2.

    If the gas-like particles are faster, p goes up, up to p=+rho/3 that you get for radiation at v=c.

    Environments composed of something localized may only be negative if the "something" is extended. The values for cosmic strings and cosmic domain walls (long 1D, 2D objects) are p=-1rho/3 and p=-2rho/3.

    Only the cosmological constant - dark energy - has p=-rho. It can't be "composed" of anything - it must be a 3D environment uniformly filling the space. You may write it as p=-3rho/3, and add it to the examples in the previous paragraph. It's no coincidence. The derivation of the -1/3 and -2/3 constants may be done "just like" the derivation for the cosmological constant.

    Inside particles, e.g. inside protons and neutrons where quarks and gluons live, the stress-energy tensor is a fluctuating quantum variable, so you may find regions both with positive and negative values. The averaging over space gives you those p=0 through p=+rho/3 I mentioned previously.


    for an extended version of this comment.

  6. kashyap vasavadaNov 18, 2013, 8:04:00 PM

    Probably a dumb question. Is there any relation between cosmic strings and strings of string theory. Or it is just in the name?

  7. They might be the same thing.

    A cosmic string is a 1-dimensional object without any internal "atomic" structure. Such objects are predicted by unified theories - by grand unified theories and string theory in particular.

    In string theory, one may typically have several types of cosmic strings that are predicted. They include the fundamental strings - they're the very same strings that elementary particles are containing inside - they are just stretched to astronomical sizes!

    Other cosmic strings may be p-dimensional branes wrapped on (p-1)-dimensional manifolds inside the compact dimensions so that 1 spatial dimension (plus 1 time) is left.

  8. If the overall shape of space (re the sum of the interior angles of a triangle defined by three intersecting light beams in free fall) averages slightly hyperbolic rather than exactly Euclidean, doesn't the balloon inflate itself?

  9. Lubos:

    You've confused me. Your write that (rho + 3 p)>0 for "attraction" (gravity), and then you write that when p<-rho/3 you get "accelerated expansion". Shouldn't you have written p>-rho/3?

  10. No, Lino, there's no bug here. rho+3p > 0 is *equivalent* to p > -rho/3 - it's the attractive gravity. But an accelerating expansion is equivalent to a *repulsive* gravity which is the opposite of the attractive one, so it has the opposite condition p < -rho/3.

  11. Umm - wow. Does Sean not realize that the 'first' form of the Friedman equation follows from the 'second' form (in the limit he mentions with spatial curvature set to zero) from the same trick we teach first-year students to get the work-energy theorem from F=ma? That is,

    d^2a/dt^2 = adot d(adot)/da

    from which a correct first form of the Friedman equation follows directly with rho + 3p, not just rho, on the RHS? As Lubos emphasizes, this is completely analogous with first-year physics calculations.

    OK, after writing this, I remain puzzled by how Sean could make such an elementary mistake. After consulting my trusted copy of Kolb and Turner's The Early Universe, it would appear that this follows precisely FROM neglecting the curvature term k/R^2 in the first Friedman equation.

  12. Thanks, Bill, for your understanding of my point! I wish TeX could be used in DISQUS to make the expressions prettier.

    I also think it's very important to understand that these equations are nothing else than relatives of the equations we use since the courses of classical mechanics. It's not just some vague or popular analogy - it's physically *the same thing*, just the collections of the degrees of freedom are somewhat more extended and some of the equations may have extra terms or nonlinearities and so on. But there's no cheating here.

    Throughout physics, the first derivatives of the coordinates, whatever they are, have pretty much the same meaning, and the same is true for the second derivatives of the coordinates etc. The generalizations are straightforward and when a layman/beginner *gets* this correspondence, it's good for her because there's no cheating here whatsoever.

    In the same way, the comments about the pressure (and negative pressure) are no cheating. The pressure is defined as the (when it is isotropic) spatial-spatial component of the stress-energy tensor, here in the convention in which the cosmological constant term is included to the stress-energy tensor which is set equal to the Einstein tensor (times the usual constant). The positive cosmological constant does *literally* produce negative pressure in this completely conventional, not popular, definition. The sign *is* the opposite to the sign of the pressure we have in gases.

    And general relativity implies that the pressure matters for the precise character of the curvature of the spacetime. This is no cheating, either. It *does* matter.

  13. This might be a naive layman's question but I have a hard time understanding what negative pressure is. It seems reasonable to me that particles under high pressure (and without other constraints) will move in such a way that these particles feel less pressure, but as soon as there is any matter around it seems that the best they can to is reduce to a state of equilibrium where the minimum pressure is zero. Completely relaxed (thanks to gravity being so much weaker than EM). How does negative pressure arise?

  14. Dear Rasmus, it's totally fine that you reach a contradiction when you consider particles that create negative pressure. The reason why there doesn't have to be any consistency here is that *(localized) particles cannot be the reason of negative pressure*.

    Negative pressure has to be carried something that is delocalized in the whole space - like the cosmological constant - and its effects are exactly opposite to positive pressure.

    For example, if you create a vacuum in a metalic sphere, the walls are normally being pushed to shrink by the atmospheric pressure. If you had a negative-pressure "medium" and placed it inside, the pressure on the walls would be even larger than the atmospheric pressure.

  15. Lubos:

    Yes, I was misinterpreting what p<rho/3 means.

    But I think a better mode of expression, even algebraically, should be implemented.

    For example, you say that the value for negative rho/3 should be "smaller." Well, if I have a ten foot pencil, and I keep sharpening it, it gets "smaller" and "smaller." It gets to be below one foot, then "smaller" than one inch, then "smaller" then a hundredth of an inch. As "small" as it gets, it will never become 'negative.' Something rather remarkable happens at zero. A symmetry develops.

    If we think in terms of the number line, we move right to left from positive infinity to zero. And we go from negative infinity to zero. They're mirror images of themselves.

    If 10 is 'greater' than 9, then -10 should be considered 'greater' than -9. That is "more negative".

    Consider the parabola, y=x^2. On the r.h.s. of the origin, we'd say that y is increasing as x is increasing, or, y is getting 'larger' as x gets 'larger'. Yet, on the l.h.s., we'd be saying y is getting 'larger' as x is getting 'smaller.' Aren't we better off saying 'more negative' than 'smaller'?

    I've tried a couple of stabs at looking at different ways we might write this down algebraically, but it's not so easy. Maybe we use the same symbols, but are careful about our terminology.

    Still scratching my head over this. I've looked at numerical analysis texts and some of the definitions involved in a Cauchy sequence, e.g., and was confused. I'm rather sure it's this symmetrical view of the number line that's getting me into trouble; specifically the word "smaller", and x<-10 meaning that x could be -11.

    Here's my new memory aide: "less than" means we're moving "left' on the number line.

  16. “… in which he tries to assault an elementary fact that the reason behind the acceleration of the expansion of the Universe is the negative pressure (caused by dark energy).”

    No, he really doesn’t. To quite him:

    “This explanation isn’t wrong; it does track the actual equations. But it’s not the slightest bit of help in bringing people to any real understanding. It simply replaces one question (why does dark energy cause acceleration?) with two facts that need to be taken on faith (dark energy has negative pressure, and gravity is sourced by a sum of energy and pressure). The listener goes away with, at best, the impression that something profound has just happened rather than any actual understanding.”

    He’s not saying the negative pressure explanation is *wrong*, only that laymen aren’t likely to be enlightened by it.

    “His crusade is made even more paradoxical given the apparent fact that he knows the equation and other key pieces needed to understand why it's accurate to say that the acceleration is caused by negative pressure. But like a schoolkid who has just mindlessly memorized an equation but can't understand what it means, he just can't sort out what the basic implications of the equations are”

    I disagree here, too. The reason he doesn’t use this explanation is what I quoted above.

    “These sentences of mine are no "popular presentations" and surely not "misleading popular presentations" and whoever understands them really understands what drives the acceleration etc.”

    Yes, but the problem is that the typical layman, with almost no physics education, would *not understand* this post all the way through. I would guess that is exactly why Carroll prefers his explanation *when talking to laymen*. Note the very first sentence of his post:

    “Peter Coles has issued a challenge: explain why dark energy makes the universe accelerate in terms that are understandable to non-scientists.”

    Whether your explanation is more scientifically correct is not really the point as long as the goal is to explain this to the kind of person who doesn't know basic physics. If the goal was to explain it to a second-year physics undergrad, your post might be better.

    His post also doesn't make him sound like an arrogant jerk, so there's that too. I believe it didn't include a single personal insult, actually! :-)

  17. It's wrong to lie - it's wrong to lie to the scientists and it's at least equally wrong to lie to the laymen. I've been explaining these matters at many other places of this thread.

    The negative pressure is the only correct explanation of the repulsive character of gravity during accelerated cosmological expansion. Everything else is simply a lie and every feeling that one is learning something from a totally different "explanation" is pure illusion.