Tuesday, May 22, 2012

Hartle, Hawking, Hertog: how our C.C. could be negative

A reader has pointed out that I missed a paper by Hartle, Hawking, and Hertog last week:
Accelerated Expansion from Negative Λ
They claim – and please sit down so that your stability gets improved – that the accelerated expansion of our Universe could result from a theory that has a fundamentally negative cosmological constant, like in the Anti de Sitter space (AdS).

I enjoyed reading their paper so far. They clearly have brilliant minds. Too bad that the main claim seems to boil down to a sign error so far. ;-)

Of course, it is easier to study stringy AdS vacua than dS vacua and they're related to CFTs by holography, unlike dS vacua (sorry for that comment, Andy Strominger), so I don't have to explain to you how welcome their bizarre conclusion could be from the viewpoint of string cosmology if it were right. One additional advantage of AdS vacua over dS vacua is that they may preserve SUSY but I guess that they don't claim that there is unbroken SUSY in our Universe which is just masked by their tricks, do they?

(The final section of the bulk of their paper is dedicated to string cosmology; they mention holography a few times, too.)

Although I've been promoting various types of analytical continuations and complexifications of types that are similar to theirs, I just can't see so far how it could work when someone else does it. ;-)

They claim that the Wheeler-DeWitt equation (I am sure that under the influence of the authors' names, you're tempted to write "Hweeler-Hewitt equation") may be approximated by a semiclassical one in the large-volume-of-the-cosmos limit and it has solutions where the wave function is approximated by the exponential of a classical action times the imaginary unit.

But their classical action is complex and its nonzero imaginary part is what allows one the real part to resemble a de-Sitter-like expansion even though the underlying theory has a negative cosmological constant. If this is true, it is true regardless of the boundary conditions; so despite the names of the 2/3 of the authors, we don't necessarily talk about the Hartle-Hawking wave function only. We're talking about any wave function that obeys the Wheeler-DeWitt equation.

What I still don't understand is what they actually complexify. Surely if the spacetime coordinates are real and the fields obey their usual Minkowski-signature reality conditions, the evolution of histories that result from an AdS-like fundamental theory can't resemble a dS-like expansion, can they? By their complexified (unreal) action, they seem to be affecting the overall normalization of the wave function in a different way; but the overall normalization of the wave function should be determined by the conservation of the overall probability, shouldn't it? In other words, the relevant Hamiltonian or effective Hamiltonian or even the Wheeler-DeWitt H is real, even a priori, isn't it? This must constrain the unreality of the players, mustn't it? Or do they suggest this normalization is redefined as time goes by? That would surely be interesting but I wouldn't understand how it can be done.

After all, such a rescaling would have to take place differently at different spatial slices (different moments). But there are no preferred slices in a relativistic theory with a dynamical, curved spacetime, are there?

So while I feel these are deep things to be considered, I am still not getting how they can actually fool the fact that in the large-volume, classical limit, the value of the vacuum energy may simply be decoded from the metric tensor in the vacuum, and if the space is accelerating, the cosmological constant has to be positive. If there's some loophole through complexification in these arguments, I am not getting it so far.

Can you help me? Is there one equation or two equations in the paper that really clarify what's going on and where they violate the normal rules, normal rules that – as I believe - imply that in the classical limit, the expansion must reflect the same value of the parameters including the cosmological constant as the values we call fundamental?

So far, the paper by Hawking et al. looks like a contradiction with Einstein's equations to me, classical, quantum, or otherwise. So I have embedded this Hawking-vs-Einstein rap which is hopefully no longer taboo after 45 million YouTube views. ;-)


  1. Dear Lubos,

    One cannot get (2.4) from (2.2) and (2.3)? I think (2.4) has a wrong sign. As you said a sign error.


  2. Wow, I just verified your statement. The term with \(1/l^2\) in (2.4) should have the opposite sign, right? Both the middle term \(-1\) and the last term come from \({\mathscr V}\) which has the same sign for both terms, so they can't have the opposite sign.

    The first term is OK because we have to vary the prefactor \(N\) as well as \(1/N^2\), so we get \((1-2)=(-1)\) from the variation.

    Amazing: that's pretty clearly the term linked to the cosmological constant. Thanks a lot. ;-)

  3. You are welcome.

    I thought you're going to like it.

    A good comeback Sheldon (aka Lubos).