*(This article was originally posted on blogger.com on Oct 26, 2005)*

Prof. Lisa Randall’s new book, Warped Passages: Unravelling the Mysteries of the Universe’s Hidden Dimensions, was released last month in the U.S. (I got it three months ago from amazon.co.uk , where it was already out on June 6th.). This book is intended for a popular audience, but is also a *very* interesting read for anybody with a background in theoretical physics. The first part contains an overview of modern physics – Einstein’s theories of relativity, quantum mechanics and the Standard Model of particle physics. The last part concentrates on the idea of extra dimensions beyond the standard four we know about, which is motivated by string theory and its discovery of D-branes. Specifically, she explains the work, pioneered by herself, Raman Sundrum and others, on the so-called “braneworld scenarios”. Basically, this is the idea that our four dimensional space-time is embedded in some higher dimensional space, usually called the “bulk”.

What is it with those extra dimensions? Are they not just part of a set of crazy ideas? (Like string theory, which a rather well-known person in the “blogosphere” would claim). On the contrary. You should know, that the idea of extra dimensions is actually not at all new. Already in 1884, the original book, “Flatland: A Romance of Many Dimensions”* *(written by the English mathematician Edwin Abbott) described a world of two-dimensional beings, who only have indirect knowledge of the extra third space-dimension (you can think of these beings as a group of flat ants living on an infinite plane). But, from a mathematical point of view, one can imagine as many dimensions as one wants to.

In physics, there are basically two distinct ways in which one can add extra dimensions to our four-dimensional (including three spatial dimensions and one time) universe. Already in the 1920’s, Klein suggested that our universe is five-dimensional, where the extra dimension is topologically a circle, which is so tiny, that the universe looks four-dimensional at long enough distance-scales. The motivation was to give a unified geometrical description of electromagnetism and gravitation using Einstein’s general theory of relativity. So, locally, space-time is a product R^4 x S^1 (to be more correct, it is actually R^{1,3} x S^1, but that’s not important here). However appealing, this theory was destined to fail. Today we know, that there are other forces which should be included in a unified theory: namely the weak and strong nuclear forces. Presently there is only one theory which can possibly do the work, and this is string theory. Perturbative string theory tells us, that our space-time is ten-dimensional, and that the extra six dimensions should be rolled up in a small but complicated shape. The topology is then R^4 x M^(6), where M^(6) is a six-dimensional Calabi-Yau manifold (i.e. that determines the possible shapes of the compactification manifold).

Another way to achieve hidden extra dimensions of space is to suppose, that all normal matter, as well as the light by which we see the world, is confined to a four-dimensional “brane” embedded in a five-dimensional bulk. These so-called braneworld theories are the ones of Lisa Randall, Raman Sundrum (and others – most notably Nima Arkani-Hamed of Harvard University). Warped Passages explains the logic behind these seemingly fancy ideas.

What I particularly liked about the first part of this book is how Lisa makes people envision extra dimensions. As Lisa writes, “we are not physiologically equipped to envision more than three dimensions of space”. But, as Lisa also explains, readers need not imagine a dimension only in spatial terms. Here is an example from the book: If you are buying a house, the factors you might consider include its location (specified by three numbers), price (one number), size (one number), and possibly many other things. So, the number of dimensions in your house search simply equals “the number of quantities you find worth investigating”.

What Lisa describes in the last part is the work concentrating on the following papers:

1. The idea of Arkani-Hamed, Dimopoulos and Dvali on large extra dimensions, which explains the weakness of gravity (the hierarchy problem) as due to the large size (originally ~1 mm, depending on the number of extra dimensions) of some of the extra dimensions, with only gravity propagating in them (hep-ph/9803315).

2. Work with Raman Sundrum on solving the flavor-changing problem, the gaugino mass problem (and other things) that occur in supersymmetric models with the supersymmetry breaking sector on another brane, separated from ours, or in the bulk (hep-th/9810155).

3. The Randall-Sundrum warped geometry with two branes (a so-called weak-brane, where we are supposed to live, and a gravity-brane) (hep-ph/9905221).

4. The Randall-Sundrum warped geometry with an infinite extra dimension, using AdS geometry (hep-th/9906064).

5. Work with Andreas Karch on “localized gravity” (hep-th/0011156).

The main point guiding Lisa’s research is the fact that gravity is such a profoundly weak force. Indeed, gravity is the puniest of the fundamental forces governing the matter in the universe, by a huge margin (typically a 10^36 times weaker than the electromagnetic force between two charged particles). Why is this so?

Lisa Randall suggests, that we live in a universe containing at least one extra dimension beyond those we can perceive. And gravity is weak because it has been diluted into this extra space. This is indeed a very simple and persuasive idea. (And also from the point of view of string theory a very compelling idea: we live on a three-brane embedded in a higher dimensional space with gauge interactions described by open strings attached to this brane, while gravity is interpreted as excitations of closed strings, which are free to leave the three-brane. But note that Lisa’s original models are not inherently string-theoretical; it is just that her models have an elegant and simple interpretation in string theory).

The breakthrough paper #3 above by Randall and Sundrum proposed that gravity’s dilution can be explained in terms of a cosmic configuration featuring two branes, separated by a higher dimensional bulk space. Roughly speaking, the “center of gravity” is on the “gravity-brane” – and some gravity leaks out of this brane, through the bulk, and onto the other brane, usually called the “weak-brane”, which is where we live, and which contain the Standard Model fields. In paper #4 above, Randall and Sundrum found that their concept is also theoretically consistent with a configuration which includes only one brane (and incidentally, this also sounds like a much simpler idea). Usually, one would think that Newton’s 1/r^2 law of gravitation implies that there are four and only four non-compact dimensions. Randall and Sundrum showed that this is not correct.

The fact that branes are an important part of modern string theory (thanks to Joe Polchinski and others) meant that string theorists took an early interest in the Randall-Sundrum models. Furthermore, since Lisa’s research didn’t directly challenge string theory models, the string theory community actually accepted and recognized the significance of her work very quickly. In particular, the fact that the Randall-Sundrum model uses the AdS geometry (as Juan Maldacena did), and has interesting relations to the AdS/CFT correspondence, has drawn a lot of interest from string theorists.

One of the long outstanding problems of the Standard Model that braneworlds do provide an interesting answer for is the “hierarchy problem”, or why the weak and Planck scales are so disparate (10^2 GeV compared with 10^19 GeV). In these scenarios, the fundamental gravitational scale is not the Planck scale, but something closer to the weak scale. The conjecture is that gravity is not weak because the Planck scale is so large, but because braneworlds provide various geometrical mechanisms for making the gravitational force much weaker than the others. (In technical terms it is because their bulk space-time metric is not factorizable, so the metric in the four familiar dimensions depends on the coordinate in the extra dimension).

The idea that the gravitational scale may be closer to and maybe even directly related to the weak scale is an appealing one, which is one of the reasons why Randall and Sundrum’s paper #3 has been cited 2640 times as of today.

All this would of course be pointless speculation unless there was some way for the extra dimensions to manifest themselves. So, can these ideas be experimentally tested? Yes indeed. It is possible that evidence supporting the braneworld models could appear within a decade or so, after the Large Hadron Collider (LHC), currently being built at the particle-physics laboratory CERN, starts operating in 2007. Since gravitons are not confined to branes, but can escape into the bulk, the traces of extra dimensions could come from such gravitons moving into the extra dimension(s) (i.e. missing transverse energy).

Some versions of the theory even predict that, as a result, small black holes could conceivably be created from the high-speed collisions between the LHC’s protons and antiprotons (but don’t worry, the black holes will only exist for ~10^{-26} sec due to Hawking radiation and will not soak up the Earth or the entire Solar System). If the LHC sees the kinds of effects predicted by these models, be sure that there will be some well-earned Nobel prizes for the people involved in this story.

Let me say a few words about the style of the book. The book’s central point – the possible existence of extra dimensions in space – is easy enough to explain; at least if the reader can comprehend that our universe has more than four space-time dimensions, which might not be easy. However, to motivate the conjecture of higher dimensions, Lisa must first explain the General Theory of Relativity, quantum mechanics and the Standard Model of particle physics, with its zoo of subatomic objects – quarks, leptons, bosons of various sorts – and the details of the forces that act between them. To ensure the convergence of her ideas in the readers mind, she then has to go into more advanced topics such as spontaneous symmetry breaking, the Higgs potentials and supersymmetry. Because unification physics is formulated in terms of quantum field theory – that has to be covered too.

All this is a prerequisite for being able to properly describe string theory, and must then proceed to the less understood generalization called M-theory (which is an eleven-dimensional theory containing two-branes and five-branes). Only then can Lisa explain how branes emerge from a jumble of concepts and ideas, that most likely might be is unfamiliar to the general reader. But as far as I can judge, she accomplish this tour de force with much success; still, I wonder how many non-physicists will be able to stay with the story to the very end. The immanent “problem” with such theories of unification is that one cannot leave anything out and therefore – to be explained and understood – they require background knowledge in virtually all of physics.

Finally I would like to stress that this book is very different from many of the random popular books about physics because Lisa Randall actually knows what she is talking about (and therefore I’ll suggest that you steer away from popular books on physics with suggestive titles as: “The Final Theory”, “The Road to Reality : A Complete Guide to the Laws of the Universe”, “Parallel Worlds: A Journey Through Creation, Higher Dimensions, and the Future of the Cosmos” and “Our Superstring Universe: Strings, Branes, Extra Dimensions and Superstring M-Theory”). And you should know, that Lisa is not a random person: she has become the most cited high-energy physicist since 1999.

A nice introduction to extra dimensions is the review, “To the fifth dimension and back”, by Raman Sundrum.

There are other reviews/comments about Lisa’s book – one by Lubos Motl, one by the anti-stringy Peter Woit, one by Paul Davies and one by John Gribbin (though, I’m not sure I would call the last one an honest review; I’ll let you judge for yourself), and reviews at amazon.com and

at amazon.co.uk.

**Update**: *Warped Passages* now has its own website (March 26’th, 2006).

Papers are accepted upon payment of some unknown amount of money, which immediately makes you think that the scientific value of such articles is comparable to that of commercials during the Super Bowl.

SWJ er ikke et betalt tidsskrift, du må altså rette det

I don’t understand how this comment should be relevant to Prof. Randall’s book “warped passages” ??

I guess “Ventegodt” is referring to my post on “More Dangerous Pseudo-Science?”

“Ventegodt” should post his comment at the relevant place.

Kasper

[…] I have reviewed Lisa’s excellent book, “Warped Passages”, here. […]

I agree about the Gribbin review. I bought the book in spite of it. There appears to be something personal going on in that review. Shame, really.