Brian Greene: The Elegant Universe.
Part 2, hidden dimensions.

Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory.

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Hidden dimensions.

In 1919, Theodor Kaluza showed Einstein that general relativity could unite his equation of gravity with Maxwell's electro-magnetic equations, by assuming a fourth dimension of space. ( With time, this made five dimensions in all. ) Oskar Klein suggested the fourth dimension could exist as a curled-up space too small to be observable, perhaps being only of Planck length. A simple analogy is that a garden hose looks like a single dimension from a distance. But close up, the line has thickness admitting of another circular dimension that can be traveled round by an insect.

Kaluza's findings didnt fit the experimental data about the electron's mass and charge. Eventually, as more particles and the strong and weak forces became known, theorists wondered whether the fault with Kaluza-Klein theory had been too few dimensions rather than too many.

This turned out to be the case for string theory. It had resolved the infinite probabilities, thrown up by elementary point particles, in an attempted quantum gravity theory. But negative probabilities also kept turning up. And these could only be removed by letting the strings vibrate in nine dimensions. ( A tenth spatial dimension was later infered, making eleven, including time. )

Just as an ordinary string may be allowed to vibrate in three independent directions, a theoretical string may vibrate in nine independent directions. The curled-up six dimensions, that fulfill the equations of string theory, are called Calabi-Yau spaces ( or shapes ).
These shapes may be likened to musical instruments that create particular vibration patterns. The testing question is: how well do these patterns match with the elementary particles found, or capable of being found, by experiment?

Calabi-Yau shapes contain various holes, which themselves have various dimensions, analgous to a do-nut and a double or triple do-nut. A family of lowest energy string patterns is associated with such holes. Multiple holes imply multiple families, like the three families of elementary particles. Just the right shapes are currently still being sought.

String theory predicts other fractional charges than those of the quarks. And experiments finding super-partners would also be relevant to super-strings.

Possible behavior of strings can be described in a simplified form of the large and the hidden dimensions, the afore-mentioned 'garden hose' universe, with one familiar line dimension and a hidden, curled-up dimension. The universe may collapse back on itself, from a big bang to a big crunch. This depends on whether there is enough mass in the cosmos to pull it back. The big crunch may resemble the formation of a singularity at the heart of a black hole. All the cosmic mass may be crunching into a single linear stream. It looks to be of one dimension only but has a cylindrical dimension also, like a garden hose.

The difference from point particle physics is that strings can not only move about on this cylinder. They can also wrap around it: they have a winding mode. So, strings have two sources of energy, winding energy, as well as vibrational motion. The latter consist of uniform vibrations and ordinary vibrations. Ordinary vibrations are the kind of oscillations considered above, and are not decisive in this context. Uniform vibrations are 'the overall motion of a string as it slides from one position to another without changing its shape.'

Uniform vibrations' string energies are inversely proportional to the radius of the circular dimension. The uncertainty relation ensures that a constricting hose radius, confining the string, increases its energy.
But the winding energy is proportional to the radius. The greater radius and circumference, the longer the string and the greater its mass, when wrapped around the 'hose', and according to how many times wrapped round, giving the 'winding number'. There are also multiple vibration numbers.
The units are on the Planck scale of length and energy.

The winding energies and vibration energies of the strings compensate each other. You could have a table of winding numbers and vibration numbers for a given radius, and another table of the same, for its inverse radius, giving an over-all correspondence of entries. You could have one universe with a small radius and large vibration energy that corresponded exactly in total energy with another universe, having a large radius and a smal winding energy. The two universes are effectively the same, having the same allowed quantum particle energies and charges.

We do not know whether our own universe has a hidden curvature, in the sense that it is too large, rather than too small, for us to see. Space might be traversed as Magellan's expedition circumnavigated the globe. If the universe has a 15 billion light year's expansion age to put a radius to, say 10, to the power of 61, Planck lengths, then string theory provides an alternative inverse radius of the universe ( 10 to the power of minus 61 ), a radius that is miniscule and contracting, but just as valid in its own terms.

Measuring distance, the familiar way, by light amounts to using light ( meaning not-heavy ) string modes as probes. In principle, if they were technically able, astronomers might equally well measure distance by heavy wound-string modes. But such probes, being proportional to a cosmic radius would have to be incredibly massive.

Whichever string mode happened to be the light or 'easy' mode, it never measures below Planck length. Even if the non-standard measure of distance were adopted, so the radius is below Planck length, the physics is the same as for the complementary table in which the radius is more than Planck length in the conventional measure of distance.

Having discovered that geometrical forms could differ in size, yet be physically indistinguishable, physicists, including Greene, found that the same could be true of different shapes, by orbifolding Calabi-Yau spaces. The number of odd-dimensional holes equaled the number of even ones, in the original, and vice versa. Their totals of holes is equal, implying the same number of particle families, tho their shapes and structures differ.

The shapes agreed on the rest of their physical properties. The beauty of these 'mirror manifolds' was that one might be chosen as the possible hidden dimensions creating the sought-after particle masses and force charges. The calculations involved had often been impossible. This had also been the case for the pure mathematical study of Calabi-Yau spaces. But it turns out that the 'mirror' partners, figuratively speaking, are often easy to calculate, a source of progress in string theory, and a return by physicists for what they'd learned from pure mathematics.

In 1987, Calabi-Yau spaces were found to be transformable into each other, according to a mathematical pattern of puncturing and sewing their surfaces. ( This was a space-tearing 'flop transition', which is sometimes 'topologically distinct'. ) Considering such processes as possible physical tearings of space, mirror symmetry of Calabi-Yau spaces was used to give fuller grounds for the suspicion. An absence of catastrophe in the 'mirror' partner would make the space-tearing original physically allowable.

Edward Witten showed that travelling strings, unlike point particles, could protectively encircle spatial tears, with relative possibilities ( calculated from Feynmann's sum-over-paths ) that would cancel out a 'cosmic calamity'.
He and colleagues, including Brian Greene, also showed that spatial tears would leave types and families of particles unaffected. But the energies of the possible string vibration patterns, meaning the individual particle masses, could change. Experiment shows these to be stable. If there is any spatial tearing in the universe at large, it is too slow to be noticable.

Space tearing opens the way for the possibility of worm-holes, the creation of new space joining previously unconnected parts of the universe.

Beyond strings: M-theory.

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Up till 1995, five string theories seemed to be at odds with each other. Only approximate string equations could be found and each of the five theories differed from each other. Their difficulty meant that perturbation theory had to be used, that is a method of successive approximations. A classic example of this is how the gravitational interactions of the solar system are worked out. The sun is by far the most important gravitational mass. So, its effect in relation to the earth is calculated first. This result has to take into account the next most important effect, the moon in gravitational relation to the earth, and so on, until all the significant planetary masses have been allowed for.

But the success of perturbation theory depends on being able to order the importance of the effects. Then, dealing with each in turn, one has some idea of how the margin of error should diminish in each successive approximation. Using Feynmann diagrams, Richard Feynmann's popular lectures, QED, give examples of this process of adding successively smaller corrections for all the possible ways a given particle inter-action might take place. Experiment confirmed this quantum electro-dynamics as the most accurate theory in history.

String theory has Feynmann diagrams for strings instead of point particles. The Heisenberg uncertainty principle's allowance for the creation and annihilation of virtual string pairs, in a string inter-action, is diagrammed as a series of loops between in-coming and out-going strings.

The likelihood of such temporary energy incursions is measured by the size of a 'string coupling constant'. It would determine masses and charges of string vibrations. Strongly or weakly coupled values, above and below unity, respectively, determine whether it is increasingly likely or unlikely for more and more virtual particles to appear. Therefore, values, above one, for any of the five string theories, would invalidate the use of perturbation theory.

In 1995, Witten introduced 'duality' to get beyond perturbation theory. Of the five string theories, two pairs of them get exchanged by the large / small radius duality, discussed in the previous section.
Instead of assuming the five theories were independent competitors, all amenable to perturbation theory, by being weakly coupled, it was found that two of the theories could be transmuted into each other, because of a strong-weak duality. Their physics appeared the same, when one theory was weakly coupled and the other strongly coupled.

To this end, use was made of super-symmetry constraints and minimum mass constraints to give clues about particle states ( BPS states ), for the string theory with a strong coupling constant.

Another of the five theories appeared to correspond to itself when weakly and strongly coupled: it was self-dual.

To complete the link-up of the five theories required a further insight. Super-gravity theories attempted to use super-symmetry to unify quantum field theories with general relativity. It turns out that these point particle theories were approximations to various of the five string theories.

One of the super-gravity theories was in eleven dimensions, rather than ten, and didnt fit in with the existing 10-D string theories. But a string theory, by gradually increasing its coupling constant, and with respect to its BPS states, showed 11-D super-gravity to be a low-energy approximation. The extra dimension emerges with the increasing coupling constant and a string loop turns into a two dimensional cylindrical membrane or a hoop, depending on the string theory.

Higher dimensional membranes, than two, are also possible. But with weak string coupling, all but the strings would be too massive to be produced without enormous energies.

Witten provisionally named the 11-D theory as M-theory, still something of a mystery, but the supposed under-lying theory to the five string theories, incorporating 11-D super-gravity. What was previously an embarrassment of theories, as to the truth, has become an inter-related variety of approaches to make the problems of theoretical prediction more tractable.

Witten demonstrated a primal emergence of the gravitational force from the other three forces, according to their varying strengths when the string coupling constant need not be small.

Black holes as elementary particles.

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Elementary particles and black holes have in common that they are distinguished only by their mass, force charges and spin. A black hole might be a huge elementary particle. A small enough black hole should resemble an elementary particle. But this brought into play the big versus small theory incompatibility between general relativity and quantum mechanics - until string theory, or M-theory.

In the context of space-tearing flop transitions ( discussed above ) string equations show three-dimensional surfaces, as well as beach-ball-like two-dimensional surfaces, embedded in a Calabi-Yau shape, are likely to vanishingly collapse. A one-dimensional string, moving in time, could 'lasso' a 2-D sphere, preventing a cataclysmic spatial tear. In this respect, at any instant, a 1-D string ( or one-brane ) could only surround a circle; a 2-D string membrane, or 'two-brane' wrap round a two-dimensional sphere ( like an orange ); and a three-brane wrap round a three-dimensional sphere.

Following-up the flop transition for the 3-D sphere ( called a conifold transition ), it was found that the sphere repairs and reinflates only as a 2-D sphere. This can only be imagined in lower dimensions. A two-dimensional sphere is 'a collection of points in three-dimensional space that are the same distance from a chosen center.' Its reduction, to a one-dimensional sphere, would be to the points making up the circumference of a circle, which is in two spatial dimensions.
A further reduction would be to a zero-dimensional sphere, 'the collection of points in a one-dimensional space ( a line ) that are the same distance from a chosen center.'

The replacement of a 1-D sphere ( a circle ) with a 0-D sphere ( two points ) can create a different topological shape. A do-nut has a circle, round its lesser diameter, which is pinched to nothing. The do-nut turns into a cresent or banana-shape, with the two end-points repaired by the two points of a zero-dimensional sphere. The torus cum cresent can now transform into a ball, without further tearing.
This is as if Klein's hidden extra dimensions of space transformed from the one curled-up shape to another, comparably to the normal extended three dimensions changing the shape of the universe from a torus to a ball.
The evolution of the universe may involve such transmutations between curled-up Calabi-Yau spaces.

Equations governing the 'branes' showed that, from our limited three-dimensional view-point, the three-brane "smeared" around a three-dimensional sphere, within a ( curled-up ) Calabi-Yau space, sets up a gravitational field like a black hole.
The space tearing conifold transition from three to two dimensional sphere happens to increase the number of holes by one. These holes determine the number of low mass particles, considered as low energy string vibration patterns. The shrinking volume of the 3-D sphere goes with a proportionate mass decrease to zero: a massless black hole.

The black hole is considered to have under-gone a phase transition to a massless elementary particle, like a photon. String theory has identified them as being made of the same 'stringy material'. Much as ice under-goes a phase transition to water, they look different but their make-up is the same.

'Hawking radiation' established the 'entropy' of black holes. To solve what this disorder was of, string theorists theoretically built

certain extremal black holes by starting with a particular collection of BPS branes ( of certain specified dimensions ) and binding them together according to a precise mathematical blueprint...

Strominger and Vafa could easily and directly count the number of rearrangements of the black hole's microscopic constituents that would leave its overall observable properties, its mass and force charges, unchanged. They could then compare this number with the area of the black hole's horizon -- the entropy predicted by Bekenstein and Hawking.

Hawking radiation implies the eventual evaporation of black holes. With the gradual shrinking of their areas, their entropy decreases. A current research question is whether order or 'information', lost to the black hole's gravitational suction, could be recovered from the surrounding area that the shrinking event-horizon has given up.

If the answer is 'no', this would further take the edge off a deterministic physics. Quantum mechanics had made, only probabilistic, Laplace's totally determinist conception of mechanics.

However, the author doesnt mention chaos theory, which requires infinite accuracy in initial conditions, to predict the oscillations of so simple a classical system as the force-driven pendulum. Laplace's school thought these information feed-ins, to apply physical laws to particular circumstances, would, in principle, determine the evolution of the universe.

Brian Greene discusses other questions, mainly to do with the new subject of super-string cosmology. Already, some possible answers have been put forward. Pending the big bang, all the eleven dimensions of space and time were supposed to be curled up in a universe of Planck scale size. Why did only three dimensions of space extend ( thru 'inflation' and so forth )?
As related, above, strings can wrap round these dimensions. But there are anti-strings, wrapping round 'the other way', which annihilate them on contact, producing an unwrapped string, and releasing the dimension to expand. These releasing collisions are most likely in one dimension. At different speeds, two marbles, confined to a line, are sooner or later going to hit. This is less likely of two objects moving freely on a surface, and less likely still for objects moving freely in three dimensions. Thus, the chances were that the fourth and higher dimensions of space were not released from their string wrappings by string pair annihilations.

Alan Guth's popular book, 'The Inflationary Universe', mentions there being about fifty versions of inflation theory, which explains several discrepancies in the earlier big bang model. Greene refers to a controversial pre-big bang version, derived from string theory, by Gasperini and Veneziano, which they hope presages a more inevitable development to inflation.

Closing remark.

Brian Greene's explanation of string theory may be likened to a series of beacon hills that trail off to who knows where. They give you an idea of the general direction string theory is going but they leave in darkness the maze of vallies below, which only whiz mathematicians and physicists can follow, to light up more beacons.

Postscript: Parallel universes.

BBC tv's Horizon ( 14 feb. 2002 ) featured an astonishing development in linking string theory to cosmology via the concept of parallel universes. The program followed the implication of a unified string theory or M-theory featuring an eleventh dimension and, beyond strings, the existence of membranes of various dimensions.

One of the scientists involved described the arrival of an Italian liner in New York, damaged by a rogue wave. It so happened that a study of the mathematical possibilities of what might happen when the membranes collide in their hyper-space also yielded catastrophic results of the order of the Big Bang itself, or innumerable big bangs.

Classical cosmology closes off possible events, before the big bang, with an infinitely small beginning, a singularity. But quantum theory of the Planck scale of events transcends the big bang, as the outcome of these colliding membranes. As they move, they ripple, so that collisions yield the clumps of matter after the big bang. That is the material universe.

This implies that time precedes the big bang, which is indeed one of an infinite number of different big bangs resulting in an infinite number of possible universes, with different laws of physics.

Hence, string theory has theoretically explained the origin of the big bang by implying parallel universes.
At the time of writing, this is a new theory, which the physics community has yet to decide whether to accept or not. As yet, parallel universes have not been the majority view.

Richard Lung.

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