Brian Greene: The Elegant Universe
with Related Information from Other Sources
Summary by Michael McGoodwin, prepared 2000

Cosmic microwave background radiation fluctuation
Cosmic microwave background radiation fluctuation after subtraction of dipole and galactic emission
(This image is modified from a COBE satellite image at http://aether.lbl.gov/www/projects/cobe/COBE_Home/DMR_Images.html.
For more current images, see the WMAP project.)

Notes: I prepared this summary rather hastily and for my own benefit to improve at least my retention (if not my understanding) of the concepts discussed in the book and have added related subject matter gathered from various books and some web sites I came across.  I offer this summary to assist others also interested in reading this interesting and challenging book.  I have not attempted to present a comprehensive summary.  Greek letters and other symbols that are not available with standard HTML are represented spelled out in curly braces, e.g. {tau}.  The usage "1E44" means 1 X 1044 .  I would be pleased to be notified of any errors.


Chapter 1
Tied Up With String

The pillars of 20C physics, quantum mechanics and general relativity are mutually incompatible particularly at the subatomic level at which quantum effects are most apparent. BG recalls earlier conflicts in theories pertaining (1) to the speed of light for moving observers, resolved by Einstein's special theory of relativity, and (2) the problem of the apparent instantaneity of gravitational effects resolved by general relativity's warp of space. 

Protons are not fundamental but composed of quarks (confirmed at SLAC 1968), named by Murray Gell-Mann after Finnegan's Wake: protons have 2 up-quarks and a down-quark, and neutrons 2 down-quarks and an up-quark. Reines and Cowan 1950 found evidence for neutrinos (now termed electron-neutrinos), and the muon was found in the 1930s. Charm and strange quarks  also found, along with muon neutrino and tau neutrino. All of these particles have antiparticles. The family of matter are now grouped in 3 families (p. 9) with analogous particles (e.g., electron, muon, and tau occupying similar positions in each family). All matter found to date is in one of these 3 families and their antimatter counterparts.

The fundamental forces and their force charges are the strong force (they glue quarks together, possible charges are red, green, and blue), electromagnetic (e.g., the electric charge of quarks can be 2/3 or -1/3 ), weak force (related to beta decay, charge can be 1/2 or -1/2), and gravity.  These forces are mediated by the gluon, the photon, the weak gauge bosons W and Z, and the graviton resp. (with relative strengths of 1E44, 1E42, 1E39, and 1, resp.)

String theory posits that quarks are not point-particles but contain a loop of vibrating energy that resolves the conflict between General Relativity and Quantum Mechanics (QM). The various resonant vibrational modes account for the different forms matter takes in terms of mass and force charge (e.g., electrical charge, weak charge, strong charge). String theory has been suggested as a candidate for the "Theory of Everything (TOE)" on which we might build our understanding of the world [i.e., it would unite the gravitational force with the electric, weak, and strong forces]. However, the computations may be impossibly difficult and thus cannot justify a completely reductionist view. String theory is in its early stages, "a part of 21C physics that fell by chance into the 20C" (Edward Witten). No one knows the exact equations. It is part of a grander synthesis called M-theory.

Chapter 2
Space, Time, and the Eye of the Beholder

Reviews the Maxwell equations mid 1800s uniting the electrical and magnetic forces. He found that light travels at a fixed speed. Einstein 1905 studied the speed of light as observed by observers [here arbitrarily labeled by me M and S] moving relative to each other and found that each would measure it to be the same. However, the uniformly moving observer M experiences time slower relative to a "stationary" observer S ("time dilation") and also S measures M's lengths to be shorter ("Lorentz contraction"). But concept of motion is relative and each observer will make these observations about the other, with there being no absolute frame of reference for force-free motion. Einstein concluded that the laws of physics for the 2 observers must be the same ("invariant") and that therefore they must experience the same light ray as having the same speed in vacuum (670 million mph)—a symmetry. 

Discusses consequences of light speed being constant, including the downfall of Newtonian mechanics (developed by Isaac Newton 1643 - 1727). Things that are simultaneous to M are not necessarily so to S, due to finite time for light to travel. A universal clock does not exist. Time is what is measured by clocks, something undergoing regular cycles of motion. Posits the "light clock" though experiment, 2 mirrors with a photon traveling back and forth held by M. The photon will be seen by S to travel further and therefore S's measure of M's time must be slowed down for the photon to travel at a uniform speed to S. The rate of ticking becomes slower and slower as the relative speed increases. This is an intrinsic property of time and not dependent on the means used to measure it. Tm = Ts/sqrt(1-v2/c2). In real life, high speed muons appear to disintegrate slower that those at low speed, though they experience the same amount of life. Each observer experiences that the other's clock is running slower (and this holds even if one takes into account that a transmitted timing signal is delayed by the finite speed of the photons).

Minkowski and Einstein viewed motion as occurring therefore in both space and time, or 4D spacetime. Einstein declared all objects travel through spacetime at the speed of light. [spacetime position x = (ct, x1, x2, x3); velocity = dx/d{tua} which has magnitude c, here {tau} is the "proper time" defined by d{tau}2 = dt2 - c-2(dx12+dx22+dx32]. He also concluded the equivalence of matter and energy through E=mc2

Chapter 3
Of Warps and Ripple

The invariance of the speed of light posited by special relativity was incompatible with Newton's universal theory of gravity F = G*m1*m2/r2 because it assumes an infinite speed of transmitted force changes when the mass of an object changes. Even Newton suggested gravity must be caused by an agent and not by a force acting at a distance. Einstein resolved this with general relativity, after realizing the equivalence principle, the close interconnectedness of the experience of gravity to uniformly accelerated motion. He viewed matter as curving or warping spacetime as if on a saddle-like surface, so that for example a spinning circle [drawn on a such a surface] has a longer circumference than if there were no curvature. Both time and space are warped and gravity is this warpage. The analogy of a bowling ball depressing a plane surface is given. The warpage is transmitted by gravity waves which move at the speed of light. Objects move through spacetime according to the shortest or easiest possible path [geodesic], and these paths will be curved near masses. In John Wheeler's words, "Mass grips space by telling it how to curve, and space grips mass by telling it how to move." An observer near a large mass would find his clock ticking much slower, and there is no symmetry with respect to an observer who is not near the mass—there is no equally valid perspective when acceleration is present.

General relativity was thought to have been verified by the famous prediction of the bending by 1.75 arc seconds of starlight passing by the sun during a solar eclipse, confirmed by Eddington 1919 (an experiment whose reproducibility was subsequently questioned).

Karl Schwarzschild predicted the existence of black holes 1916 (named such by Wheeler), in which matter inside the event horizon would fall in and nothing would escape, not even light. At the center of the Milky Way is prob. a c. 2.5E6 solar mass black hole, and black holes up to billions of solar masses may exist. Einstein was disturbed when he calculated that the universe must be expanding and fudged his calculation with a "cosmological constant". But Hubble confirmed the expansion. Backward extrapolation predicts 15 billion years ago an infinitely squeezed state of the universe, at which time the "Big Bang" occurred. The calculations of general relativity are quite accurate but cannot cope with the compressed state preceding the Big Bang, and is incompatible with quantum mechanics.

Chapter 4
Microscopic Weirdness

Reviews quantum mechanical phenomena at the microscopic level. By 1928, most of its equations were in place, but it is difficult to understand why it works!

Reviews the problem of the spectrum of the black body radiation, which by classical computation should have infinite energy, since an infinite number of possible resonant electromagnetic (EM) waves must each carry the same energy. By Planck's hypothesis 1900 that EM wave energy occurs in quantized increments (h*f where f= frequency Hz and h = Planck constant), the total energy combining the various vibrational modes is finite because above a certain threshold of energy, there can be no contribution. The predicted spectrum is in accord with experiment. Here the Planck constant h = 662 E-36 Joule sec or 6.624 x 1E-27 erg sec. Einstein deduced that the photoelectric effect found by Heinrich Hertz 1885 was due to light occurring as individual particle-like packets or quanta of energy (later termed photons by Gilbert Lewis), and that the color of the light and not its intensity primarily determined whether the PE occurred at all [and the ejection velocity of the photoelectrons: beam intensity only affects the number of electrons ejected, but does not change the ejection velocity; beam color is determined by its photon energy h*f or h*{nu} or h-bar * {omega} where {omega} equals angular frequency in radian/sec and h-bar = h/2{pi} = 1.06 E-34 kg m^2 sec^-1]. Thus light seemed to have both particle and wave properties. Young's double-slit experiment in early 1800s had shown light to have wave properties by exhibiting interference pattern etc. This pattern was later found to persist even if the photons pass through one at a time. Thus a wave-particle duality for light existed. 

Louis de Broglie 1923 proposed this duality for all matter, shown for the electron by Davisson and Germer 1920s. [Schiff Quantum Mechanics 2nd Ed: In scalar form, wavelength {lambda} = h/momentum. Also momentum = h-bar*k where k is the wave vector, and rest energy E = mc2 = h-bar*{mu} where {mu} is the rest frequency of a particle.]

The Schrödinger wave equation [1925-6] was refined and interpreted by Max Born 1926, who proposed that an electron and matter in general must be viewed from the standpoint of probability.  [I.e., the position probability density, which equals the square of the absolute value (magnitude or amplitude) of the complex {Psi} function, is such that P(r,t)dxdydz  gives the probability of finding a particle in this infinitesimal voxel, and the integral over all space of P(r,t) equals 1]. Exact outcomes of experiments thus cannot be predicted in QM. Einstein objected to this vagueness and randomness but was proved wrong. There is still debate as to what this all means however. Feynmann emphasized how attempts to localize an electron perturbs it and changes the outcome. He suggested electrons pass through all slits in an experiment and traverse every trajectory, but that the sum-over-paths is the resultant probability distribution, wherein for large objects all paths but one cancel each other out [this method is an alternative to calculating the probability wave function].

Werner Heisenberg's uncertainty [or indeterminacy] principle 1927 showed that certain conjugate physical quantities [Schiff: "canonically conjugate in the Hamiltonian sense"] could not be simultaneously known precisely: e.g., the order of magnitude of the product of the uncertainties of position {delta}x and momentum {delta}px >= h-bar [some authors state one-half h-bar]. Other canonically conjugate pairs are angular position/angular momentum, and time/energy. Bohr termed these quantities complementary and this effect the complementarity principle, concluding that atomic phenomena cannot be described with the same accuracy as in the macroscopic world inasmuch as more precise measurements for any experiment cannot be made than the limits set by the uncertainty principle (for example, the measuring photon perturbs the position or momentum of the particle). A photon can only pinpoint the location of an object to within one photon wavelength. If an electron is confined to a space of decreasing size, its motion (momentum) increases wildly due to "quantum claustrophobia" since {delta}p*{delta}x>= h-bar and as {delta}x decreases, the momentum must increase.  There is also quantum tunneling, in which a particle classically lacking the requisite energy to overcome a barrier is able to borrow the energy to do so [or at least to tunnel through the barrier if it is narrow enough].

Chapter 5
The Need for a New Theory; 
General Relativity vs. Quantum Mechanics

Under extreme conditions of great mass or very small scale, the success of General Relativity breaks down and nonsensical predictions occur--for instance, at the time of the Big Bang or at the center of black holes.  In these situations, we need a quantum mechanical version of general relativity.

The uncertainty of the Heisenberg principle is intrinsic to nature and not a result merely of attempting to make measurements--e.g., the quantum claustrophobia occurs even in the absence of a measurement.  Even in so-called empty space, the uncertainty principle tells us that there is tremendous activity: energy fluctuations cause an electron and antielectron to erupt into existence and to subsequently annihilate one another.  "Macroscopic averaging obscures a wealth of microscopic activity."  The mathematical technique developed by Dirac, Pauli, etc. to deal with this disorder is called quantum field theory.  The Schrödinger wave equation was found to be only approximate and unsuitable for very small microscopic domains because it ignored special relativity.  

The body of theory incorporating special relativity is called quantum electrodynamics (QED), an example of a relativistic quantum field theory or a quantum field theory QFT.  QED has been extremely successful in predicting natural phenomena.  E.g., Hinoshita has made electron predictions accurate to better than one part in one billion.  Quantum electroweak theory was developed to unite the weak and the electromagnetic forces (Glashow, Salam, and Weinberg) in a common form at high temperatures and energies; at lower energies the individual forces "crystallize out" (analogous to a phase diagram) in a symmetry-breaking process.  Quantum chromodynamics (QCD) was developed to quantify strong force interactions--it presents extremely complex computations.  [A theory unifying the strong force with electromagnetic and weak forces is called a Grand Unified Theory or GUT]... This combined body of highly successful theories is termed the Standard Model (SM)and has been very successful in making predictions.  However, the Standard Model does not explicitly include quantum gravity or the graviton. 

In the Standard Model, messenger particles carry the forces (i.e., gluon, photon, weak gauge bosons W & Z [and the graviton]).  An electromagnetic field is conceived as a swarm of photons which shoot back and forth between the two objects that are interacting.  When protons repel each other, it is as if the messenger carries the message as to how they should interact.

Discussion of the importance of symmetry in physics.  E.g., the symmetry of quarks means that the interaction of 2 red quarks is the same if 2 green quarks are substituted for the red.  The universe thus exhibits strong force symmetry in that the strong force is not affected by force-charge shifts.  The strong force is an example of a gauge symmetry...

Because of quantum fluctuations, QM predicts empty space must have undulating gravitational field, an effect that becomes more and more apparent as the scale decreases.  At tiny scales, there is violent warping and turbulence, so called quantum foam (Wheeler).  Thus at very small scales, the smooth spatial geometry required of general relativity is destroyed.  Calculations attempting to merge general relativity and gravitation with quantum mechanics thus fail due to resultant infinities (singularities).  The fabric of space appears smooth except at the ultramicroscopic level.  The scale on which this problem emerges is the Planck length = sqrt(h-bar*G/c3) = 1.616x10-33 cm (some authors use h and obtain 4x10-33 cm).  This is an important (though controversial) incompatibility, despite the fact that it only occurs at this tiny scale, and the resolution of this lies in superstring theory.

Chapter 6
Nothing but Music
The Essentials of Superstring Theory

String theory invites a musical metaphor of vibrating strings (in the standard model, fundamental particles are point-like).  The strings are on the order of the Planck length in size.  All matter and forces are proposed to arise from strings.  The theory began with Veneziano 1968, who found that certain properties of the strong nuclear force followed the Euler beta-function and in 1970 Nambu et al. suggested this meant that they could be represented as vibrating strings. Certain predictions of this model were successful, but it was overwhelmed by the success of the standard model.  String theory postulated additional messenger-like particles, and in 1974 Schwarz and Scherk realized that they matched with the expected properties of the graviton.  They suggested the theory included quantum gravity but was not confined to this.  But there were unresolved conflicts between string theory and QM until 1984, when Green and Schwarz found a way to resolve these, and that the resultant theory could incorporate all 4 forces and all of matter as well.  The "first superstring revolution" resulted, in 1984 - 1986.  "Numerous features of the standard model ...  emerged naturally and simply from the grand structure of string theory."  But so far even the equations have not been precisely determined, much less solved.  Edward Witten 1995 began the "second superstring revolution" by introducing certain ways of partially dealing with the complexity.

Strings are either truly fundamental and indivisible or composed of smaller constituents--this is an unanswered question, but the author for discussion purposes "will take strings to be nature's most fundamental ingredient".  Numeric predictions are possible from string theory that must simply be presumed in standard model to be measured input.  The standard model is too flexible and cannot be used to explain the properties of fundamental elementary particles.  In string theory, these properties (such as mass and force charges) emerge as resonances, specific vibrational patterns [note 7 p. 398 provides a hint of some of the details].  Discusses vibrational modes for open and closed strings.  Greater energy means greater mass (gravitational charge) and indicates a higher vibrational mode.  Similarly, electric charge, weak charge, and strong charge are carried by a particular string vibration mode.  One mode matches the graviton exactly.  Each elementary particle consists of a single string (all of which are fundamentally identical) undergoing a characteristic resonant vibration ("different notes").  Thus the promise of a Theory of Everything, in which fundamental properties could theoretically predicted--but such euphoria is premature and these predictions have not yet been made because of the computational complexity.

String tension varies but is very high at around 1x1039 tons (the Planck tension)--transmitted force is inversely proportional to the string tension.  This causes them to contract to very  small size (c. Planck length).  The vibrational energy is high and occurs as whole number multiples of a minimal energy denomination, the Planck energy corresponding to a Planck mass of 1E19 times the rest mass of the proton (thus Planck mass is ~ 5.5E-8 kg).  Quantum jitters is a form of negative energy that cancel out all of this huge mass except the relatively low net energy.  In particular, the graviton candidate has a perfect jitter cancellation resulting in a zero mass particle (as expected by its capability of traveling at the speed of light).  Actually proving through computations that these cancellations occur is not currently feasible.  Of the infinite potentially possible vibratory string modes, only a few are light enough to be identifiable at typical current experimental energy levels obtained in accelerators (correspondingly, the Planck mass cannot be attained experimentally currently, and heavy particles in any event are probably unstable).

"String theory softens the violent quantum undulations by "smearing" out the short distance properties of space."  When taking measurements in a system, the size of the probe particle determines the potential maximal resolution, i.e., smaller probes can determine finer detail.  For small probes such as photons, its "size" is its quantum wavelength.  But the string's inherent spatial extent prevents is from probing the structure of anything substantially smaller than its own size.  Moreover, Gross and Mende 1988 showed that increasing the energy of a string does not increase its ability to probe smaller sizes, since in fact they grow in size.  This smearing is due to quantum jitter and to its own inherent spatial extent.  "There is a limit to how finely we can probe the universe."  In the point-particle paradigm, particle interactions occur at a precise location and time regardless of the state of motion of two observers.  But interactions of strings, which can be represented as a "world sheet" are more spread out, and separate observers in different states of motion may observe the time of first contact to be different (p. 162).  This "spreads out the force's punch" and "dilutes its ultramicroscopic properties" so that "calculations yield well-behaved finite answers."  I.e., the smearing smoothes out the ultramicroscopic jitters of space at sub-Planck length distances--the smearing cannot be removed.  The incompatibility of gen. relativity. and QM is resolved.

String theory predicts the exact properties of the graviton and gravity is an intrinsic component of string theory.  With the 2nd string revolution of 1995, strings are now conceived as having more than 1 or 2 dimensions

Chapter 7
The "Super" in Superstrings

 This chapter discusses the role of aesthetics in discovery in physics, and in particular the search for and desirability of symmetries.  Physicists assume the laws of nature are fixed [at least in our local universe within a multiverse], that every location in space and time exhibits the same laws.  Special relativity says the laws are the same for observers in uniform force-free motion with respect to each other. The equivalence principle of general relativity extended the symmetry of special relativity to make the laws of physics identical for observers moving and accelerating with respect to each other (motion symmetry).  The laws of physics should also not care about the angle from which observations are made, so rotations should not change them (rotational symmetry).  

The spin intrinsic angular momentum of electrons was proposed by Uhlenbeck and Goudsmit 1925, who used this to explain spectral line-splitting in the presence of a magnetic field.  This spin was inherently quantum mechanical and intrinsic to the electron, and occurred at only one rate, up or down.  All fundamental matter particles (i.e., fermions such as quarks plus the leptons such as electron, muon, and tau) have half-integral spin (i.e., with magnitude h-bar*1/2 or "1/2").  The force carriers are bosons with integral spin=1 (except the graviton, which is postulated to have spin 2).  In string theory, the spin is associated with the pattern of vibration: the graviton was predicted by Scherk and Schwarz to have spin-2 and no mass.

When spin is considered, an additional symmetry ("supersymmetry") becomes mathematically possible.  Supersymmetry is not associated with any easily expressible quantity [The author does not attempt to define it.]  In the 1970s, supersymmetry was found to predict the existence of superpartners for each known particle, with spins predicted to differ by 1/2 (none of these have yet been demonstrated).  [One website states "every fundamental matter particle should have a massive "shadow" force carrier particle, and every force carrier should have a massive "shadow" matter particle. This relationship between matter particles and force carriers is called supersymmetry."  The superpartners of fundamental fermions have been tentatively named by adding s-: thus, the squark, selectron, and sneutrino (these superpartners all have spin 1).  The superpartners of the force particles are named by adding -ino: photino, gluino, wino, and zino (these superpartners will have spin 1/2)

Even without string theory, there are compelling reasons to suggest the existence of supersymmetry.  (1) It is aesthetic.  (2) The standard model requires extremely finely tuned particle parameters, whereas in supersymmetry there is a canceling out by bosons and fermions occurring in pairs that removes this extreme sensitivity.  (3) It can lead to a grand unification of strong and other forces with gravity...  Georgi, Quinn, and Weinberg computed that the strengths of the 3 nongravitational forces become approximately equal at the scale of 10,000 times the Planck length, exactly equal if supersymmetry is incorporated.

But supersymmetry is especially appealing in string theory, and makes it "superstring" theory--contributors have been Ramond, Gliozzi, Scherk, Olive...  Superstring theory avoids problems with tachyons (particles traveling faster than light)...  However, there is an embarrassment of riches, inasmuch as there are 5 seemingly separate supersymmetric string theory candidates: Type I theory, Type IIA, Type IIB, the Heterotic Type O(32) theory, and the Heterotic type E8 X E8 theory.  Witten has shown that these types are all parts of a larger synthesis.

Chapter 8
More Dimensions Than Meet the Eye

 In 1919, the Polish mathematician Theodor Kaluza proposed the existence of additional spatial dimensions.  In the same way that an ant walking on a garden hose seen at great distance appears to be on a one-dimensional (line) surface, the extra dimensions can be conceived as being tightly rolled up so they do not appear obvious to the macroscopic observer...  This concept was made more explicit by Oskar Klein 1926...  The extra dimension might have the form of circles or other forms having sizes as small as the Planck length.  The idea of extra tiny dimensions is called Kaluza-Klein theory.

Analogy to Flatland organisms on a garden hose.  Kaluza analyzed general relativity with one extra dimension and produced the Maxwell equations, uniting general relativity with Maxwell's theory of light.  His ideas were ahead of his time and were abandoned until revived in the 1970s.  The extra dimensions were assumed to be smaller than those which we can currently probe.  The most promising formulations incorporate supersymmetry.  Strings inhabiting a universe with higher dimensions have more independent vibrational directions [?modes].  When 9 spatial dimensions were assumed, problematic negative probabilities dropped out, so nine spatial and one time dimension rendered 10 spacetime dimensions.  In the 1990s, Witten showed that 10 was an approximation and that the actual value should be 11 spacetime dimensions.  (There is nothing to prevent additional time dimensions.)  

The extra dimensions probably take the form of 6-dimensional geometrical shapes called a Calabi-Yau space or shape, of which there are many thousands of members.

Chapter 9
The Smoking Gun: Experimental Signatures

Greene cites difficulty in obtaining any experimental verification, and the opposition that has been expressed to such an unprovable theory (e.g., Glashow), along the same lines that theology undermined medieval science...  It is not possible to do research at the Planck length level.  The necessary accelerator would need to be the size of the universe...  It will be necessary to search for indirect confirmations of the theory.  E.g., Candelas, Horowitz, Strominger, and Witten suggested that there is a family of particles corresponding with each additional hole in the Calabi-Yau space.  Perturbation theory has been the traditional way to make difficult calculations involving multiple variables, especially when one effect (like the sun in computing planetary orbits) dominates...   

The existence of the superparticles predicted by superstring theory will be sought with the CERN Large Hadron Collider under construction in Geneva for operation in 2010.  Also, fractionally charged particles have been predicted and will be sought, though likely to have very large mass ~Planck mass.  Even less likely possibly related phenomena to be found are macroscopic strings, nonzero neutrino mass, disintegration of the proton, decays of quarks, new forces, dark matter, a revised cosmological constant, etc.  Whole generations of physicists will labor with superstring theory without the benefit of constant experimental results to correct erroneous conclusions.

Chapter 10
Quantum Geometry

Reviews Georg Riemann's view of curved spaces, which is the mathematical core of general relativity.  Quantum geometry is the mathematical core of  string theory, though it is not as ready-made as was Riemann's geometry for Einstein.  Riemann drew on Gauss, Lobachevsky, Bolyai etc. and evaluated the measure of distances in curved space.  Einstein concluded the curvature of space is gravity.

The Big Bang occurred 15 billion years ago.  If the critical density of the universe (1E-29 gram per cc) is exceeded, the expanding universe will halt and reverse and collapse.  String theory is needed to evaluate the extremely compressed early state, and sets a lower limit to the ultimate size of the Big Crunch (the Planck length).  This limit arises from string wrapped configurations called winding modes, in which there is a minimum length and mass...  Strings that attempt to collapse smaller than the Planck length begin to expand as follows.  For every large circular radius, there is a corresponding small circular radius for which the winding energies of the corresponding strings are equal, and there is no physical distinction between these geometrically distinct forms...  Thus when one tries to make a measurement using the lightest (easiest) of the string modes, the measured result will always be larger than the Planck length.  When shrinkage to below the Planck length is attempted, the crunch becomes a bounce.  Ultrashort distances and their infinities are thereby avoided--sub-Planck-scale distances are simply inaccessible and meaningless...  Brandenberger, Vafa, etc. used these results to suggest a new cosmology where the big crunch is Planck length in size.  Regardless of the shape of the compact dimensions, there is probably a limiting size.

Greene discusses his, Plesser, and Candela's discovery of a key string property, mirror symmetry.  Mirror manifolds are physically equivalent yet geometrically distinct Calabi-Yau spaces which exist for certain Calabi-Yau spaces that were previously thought to be unrelated but are connected through string theory.  This pairing allows what would be a very difficult calculation of a particular Calabi-Yau space to be made on the simpler mirror symmetric Calabi-Yau space.  One example is the counting of spheres that can be packed inside the Calabi-Yau space...

Chapter 11
Tearing the Fabric of Space

Greene discusses concept of wormhole, "a bridge or tunnel that provides a shortcut from one region of the universe to another".  Can space tear?  In 1987, Yau and Tian showed that certain Calabi-Yau shapes could be transformed by puncturing their surfaces and sewing them back up according to a pattern...  They envisioned an internal sphere which is pinched and torn leading to a flop-transition as if the sphere is flopped into a new orientation...  By adding the mirror perspective, Greene showed that a space-tearing flop transition for one Calabi-Yau space was perfectly well behaved in the mirror, excluding the possibility of a catastrophe in the original...

Homage to Edward Witten's brilliance.  

Greene describes his successful effort to verify using a computer program that space tearing transitions (topology-changing transitions) are part of string theory.  Witten added that strings encircle the tear, shielding the universe from catastrophe...  Papers published 1993...

Chapter 12
Beyond Strings: In Search of M-Theory

Greene summarizes the problems with string theory apparent by the 1980s.  There were too many candidates.  The exact form of the equations was not known.  In 1995, the second superstring revolution began by Witten's suggestion that the 5 candidates are all related and part of an overall synthesis or framework called M-theory.  M-theory requires 10 space dimensions, 11 overall dimensions including time.  Previously, one space dimension had been "overlooked" because the calculations made were only approximate.  M-theory not only includes vibrating one-dimensional strings ("one-branes") but more extended objects such as 2-dimensional membranes ("two-branes"), 3-D blobs ("three-branes"), and other even more complex objects.  Moreover, the techniques commonly employed to facilitate approximate calculations, perturbation theory (originally applied to astronomical orbits), failed in some cases of string theory.  Perturbation theory helped in dealing with virtual string pairs that transiently materialize, even in multiple numbers, but are the computations correct?  The answer depends on the value of the "string coupling constant": if less than one (weak coupling), the perturbative approach is likely to be valid, but if the coupling constant is one or greater (strong coupling), the likelihood or higher order virtual pairs affecting the calculations is greater and perturbative theory cannot be used.  The value of this constant is currently not known, and approximate string equations have proven too flexible to be useful in aiding this determination.

Witten introduced the concept of duality at the 1995 string conference at USC.  This concept allows the use of perturbative theory for a wider range of problems.  Examples of duality are (1) the string pairs resulting from mirror symmetry and (2) the equivalence of string computations at circular dimensions of R and 1/R.  Witten suggested the five competing theories are actually dual (each has an equivalent string in at least one other theory), and that a string exhibiting strongly coupled behavior in one theory has a dual description with weak coupling under another theory, thereby allowing calculations to be made with perturbative theory.  He also included a 6th theory to complete the dualities: 11-D supergravity.  Through the use of supersymmetry and this dualism, previously intractable properties can theoretically be reduced to fewer possibilities and can be calculated indirectly...  A set of states proposed by Bogomol'nyi, Prasad, and Sommerfield ("BPS states") specify a minimum mass condition which is conducive to calculation without use of perturbative approach, i.e. the masses and charges calculated are nonperturbative yet involving strong coupling and provide useful known results.

"These strong coupling characteristics of Type I string theory exactly agree with known properties of Heterotic-O string theory when the latter has a small value for its string coupling constant."  This "suggests that the physics of the Type I theory for large values of its coupling constant is identical to the physics of the Heterotic-O theory for small values of its coupling constant."  This is a strong-weak duality.  Proof of this will be difficult to achieve, but theorist believe duality is likely to be correct as well as helpful in making calculations.  Type IIB strings are self-dual.  Moreover, Type IIA strings in a universe of radius R have identical physics to Type IIB strings in a universe of radius 1/R. 

Supergravity theory preceded string theory: "supersymmetric quantum field theories that try to incorporate general relativity", but these met with failure, though the most nearly successful were those incorporating 10 (or 11) dimensions.  Witten 1995 argued that when its coupling constant is increased, the 10-dimensional Type IIA string theory  approximates 11-dimensional supergravity, the added dimension becoming more and more apparent as the coupling constant rises... Witten and Horava suggested also that as Heterotic-E theory becomes more and more strongly coupled (i.e., the coupling constant increases), an 11-th dimension becomes increasingly apparent as the strings get stretched into cylindrical membranes.  This added dimension is not one in which the vibration can occur.  A similar description applies to Type IIA strings, though these stretch into a 2D toroid surface.

For low energies, the postulated but as yet undefined 11-D theory is the former supergravity quantum field theory.  Witten has named the overarching theory M-theory (for Mystery, Mother, Membrane, Matrix, etc.).   All 5 string theories are joined together by dualities as is 11-D supergravity--we can pass between any one theory to any other via the central M-theory (as with a hub and spoke analogy) by means of varying the coupling constant and various geometric parameters.  In each of the 5 string theories, the strings can actually be 2-dimensional membranes.  In fact, they can theoretically have up to 9 dimensions (nine-branes, generically "p-branes").  However, all but the 1-dimensional strings will be very massive and are unlikely to play much of a role in routine conditions and can probably be ignored.  A key unanswered question is what is the actual position in the synthesis of the 6 theories that actually describes our universe.

Chapter 13
Black Holes: A String/M-Theory Perspective

Reviews contributions of Hawking, Penrose, etc.  Black holes are similar to elementary particles inasmuch as they have no identifiable internal structure: they can be completely characterized by their mass, force charge, and spin angular momentum.  Discussion of problem of collapse of space in the region of a black hole and possible cataclysm--Andrew Strominger proved this will not happen due to wrapping of three-branes about 3-D sphere, etc...  Greene and others realized that a collapsing 3D sphere can lead to a tear in a Calabi-Yau space with reinflation of the sphere now as a 2D sphere (a "conifold transition")...  "One Calabi-Yau shape could, in essence, transform itself into a completely different Calabi-Yau shape." with a different number of holes (in fact, a Calabi-Yau space can transform into any other Calabi-Yau space)...  Regarding black holes, the author suggests that string theory predicts they can undergo this transition to a massless elementary particle, a photon, through what is essentially a phase transition.  Black holes and photons are simply "two phases of the same underlying stringy material".

What happens to entropy when matter is consumed by a black hole?  Bekenstein with Wheeler 1970 addressed this question, suggesting that they must have entropy to satisfy the 2nd law of thermodynamics, and that the area of the event horizon would be a measure of the entropy of the black hole.  Hawking 1974 has shown that black holes have a temperature, and radiate black-body radiation due to consumption of one but not the other of a pair of virtual photons that come into existence just outside the event horizon.  The apparent temperature of a black hole implies there is entropy associated with the black hole.  [The predicted temperature is (6 x 10-8/M) Kelvin, where M = number of solar masses.  Thus small black holes have higher temperature and more rapidly radiate away their energy and even eventually evaporate.]

String theory (via Strominger and Vafa 1996 publication) answers the question: where is the disorder in the high entropy of a large black hole?...  Their calculation, made on idealized "designer" black holes, agreed with the Hawking/Bekenstein prediction, an important verification of string theory.

The determinism of Pierre-Simon Laplace was replaced by a lesser quantum determinism, in which future probabilities are determined via wave functions but actual outcomes are not precisely predictable.  Hawking showed that anything sucked into a black hole has its wave function sucked in as well, so information is apparently lost forever (unappealing to physicists), but can it ever reemerge occurs from the black hole?  Hawking maintains no, but Strominger and Vafa have suggested that it might as radiation.

Evaluation of what has previously been thought to be a singularity at the center of a black hole is also a subject for string theory investigation, but so far has been unsuccessful.

Chapter 14
Reflections on Cosmology

Greene reviews standard model of cosmology based on Big Bang 15 billion years ago, etc. (see Fig. 14.1 p. 356 or this timeline).   At the Planck time (1.35E-43 sec) after the Big Bang ("ATB"), the temperature was 1E32 Kelvin.  At 1E-5 sec ATB, it had cooled to 1E12 Kelvin and allowed quarks to clump into neutrons and protons.  At 1E-2 sec ATB, primordial nucleosynthesis began.  In several thousand years, neutral atoms formed and the universe became transparent.  From then on, photons have traversed freely.  At 1 billion years ATB, galaxies and stars formed...  The cosmic background radiation is the residual black body radiation cooled by the expansion of the universe and is now c. 2.7 Kelvin at a current density in space of c. 4E8 photons/m3.  Point particle physics cannot be used to deal with calculations prior to the Planck time, whereas string theory can.  The time prior to 1E-35 sec ATB (temperatures exceeding 1E28 Kelvin) is a realm where the 3 nongravity forces were united prior to symmetry breaking.  

However, the "horizon problem" exists: The observed degree of apparent homogeneity of the cosmic background radiation is not predicted by the expected spatial proximity and temporal duration prevailing in the Big Bang.  I.e., there would not have been adequate time for thermal equilibrium to form between regions over the horizon, as it were, given the finite speed of light.  The observed facts can be reconciled via a period of inflation as proposed by Alan Guth et al 1979 and refined by Linde, Steinhardt, Albrecht, and others.  This solution is compatible with Einstein's general relativity equations and posits a period during which the universe inflated at an exponentially accelerating rate.  This provides more time in the early stages for communication and equilibration to occur.   Specifically, during the interval 1E-36 ATB to 1E-34 ATB, the universe would have expanded 1E30 times, and before that, matter was sufficiently close together to be in equilibrium.  [During the inflationary time, "the energy density of the universe was dominated by a cosmological constant term that later decayed to produce the matter and radiation that fill the universe today" (quote is from the linked location).]  String theory limits the lower limit of the size of the universe to the Planck length.  Brandenberger and Vafa suggested that at about the Planck time, three of the previously tightly curled up dimensions are singled out (or less teleologically, randomly become favorably positioned) for rapid inflationary expansion to extended spatial dimensions.  The number three is favored by string theory on probability grounds having to do with the likelihood of particle interactions (which are unlikely if more than three).  

Greene offers additional speculation on alternate pre-big bang scenarios and points out that Gabriele Veneziano has concluded that string theory allows the inflationary sequence to follow naturally. 

M-theory may also allows a less artificially forced merging of gravity with the three nongravitational forces and avoids the extreme states of infinite compression and energy.  Greene discusses speculation about a larger multiverse, in which our universe is simply a local manifestation or island universe for which an era of inflationary expansion occurred.  Other universes may experience periods of inflation at other "times" and might have different laws of physics or particle properties, number of dimensions, etc.  But our universe has the properties that have made life and therefore us as observers possible--the so-called [weak] anthropic principle.  The multiverse may exhibit chaotic variation among the various universes.  Smolin has suggested that every black hole is the seed for a new universe, and that universes with parameters optimized for forming black holes have a [reproductive-like] advantage, and come to dominate the population of universes within the multiverse.

Chapter 15
Prospects [for Unification on the Twenty-First Century]

There are five central unanswered questions of greatest importance facing string theory:  

We are in for even grander surprises in the 21C, but have been greatly enriched by this quest so far.