The Elegant Universe
Chapter Summaries
Editor: Ross Cheung, minor corrections by Wing Ning Yung
Chapter 1, part 1: “String Theory: The Basic Idea"
Kumar Jeev
The first chapter "Tied up with Strings" starts off with a description of the
inherent antagonism between the two "foundational pillars of modern physics - quantum
mechanics and theory of general relativity. Quantum mechanics provides the set of laws
explaining phenomena at the small scale while general relativity does the same for the
large scale. Unfortunately, these two theories are incompatible. This provides the
motivation for further research to search for some kind of unification of the two using the
string theory. The author then begins a brief overview of matter at the microscopic level
and describes the search of the Greek "atom" (last indivisible part of all matter). In this
process the author brings us up-to-date on the most recently discovered three families of
fundamental particles. To complete the overview the author finally describes the four
fundamental forces of nature - gravity, electromagnetism, strong, and weak and the
particles associated with them.
Chapter 1, part 2
Ross Cheung
This section introduces the concept of string theory, the theory that all electrons
and quarks are composed of a "string," a one-dimensional infinitesimally thin loop.
String theory suggests that all matter and all forces are determined solely by the
oscillations and vibrations of this string. For this reason, many scientists, with varying
degrees of skepticism, see string theory as perhaps the unified field theory, or "Theory of
Everything", that has been the Holy Grail of physics for much of the 20th century.
This is, however, a controversial school of thought. Staunch reductionists claim that
virtually everything in the world can be explained in terms of the properties of this "string
theory", an outlook that many are uncomfortable with. Opponents of reductionism claim
that new developments such as chaos theory state that at every increasing level of
complexity, new laws come into play, which cannot be reduced to a simpler set of laws.
The author suggests that the new laws that are formed with each increasing level of
complexity aren't necessarily independent of each other, but can be reduced to a simpler
level; they exist because to reduce all the laws to a basic set of principles would be
virtually impossible due to the level of complexity that would result. Finally this section
ends with the current state of string theory. The problem with the theory is that due to
technological limitations, the theory hasn't yet been confirmed with any rigorous
experiments and has yet to be accepted by the scientific community. The theory has
revealed numerous insights into the nature of our universe, and presents solutions to
certain conflicts which have troubled physicists for some time. However, it can at best be
viewed as a work in progress.
Chapter 2, Part 1: “Space, Time, and the Eye of the Beholder”
Kancy Lee
In the mid 1800s, after the successful work of the physicists Michael Faraday and
James Maxwell that united electricity and magnetism in the framework of the
electromagnetic field, the paradox about light greatly troubled Einstein and many other
physicists. Maxwell’s theory showed that visible light is a kind of electromagnetic wave
that always travels at light speed, never stopping or slowing down. However, according
to Newton’s law of motion, light will appear stationary if we travel at the same speed.
Not until 1905 did Einstein resolve the conflict through his special theory of relativity,
introducing new conceptions of space and time.
Special relativity claims that observers in relative motion will have different
perceptions of distance and time. In other words, individuals who are moving with
respect to each other will not agree on their observations of either space or time. The
effects of special relativity depend on how fast one moves: as the relative velocity of
individuals gets larger, the differences in measurements of time and distance become
increasingly apparent. These phenomena are called “time dilation” and “Lorentz
contraction”. The principle of relativity states that all constant-velocity, force-free motion
is relative, and that the laws of physics are absolutely identical for all observers
undergoing constant-velocity motion.
Chapter 2, Part 2
Daniel Ho
Time is a very abstract concept, but for simplicity, we may consider it as
something measured by clocks. However, motion can affect the passage of time, i.e. the
clocks, which even include the biological clocks in our bodies, making them run slower
or faster due to their motions. Not only time but space is also affected by motion. For
example, the length of a moving car measured by a stationary observer standing on the
ground is shorter than its original length. However, all these effects are hard to detect in
our daily lives as the speeds we usually encounter are very slow compared to light speed.
Einstein interprets all these phenomena by just considering space and time as
different dimensions of spacetime. He thinks that all objects are always moving in
spacetime at light speed, and the phenomena of special relativity are due to the sharing of
motion between the space dimension and the time dimension when an object is moving.
Not only are space and time interwoven and twisted with each other, another example is
energy and mass, which leads to Einstein’s most famous equation: E = mc2. Energy and
mass are not independent. They are convertible to each other, just like coins and
banknotes.
Lastly, nothing can travel as fast as light. The reason is that the faster an object is
travelling, the greater its energy will be. According to Einstein’s most famous equation,
the object mass will increase as the energy increases. As a result, it is harder and harder
to push the object. In fact, if the object’s speed is at the light speed, its energy is infinite,
which is certainly impossible to achieve. This is why nothing can travel as fast as light.
Chapter 3: “Of Warps and Ripples”
Ting Liao
In chapter 2, the author explained how special relativity resolved the conflict
between the classical intuition about motion and the constancy of the speed of light. In
chapter 3, he turns to the conflict resulting from special relativity and the Newtonian
gravitational law.
According to the Newtonian gravitational law, the gravitational force depends
only on the masses of the objects and the distance between them. That means, if I move
one of the objects, the other object will feel the change in gravitational force
instantaneously since the distance is changed. However, according to special relativity,
no information can travel faster than light, so the object cannot feel the change in the
force instantaneously. Something is inconsistent.
Einstein, who had a lot of faith in special relativity, decided to find out what
gravity is. Gravity had been a black box to human beings before Einstein. No one knew
how it worked but only what it did, and this made it very hard for Einstein to start. On a
day in 1907, while Einstein was working on this issue in the patent office in Bern,
Switzerland, he had an idea about the link between gravity and acceleration. The author
used an example to explain the idea. If we want to send something to outer space without
changing the down-pull force that it is feeling, we can use the acceleration of the rocket
to make the object feel the force that is equal to the gravitational pull when it is on the
ground. This idea led Einstein to think of gravity and acceleration as equivalent.
Therefore, to work out what gravity is really doing, he started out with acceleration. This
was incredibly difficult even for one of the brightest minds in history.
Using special relativity and the link between gravity and accelerated motion,
Einstein made another breakthrough. To explain this, the author used an example of
motion in a circle. In a circular motion, the speed is constant and the acceleration only
affects the direction of the motion. If a person is inside the object that is rotating, say a
tornado, and he tries to measure the ratio of the circumference to the radius, what would
he find? For us, the bystanders, we will find a ratio of 2*pi. However, due to special
relativity, the person’s ruler will be contracted (shorter) due to Lorentz transformation
since his ruler will be in the direction of the velocity. To him, the ruler is still the same
length because they are in the same reference frame. Therefore, the circumference length
he measures will be longer. On the other hand, the radius is perpendicular to the velocity,
so there is no contraction and the radius measured will be the same as what we measure.
As a result, the ratio will not agree. Why is the ratio different? To answer this question,
Einstein proposed that space is not flat when there is acceleration. In other words, what
gravity does is actually curve the space. When things are traveling, they travel in the
shortest path. According to general relativity, the “pull” is just the curvature leading
objects to accelerate. The image is very hard for us to imagine since we live in a three
dimensional of apparently non-curving space, curvature of space is hard to visualize.
Besides, the time is also curved since time is just another dimension, but this is so
complex to understand so the author did not spend much time on this.
For some time there was no confirmation of general relativity. Finally, using the
eclipse in 1919, scientists showed that general relativity predicted the angle of the
incoming light precisely. Since then, different experiments have successfully confirmed
general relativity. Yet, there is still a problem. The original form of general relativity
predicts the universe to be expanding, thus Einstein added a cosmological constant to
make it satisfy a static universe. When Hubble found proofs that the universe is in fact
expanding, Einstein removed that term, (although we now believe that that term might
actually be needed to describe the acceleration in the expansion of the universe). General
relativity deals with objects with great mass, such as stars and black hole. The problem is
that black holes can be really small but really massive, and we have to use quantum
mechanics and general relativity to explain them at the same time. There is where the
conflict between the two theorems arises.
Chapter 4, Part I: Microscopic Weirdness
The chapter "Microscopic Weirdness" is an introduction to quantum mechanics.
The 'weirdness' of the quantum phenomena is a highlight of the H-bar where the fictional
trans-solar system explorers Gracie and George share a drink in the introduction of this
chapter. The author then quickly moves further to establish quantum mechanics as the
"conceptual framework" to understand the microscopic universe. He introduces the
central problem of nineteenth century physics, that of the lack of understanding of the
dual nature of electromagnetic radiation. Planck's hypothesis, about the lumpy nature of
energy and the existence of Planck's constant as the proportionality constant between
frequency of a wave and its energy, bridges the gap in this lack of understanding. Plank's
theory generated a new question about the reason of the existence of these energy lumps.
This was answered by Einstein in his paper on the Photoelectric Effect who proposed the
existence of photons, the light quanta. The author then describes Young's double slit
experiment which arguably proves the wave-nature of light. This in turn brings the
"wave-particle duality" to the forefront which sets the need for a theoretical
understanding of the 'weird' micro-cosmos.
Chapter 4, Part II: Microscopic Weirdness
Kancy Lee
In 1926 German physicist Max Born suggested that the wave nature of matter can
only be interpreted in a probabilistic manner: we can only predict that the location of an
electron may be found with a non-negligible probability, but we cannot predict the exact
location of that electron. Although Schrödinger’s equation, that governs the shape and the
evolution of probability waves, can make accurate predictions which agree with
experimental results, many still find this idea unacceptable and there is no consensus on
what it means to have probability waves.
Richard Feynman was one of the greatest theoretical physicists. He fully accepted
the probabilistic interpretation, but argued that any individual electron traveling from one
location to another actually traverses every possible trajectory simultaneously. By
assigning a particular number to each of these trajectories, Feynman’s formulation shows
that the combined average of these paths can give results that agree with those of
Schrödinger’s equation and experiments. Therefore, Feynman’s perspective, which does
not require the probability wave, has provided a different approach to quantum
mechanics.
The hallmark feature that fundamentally differentiates quantum from classical
reasoning is the uncertainty principle. Discovered by German physicist Werner
Heisenberg in 1927, the uncertainty principle states that there is always some disruption
to the electron’s velocity through our measurement of its position. Heisenberg found that
the mathematical relationship between the two is inversely proportional to each other.
That is, the greater precision in a position measurement entails greater imprecision in a
velocity measurement, and vice versa. Likewise, he also found that the increasing
precision of energy measurements require longer durations to carry them out. Hence,
according to Heisenberg’s uncertainty principle, quantum mechanics allows a particle to
borrow energy within a certain time frame.
Chapter 5
Daniel Ho
The heart of quantum mechanics is the uncertainty principle, and it reveals the
uncertainty and frenzy in the quantum world. The Schrodinger Equation is also another
important description of the quantum world. However, physicists found that it is only an
approximation, as it does not include special relativity. As a result, physicists started to
seek a new theory that includes both of them. They created the quantum field theory,
which helps to describe three fundamental interactions. Then they generalized all they
know about these three interactions into a theory called the standard model. According to
the standard model, there are messenger particles responsible for different types of
interactions. They help to transmit the message of how different particles should interact
with others.
There is still one fundamental interaction that has not been described with its own
quantum field theory: gravitation. In order to create this quantum field theory, physicists
have to combine quantum mechanics and general relativity. However, physicists found
that they are not compatible. The reason is that the smooth spatial geometry, which is the
central principle of General Relativity, is destroyed by the violent quantum fluctuation
when the spacetime is examined in a very small scale. Mathematically, the combination
of equations of General Relativity and quantum mechanics yields a non-physical
solution: infinity. As a result, a new theory found to combine them is the superstring
theory.
Chapter 6 : “Nothing but Music : The Essentials of Superstring Theory”
Ting Liao
The chapter started off explaining why we need string theory. The standard model
(before string theory) views the elementary constituents of the universe as point-like
ingredients with no internal structure. It can explain almost everything, except gravity.
Moreover, attempts to incorporate gravity into its quantum-mechanical framework have
failed due to the violent fluctuations in the spatial fabric that appear at ultramicroscopic
distances (shorter than the Planck length). With these problems present, physicists started
to search for a deeper understanding of nature. Michael Green and John Schwarz
provided the first piece of convincing evidence that superstring theory might provide this
understanding.
The author then turned to talk about the history of string theory. Physicists first
proposed the string theory because it can explain why the nuclear interactions are
described by the Euler beta-function. However, the string model led to a number of
wrong predictions and was abandoned by most physicists in the 1970s. Yet a few
physicists kept working on it. Their work was ignored by most people, partly due to the
conflicts between string theory and other model, such as quantum mechanics. Such was
the case until 1984, when Green and Schwarz found that the subtle quantum conflict
afflicting string theory could be resolved. They also showed that the resulting theory had
sufficient breadth to encompass all of the four forces and all of matter.
At that point, physicists around the world all tried to work on string theory, a
theory that appeared to be the theory of everything. The period from 1984 to 1986 is
known as the first superstring revolution. There were great developments on string theory
in those 3 years. Unfortunately, the mathematics for string theory is so complex that
everyone got stuck at some point thus many decided to give up. Finally in 1995, Edward
Witten announced a plan for taking the next step. The is the beginning of the second
superstring revolution.
We used to have no idea why particles have the properties they have. String
theory offers the explanation that these properties are the allowed vibrational patterns of
the strings. With the calculation on the force of gravity, physicists found that the tension
of the string is in the order of 10^39 tons. That led to three consequences. Firstly, the
huge string tension causes the loops of string theory to contract to a minuscule size.
Secondly, the typical energy of a vibrating loop in string theory is extremely high. With
that said, how do the strings build up the particles with low energy? The author explained
due to quantum mechanics, there is a quantum fluctuation that cancels the energy and
give rise to the low energy particles. The third consequence is that the strings can execute
an infinite number of different vibrational patterns. However, all but a few of these
vibrational patterns will correspond to extremely heavy particles. With our technology, it
is very hard to search for these heavier particles.
How, then, does the string theory solve the conflict between general relativity and
quantum mechanics? Basically string theory softens the violent quantum undulations by
“smearing” out the short-distance properties of space. The author gave a rough and a
precise answer about what this means. The basic idea is the same. To probe something,
the “object” we use as a probe cannot be larger than the feature we are looking at. The
most appropriate measure of a particle’s probing sensitivity is its quantum wavelength. In
1988, David Gross and his student Paul Mende showed that when quantum mechanics is
taken into account, continually increasing the energy of a string does not continually
increase its ability to probe finer structures, in direct contrast with what happens for a
point particle. The upshot is that no matter how hard you try, the extended nature of a
string prevents you from using it to probe phenomena on sub-Planck-length distances.
The conflict between general relativity and quantum mechanics arises when looking at
sub-Planck-length distances. Now that this tiny length is smeared out, the fluctuation
according to quantum mechanics is smoothed out just enough to cure the incompatibility
between the two theories.
Chapter 7
Kumar Jeev
The chapter "The Super in Superstrings" is concerned primarily with the principle
of supersymmetry in string theory. The author starts off with an introduction of the
various symmetries that exist in nature and then talks about the most recently discovered
symmetry of nature - that related to the spin of a particle (Supersymmetry).
Supersymmetry however required the existence of superpartners, whose existence has not
yet been experimentally verified. Before string theory came in, the case for
supersymmetry was weak. It was supported merely by the aesthetics of symmetry, ease of
fitting gravity into the quantum model, and its support of the grand unification of nature's
forces. These justifications although were significant were not rigorous. It was only after
the advent of string theory that the profound concept of super symmetry got integrated
into physics. However, this integration created problems for string theory because it
resulted in the creation of five different and totally consistent string theories. The author
ends the chapter with a note that these seemingly different theories have now been shown
to be different ways of describing "the one and the same overarching theory".
Chapter 8: More Dimensions Than Meet the Eye
Kancy Lee
Perceptions are based on experiences. Beliefs and expectations from common
experiences are the most difficult to challenge. In 1919, a Polish mathematician Theodor
Kaluza had the temerity to take the challenge. He suggested that the universe might have
more than three spatial dimensions. Later in 1926, a Swedish mathematician Oskar Klein
refined Kaluza’s work, and proposed that the universe may have additional spatial
dimensions that are tightly curled up into tiny spaces which even the most refined
experimental equipment cannot detect. Just like a garden hose which may look like a thin,
one dimensional line from a substantial distance, we can see its curled-up dimension
when we move closer. Thus the universe should have at least four spatial dimensions:
three large, extended dimensions, and one small, circular dimension.
Based on the possibility of having an extra spatial dimension, Kaluza argued that
both gravity and electromagnetism are associated with ripples in the fabric of space.
Gravity is carried by the familiar large and extended dimensions, while electromagnetic
force is carried by the new curled-up dimension. Although Kaluza’s equations, which
resulted from analyzing the general relativity with the extra dimensions, were strikingly
similar to those used to describe electromagnetic force, his theory was unable to
incorporate all features of forces and matter. But nonetheless, Kaluza’s and Klein’s
reasoning had indicated that the universe may very well have more dimensions than meet
the eye, and hence provided a compelling framework to relate Einstein’s general
relativity and Maxwell’s electromagnetic theory.
Chapter 9: “The Smoking Gun: Experimental Signatures”
Daniel Ho
However elegant the string theory is, it is not easy for physicists to accept it
without experimental verification of its predictions. Also, it should provide an
explanation of phenomena not understood by any other current theory. In the past, many
physicists did not support the string theory just because it is not easy to find experimental
results of verification. As a result, they debated whether they should do any further
research on string theory. Strings theorists believe the lack of verifying experimental
results is due to a lack of sufficient technology. This does not mean that string theory is
fundamentally divorced from experiment.
However, string theory can be verified in an indirect manner, using things like
superparticles, fractional charged particles, etc. In addition, string theory gives the
framework to explain the properties of the elementary particles and messenger particles.
Nevertheless, the story is not that simple. There are many Calabi-Yau shapes, which is a
very important concept in string theory, making it possible for us to explain our world.
The equations of string theory are only an approximation and they are not strong enough
to tell which shapes are most suitable. Furthermore, string theory may be able to explain
the data and properties of neutrinos, especially if neutrinos are found to be of non-zero
mass, since according to standard model, neutrinos have zero mass. String theory may
also be able to find the possible candidates for dark matter, which is important in
cosmology. Finally, compelling evidence may come if string theory can help improve
the mismatch between the calculated value and the observed value of the cosmological
constant.
Chapter 10: “quantum geometry”
Ting Liao
Einstein’s General Relativity changed the world’s view about gravity, but it was
Riemann who broke the chains of flat-space Euclidean thought and paved the way for a
democratic mathematical treatment of geometry on all varieties of curved surfaces. It was
Riemann’s insights that provide the mathematics for quantitatively analyzing warped
spaces. While in string theory we need to modify general relativity in small distances in
the Planck’s scale, we need another geometry to describe the physics, and that is what we
call quantum geometry.
In general relativity, the curvature of spacetime reflects the distorted distance
relations between its points. By making the object smaller and smaller, the physics and
the mathematics align ever more precisely as we get closer and closer to physically
realizing the abstract mathematics of a point. However, in string theory, we cannot get
any smaller than a string, so the geometry is modified on ultramicroscopic scales.
According to the big bang model of cosmology, the universe violently emerged
from a singular cosmic explosion. Today, the “debris” from the explosion is still
streaming outward. If the total mass in the universe is great enough, the gravity may be so
strong that all the mass collapses to one singular point again according to the theory of
general relativity. However, when the distance scales involved are around the Planck
length or less, quantum mechanics invalidates the equations of general relativity. String
theory again gives a limit on how small the universe can be. We now turn to how string
theory changes the picture.
For simplicity, we think about the problem in a garden-hose universe. The main
difference between a garden-hose universe and a flat universe is that string can be
wrapped around the universe in a garden-hose universe. That implies that the energy of a
string in the garden-hose universe comes from two sources: vibrational motion and
winding energy. Vibrational motion is further split into ordinary vibrations (usual
oscillations) and uniform vibrations (the overall motion of string as it slides from one
position to another without changing its shape). Uniform vibrational excitations of a
string have energies that are inversely proportional to the radius of the circular dimension
according to quantum mechanics. On the other hand, the winding mode energies are
directly proportional to the radius. So when a universe is becoming smaller, the uniform
vibrational energy is increasing and the winding mode energies are decreasing. Since we
care only about the total energy of a string, as long as the sum of the two energies is the
same, it is the exact same configuration. Also if we exchange the energy between the
vibrational energy and the winding mode energy we can get the same state. But the
difference in the distribution of energy means the universe has a different size. What that
means is that for any size of the universe, there is a corresponding size that gives the
exact same physics, so when the universe is crushing down, it can be looked at as a
collapsing to the planck’s length and then expanding again.
Before we get any further, we have to understand what distance is. There are two
different yet related operational definitions of distance in string theory. Each lays out a
distinct experimental procedure for measuring distance and is based, roughly speaking,
on the simple principle that if a probe travels at a fixed and known speed then we can
measure a given distance by determining how long the probe takes to traverse it. The
difference is in the choice of probe used. The first definition uses strings that are not
wound around a circular dimension where the second uses strings that are. Unwound
strings can move around freely and probe the full circumference of the circle, a length
proportional to R. By the uncertainty principle, their energies are proportional to 1/R. On
the other hand, wound strings have minimum energy proportional to R, and are sensitive
to the reciprocal of this value, 1/R. Therefore, using different probes to measure distance
can result in different answers. This property extends to all measurements of lengths and
distances. The results obtained by wound and unwound string probes will be inversely
related to one another, and they are equivalent. However, no one knows for sure if these
conclusions hold if the spatial dimensions are not circular in shape.
Now that we have seen radius of R and 1/R are indistinguishable in string theory,
we want to know if it is possible to have equivalent strings with different shapes. In 1988,
physicists, based upon aesthetic arguments rooted in considerations of symmetry, made
the audacious suggestion that it might be possible. A key fact is that the number of
families of particles arising from string vibrations is sensitive only to the total number of
holes, not to the number of holes of each particular dimension. So we can have calabi-yau
spaces with different configurations but that give rise to same number of families.
Although this is just one property, it shows how the hypothesis can be true. Later, people
used orbifolding to look at the changes in calabi-yau shapes. They found that if particular
groups of points were glued together in just the right way, the Calabi-Yau shape
produced differed from the initial one in a startling manner. The number of odd
dimensional holes and even dimensional holes were exchanged. The number of holes is
thus the same, but the even-odd interchanges means the shapes and fundamental
geometrical structures are quite different. After that, people started asking about if other
physical properties will be the same in these different shapes. The answer was very likely
affirmative.
This symmetry between different shapes, called mirror symmetry, helps in finding
certain aspects of calabi-yau spaces. For a problem in one space, there may be an easy
way of solving it in the symmetric space. Using this method, physicists have been able to
solve difficult mathematical problems. String theory not only provides a unifying
framework for physics, it may well forge an equally deep union with mathematics as
well.
Chapter 11: Tearing the Fabric of Space
Kumar Jeev
This chapter is the success story of string theory over the other existing theories in
discovering the nature of our universe. Although the tearing of the spatial fabric is
impossible according to Einstien's general relativity, this chapter talks about how string
theory shows the converse. Yau and Tian's seminal work on constructing new Calabi-Yau
spaces by "tearing" and "stitching" existing Calabi-Yau spaces opened the avenue of
research in relating flop-transitions to the "real" spatial fabric. The author in the sections
that follow writes about the events of the fall of 1992 when at the Institute of Advanced
Study, Green, Aspinwall, and Morrison on one side and Witten on the other, showed
independently that the space in fact can undergo tearing and probably is undergoing an
elongated rupture. While Green and Aspinwall's approach involved the mathematics
behind mirror symmetry of Calabi-Yau spaces, Witten's approach handled the problem
from a microscopic point of view.
Chapter 12: “Beyond Strings: In Search of M-Theory”
Kancy Lee
Although string theory appears to be able to provide a unique picture of the
universe in the late 1980s, there are actually five different versions of string theory: Type
I, Type IIA, Type IIB, Heterotic-O, and Heterotic-E. Many string theorists had been
trying to combine all five distinct string theories into a single, all-encompassing
framework. During a conference at the University of Southern California in March 1995,
Edward Witten announced a new strategy for transcending the perturbative understanding
of string theory, introducing a new approach that is rooted in the power of symmetry and
the concept of duality. He suggested that the five string theories, although apparently
different in their basic construction, are just different ways of describing the same
underlying physics. Witten was the first to come upon a unique framework, which has
eleven dimensions for a more satisfying synthesis of the theory, called the M-theory.
Even though many issues remain unresolved, and we have only a scant understanding of
Witten’s discovery, the M-theory has provided a substrate for a far grander unifying
framework of string theory and has opened our eyes to a whole slew of different spatial
dimensions.
Chapter 13
Daniel Ho
Both black holes and elementary particles can be described by their mass, force,
charges and spins. It seems that a black hole is just a huge elementary particle. In order
to understand this, string theory is needed. This chapter is about the development of
string theory and how it eventually explains this relationship. It begins with a problem
about the collapse of a three-dimensional sphere surrounded by a Calabi-Yau shape’s
fabric. Not until 1995 did the string theorists think this would not lead to catastrophes.
At that time, Strominger showed that the three branes, which were discovered just before,
provide a shield that completely cancels out the cataclysmic effects arising from the
collapse of the three-dimensional sphere.
Then, other physicists started to study the collapse of this sphere and they found
out that it would tear and subsequently repair itself, becoming a two dimensional sphere.
This means one Calabi-Yau shape can transform into another shape. This has something
to do with black holes and elementary particles. String theorists found out that the
gravitational field of a black hole is similar to that set up by the three-brane smeared
around the three dimensional sphere. They also found out that the mass of a black hole,
which is just the mass of the three-brane, is proportional to the volume of the three
dimensional sphere it wraps. To conclude, a massive black hole becomes ever lighter, as
the three dimensional sphere is collapsing, until it is massless and then it transmutes into
a massless particle. This phenomenon shows that black holes and elementary particles
are just different phases of the same underlying stringy material.
Physicists also asked what the entropy of a black hole is. In 1970, Jacob
Bekenstrin, then a graduate student in Princeton, thought that the total area of the event
horizon of a black hole provides a precise measurement of the entropy. His suggestion
was not accepted until Stephen Hawking found that black holes emit radiation quantum
mechanically. From their prediction, they found that the entropy of a black hole is
extremely large. This confused physicists very much as they thought that black holes are
very ordered object and so the entropy should be small. However, string theorists solved
this problem. They set up a theoretical black hole, which is based on the string theory,
and they calculate the entropy of this black hole. It turns out the result agrees perfectly
with Stephen Hawking and Bekenstein’s result.
Chapter 14
Ting Liao
Humans have been asking questions about the origin of the universe for centuries,
and we have now come to a point where a framework is emerging for answering some of
these questions scientifically. The currently accepted model is the big bang model. The
model, which is usually referred as the standard model of cosmology, says that the
universe was initially in an extreme condition with infinite mass, energy and density. To
understand the infinitely massive universe with an infinitely small volume, we had to use
both the theory of general relativity and quantum mechanics. String theory changes our
view on the standard model by changing our views on the two physics.
In the standard model, the universe erupted from an enormously energetic,
singular event, which spewed forth all of space and all of matter. As time passed, the
universe expanded and cooled, and particles started to come together and formed bigger
particles or substances as we see today. When charged particles (electron and proton)
merged, they formed a neutral particle. Since neutral particles do not affect light, which is
basically electromagnetic wave, some time after the big bang, the photons were free to
travel through the universe. The detections of these photons have given us evidences
supporting the standard model. Furthermore, the predictions about the relative abundance
of the light elements produced during the period of primordial nucleosynthesis have
largely agreed with experiments. However, both these results correspond to conditions
some time (about a hundredth of a second) after the big bang. Physicists believe that
between the Planck time (10^-43 seconds) and a hundredth of a second after the big bang,
the universe passed through at least two phase transitions. At temperature above 10^28
Kelvin, the three nongravitational forces appeared as one, as symmetric as they could
possibly be. But as the temperature dropped below that value, the three forces broke out
from their common union in different ways. At this point the weak and electromagnetic
forces were still deeply interwoven. When the universe reached 10^15 Kelvin, these two
forces also broke out from the union and became more different as the universe cooled
down. This is why the forces are so different to us nowadays.
Now we turn to the horizon problem. The problem is that when we look at the
cosmic background radiation, no matter which direction we look at, we get the same
result. However, in the standard model, the separation between the two ends of the
universe has always been so far that, information has never had enough time to travel
from one end to another. Therefore they could not have had exchanged information any
time in the past. How could they be the same without any connection? The explanation is
that at around 10^-36 seconds after the big bang, the universe underwent a process called
the inflation. During that time, the universe expands much quicker than normal. What
that means is that the size of the universe before the inflation is much smaller than we
expected, so there was enough time for information to travel allover the universe.
We have just talked about the evolution of the universe after Planck’s time, now
we should take a look at what was going on before Planck’s time. As mentioned, since
the size of the universe was so small yet so massive, we have to merge general relativity
and quantum mechanics, and this is where string theory comes into play. There are three
essential ways in which string theory modifies the standard cosmological model. First,
string theory implies that the universe has what amounts to a smallest possible size,
instead of a infinitely tiny size. Second, string theory has a small-radius/large-radius
duality. What that means is when the universe collapses to the Planck’s length, any
further collapse will be equivalent to an expansion. In other words, we can say that the
universe is in a cycle of expanding and collapsing to the Planck’s size. Therefore, some
physicists suggest that at the beginning, the universe was a nugget with Planck’s length in
every dimension. Finally, string theory has more than four space-time dimensions, and
from a cosmological standpoint, we must address the evolution of them all. The first
question comes into mind is that why do only three of the spatial dimensions expand. The
basic answer is that in four or more space dimensions wrapped strings are less and less
likely to collide. Since half of the collisions will involve string/antistring pairs, leading to
annihilations that continually lessen the construction, allowing these three dimensions to
continue to expand. There are a lot of other different insights raised by string theory, but
they’re too complicated to explain.
Although string theory gives a lot of new insights, there are just too many
possibilities on how the universe actually is. Right now it is too hard to tell if string
theory is really the theory of everything. Nevertheless, the cosmological implications of
string/M-theory will be a major field of study in this century. Without accelerators
capable of producing Planck-scale energies, we will increasingly have to rely on the
cosmological accelerator of the big bang, and the relics it has left for us throughout the
universe, for our experimental data.
Chapter 15
Ross Cheung
This chapter exists to conclude and tie together many of the ideas presented in this
book. To do so, it presents numerous questions about the nature of the universe.
The first one asks, is string theory the ultimate fundamental principle that explains the
laws of the universe, or is it just the next step, the next deepest level of understanding? In
the past, whenever we believed we had figured out everything, it turned out there are
more beneath it all. However he points out that all of the principles of symmetry, which
are associated with all the known laws of Physics, emerge from the structure of string
theory.
The second question deals with the nature of space and time, and what it is.
While there are numerous ways of visualizing it, the puzzling question is what is really
meant by the fabric of the universe? Isaac Newton believed that the universe is constant
and unchanging, and thus space and time were absolute. Gottfried Leibniz and many
others argued that space and time are merely bookkeeping devices used to summarize
relationships between objects and events in the universe, a viewpoint that has been
supported by developments in string theory.
The third question considers whether or not string theory will reformulate
quantum mechanics. So far, developments in string theory have followed that of many
classical principles, mainly that string are first described in classical terms, and are then
converted into a quantum-mechanical framework. However, there are suggestions that
this method is inherently flawed, and if the universe is governed by quantum mechanical
principles, the theories to explain it should be quantum mechanical from the start. So far
scientists have been able to get away with it, but as string and M-theory gets deeper, it
may not be possible to continue doing so.
The fourth question deals with whether or not string theory can be experimentally
tested. While the theory fits with all of the other principles of Physics, it has yet to be
tested experimentally because of technological limitations. Larger and larger particle
accelerators are being built right now, but whether they will be sufficient remains to be
seen.
The fifth and final question is whether there are limits to explanation. Even if we
were to have perfect understanding of string and M-theory, what if we were still unable to
calculate particle masses and force strength? More importantly, would the failure of
string theory reflect that those observed properties have no explanation? The answer
hinges on whether string theory is ultimately the deepest level of understanding in the
universe. If so, then its failure means that there are just some things in the universe that
science can not explain.