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Authors: Lawrence M. Krauss

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Beyond its profound underlying physical
implications, this is precisely what Einstein’s general theory
does, making a small correction to Newton’s law. It turns out that
when the force law is no longer precisely as Newton described it,
then a precession is predicted. Einstein, to his credit, was able
to derive an approximate solution to his equations that was
accurate enough to predict the precession of Mercury’s orbit, and
to his immense surprise and satisfaction, the prediction was
precisely in agreement with this half-century-old puzzling
result.

Years afterward Einstein recalled that, upon
discovering this agreement between prediction and observation, he
had the feeling that something had actually snapped within him. He
suddenly realized that his journey of the mind had led him to more
than mathematical fantasies. He said he was so excited that he had
palpitations of the heart. Later in his career, Einstein would
become more enamored with the simplicity of the mathematical
principles that were the foundation for general relativity. But I
think it is crucially important to recognize—and I shall have cause
to return to this theme—that what distinguished Einstein the
physicist from Hilbert the mathematician was that what Einstein
wanted to do was explain the way nature worked, not merely derive
beautiful equations. It was the excitement of seeing that, even by
such a small effect, nature obeyed the laws he discovered in his
mind that made Einstein weak with excitement.

In the same paper in which he derived the
precession of Mercury’s orbit, written a week before the paper that
presented the final form of general relativity, Einstein made
another prediction. He calculated that light would indeed bend in a
gravitational field, as he had realized almost a decade earlier, but
that the actual magnitude of the bending would be twice as large as
he had previously estimated, and twice as large as the value one
might get by simply pretending that light had mass and then using
Newton’s theory to calculate the effect of gravity on its
trajectory. He thus predicted that light passing near the sun would
be deflected by approximately 1/2000 of a degree. As small as this
value was, its predicted effect would be measurable, as Einstein
realized as early as 1911, when he was still in fact predicting the
wrong value. If one observed stars near the sun during a solar
eclipse, their position would be shifted by this very small amount
compared to where one would otherwise predict them to lie.
Fortunately for Einstein, war and other human idiocies prevented a
successful eclipse expedition to test his ideas until three years
after he had indeed made the correct prediction. In November 1919,
two British expeditions reported on their observations of a May
1919 eclipse: Einstein, not Newton, was correct.

This discovery forever changed Einstein’s life
and, with it, the world of physics. News of the eclipse
observations spread across the headlines of papers throughout the
world, and within weeks, Einstein attained a celebrity that would
remain with him for the rest of his life. Special relativity had
made him famous among physicists and perhaps even among educated
intellectuals; general relativity made him a household name. His
discovery that we are living in a possibly curved three-dimensional
space had an immediate popular impact that might be akin to the
revelation in Renaissance Europe that the earth wasn’t flat. In a
single moment, everything changed, and Einstein’s fame would soon
rival that of Columbus.

Part of the reason for his fame was surely the
fact that he had now supplanted Newton as the father of gravity.
But I think the general excitement that greeted his discovery was
more deeply based, and for good reason. While special relativity
had connected space and time in a new way that made separate
measurements of length and time observer dependent, space-time
itself nevertheless remained a fixed background in which the events
of the universe played out. In Einstein’s general relativity,
however, space and time become truly dynamic quantities. They are
no longer mere backdrops in which the drama of life ensues, but
respond
to the presence of matter and
energy, bending, contracting, or even expanding in the presence of
appropriate forms of matter or energy. One of the predictions of
general relativity that took almost half a century to verify
empirically was that clocks tick more slowly in a gravitational
field. Normally the effect is truly minuscule, and to measure it
required careful optical techniques, unstable radioactive
compounds, and ultimately the use of atomic clocks.

However, sometimes, if we take into account the
fact that we live in a large universe, small effects can be
magnified tremendously. One of my favorite examples of this (for
reasons that will become obvious in a moment) involves some work a
colleague of mine and I did shortly after the discovery, on
February 23, 1987, of an exploding star on the outskirts of our
galaxy, the first such event seen in almost four hundred years. Its
demise was observed both via the light emitted by the star, which
shined with a brightness approaching that of a billion stars for
days, and also via the almost simultaneous detection of ghostlike
elementary particles called neutrinos, which are in fact the
dominant form of radiation emitted by exploding stars. Within a few
weeks of the event, there were literally scores of scientific papers
(including some by me) analyzing every aspect of these signals.

About two months after this flurry, Scott
Tremaine (who is now at Princeton, but at the time was at the
Canadian Institute for Theoretical Astrophysics in Toronto) and I
were at a meeting in Halifax, Canada, when we suddenly realized
that one could calculate the extra time it would have taken for
both the light and the neutrinos to travel from the distant star to
Earth, due to the fact that both bursts were traveling in the
gravitational field of our galaxy and hence not in a flat background.
The result surprised both of us: The gravitational time delay was
about six months. If it hadn’t been for the warping of both space
and time as predicted in general relativity, Supernova 1987a, as it
became known, would have been called Supernova 1986d, as it would
have been observed sometime around the middle of the previous
year!

It is virtually impossible for us, who are
confined to live within a curved three-dimensional space, to
physically picture what such a curvature implies. We can
intuitively grasp a curved two-dimensional object, such as the
surface of the earth, because we can embed it in a threedimensional
background for viewing. But the possibility that a curved space can
exist in any number of dimensions without being embedded in a
higher-dimensional space is so foreign to our intuition that I am
frequently asked, “If space is curved, what is it curving
into
?” There are, however, mathematical ways to
define the geometry of a space without the existence of extrinsic
quantities. The simplest example involves something with which we
are all familiar. Consider a triangle drawn on this piece of
paper.

As any European high school student could tell
you, the sum of the angles inside this triangle is 180 degrees,
independent of the shape or size of the triangle.

Now, however, consider the following figure:

All three angles of this triangle are right
angles, adding up to a sum of 270 degrees. Were we intelligent ants
living on this curved surface, even if we could never
circumnavigate it or view it from above, by drawing a large enough
triangle and measuring the sum of its internal angles, we could
nevertheless infer that we were living on a spherical surface.
Another factor distinguishes a sphere from a flat piece of paper,
which I alluded to earlier. Lines of longitude, extending from the
North to the South Pole, are all parallel lines, yet all of these
lines meet at both poles:

As obvious as all this might seem in
retrospect, the notion that it might be possible to have a geometry
where Euclid’s axioms about parallel lines or about the sum of
angles in a triangle might not hold caused a revolution in
philosophy. Euclid’s axioms had remained unchallenged for two
thousand years when the mathematicians Gauss, Lobachevsky, and
Bolyai independently discovered between the years 1824 and 1832
that one could build a consistent mathematical framework in which
the axiom about parallel lines could be violated. So great was the
resistance to these notions that the famous physicist Helmholtz
felt it necessary to incorporate precisely the examples I have
given here in his 1881
Popular Lectures on
Scientific
Subjects
(published three
years before
Flatland
), in which he
described a hypothetical world of two-dimensional beings living on
the surface of a sphere, in order to convince people that the
abstract mathematical notions of Gauss and others could be
manifested in a consistent physical reality. Interestingly enough,
both Gauss and Lobachevsky realized that if non-Euclidean geometry
was possible in principle, it might also be possible in practice,
and both conducted independent experiments to see if our
three-dimensional space might be curved. Gauss was more modest in
his attempts, merely measuring the sum of the angles in a large
triangle formed by three distant mountain peaks. Lobachevsky, in
contrast, performed a far more modern experiment. He observed the
parallax of various distant stars, that is, the angle by which they
shift compared to background objects when the earth is on one side
of the sun, compared to when the earth is on the other side of the
sun a half-year later. Plane geometry gives a straightforward
prediction for what this shift should be for stars at a fixed
distance, at least in a flat space. The shift would be different,
however, if space was curved. Given the limited sensitivity of
their observations, neither Gauss nor Lobachevsky was able to
obtain any evidence whatsoever for the nonEuclidean nature of
space. That evidence would have to wait for almost a century, until
after Einstein had made it clear what to look for. While the
British solar eclipse expeditions were able to detect the curvature
of space in the vicinity of the sun, general relativity posed a
much, much bigger challenge. This was a theory not merely of how
objects might move throughout space and time, but of how space and
time themselves might evolve. Einstein opened up the possibility of
describing the dynamics of the universe itself, and since general
relativity is a geometric theory, the central question of
twentieth-century cosmology soon became: Is the geometry of the
universe, on its largest scales, described by Euclid?

C H A P T E R 6
THE MEASURE OF ALL THINGS

I believe with Schopenhauer
that one of the strongest motives that
leads men to art and science is escape from everyday
life with its
painful crudity and hopeless
dreariness from the fetters of one’s own
everyday desires. . . . A finely tempered nature longs to
escape from
personal life into the world of
objective perception and thought.

—Albert Einstein

I
am not sure that I
completely agree with Einstein’s romantic view of the scientific (or
artistic) enterprise, having always felt that these activities, as
human endeavors, are intimately connected with the rest of our
existence, dreary or otherwise. But perhaps that is one of the many
reasons why Einstein was Einstein, and I am me. In any case, for
over twenty years I have devoted most of my scientific effort to
questions about the origin, nature, and future of our expanding
universe that are about as far removed from the world of my
everyday experience as can be. While I like to think that my
contributions have helped us move forward in our knowledge, nothing
truly prepared me for the revolutionary developments of the past
decade, which is why I want to make a brief digression from the
historical presentation thus far, and jump to a present-day result
that has finally addressed the question first asked by Einstein
almost a hundred years ago. I remember, from the time I taught at
Yale, a conversation with a senior member of its astronomy
department, Gus Oemler. I used to visit him regularly with crazy
ideas about how one might measure such fundamental quantities as
the expansion rate and the geometry of the universe. With his
wealth of experience, he brought valuable skepticism to any
discussions we had. On this particular day we were discussing how
to measure what has become known as the Hubble constant, a quantity
that describes the expansion rate of our universe and which, in a
manner characteristic of much of astronomical nomenclature, is
actually not in general a constant quantity at all, but varies over
cosmological time. In any case, in the course of our talk Gus
revealed to me a theorem he had about the universe: “I believe that
the universe will always conspire to make any fundamental and
precise measurement of cosmological quantities such as the Hubble
constant impossible.” As outrageous as this claim might seem, it
was based on decades of experience in astronomy. On many occasions
over the past thirty to forty years, astronomers had claimed to
make definitive measurements about such quantities as the curvature
of the universe or its expansion rate, and each time it turned out
that subtle uncertainties that had not been anticipated by the
observers clouded their results, ultimately invalidating many of
them.

Thus it was that in 1999 I was unprepared for a
totally clean and unambiguous observation, using a method that I
had in fact written about in a slightly different context almost a
decade earlier: a profound and direct measurement of the geometry
of the universe in which we live. Equally remarkable is the fact
that the method used is almost identical, at least in principle, to
that used by Lobachevsky over 150 years earlier to first explore for
a possible curvature of space. The only difference is that the
triangles we now use “as reference points” span not the distances
to the nearest stars, but rather the distance across the entire
visible universe. This observation became possible because of the
accidental discovery, forty years ago, of a then-mysterious thermal
bath of radiation bombarding us from all directions, with a
temperature of about three degrees above absolute zero (on the
Kelvin temperature scale, in which absolute zero, the coldest
temperature possible, is labeled zero, unlike the Fahrenheit scale,
where absolute zero is minus 459.67 degrees). It didn’t remain
mysterious for long, however. When the perplexed scientists at Bell
Laboratories who had found this excess “noise” in their antennas
went down the road with their findings to Princeton University, the
scientists there informed them that they had discovered the
afterglow of the big bang. Shortly after Edwin Hubble’s discovery
in 1929 that the universe is expanding, it was realized that by
following this expansion backward in time one might hope to trace
out the thermal history of the universe. By going back over ten
billion years, the universe one would encounter would have
consisted of a hot, dense gas of particles and radiation in thermal
equilibrium. Such an extrapolation was, of course, bold, but it did
make many theoretical predictions possible, all of which could be
tested against observations. The most robust of them, perhaps,
involved the prediction of a background of radiation left over from
the big bang that would have permeated the universe, and would have
been cooling as the universe expanded over the billions of years
between the big bang and now.

We can understand why this microwave radiation
bath exists and what its origin is by remembering one of the
fundamental facts of electromagnetism: Light travels at a finite
velocity through space, so that the farther out we look in the
universe, the further back in time we are looking. Every time we
peer through a telescope, we are doing cosmic archaeology. Pushing
this idea to its logical limit means that in principle, if the
universe had a beginning a finite time ago in the past, if we look
out far enough with sufficiently powerful telescopes, we should be
able to witness the big bang itself! Unfortunately, however, there
is a fundamental roadblock to actually achieving this goal. Between
the big bang and now, the universe went through an opaque period
when it was so hot and dense that light could not travel unimpeded
throughout space, unlike the present time, when it can traverse the
vast distances between stars and galaxies.

Using well-known laws of physics, we can
actually calculate the precise time before which the universe was
opaque. At a temperature of greater than about three thousand
degrees Kelvin above absolute zero the ambient radiation present is
sufficiently energetic to break apart the bonds that hold atoms such
as hydrogen together. Hydrogen is the simplest atom, made up of a
single proton, surrounded by an electron. At extremely high
temperatures, absorption of energy from a radiation bath is
sufficiently great to allow the electron to be knocked free of its
electronic bond to its host proton. While it could be captured
again by another bare proton, the radiation would once again knock
it free. At the high temperatures of the early universe, therefore,
hydrogen was ionized, meaning that its charged particles (protons
and electrons) were separated and not bound together into neutral
atoms. Now, ionized matter, being charged, interacts very strongly
with electromagnetic radiation. Thus, a light ray cannot permeate a
configuration of ionized atoms, which we call a plasma, without
being constantly absorbed and reemitted. This means that as we
attempt to look back farther and farther we eventually hit a
metaphorical wall. If we try to look back to earlier times, we
simply cannot do so using electromagnetic radiation, just as we
cannot look behind the walls in the room that surround us, because
the radiation cannot penetrate their surface. Indeed, when we look
at a wall, we are seeing radiation that has been absorbed at the
surface, and later reemitted into the room, making its way through
the transparent air to our eyes.

Similarly, as the universe cooled below three
thousand degrees, and neutral atoms could finally form, space became
transparent to radiation. Thus, we should expect to be able to see
a “surface” located billions of light years away from us that
represents the time when the universe first became neutral, when it
was about three hundred thousands years old. From this surface we
should expect to receive a bath of radiation coming at us from all
directions. Since the universe has been expanding and cooling since
the time that that surface originally emitted the radiation, by the
time it gets to our sensors the radiation should have cooled
considerably. The first people to propose that such a radiation
background should exist were a research group associated with the
scientist and writer George Gamow, whose many popular books
inspired generations of young people (including me) to think about
science. At the time that Gamow’s colleagues Robert Alpher and
Robert Hermann made their proposal, no one really took the big bang
picture seriously. However, as I mentioned previously, twenty years
after his prediction two young wouldbe radio astronomers at Bell
Laboratories in New Jersey discovered an unusual source of noise in
a sensitive radio receiver they planned to use to scan the heavens.
The noise was characteristic of a background of radiation at a
temperature of about three degrees above absolute zero. While they
had no idea of it at the time, this was more or less precisely the
temperature such a radiation bath remnant of the big bang was
predicted to now possess.

Because this radiation emanates from within the
first three hundred thousand years after the big bang, it has become
one of the most important probes of cosmology. By carefully
measuring its properties, one can hope to glean a wealth of
information about the early universe. In 1999 an experiment was
launched near the South Pole to measure this background radiation
with unprecedented accuracy. A microwave radiation detector was set
aloft on a huge balloon that would rise to a hundred thousand feet
above the earth, well above most of the atmosphere that would
otherwise absorb some of the radiation before it could reach the
earth. The balloon with its important payload took almost two weeks
to circle Antarctica, returning close to the spot from where it had
been launched (which is why it was called the boomerang
experiment), and during this time the microwave radiometer focused
on a small patch of the sky, measuring the temperature of the
background radiation across the patch to an accuracy of better than
one part in one hundred thousand.

What the experimenters who built and operated
the device were looking for was a very particular distribution of
hot and cold spots about one degree across in the microwave sky.
This angular size has a special significance, for it represents the
distance light could have traveled across points on the “surface”
from which the microwave background emanates, about three hundred
thousand years following the big bang. Since no signal can be
transmitted faster than light, this distance, about three hundred
thousand light years, thus represents the largest distance over
which the effects of any physical disturbance located at one place
could propagate.

Put another way, this scale is the largest
scale over which local physical processes could respond to
macroscopic conditions. For example, a bit of excess mass in some
region might, by its gravitational selfattraction, begin to
collapse. The increased density in this region would then cause a
corresponding increase in pressure. Such effects of pressure
responding to gravity could only occur across regions smaller than
or equal to three hundred thousand light years across, however,
because on larger scales lumps of excess mass do not even know they
are lumps—light cannot have traveled across them. This is why the
angular scale associated with this distance is special—it is
associated with the largest size regions within which there is
causal contact. For this reason, one would expect to see a residual
imprint on the microwave background on such scales.

Such a situation in principle provides us with
all the ingredients we need to be able to directly probe the
geometry of the universe, by giving us a large triangle, as shown
below. Two of the sides of the triangle represent the distance from
Earth out to the surface from which the microwave background
emanates. The third side is this special distance across the
surface, representing the maximum distance a physical signal could
have propagated at that time, about three hundred thousand
lightyears. General relativity implies that light rays travel in
space in straight lines, but if the underlying space is curved, the
trajectories of the light rays themselves will be curved. Thus, the
light rays emanating from the edges of a region spanning a distance
of three hundred thousand light-years across would follow one of
three different kinds of trajectories on their way to the earth. If
the universe is positively curved, then the light rays would bend
inward on their travels. If it were negatively curved, the light
rays would bend outward. And if the universe is flat, the light rays
would follow straight lines.

 

From the point of view of an observer on Earth,
then, the angular size of these regions will depend upon what the
geometry of the universe is. If space is negatively curved across
the universe, the apparent angular size of these hot spots and cold
spots will be reduced. If it is positively curved, the hot and cold
spots will appear enlarged. If it is flat, the size will be
somewhere in between. In 1999 the boomerang experiment released its
results, with complex charts demonstrating the quantitative
features of the temperature variations across the region of the
microwave sky that it observed. However, in the spirit of the
statement that a picture is worth a thousand words, the
experimental team also produced a graphical representation of their
findings. Here is an actual false color image (rendered here in
shades of gray) of the data, with hot spots one shade and cold
spots another, compared to three computer-generated versions of
what you might expect for a positively curved, flat, and negatively
curved space.

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