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The Reluctant Father of Black Holes

PRO AND CON: In 1939 J. Robert Oppenheimer
(right) argued for the existence of black holes, at the same time Albert
Einstein tried to disprove them. Their careers crossed paths at the
Institute for Advanced Study in Princeton, N.J., in the late 1940s, when
this photograph was taken, but it is unknown whether they ever discussed
black holes.

by Jeremy Bernstein

Albert Einstein's equations of gravity are the
foundation of the modern view of black holes; ironically, he used the equations
in trying to prove these objects cannot exist.

Great science sometimes produces a legacy that outstrips not only the
imagination of its practitioners but also their intentions. A case in point is
the early development of the theory of black holes and, above all, the role
played in it by Albert
Einstein. In 1939 Einstein published a paper in the journal Annals of
Mathematics with the daunting title "On a Stationary System with Spherical
Symmetry Consisting of Many Gravitating Masses." With it, Einstein sought to
prove that black holes-celestial objects so dense that their gravity prevents
even light from escaping-were impossible.

WAV 135K

The irony is that, to make his case, he used his own general
theory of relativity and gravitation, published in 1916-the very theory that is
now used to argue that black holes are not only possible but, for many
astronomical objects, inevitable. Indeed, a few months after Einstein's
rejection of black holes appeared-and with no reference to it- J. Robert
Oppenheimer and his student Hartland S. Snyder published a paper entitled "On
Continued Gravitational Contraction." That work used Einstein's general theory
of relativity to show, for the first time in the context of modern physics, how
black holes could form.

Perhaps even more ironically, the modern study of black holes, and more
generally that of collapsing stars, builds on a completely different aspect of
Einstein's legacy-namely, his invention of quantum-statistical mechanics.
Without the effects predicted by quantum statistics, every astronomical object
would eventually collapse into a black hole, yielding a universe that would bear
no resemblance to the one we actually live in.

Bose, Einstein and Statistics

Einstein's creation of quantum statistics was inspired by a letter he
received in June 1924 from a then unknown young Indian physicist named Satyendra
Nath Bose. Along with Bose's letter came a manuscript that had already been
rejected by one British scientific publication. After reading the manuscript,
Einstein translated it himself into German and arranged to have it published in
the prestigious journal Zeitschrift für Physik.

Why did Einstein think that this manuscript was so important? For two
decades, he had been struggling with the nature of electromagnetic radiation-
especially the radiation trapped inside a heated container that attains the same
temperature as its walls. At the turn of the century the German physicist Max Planck
had discovered the mathematical function that describes how the various
wavelengths, or colors, of this "black body" radiation vary in intensity. It
turns out that the form of this spectrum does not depend on the material of the
container walls. Only the temperature of the radiation matters. (A striking
example of black-body radiation is the photons left over from the big bang,
in which case the entire universe is the "container." The temperature of these
photons was recently measured at 2.726 ± 0.002 kelvins.)

Somewhat serendipitously, Bose had worked out the statistical mechanics of
black-body radiation-that is, he derived the Planck law from a mathematical,
quantum-mechanical perspective. That outcome caught Einstein's attention. But
being Einstein, he took the matter a step further. He used the same methods to
examine the statistical mechanics of a gas of massive molecules obeying the same
kinds of rules that Bose had used for the photons. He derived the analogue of
the Planck law for this case and noticed something absolutely remarkable. If one
cools the gas of particles obeying so-called Bose-Einstein statistics, then at a
certain critical temperature all the molecules suddenly collect themselves into
a "degenerate", or single, state. That state is now known as Bose-Einstein
condensation (although Bose had nothing to do with it).

An Early History of Black Holes

1900

Max Planck discovers black body
radiation.

1905

In a paper on black body radiation,Albert Einstein
shows that light can be viewed as particles (photons).

1905

Through spectroscopic studies,astronomer Walter S.Adams
identifies Sirius's faint companion (which causes Sirius to wobble
slightly as it moves) as a small,hot,dense star - a white
dwarf.

1916

Einstein publishes his general theory of
relativity,producing equations that describe
gravity.

An interesting example is a gas made up of the common isotope helium 4, whose
nucleus consists of two protons and two neutrons. At a temperature of 2.18
kelvins, this gas turns into a liquid that has the most uncanny properties one
can imagine, including frictionless flow (that is, superfluidity). U.S.
researchers in the past year accomplished the difficult task of cooling other
kinds of atoms to several billionths of a kelvin to achieve a Bose-Einstein
condensate.

Not all the particles
in nature, however, show this condensation. In 1925, just after Einstein
published his papers on the condensation, the Austrian-born physicist
Wolfgang Pauli identified a second class of particles, which includes the
electron, proton and neutron, that obeyed different properties. He found that no
two such identical particles-two electrons, for example - can ever be in exactly
the same quantum-mechanical state, a property that has since become known as the
Pauli exclusion principle. In 1926 Enrico Fermi and P.A.M.
Dirac invented the quantum statistics of these particles, making them the
analogue of the Bose-Einstein statistics.

Because of the Pauli principle, the last thing in the world these particles
want to do at low temperatures is to condense. In fact, they exhibit just the
opposite tendency. If you compress, say, a gas of electrons, cooling it to very
low temperatures and shrinking its volume, the electrons are forced to begin
invading one another's space. But Pauli's principle forbids this, so they dart
away from one another at speeds that can approach that of light. For electrons
and the other Pauli particles, the pressure created by these fleeing
particles-the "degeneracy pressure " -persists even if the gas is cooled to
absolute zero. It has nothing to do with the fact that the electrons repel one
another electrically. Neutrons, which have no charge, do the same thing. It is
pure quantum physics.

Quantum Statistics and White Dwarfs

But what has quantum statistics got to do with the stars? Before the turn of
the century, astronomers had begun to identify a class of peculiar stars that
are small and dim: white dwarfs. The one that accompanies Sirius, the brightest
star in the heavens, has the mass of the sun but emits about 1/360 the light.
Given their mass and size, white dwarfs most be humongously dense. Sirius's
companion is some 61,000 times denser than water. What
are these bizarre objects? Enter Sir Arthur Eddington.

1916

Karl Swarzschild shows that a radius of a
collapsing object exists at which Einstein's gravity equations become
"singular" - time vanishes,and space becomes infinite.

1924

Einstein publishes Satyendra Nath Bose's work on
black-body radiation,developing so-called quantum statistics for one class
of particles (such as photons).

1924

Sir Arthur Eddington proposes that gravity can
strip away electrons from protons in a white dwarf.

1925

Wolfgang Pauli formulates the exclusion
principle,which states that certain particles cannot be in exactly the
same quantum mechanical state.

When I began studying physics in the late 1940s, Eddington
was a hero of mine but for the wrong reasons. I knew nothing about his great
work in astronomy. I admired his popular books (which, since I have learned more
about physics, now seem rather silly to me). Eddington, who died in 1944, was a
neo-Kantian
who believed that everything of significance about the universe could be
learned by examining what went on inside one's head. But starting in the late
1910s, when Eddington led one of the two expeditions that confirmed Einstein's
prediction that the sun bends starlight, until the late 1930s, when Eddington
really started going off the deep end, he was truly one of the giants of
20th-century science. He practically created the discipline that led to the
first understanding of the internal constitution of stars, the title of his
classic 1926 book. To him, white dwarfs were an affront, at least from an
aesthetic point of view. But he studied them nonetheless and came up with a
liberating idea.

In 1924 Eddington proposed that the gravitational pressure that was squeezing
the dwarf might strip some of the electrons off protons. The atoms would then
lose their "boundaries" and might be squeezed together into a small, dense
package. The dwarf would eventually stop collapsing because of the Fermi- Dirac
degeneracy pressure-that is, when the Pauli exclusion principle forced the
electrons to recoil from one another.

The understanding of white dwarfs took another step forward in July 1930,
when Subrahmanyan Chandrasekhar, who was 19, was on board a ship sailing
from Madras to Southampton. He had been accepted by the British physicist R. H.
Fowler to study with him at the University of Cambridge (where Eddington was,
too). Having read Eddington's book on the stars and Fowler's book on
quantum-statistical mechanics, Chandrasekhar had become fascinated by white
dwarfs. To pass the time during the voyage, Chandrasekhar asked himself: Is
there any upper limit to how massive a white dwarf can be before it collapses
under the force of its own gravitation? His answer set off a revolution.

A white dwarf as a whole is electrically neutral, so all the electrons must
have a corresponding proton, which is some 2,000 times more massive.
Consequently, protons must supply the bulk of the gravitational compression. If
the dwarf is not collapsing, the degeneracy pressure of the electrons and the
gravitational collapse of the protons must just balance. This balance, it turns
out, limits the number of protons and hence the mass of the dwarf. This maximum
is known as the Chandrasekhar limit and equals about 1.4 times the mass of the
sun. Any dwarf more massive than this number cannot be stable.

Chandrasekhar's result deeply disturbed Eddington. What happens if the mass
is more than 1.4 times that of the sun? He was not pleased with the answer.
Unless some mechanism could be found for limiting the mass of any star that was
eventually going to compress itself into a dwarf, or unless Chandrasekhar's
result was wrong, massive stars were fated to collapse gravitationally into
oblivion.

Eddington found this intolerable and proceeded to attack Chandrasekhar's use
of quantum statistics-both publicly and privately. The criticism devastated
Chandrasekhar. But he held his ground, bolstered by people such as the Danish
physicist Niels Bohr, who assured him that Eddington was simply wrong and
should be ignored.

A Singular Sensation

As researchers explored quantum statistics and white dwarfs, others tackled
Einstein's work on gravitation, his general theory of relativity. As far as I
know, Einstein never spent a great deal of time looking for exact solutions to
his gravitational equations. The part that described gravity around matter was
extremely complicated, because gravity distorts the geometry of space and
time, causing a particle to move from point to point along a curved path.
More important to Einstein, the source of gravity-matter could not be described
by the gravitational equations alone. It had to be put in by hand, leaving
Einstein to feel the equations were incomplete. Still, approximate solutions
could describe with sufficient accuracy phenomena such as the bending of
starlight. Nevertheless, he was impressed when, in 1916, the German astronomer
Karl Schwarzschild came up with an exact solution for a realistic
situation in particular, the case of a planet orbiting a star.

1926

Enrico Fermi and P.A.M. Dirac develop the
statistics for particles that obey Pauli's exclusion principle (such as
electrons and protons). When compressed,such particles fly away from one
another,creating a so-called degeneracy pressure.

1930

Using quantum statistics and Eddington's work on stars,
Subrahmanyan Chandrasekhar finds that the upper limit for white dwarfs
is 1.4 times the mass of the sun, suggesting that more massive stars
collapse into oblivion.Eddington makes fun of
him.

In the process, Schwarzschild found something disturbing. There is a distance
from the center of the star at which the mathematics goes berserk. At this
distance, now known as the Schwarzschild radius, time vanishes, and space
becomes infinite. The equation becomes what mathematicians call singular.
The Schwarzschild radius is usually much smaller than the radius of the object.
For the sun, for example, it is three kilometers, whereas for a one-gram marble
it is 10-28 centimeter.

Schwarzschild was, of course, aware that his formula went crazy at this
radius, but he decided that it did not matter. He constructed a simplified model
of a star and showed that it would take an infinite gradient of pressure to
compress it to his radius. The finding, he argued, served no practical interest.

But his analysis did not appease everybody. It bothered Einstein, because
Schwarzschild's model star did not satisfy certain technical requirements of
relativity theory. Various people, however, showed that one could rewrite
Schwarzschild's solutions so that they avoided the singularity. But was the
result really nonsingular? It would be incorrect to say that a debate raged,
because most physicists had rather little regard for these matters-at least
until 1939.

In his 1939 paper Einstein credits his renewed concern about the
Schwarzschild radius to discussions with the Princeton cosmologist Harold P.
Robertson and with his assistant Peter G. Bergmann, who is now professor
emeritus at Syracuse University. It was certainly Einstein's intention in this
paper to kill off the Schwarzschild singularity once and for all. At the end of
it he writes, "The essential result of this investigation is a clear
understanding as to why 'Schwarzschild singularities' do not exist in physical
reality." In other words, black holes cannot exist.

To make his point, Einstein focused on a collection of small particles moving
in circular orbits under the influence of one another's gravitation-in effect, a
system resembling a spherical star cluster. He then asked whether such a
configuration could collapse under its own gravity into a stable star with a
radius equal to its Schwarzschild radius. He concluded that it could not,
because at a somewhat larger radius the stars in the cluster would have to move
faster than light in order to keep the configuration stable. Although Einstein's
reasoning is correct, his point is irrelevant: it does not matter that a
collapsing star at the Schwarzschild radius is unstable, because the star
collapses past that radius anyway. I was much taken by the fact that the then
60-year-old Einstein presents in this paper tables of numerical results, which
he must have gotten by using a slide rule. But the paper, like the slide rule,
is now a historical artifact.

From Neutrons to Black holes

While Einstein was doing this research, an entirely different enterprise was
unfolding in California. Oppenheimer and his students were creating the modern
theory of black holes [see "J. Robert Oppenheimer: Before the War," by John S.
Rigden; SCIENTIFIC AMERICAN, July 1995]. The curious thing about the black-hole
research is that it was inspired by an idea that turned not to be entirely
wrong. In 1932 the British experimental physicist James Chadwick found the
neutron, the neutral component of the atomic nucleus. Soon thereafter
speculation began- most notably by Fritz Zwicky of the California Institute of
Technology and independently by the brilliant Soviet theoretical physicist Lev
D. Landau-that neutrons could lead to an alternative to white dwarfs.

1932

James Chadwick discovers the neutron.Its existence
leads researchers to wonder if "neutron stars" could be an alternative to
white dwarfs.

1939

Sparked by conversations with colleagues,Einstein tries to
kill off the Schwarzschild radius once and for all:he concludes that black
holes are impossible in a paper published in Annals of
Mathematics.

1939

Using ideas of collapsing neutron stars and white
dwarfs J.Robert Oppenheimer and his student Hartland S.Snyder show
how a black hole can form.

When the gravitational pressure got large enough, they argued, an electron in
a star could react with a proton to produce a neutron. (Zwicky even conjectured
that this process would happen in supernova explosions; he was right, and these
"neutron stars" we now identify as pulsars.) At the time of this work, the
actual mechanism for generating the energy in ordinary stars was not known. One
solution placed a neutron star at the center of ordinary stars, in somewhat the
same spirit that many astrophysicists now conjecture that black holes power
quasars.

The question then arose: What was the equivalent of the Chandrasekhar mass
limit for these stars? Determining this answer is much harder than finding the
limit for the white dwarfs. The reason is that the neutrons interact with one
another with a strong force whose specifics we still do not fully understand.
Gravity will eventually overcome this force, but the precise limiting mass is
sensitive to the details. Oppenheimer published two papers on this subject with
his students Robert Serber and George M. Volkoff and concluded that the mass
limit here is comparable to the Chandrasekhar limit for white dwarfs. The first
of these papers was published in 1938, and the second in 1939. (The real source
of stellar energy-fusion- was discovered in 1938 by Hans Bethe and Carl
Friedrich von Weizsitcker, but it took a few years to be accepted, and so
astrophysicists continued to pursue alternative theories.)

Oppenheimer went on to ask exactly what Eddington had wondered about white
dwarfs: What would happen if one had a collapsing star whose mass exceeded any
of the limits? Einstein's 1939 rejection of black hole- to which Oppenheimer and
his students were certainly oblivious, for they were working concurrently, 3,000
miles away-was of no relevance. But Oppenheimer did not want to construct a
stable star with a radius equal to its Schwarzschild radius. He wanted to see
what would happen if one let the star collapse through its Schwarzschild radius.
He suggested that Snyder work out this problem in detail.

To simplify matters, Oppenheimer told Snyder to make certain assumptions and
to neglect technical considerations such as the degeneracy pressure or the
possible rotation of the star. Oppenheimer's intuition told him that these
factors would not change anything essential. (These assumptions were challenged
many years later by a new generation of researchers using sophisticated
high-speed computers-poor Snyder had an old-fashioned mechanical desk
calculator-but Oppenheimer was right. Nothing essential changes.) With the
simplified assumptions, Snyder found out that what happens to a collapsing star
depends dramatically on the vantage point of the observer.

Two Views of a Collapse

Let us start with an observer at rest a safe distance from the star. Let us
also suppose that there is another observer attached to the surface of the star
"co-moving" with its collapse-who can send light signals back to his stationary
colleague. The stationary observer will see the signals from his moving
counterpart gradually shift to the red end of the electromagnetic spectrum. If
the frequency of the signals is thought of as a clock, the stationary observer
will say that the moving observer's clock is gradually slowing down.

Indeed, at the Schwarzschild radius the clock will slow down to zero.
The stationary observer will argue that it took an infinite amount of time for
the star to collapse to its Schwarzschild radius. What happens after that we
cannot say, because, according to the stationary observer, there is no
"after." As far as this observer is concerned, the star is frozen at its
Schwarzschild radius.

Indeed, until December 1967, when the physicist John A.
Wheeler, now at Princeton University, coined the name "black hole" in a
lecture he presented, these objects were often referred to in the literature as
frozen stars. This frozen state is the real significance of the singularity in
the Schwarzschild geometry. As Oppenheimer and Snyder observed in their paper,
the collapsing star "tends to close itself off from any communication with, a
distant observer; only its gravitational field persists." In other words, a
black hole has been formed.

But what about observers riding with collapsing stars? These observers,
Oppenheimer and Snyder pointed out, have a completely different sense of things.
To them, the Schwarzschild radius has no special significance. They pass right
through it and on to the center in a matter of hours, as measured by their
watches. They would, however, be subject to monstrous tidal gravitational forces
that would tear them to pieces.

The year was 1939, and the world itself was about to he torn to pieces.
Oppenheimer was soon to go off to war to build the most destructive weapon ever
devised by humans. He never worked on the subject of black holes again. As far
as I know, Einstein never did, either. In peacetime, in 1947, Oppenheimer became
the director of the Institute for Advanced Study in Princeton, NJ., where
Einstein was still a professor. From time to time they talked. There is no
record of their ever having discussed black holes. Further progress would have
to wait until the 1960s, when discoveries of quasars, pulsars and compact x-ray
sources reinvigorated thinking about the mysterious fate of stars.

The Author

JEREMY BERNSTEIN is professor emeritus of physics at the
Stevens Institute of Technology, an adjunct professor at the Rockefeller
University and a vice president of the board of trustees of the Aspen
Center for Physics. He is a former staff writer for the New Yorker and the
recipient of many science-writing awards. This article is adapted from his
collection of essays, A Theory for Everything, to be published in
August by Copernicus (Springer- Verlag). He wrote about Niels Bohr and the
atomic bomb in the May 1995 issue of Scientific
American.

Further Reading

SUBTLE IS THE LORD: THE SCIENCE AND THE LIFE OF ALBERT
EINSTEIN. Abraham Pais. Oxford University Press, 1982.
DARK STARS: THE
EVOLUTION OF AN IDEA. Werner Israel in 300 Years of Gravitation.
Edited by S. W. Hawking and W. Israel. Cambridge University Press, 1987.

CHANDRA: A BIOGRAPHY OF S. CHANDRASEKHAR. Kameshwar C. Wali.
University of Chicago Press, 1991.
BLACK HOLES. J.-P. Luminet et al.
Cambridge University Press, 1992.
BLACK HOLES AND TIME WARPS. Kip
Thorne. W. W. Norton, 1994.

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