Albert
Einstein was certainly the greatest physicist of the 20th century, and one of the greatest scientists
of all time. It may seem presumptuous to talk of mistakes made by such a towering Figure, especially
in the centenary of his annus mirabilis. But the mistakes made by leading scientists often provide
a better insight into the spirit and presuppositions of their times than do their successes.1
Also, for those of us who have made our share of scientific errors, it is mildly consoling to note
that even Einstein made mistakes. Perhaps most important, by showing that we are aware of mistakes
made by even the greatest scientists, we set a good example to those who follow other supposed paths
to truth. We recognize that our most important scientific forerunners were not prophets whose
writings must be studied as infallible guides—they were simply great men and women who prepared
the ground for the better understandings we have now achieved.
The cosmological constant
In thinking of Einstein's mistakes,
one immediately recalls what Einstein (in a conversation with George Gamow2) called
the biggest blunder he had made in his life: the introduction of the cosmological constant. After
Einstein had completed the formulation of his theory of space, time, and gravitation—the
general theory of relativity—he turned in 1917 to a consideration of the spacetime structure
of the whole universe. He then encountered a problem. Einstein was assuming that, when suitably
averaged over many stars, the universe is uniform and essentially static, but the equations of
general relativity did not seem to allow a time-independent solution for a universe with a uniform
distribution of matter. So Einstein modified his equations, by including a new term involving
a quantity that he called the cosmological constant. Then it was discovered that the universe is
not static, but expanding. Einstein came to regret that he had needlessly mutilated his original
theory. It may also have bothered him that he had missed predicting the expansion of the universe.
This story involves a tangle
of mistakes, but not the one that Einstein thought he had made. First, I don't think that it can count
against Einstein that he had assumed the universe is static. With rare exceptions, theorists have
to take the world as it is presented to them by observers. The relatively low observed velocities
of stars made it almost irresistible in 1917 to suppose that the universe is static. Thus when Willem
de Sitter proposed an alternative solution to the Einstein equations in 1917, he took care to use
coordinates for which the metric tensor is time-independent. However, the physical meaning of
those coordinates is not transparent, and the realization that de Sitter's alternate cosmology
was not static—that matter particles in his model would accelerate away from each other—was
considered to be a drawback of the theory.
It is true that Vesto Melvin
Slipher, while observing the spectra of spiral nebulae in the 1910s, had found a preponderance
of redshifts, of the sort that would be produced in an expansion by the Doppler effect, but no one
then knew what the spiral nebulae were; it was not until Edwin Hubble found faint Cepheid variables
in the Andromeda Nebula in 1923 that it became clear that spiral nebulae were distant galaxies,
clusters of stars far outside our own galaxy. I don't know if Einstein had heard of Slipher's redshifts
by 1917, but in any case he knew very well about at least one other thing that could produce a redshift
of spectral lines: a gravitational field. It should be acknowledged here that Arthur Eddington,
who had learned about general relativity during World War I from de Sitter, did in 1923 interpret
Slipher's redshifts as due to the expansion of the universe in the de Sitter model. (The two scientists
are pictured with Einstein and others in Figure 1.) Nevertheless, the expansion of the universe
was not generally accepted until Hubble announced in 1929—and actually showed in 1931—that
the redshifts of distant galaxies increase in proportion to their distance, as would be expected
for a uniform expansion (see Figure 2). Only then was much attention given to the expanding-universe
models introduced in 1922 by Alexander Friedmann, in which no cosmological constant is needed.
In 1917 it was quite reasonable for Einstein to assume that the universe is static.
Einstein did make a surprisingly
trivial mistake in introducing the cosmological constant. Although that step made possible a
time-independent solution of the Einstein field equations, the solution described a state of
unstable equilibrium. The cosmological constant acts like a repulsive force that increases with
distance, while the ordinary attractive force of gravitation decreases with distance. Although
there is a critical mass density at which this repulsive force just balances the attractive force
of gravitation, the balance is unstable; a slight expansion will increase the repulsive force
and decrease the attractive force so that the expansion accelerates. It is hard to see how Einstein
could have missed this elementary difficulty.
Einstein was also at first
confused by an idea he had taken from the philosopher Ernst Mach: that the phenomenon of inertia
is caused by distant masses. To keep inertia finite, Einstein in 1917 supposed that the universe
must be finite, and so he assumed that its spatial geometry is that of a three-dimensional spherical
surface. It was therefore a surprise to him that when test particles are introduced into the empty
universe of de Sitter's model, they exhibit all the usual properties of inertia. In general relativity
the masses of distant bodies are not the cause of inertia, though they do affect the choice of inertial
frames. But that mistake was harmless. As Einstein pointed out in his 1917 paper, it was the assumption
that the universe is static, not that it is finite, that had made a cosmological constant necessary.
Aesthetically motivated simplicity
Einstein made what from the perspective
of today's theoretical physics is a deeper mistake in his dislike of the cosmological constant.
In developing general relativity, he had relied not only on a simple physical principle—the
principle of the equivalence of gravitation and inertia that he had developed from 1907 to 1911—but
also on a sort of Occam's razor, that the equations of the theory should be not only consistent with
this principle but also as simple as possible. In itself, the principle of equivalence would allow
field equations of almost unlimited complexity. Einstein could have included terms in the equations
involving four spacetime derivatives, or six spacetime derivatives, or any even number of spacetime
derivatives, but he limited himself to second-order differential equations.
This could have been defended
on practical grounds. Dimensional analysis shows that the terms in the field equations involving
more than two spacetime derivatives would have to be accompanied by constant factors proportional
to positive powers of some length. If this length was anything like the lengths encountered in elementary-particle
physics, or even atomic physics, then the effects of these higher derivative terms would be quite
negligible at the much larger scales at which all observations of gravitation are made. There is
just one modification of Einstein's equations that could have observable effects: the introduction
of a term involving no spacetime derivatives at all—that is, a cosmological constant.
But Einstein did not exclude
terms with higher derivatives for this or for any other practical reason, but for an aesthetic reason:
They were not needed, so why include them? And it was just this aesthetic judgment that led him to
regret that he had ever introduced the cosmological constant.
Since Einstein's time,
we have learned to distrust this sort of aesthetic criterion. Our experience in elementary-particle
physics has taught us that any term in the field equations of physics that is allowed by fundamental
principles is likely to be there in the equations. It is like the ant world in T. H. White's The
Once and Future King: Everything that is not forbidden is compulsory. Indeed, as far as we have
been able to do the calculations, quantum fluctuations by themselves would produce an infinite
effective cosmological constant, so that to cancel the infinity there would have to be an infinite
"bare" cosmological constant of the opposite sign in the field equations themselves. Occam's
razor is a fine tool, but it should be applied to principles, not equations.
It may be that Einstein was
influenced by the example of Maxwell's theory, which he had taught himself while a student at the
Zürich Polytechnic Institute. James Clerk Maxwell of course invented his equations to account
for the known phenomena of electricity and magnetism while preserving the principle of electric-charge
conservation, and in Maxwell's formulation the field equations contain terms with only a minimum
number of spacetime derivatives. Today we know that the equations governing electrodynamics
contain terms with any number of spacetime derivatives, but these terms, like the higher-derivative
terms in general relativity, have no observable consequences at macroscopic scales.
Astronomers in the decades
following 1917 occasionally sought signs of a cosmological constant, but they only succeeded
in setting an upper bound on the constant. That upper bound was vastly smaller than what would be
expected from the contribution of quantum fluctuations, and many physicists and astronomers
concluded from this that the constant must be zero. But despite our best efforts, no one could find
a satisfactory physical principle that would require a vanishing cosmological constant.
Then in 1998, measurements
of redshifts and distances of supernovae by the Supernova Cosmology Project and the High-z
Supernova Search Team showed that the expansion of the universe is accelerating, as de Sitter had
found in his model (see the article by Saul Perlmutter, PHYSICS TODAY, April 2003, page 53). As discussed
in Figure 3, it seems that about 70% of the energy density of the universe is a sort of "dark energy,"
filling all space. This was subsequently confirmed by observations of the angular size of anisotropies
in the cosmic microwave background. The density of the dark energy is not varying rapidly as the
universe expands, and if it is truly time-independent then it is just the effect that would be expected
from a cosmological constant. However this works out, it is still puzzling why the cosmological
constant is not as large as would be expected from calculations of quantum fluctuations. In recent
years the question has become a major preoccupation of theoretical physicists. Regarding his
introduction of the cosmological constant in 1917, Einstein's real mistake was that he thought
it was a mistake.
A historian, reading the
foregoing in a first draft of this article, commented that I might be accused of perpetrating Whig
history. The term "Whig history" was coined in a 1931 lecture by the historian Herbert Butterfield.
According to Butterfield, Whig historians believe that there is an unfolding logic in history,
and they judge the past by the standards of the present. But it seems to me that, although Whiggery
is to be avoided in political and social history (which is what concerned Butterfield), it has a
certain value in the history of science. Our work in science is cumulative. We really do know more
than our predecessors, and we can learn about the things that were not understood in their times
by looking at the mistakes they made.
Contra quantum mechanics
The other mistake that is widely attributed
to Einstein is that he was on the wrong side in his famous debate with Niels Bohr over quantum mechanics,
starting at the Solvay Congress of 1927 and continuing into the 1930s. In brief, Bohr had presided
over the formulation of a "Copenhagen interpretation" of quantum mechanics, in which it is only
possible to calculate the probabilities of the various possible outcomes of experiments. Einstein
rejected the notion that the laws of physics could deal with probabilities, famously decreeing
that God does not play dice with the cosmos. But history gave its verdict against Einstein—quantum
mechanics went on from success to success, leaving Einstein on the sidelines.
All this familiar story
is true, but it leaves out an irony. Bohr's version of quantum mechanics was deeply flawed, but not
for the reason Einstein thought. The Copenhagen interpretation describes what happens when an
observer makes a measurement, but the observer and the act of measurement are themselves treated
classically. This is surely wrong: Physicists and their apparatus must be governed by the same
quantum mechanical rules that govern everything else in the universe. But these rules are expressed
in terms of a wavefunction (or, more precisely, a state vector) that evolves in a perfectly deterministic
way. So where do the probabilistic rules of the Copenhagen interpretation come from?
Considerable progress
has been made in recent years toward the resolution of the problem, which I cannot go into here. It
is enough to say that neither Bohr nor Einstein had focused on the real problem with quantum mechanics.
The Copenhagen rules clearly work, so they have to be accepted. But this leaves the task of explaining
them by applying the deterministic equation for the evolution of the wavefunction, the Schrödinger
equation, to observers and their apparatus. The difficulty is not that quantum mechanics is probabilistic—that
is something we apparently just have to live with. The real difficulty is that it is also deterministic,
or more precisely, that it combines a probabilistic interpretation with deterministic dynamics.
Attempts at unification
Einstein's rejection of quantum mechanics
contributed, in the years from the 1930s to his death in 1955, to his isolation from other research
in physics, but there was another factor. Perhaps Einstein's greatest mistake was that he became
the prisoner of his own successes. It is the most natural thing in the world, when one has scored great
victories in the past, to try to go on to further victories by repeating the tactics that previously
worked so well. Think of the advice given to Egypt's President Gamal Abd al-Nasser by an apocryphal
Soviet military attaché at the time of the 1956 Suez crisis: "Withdraw your troops to the center
of the country, and wait for winter."
And what physicist had scored
greater victories than Einstein? After his tremendous success in finding an explanation of gravitation
in the geometry of space and time, it was natural that he should try to bring other forces along with
gravitation into a "unified field theory" based on geometrical principles. About other things
going on in physics, he commented3 in 1950 that "all attempts to obtain a deeper knowledge
of the foundations of physics seem doomed to me unless the basic concepts are in accordance with
general relativity from the beginning." Since electromagnetism was the only other force that
in its macroscopic effects seemed to bear any resemblance to gravitation, it was the hope of a unification
of gravitation and electromagnetism that drove Einstein in his later years.
I will mention only two of
the many approaches taken by Einstein in this work. One was based on the idea of a fifth dimension,
proposed in 1921 by Theodore Kaluza. Suppose you write the equations of general relativity in five
rather than four spacetime dimensions, and arbitrarily assume that the 5D metric tensor does not
depend on the fifth coordinate. Then it turns out that the part of the metric tensor that links the
usual four spacetime dimensions with the fifth dimension satisfies the same field equation as
the vector potential in the Maxwell theory of electromagnetism, and the part of the metric tensor
that only links the usual four spacetime dimensions to each other satisfies the field equations
of 4D general relativity.
The idea of an additional
dimension became even more attractive in 1926, when Oskar Klein relaxed the condition that the
fields are independent of the fifth coordinate, and assumed instead that the fifth dimension is
rolled up in a tiny circle so that the fields are periodic in that coordinate. Klein found that in
this theory the part of the metric tensor that links the fifth dimension to itself behaves like the
wavefunction of an electrically charged particle, so for a moment it seemed to Einstein that there
was a chance that not only gravitation and electromagnetism but also matter would be governed by
a unified geometrical theory. Alas, it turned out that if the electric charge of the particle is
identified with the charge of the electron, then the particle's mass comes out too large by a factor
of about 1018.
It is a pity that Einstein
gave up on the Kaluza–Klein idea. If he had extended it from five to six or more spacetime dimensions,
he might have discovered the field theory constructed in 1954 by C. N. Yang and Robert Mills, and
its generalizations, some of which later appeared as parts of our modern theories of strong, weak,
and electromagnetic interactions.4 Einstein apparently gave no thought to strong
or weak nuclear forces, I suppose because they seem so different from gravitation and electromagnetism.
Today we realize that the equations underlying all known forces aside from gravitation are actually
quite similar, the difference in the phenomena arising from color trapping for strong interactions
and spontaneous symmetry breaking for weak interactions. Even so, Einstein would still probably
be unhappy with today's theories, because they are not unified with gravitation and because matter—electrons,
quarks, and so on—still has to be put in by hand.
Even before Klein's work,
Einstein had started on a different approach, based on a simple bit of counting. If you give up the
condition that the 4 × 4
metric tensor should be symmetric, then it will have 16 rather than 10 independent components,
and the extra 6 components will have the right properties to be identified with the electric and
magnetic fields. Equivalently, one can assume that the metric is complex, but Hermitian. The trouble
with this idea, as Einstein became painfully aware, is that there really is nothing in it that ties
the 6 components of the electric and magnetic fields to the 10 components of the ordinary metric
tensor that describes gravitation, other than that one is using the same letter of the alphabet
for all these fields. A Lorentz transformation or any other coordinate transformation will convert
electric or magnetic fields into mixtures of electric and magnetic fields, but no transformation
mixes them with the gravitational field. This purely formal approach, unlike the Kaluza–Klein
idea, has left no significant trace in current research. The faith in mathematics as a source of
physical inspiration, which had served Einstein so well in his development of general relativity,
was now betraying him.
Even though it was a mistake
for Einstein to turn away from the exciting progress being made in the 1930s and 1940s by younger
physicists, it revealed one admirable feature of his personality. Einstein never wanted to be
a mandarin. He never tried to induce physicists in general to give up their work on nuclear and particle
physics and follow his ideas. He never tried to fill professorships at the Institute for Advanced
Studies with his collaborators or acolytes. Einstein was not only a great man, but a good one. His
moral sense guided him in other matters: He opposed militarism during World War I; he refused to
support the Soviet Union in the Stalin years; he became an enthusiastic Zionist; he gave up his earlier
pacifism when Europe was threatened by Nazi Germany, for instance urging the Belgians to rearm;
and he publicly opposed McCarthyism. About these great public issues, Einstein made no mistakes.
Steven Weinberg
holds the Josey Chair in Science at the University of Texas at Austin, where he is a member of the physics
and astronomy departments and heads the physics department's Theory Group.
References
1. The set of mistakes discussed in this article is not intended to be exhaustive. They are a selection, mostly chosen because they seemed to me to reveal something of the intellectual environment in which Einstein worked. In PHYSICS TODAY, March 2005, page 34, Alex Harvey and Engelbert Schucking have described an erroneous prediction of Einstein regarding the rates of clocks on Earth's surface, and in his book Albert Einstein's Special Theory of Relativity, Addison-Wesley, Reading, PA (1981), p. 328, Arthur I. Miller has discussed an error in Einstein's calculation of the electron's transverse mass.
2. G. Gamow, My World Line—An Informal Autobiography, Viking Press, New York (1970), p. 44. I thank Lawrence Krauss for this reference.
3. A. Einstein, Sci. Am., April 1950, p. 13.
4. Oddly enough, at a conference in Warsaw in 1939, Klein presented something very like the Yang–Mills theory, on the basis of his five-dimensional generalization of general relativity. I have tried and failed to follow Klein's argument, and I do not believe his derivation makes sense; it takes at least two extra dimensions to get the Yang–Mills theory. It seems that scientists are often attracted to beautiful theories in the way that insects are attracted to flowers—not by logical deduction, but by something like a sense of smell.
5. E. Hubble, Proc. Natl. Acad. Sci. USA15, 168 (1929).
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