[Paleopsych] Physics Today: Einstein's Mistakes
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Einstein's Mistakes
http://www.physicstoday.org/vol-58/iss-11/p31.html
Science sets itself apart from other paths to truth by recognizing
that even its greatest practitioners sometimes err.
[12]Steven Weinberg
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.[13]^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 Gamow[14]^2) 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.
[15]Einstein, de Sitter, Eddington, Lorentz, and Ehrenfest
[16]Figure 1
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 [17]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 [18]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.
[19]Graph of Recessional velocities of nearby galaxies
[20]Figure 2
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.
[21]Graph of Measurements on distant supernovae
[22]Figure 3
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, [23]page 53). As discussed in [24]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 commented[25]^3 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
10^18.
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.[26]^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. 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, [27]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. 2. G. Gamow, My World Line--An Informal Autobiography, Viking
Press, New York (1970), p. 44. I thank Lawrence Krauss for this
reference.
3. 3. A. Einstein, Sci. Am., April 1950, p. 13.
4. 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. 5. E. Hubble, Proc. Natl. Acad. Sci. USA 15, 168 (1929).
6. 6. A. G. Riess et al., [28]Astrophys. J. 607, 665
(2004) [29][SPIN].
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