[Paleopsych] Physics Today: Einstein's Mistakes

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Einstein's Mistakes

    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

    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

    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.


     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
     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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