Lecture 6: Interrogating the legend of Einstein’s “biggest blunder”
Cormac O’Raifeartaigh (Cormac O’Rafferty) is an Irish physicist based at Waterford Institute of Technology in Ireland. A solid-state physicist by training, he is best known for several contributions to the study of the history and philosophy of 20th century science, including the discovery that Albert Einstein once attempted a steady-state model of the expanding universe, many years before Fred Hoyle.
He is a graduate of University College Dublin (BSc Hons) and Trinity College Dublin (PhD). He is a Fellow of the Royal Astronomical Society, a Fellow of the Institute of Physics and has been a Visiting Fellow at the Science, Technology and Society Program of Harvard University. A solid-state physicist by training, his research has recently focused on the history and philosophy of 20th century physics.
My notes from the lecture (if they don’t make sense then it is entirely my fault)
In 1917 Einstein presented the paper Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie (Cosmological Considerations in the General Theory of Relativity) to the Prussian Academy of Sciences in Berlin. This paper laid the foundations for today’s Big Bang model of the universe.
The paper assumed a static universe, closed spatial geometry, introduced a new term to the field equations of relativity and produced a quantitative model of Universe.
The new term was called the cosmological constant and was regarded as bit of a fudge factor. Einstein’s interpretation of it was introduced as an analogy with Newtonian cosmology, modifying Newton’s gravity at the cosmic scale.
Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).
A de Sitter universe is a cosmological solution to the Einstein field equations of general relativity, named after Willem de Sitter. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominated by the cosmological constant, thought to correspond to dark energy in our universe or the inflaton field in the early universe. According to the models of inflation and current observations of the accelerating universe, the concordance models of physical cosmology are converging on a consistent model where our universe was best described as a de Sitter universe at about a time t=10-33 seconds after the fiducial Big Bang singularity, and far into the future.
Willem de Sitter (6 May 1872 – 20 November 1934) was a Dutch mathematician, physicist, and astronomer.
Einstein was greatly perturbed by de Sitter’s model universe.
A long debate between the two physicists ensued. In compiling research for a review of “Cosmological considerations,” no evidence was found in Einstein’s writings that he ever accepted de Sitter’s solution as a realistic model of the universe.
However the existence of a mathematically viable vacuum solution to the modified field equations may have marked the beginning of Einstein’s distrust of the cosmological constant.
Einstein’s view of the cosmological constant was evolving:
Not an energy of space;
A necessary evil;
In March 1918, the Austrian physicist Erwin Schrödinger suggested that a consistent model of a static, matter-filled cosmos could be obtained from Einstein’s field equations without the introduction of the cosmological constant. Einstein’s response was that Schrödinger’s formulation was entirely equivalent to that of his 1917 memoir, provided the negative-pressure term was constant (Einstein 1918a). Within a year, Einstein proposed a slightly different interpretation of the cosmic constant, namely that of a constant of integration, rather than a universal constant associated with cosmology.
In 1922 the young Russian physicist Alexander Friedmann suggested that nonstatic solutions of the Einstein field equations should be considered in relativistic models of the universe. Einstein did not welcome his contribution and his first reaction was that the Russian had made a mathematical error.
Alexander Alexandrovich Friedmann (June 16 [O.S. 4], 1888 – September 16, 1925) was a Russian and Soviet physicist and mathematician.
In 1927 the Belgian physicist Georges Lemaître independently derived differential equations for the radius of the universe that were almost identical to the Friedmann equations. Einstein did not view this work favourably either; in fact, Lemaître later reported that Einstein described his expanding model as “abominable.”
Georges Henri Joseph Édouard Lemaître, RAS Associate (17 July 1894 – 20 June 1966) was a Belgian Roman Catholic priest, mathematician, astronomer, and professor of physics at the Catholic University of Louvain. He proposed on theoretical grounds that the universe is expanding, which was observationally confirmed soon afterwards by Edwin Hubble.
Some of the earliest models of the Universe to incorporate Einstein’s general relativity were proposed by Aleksandr Friedmann (1888-1925) and Abbe Georges Lemaitre (1894-1966). They derived solutions to Einstein’s field equations (Friedmann in 1922 and Lemaitre in 1927) which predicted that galaxies should be receding from each other due to cosmic expansion.
They found three distinct classes of models:
1) the open Universe: a low matter-density Universe which expands forever
2) the closed Universe: a high matter-density Universe in which gravity eventually stops the expansion and the Universe re-collapses
3) the critical density (flat Universe) Universe: in which the density of matter is just enough to theoretically stop the expansion, but only after an infinite length of time.
Things changed in 1929, when American astronomer Edwin Hubble published the first evidence of a linear relation between the redshifts of the spiral nebulae and their radial distance.
Edwin Powell Hubble (November 20, 1889 – September 28, 1953) was an American astronomer. He played a crucial role in establishing the fields of extragalactic astronomy and observational cosmology and is regarded as one of the most important astronomers of all time.
The discovery marked a turning point in modern cosmology as the data could not be explained in the context of Einstein’s static matter-filled world (solution A) or de Sitter’s empty universe (solution B). As de Sitter remarked at a meeting of the Royal Astronomical Society (RAS) in January 1930: “It would be desirable to know what happens when we insert matter into the empty world represented by solution B. The difficulty in the investigation of this problem lies in the fact that it is not static” (de Sitter 1930a). A report of the meeting was read by Georges Lemaître and, following a communication from him, Arthur Stanley Eddington arranged for Lemaître’s 1927 paper to be republished in English translation in the Monthly Notices of the RAS, bringing the work to a wider audience (Lemaître 1931a). Soon, a number of papers had emerged that explored expanding models of the Friedman-Lemaître type for diverse values of cosmic parameters such as spatial curvature and the cosmological constant (Eddington 1930, 1931a: de Sitter 1930b, 1930c; Einstein 1931a; Einstein and de Sitter 1932; Tolman 1930, 1931a, 1932; Heckmann 1931, 1932; Robertson 1932).
Sir Arthur Stanley Eddington OM FRS (28 December 1882 – 22 November 1944) was an English astronomer, physicist, and mathematician of the early 20th century who did his greatest work in astrophysics.
Einstein lost little time in abandoning his static cosmology at that point (although he did attempt a steady-state model of the expanding universe). In the early 1930s, he published two distinct models of the expanding universe, one of positive spatial curvature and one of Euclidean geometry. In each case, he abandoned the cosmological constant, stating that the term was both unsatisfactory (it gave an unstable solution) and redundant (relativity could describe expanding models of the universe without the term).
The Friedmann–Einstein universe is a model of the universe published by Albert Einstein in 1931. The model is of historic significance as the first scientific publication in which Einstein embraced the possibility of a cosmos of time-varying radius.
Interpreting Edwin Hubble’s discovery of a linear relation between the redshifts of the galaxies and their radial distance as evidence for an expanding universe, Einstein abandoned his earlier static model of the universe and embraced the dynamic cosmology of Alexander Friedmann. Removing the cosmological constant term from the Friedmann equations on the grounds that it was both unsatisfactory and unnecessary, Einstein arrived at a model of a universe that expands and then contracts; a model that was later denoted the Friedmann–Einstein model of the universe.
In the model, Einstein derived simple expressions relating the density of matter, the radius of the universe and the timespan of the expansion to the Hubble constant. With the use of the contemporaneous value of 500 km·s−1Mpc−1 for the Hubble constant, he calculated values of 10−26 cm−3, 108 light-years and 1010 years for the density of matter, the radius of the universe and the timespan of the expansion respectively. It has recently been shown that these calculations contain a slight systematic error.
However the error is a small one and it did not substantially affect a major puzzle raised by the model; if the timespan of cosmic expansion represented the age of the universe, it was strangely small in comparison with estimates of the age of stars (as calculated from astrophysics) or estimates of the age of the earth (as deduced from radioactivity). Einstein attributed this age paradox to the idealized assumptions of the model, in particular the assumption of a homogeneous distribution of matter on the largest scales (Einstein 1931a)
The Einstein–de Sitter universe is a model of the universe proposed by Albert Einstein and Willem de Sitter in 1932. On first learning of Edwin Hubble’s discovery of a linear relation between the redshift of the galaxies and their distance, Einstein set the cosmological constant to zero in the Friedmann equations, resulting in a model of the expanding universe known as the Friedmann–Einstein universe. In 1932, Einstein and de Sitter proposed an even simpler cosmic model by assuming a vanishing spatial curvature as well as a vanishing cosmological constant. In modern parlance, the Einstein–de Sitter universe can be described as a cosmological model for a flat matter-only Friedmann–Lemaître–Robertson–Walker metric (FLRW) universe.
At first, not many of Einstein’s colleagues took his lead in abandoning the cosmological constant. Some felt that the term should be retained for reasons of mathematical generality; others felt that it could be used to address cosmological puzzles such as the timespan of the expansion and the formation of galaxies in an expanding universe. Still others felt that the term had an important role to play in giving a physical cause for cosmic expansion.
Eddington pointed out that relativity allowed for an expanding universe, but it did not explain the phenomenon. In his view, the cosmological constant supplied a physical explanation for the phenomenon: “It is found similarly that the added term () gives rise to a repulsion directly proportional to the distance…It is a dispersive force like that which I imagined as scattering apart the audience in the lecture-room (Eddington 1933 p23). A similar point was made by Willem de Sitter, who asked: “What is it then that causes the expansion? Who blows up the india-rubber ball? The only possible answer is: the lambda (the cosmological constant) does it” (de Sitter 1931).
The Russian scientist George Gamow reported in his memoirs that Einstein once described the cosmological constant as his “biggest blunder”, although there is some doubt about this because it has not been found in any of Einstein’s papers.
George Gamow (March 4, 1904 – August 19, 1968), born Georgiy Antonovich Gamov, was a Soviet-American theoretical physicist and cosmologist. He was an early advocate and developer of Lemaître’s Big Bang theory. He discovered a theoretical explanation of alpha decay via quantum tunnelling, and worked on radioactive decay of the atomic nucleus, star formation, stellar nucleosynthesis and Big Bang nucleosynthesis (which he collectively called nucleocosmogenesis), and molecular genetics.
Gamow had a deserved reputation as an irreverent physicist with a well-developed sense of humour and he may have overstated his interactions with Einstein. However Gamow was in the right place at the right time.
Research has concluded that Einstein probably did make the “biggest blunder” remark. First, the remark is highly compatible with Einstein’s science—from 1931 onward, he emphatically dismissed the cosmological constant in all of his writings on cosmology (he said the term is both redundant and satisfactory).
Secondly it turns out that Gamow is not the only physicist to have reported Einstein’s “biggest blunder” remark. In the 2000 book Exploring Black Holes: Introduction to General Relativity, the famous theorist John Archibald Wheeler (1911–2008) states the following: “Going into the doorway of the Institute for Advanced Study’s Fuld Hall with Einstein and George Gamow, I heard Einstein say to Gamow about the cosmological constant, ‘That was my biggest blunder of my life.’”12 Similarly, Ralph Alpher (1921–2007), a close colleague of Gamow’s, recalled on an online message board a conversation with Einstein at Princeton about problems concerning the apparent time span of cosmic expansion: “A way to fix this was to reactivate the cosmological constant. Einstein did not like this very much and, as I recall, said his introduction of the concept in his early work was a blunder.”
In the 1940s, few physicists outside the relativity community paid attention to the Lemaîtreor Eddington-Lemaître models of the universe. While many accepted Hubble’s observations as possible evidence for an expanding universe, theories concerning cosmic origins were considered deeply speculative, an attitude that persisted for some years (Kragh 1996 pp 135-143).
Gamow suggested that Friedman-Lemaître cosmologies might offer a radical solution to the puzzle of nucleosynthesis, why is there such an abundance of light chemical elements.
In parallel with the work of Gamow et al., a new type of cosmic model was proposed in the United Kingdom known as the ‘steady-state’ universe. In this cosmology, the universe expands but remains essentially unchanged in every other respect.
Albert Einstein, The meaning of relativity Princeton, NJ: Princeton University Press, 1945. https://www.eveningstarbooks.net/pages/books/00007808/albert-einstein/the-meaning-of-relativity
Second Edition of Einstein’s explanation of his theory of relativity for a general audience. Includes a new appendix prepared by Einstein in 1944 covering new developments since the publication of the first edition in 1922.
Lemaître admired Einstein but he was perfectly aware of the limits of some of his last theoretical attempts. A great part of the Einstein-Lemaître discussions had been dedicated to the problem of the cosmological constant. Einstein wanted to suppress it but Lemaître considered that it was very important, although not yet well formulated. In a letter of 30 July 1947 to Einstein, Lemaitre told him that he considered that the introduction of the cosmological constant was one of his greatest contributions to science!
Lemaître never did convince Einstein of his interpretation of the cosmological constant. In reply to the letter Einstein wrote it would be “abominable” to suppose that gravitation is made of two logically independent terms one which is attractive (as in the classical case) and the other one being repulsive in order to keep the cosmological constant.
In 1949 a book was published in honour of Einstein’s 70th birthday. It included his autobiographical notes where he summarised his ideas and attitudes to physics throughout his career up to that point in time. It showed how uncomfortable he was with the idea of quantum theory.
The Bang to Crunch Universe is considered too simple to be wrong;
The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole
A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it.
The seeds of the Big Bang theory were laid by the Alexander Friedmann, who had the courage to examine the full implications of Einstein’s general theory of relativity for cosmology—that the universe may not be static.
It could be expanding or contracting. Few people took Friedmann seriously and his work was little known. As mentioned earlier the idea was independently proposed by Georges Lemaître whose theory that the universe began with the decay of a single atom was also found wanting because it, too, could not explain how hydrogen and helium, the building blocks of the other elements, were formed.
Ralph Asher Alpher (February 3, 1921 – August 12, 2007) was an American cosmologist, who carried out pioneering work in the early 1950s on the Big Bang model, including big bang nucleosynthesis and predictions of the cosmic microwave background radiation.
The Big Bang is a term coined initially in derision by Fred Hoyle on BBC Radio in 1950 to describe the cosmological model of the universe as expanding into its current state from a primordial condition of enormous density and temperature. Nucleosynthesis is the explanation of how more complex elements are created out of simple elements in the moments following the Big Bang.
Gamow and Alpher tackled the Big Bang problem by starting with the density of matter in the universe today and running the clock backward, imagining what would happen if everything were compressed into a denser and denser state. As pressure increased the temperature would rise, and molecules would be broken down into atoms, and atoms into their constituent parts. When they got to the smallest possible components they ran the clock forwards, envisioning how the atoms would coalesce out of the mixture of particles that that they called ylem, an old English word meaning “the primordial substance from which the elements were formed.”
The mathematical genius, Ralph Alpher, showed that the expansion of the universe from an initial ultra-dense state would indeed produce all of the elementary particles needed to assemble hydrogen and helium in the proportions seen in the universe today in just a few minutes. This was an important result because hydrogen and helium make up 99.99% of the matter in today’s universe. This time the world took notice and on April 14, 1948 the Washington Post carried the headline “World Began in 5 Minutes.”
The Modern Big Bang Model
The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parametrization of the Big Bang cosmological model in which the universe contains a cosmological constant, denoted by Lambda (Greek Λ), associated with dark energy, and cold dark matter (abbreviated CDM). It is frequently referred to as the standard model of Big Bang cosmology because it is the simplest model that provides a reasonably good account of the following properties of the cosmos:
The existence and structure of the cosmic microwave background
The large-scale structure in the distribution of galaxies
The abundances of hydrogen (including deuterium), helium, and lithium
The accelerating expansion of the universe observed in the light from distant galaxies and supernovae
The model assumes that general relativity is the correct theory of gravity on cosmological scales. It emerged in the late 1990s as a concordance cosmology, after a period of time when disparate observed properties of the universe appeared mutually inconsistent, and there was no consensus on the makeup of the energy density of the universe.
The ΛCDM model can be extended by adding cosmological inflation, quintessence and other elements that are current areas of speculation and research in cosmology.
Some alternative models challenge the assumptions of the ΛCDM model. Examples of these are modified Newtonian dynamics, entropic gravity, modified gravity, theories of large-scale variations in the matter density of the universe, bimetric gravity, and scale invariance of empty space
The standard cosmological model which predicts that the Universe is dominated by 74% dark energy (CMB, Supernovae) and 22% dark matter. The remaining 4% are the atoms of ordinary matter that we see around us (the area researched by astrophysicists). Thus in this model 96% of the Universe is dark. (Credit: NASA/WMAP Science Team)
Density Parameter, Ω
The density parameter is the ratio of the average density of matter and energy in the Universe to the critical density (the density at which the Universe would stop expanding only after an infinite time).
Although current research suggests that Ω0 is very close to 1, it is still of great importance to know whether Ω0 is slightly greater than 1, less than 1, or exactly equal to 1, as this reveals the ultimate fate of the Universe. If Ω0 is less than 1, the Universe is open and will continue to expand forever. If Ω0 is greater than 1, the Universe is closed and the will eventually halt its expansion and recollapse. If Ω0 is exactly equal to 1 (which would seem a remarkable coincidence) then the Universe is flat and contains enough matter to halt the expansion but not enough to recollapse it.
It is important to note that the ρ used in the calculation of Ω0 is the total mass/energy density of the Universe. In other words, it is the sum of a number of different components including both normal and dark matter as well as the dark energy suggested by recent observations.
We can therefore write:
where ΩB is the density parameter for normal baryonic matter, ΩD is the density parameter for dark matter and ΩΛ is the density parameter for dark energy.
Current observations suggest that we live in a dark energy dominated Universe with ΩΛ = 0.73, ΩD = 0.23, and ΩB = 0.04. To the accuracy of current cosmological observations, this means that we live in a flat, Ω0 = 1 Universe.
It has been said that Einstein’s biggest blunder was setting the cosmological constant to zero.