# How does the Universe Expands?

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Einstein’s field equations without the cosmological constant contain perfectly satisfactory solutions in which the Universe is uniformly expanding. According to George Gamow, when the expansion of the Universe was discovered, Einstein regarded the introduction of the cosmological constant as ‘the biggest blunder of my life’ (Gamow, 1970). The cosmological constant was not consigned to oblivion for long however. As Yakov Zeldovich remarked:

In Newton’s Principia Mathematica, he emphatically took the position that all motion takes place with respect to a system of absolute space and time. He fiercely rejected the idea that the motion of a body could only be described relative to those of other bodies. This position was challenged by Bishop Berkeley, Christiaan Huygens and others but, at least until the late nineteenth century, Newton’s view prevailed. The issue was revived by Ernst Mach who argued that motion can only be defined relative to other bodies. Specifically, he took the view that the local inertial frame of reference is determined by the frame of the distant stars, or galaxies in modern parlance. Thus, a freely swinging Foucault pendulum swings in a reference frame which is fixed relative to the distant galaxies. Albert Einstein gave the name Mach’s principle to this idea.

During the late eighteenth century, non-Euclidean geometries began to be taken seriously by mathematicians who realised that the fifth postulate of Euclid, that parallel lines meet only at infinity, might not be essential for the construction of a self-consistent geometry. Proposals that the global geometry of space might not be Euclidean were discussed by Girolamo Saccheri and Johann Lambert. In 1816, Carl Friedrich Gauss repeated this proposal in a letter to Christian Gerling and was aware of the fact that a test of the local geometry of space could be carried out by measuring the sum of the angles of a triangle between three high peaks, the Brocken, Hoherhagen and Inselberg. In 1818, Gauss was asked to carry out a geodetic survey of the state of Hanover and he devoted a large effort to carrying out and reducing the data himself. He was certainly aware of the fact that the sum of the angles of the triangle was 180 degrees within the limits of geodetic measurements.

The fathers of non-Euclidean geometry were Nikolai Lobachevsky, who became rector of Kazan University in Russia in 1827, and JÃ¡nos Bolyai in Transylvania, then part of Hungary. In the 1820s, they independently solved the problem of the

existence of non-Euclidean geometries and showed that Euclid’s fifth postulate could not be deduced from the other postulates (Lobachevsky, 1829, 1830; Bolyai, 1832). In his papers entitled On the Principles of Geometry, Lobachevsky also proposed an astronomical test of the geometry of space. If the geometry were hyperbolic, the minimum parallax of any object would be

existence of non-Euclidean geometries and showed that Euclid’s fifth postulate could not be deduced from the other postulates (Lobachevsky, 1829, 1830; Bolyai, 1832). In his papers entitled On the Principles of Geometry, Lobachevsky also proposed an astronomical test of the geometry of space. If the geometry were hyperbolic, the minimum parallax of any object would be

Î¸ = arctan(a/R)

where a is the radius of the Earth’s orbit and R the radius of curvature of the

geometry. He found a minimum value of R ≥ 1.66 × 10^5 AU = 2.6 light years,

using an observational upper limit of 1 arcsec for the parallax of bright stars. In

a prescient statement which will warm the hearts of observational astronomers, he remarked:

geometry. He found a minimum value of R ≥ 1.66 × 10^5 AU = 2.6 light years,

using an observational upper limit of 1 arcsec for the parallax of bright stars. In

a prescient statement which will warm the hearts of observational astronomers, he remarked:

*There is no means other than astronomical observations for judging the exactness which attaches to the calculations of ordinary geometry.*

*Non-Euclidean geometries were placed on a firm theoretical basis by Bernhard Rie-mann, who also discovered closed spherical geometries. The English-speaking world was introduced these ideas through the works of William Clifford and Arthur Cayley. Until Albert Einstein’s discovery of the General Theory of Relativity, considerations of the geometry of space and the role of gravity in defining the large-scale structure of the Universe were separate questions. After 1915, they were inextricably linked.*

In that year, after a titanic intellectual struggle, Einstein discovered the definitive

version of his General Theory of Relativity which describes how space–time is distorted by the presence of matter and how, in turn, matter moves along trajec- tories in bent space–time (Einstein, 1915, 1916). For the first time, a relativistic

theory of gravity was available which enabled self-consistent models of the Uni-

verse as a whole to be constructed and, characteristically, Einstein did not hesitate to do so.

version of his General Theory of Relativity which describes how space–time is distorted by the presence of matter and how, in turn, matter moves along trajec- tories in bent space–time (Einstein, 1915, 1916). For the first time, a relativistic

theory of gravity was available which enabled self-consistent models of the Uni-

verse as a whole to be constructed and, characteristically, Einstein did not hesitate to do so.

In seeking a solution of his field equations for the Universe as a whole, Einstein had explicitly in mind that Mach’s principle should be incorporated into any model of the large-scale structure of the Universe. He had, however, a major problem. Without modification, the field equations predicted that the Universe was unstable. He could only find static solutions by introducing what is now known as the cosmical or cosmological constant Î», which appears as a constant in Einstein’s field equations. In his great paper of 1917, Einstein showed that the introduction of the cosmological constant resulted in static solutions for the Universe as a whole which had closed, spherical geometry and a finite size (Einstein, 1917). He also believed that he had incorporated Mach’s principle into General Relativity, in the sense that no solution of the equations would exist if there were no matter present. In the same year, this was, however, shown to be incorrect by Willem de Sitter, who found solutions of the equations even if there were no matter present in the Universe (de Sitter, 1917).

For many decades, the status of the cosmological constant was the subject of debate. In 1919, Einstein realised that a term involving the cosmological constant would appear in the field equations of General Relativity, quite independent of its cosmological significance (Einstein, 1919). In the derivation of the field equations, the Î»-term appears as a constant of integration which is normally set equal to zero in the development of standard General Relativity. Einstein was not enthusiastic about the term, remarking that it ‘detracts from the formal beauty of the theory’. Willem de Sitter wrote in 1919 that the term

*. . . detracts from the symmetry and elegance of Einstein’s original theory, one of whose chief attractions was that it explained so much without introducing any new hypotheses or empirical constant.*

*Others regarded it as a constant which appears in the development of the General Relativity and its value should be determined by astronomical observation.*

The irony of the situation is that this debate took place before it was realised that the Universe is in fact non-stationary. In 1922, Aleksander Friedman published the first of two classic papers in which he discovered both static and expanding solutions of Einstein’s field equations. In the first paper, Friedman found solutions for expanding universes with closed spatial geometries, including those which expand to a maximum radius and eventually collapse to a singularity (Friedman, 1922). In the second paper of 1924, he showed that there exist expanding solutions which are unbounded and which have hyperbolic geometry (Friedman, 1924). These solutions correspond exactly to the standard world models of general relativity and are known as the Friedman world models.

In 1925, Friedman died of typhoid in Leningrad before the fundamental sig- nificance of his work was appreciated. The neglect of Friedman’s work in these early days is somewhat surprising since Einstein had commented, incorrectly as he admitted, on the first of the two papers in 1923. It was not until Georges LemaÃ®tre independently rediscovered the same solutions in 1927, and then became aware of Friedman’s papers, that the pioneering nature of Friedman’s contributions was appreciated (LemaÃ®tre, 1927).

Einstein’s field equations without the cosmological constant contain perfectly satisfactory solutions in which the Universe is uniformly expanding. According to George Gamow, when the expansion of the Universe was discovered, Einstein regarded the introduction of the cosmological constant as ‘the biggest blunder of my life’ (Gamow, 1970). The cosmological constant was not consigned to oblivion for long however. As Yakov Zeldovich remarked:

*The genie is out of the bottle and, once he is out, he is very difficult to put*

back in again.

back in again.

As the standard models of General Relativity became better understood, a major thrust of cosmological research became the determination of the large-scale dynamical and geometrical properties of the Universe – its rate of expansion, its deceleration, its mean density, its geometry and its age. These remained among the most difficult programmes of modern observational cosmology until, in the first years of the twenty-first century, precise estimates became available using techniques undreamt of by the pioneers of geometrical cosmology.

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