Humans have speculated about the nature of the world from time immemorial. From time to time there have been revolutions in their perception of the nature of the world. The profoundly revolutionary perception of modern cosmology had its origin in the startling discovery in 1887 by Michelson and Morley that the speed of light is independent of the velocity of the frame of reference used in measuring it. In Newtonian mechanics the velocity of an object with respect a moving frame of reference should add vectorially to the velocity of the frame of reference itself to give the velocity of the object with respect to an observer at rest. When the Michaelson-Morley result showed that such does not hold true for our universe scientists perceived that our universe might not be the infinite three dimensional world it was thought to be.

It is worthwhile to distinguish two fields of cosmology. One concerns the geometry and topology of the Universe. The other concerns the astronomy of the material distributed within the Universe.

Special Relativity

Hendrik Lorentz

George FitzGerald conjectured that the Michelson-Morley result could be explained by objects being contracted in the direction of the motion. In 1892 the Dutch physicist, Hendrik Lorentz, published the detailed analysis of this contraction. The Irish physicist, Joseph Larmor, published an analysis showing that there has to be a dilation of time as well a contraction of a spatial dimension to explain the Michelson-Morley result. In 1905 the French mathematical physicist, Henri Poincaré, tied together the mathematics of the contraction and dilation as a transformation group. In that same year Albert Einstein published his Special Theory of Relativity that, among other things, derived the implication that mass is equivalent to energy according to the famous equation E=mc². More generally Einstein showed that the energy of a moving object of rest mass m should be expressed as

E = mc²/[1−(v/c)²]^½

The RHS of the above may be expressed as series; i.e.,

E = mc² + ½mv² + (3/8)mc²(v/c)⁴ + higher order terms

This shows that the hallowed formula of ½mv² for kinetic energy is only a first approximation.

Albert Einstein

General Relativity

Marcel Grossmann

Einstein did not rest on his laurels but went further to develop the General Theory of Relativity which he published in 1915 which deals with gravity. Einstein excelled in mathematics but he considered mathematics as secondary to physical theorizing based on intuitive analysis of thought experiments. But to analyze gravity he had to learn some esoteric mathematics called Riemannian geometry, or more generally tensor analysis. He relied upon his friend, the mathematician Marcel Grossmann in Zurich, to coach him in this field of mathematics. Einstein then developed his general equations for gravity; i.e.,

R_μν −½Rg_μν + Λg_μν = {8πG/c⁴} T_μν

where g_μν is the metric tensor that describes space-time. R_μν is the Ricci curvature tensor for space-time and R is the scalar magnitude of curvature. G is the gravitational constant in the Newton theory of gravitational attraction. T_μν is the stress-energy. In this case the tensors are physical quantities that are represented by 4×4 matrices. These matrices are symmetric so they have only ten independent components instead of 16. Furthermore the components of these matrices have to satisfy four identities, known as Bianchi identities, so the number of independent components is reduced to 6. The scalar Λ has a story to it.

Without Λ the equations predict that the universe could be expanding or contracting based on the amount of mass it contains. Einstein included the term involving Λ to fix the scale of the universe.

Einstein's theory gave two notable predictions. One was that the elliptical orbits of the planets should revolve. It was known that the orbit of Mercury did revolve. The other was that the path of light rays passing gravitating bodies should curve. This caught the attention of astronomers because it could be tested by observing the apparent positions of stars near the Sun during a total eclipse. Such an eclipse would occur in a limited number of places in the world in 1919.

Sir Arthur Eddington

Arthur Eddington was one of those astronomers and he could organize an expedition to test the prediction of General Relativity. In 1919 he headed and expedition to the island of Principe off the west coast of Africa. Eddington reported in 1919 that the results of the expedition vindicated Einstein's prediction.

Eddington went on to write popular works and give public lectures on Relativity. After one of these lectures a man referred to Eddington as one of the three people in the world who understood Relativity. When Eddington did not immediately respond the man admonished him not to be shy. Eddington replied that he wasn't being shy; he was trying to think of who the third person would be.

Willem de Sitter

Meanwhile theorists were exploring the implications of Relativity. In the Netherlands Willem de Sitter, an astronomer well trained in mathematics, worked out a solution to the equations of Relativity. It implied that the Universe could be evolving; either expanding or contracting. De Sitter was in communication with Einstein and Einstein accepted the mathematical validity of the de Sitter model without accepting its physical reality.

Alexander Friedmann

In revolutionary Russia Alexander Friedmann also started working on possible solutions to Einstein's equations. In 1922 he published an article entitled, "On the Curvature of the Universe" in which he demonstrated that the Einstein model and the de Sitter model were just two special cases of solutions to the equations of Relativity. In general the solutions involved evolution, either expanding or contracting or cyclical sequence of both.

Friedmann was a man of many talents but primarily he was a mathematical physicist of prodigious computational skill. When Einstein read Friedmann's article he thought he found an error that when corrected implied a static Universe. Einstein then published a correction to Friedmann's article. But Friedmann found an error in Einstein's correction and informed him of it. After some delay Einstein published retraction of his "correction" of Friedmann's article and admitting that there are time-varying mathematical solutions to the equations of Relativity. But Einstein continued to believe that the only physically real solutions were static. Unfortunately Alexander Friedmann died in1925 at age 37 of typhus contracted while on a vacation in the Crimea.

George Lemaître

Others carried on the quest for understanding the nature of the Universe. One of those was an individual of a unique career. His name was George Lemaître. Lemaître began his higher education at a Jesuit school in Brussels. He studied mining engineering but World War I intervened before he could pursue that career. In the Belgian Army started out as a gunner and was eventually promoted to the rank of artillery officer.

After the war Lemaître studied physics and mathematics but he also entered a program to become a Catholic priest. In 1923 he was ordained as a Jesuit priest.

In 1923 Lemaître spent time at Cambridge University in England with Eddington where he studied the elements of Relativity. In 1924 Lemaître went to Cambridge, Massachusetts to begin a doctoral program at the Massachusetts Institute of Technology (M.I.T.). His studies there had to do with models of the Universe.

From the work of de Sitter there was the implication that an expanding Universe would have red-shifts in the spectra of light emitted by stars. De Sitter's model had a flaw. In it there was a center of the Universe which was in the Universe. A two dimensional universe of the surface of a sphere does not have a center that is in that spherical surface. Likewise a three dimensional curved universe does not have a center within itself. All of the points in such a three dimensional curved universe can be equivalent.

In 1915 an American astronomer, Vesto Slipher, working at the Lowell Observatory in Flagstaff, Arizona observed a red-shift in the spectra of light from what were known at that time as nebulae. The term nebula meant cloud. It was not known at that time these nebulae were other galaxies analogous to our galaxy of the Milky Way. Thus it was not known to Slipher that the nebulae he was observing were much more distant than the stars of our galaxy. Slipher did recognize that the red-shifts were probably due to the Doppler Effect and that the nebulae were moving away from us.

It took Lemaître to put together the theory of an expanding universe of de Sitter with the red-shifts of Slipher to predict a linear relationship between the distance of radiating objects and the observed red-shifts in their spectra. He published this in his dissertation for his doctorate from M.I.T. in 1927. Einstein and other top physicists paid little or no attention to Lemaître's work.

In 1925 Edwin Hubble carried out some research that drastically changed the perception of the Universe. The distance to nearby stars can be estimated by observing the change in their positions from opposite points in the orbit of Earth. The method has very limited application. There is another way based upon a type of stars called Cepheids.

Cepheid Variable Stars

A Cepheid is star of pulsating brightness and the maximum brightness and the cycle period of the pulsations are correlated. So the measured cycle period determines the maximum luminosity. The apparent brightness of a Cepheid comes from the distance of its observation. This was discovered in1908 by Henrietta Leavitt.

Edwin Hubble

Edwin Hubble

Hubble was able to identify Cepheid emissions from the Andromeda nebula. He found the brightness of the Cepheids in Andromeda implied that it was almost a million light-years away. This was five to ten times greater than the estimated diameter of our Milky Way and therefore Andromeda cannot be part of the Milky Way. It must be a separate galaxy. And if Andromeda is a separate galaxy then so must be all of the other nebulae.

In 1927 Hubble attended the conference in the Netherlands of the International Astronomical Union. There was much concern over the red-shifts being evidence of an expanding universe. Suppose there are two points on a sphere of radius R separated by an angle θ. The distance between the points is then S=Rθ. The rate of change of S is given by (dS/dt)=(dR/dt)θ. The rate of change of S is perceived as the velocity v at which the other point is receding so v=(dR/dt)θ. But θ is equal to S/R so

v = (dR/dt)S/R = [(dR/dt)/R]S

This relationship is what is known as Hubble's Law and [(dR/dt)/R] is known as Hubble's constant and denoted as H. The dimension of H is inverse time and its reciprocal is about 14.4 billion years, somewhat larger than the estimated age of the Universe as 13.8 billion years.

If 1/H is 14.4×10⁹ years then H is about 7×10⁻¹¹ per year or 2.2×10⁻¹⁸ per second.

The redshift Δλ is proportional to v, say Δλ=kv, so

Δλ = k[(dR/dt)/R]S = kH
or, equivalently
S = Δλ/ kH

Hubble set out to find such a relationship. He collaborated with Milton Humason on measuring the distance and red-shift for a collection of nebulae (galaxies). Hubble measured the cycle period and luminosities of the Cepheids in the galaxies and Humason measured their spectral red-shifts. Hubble estimated the distances for the 24 nebulae (galaxies) for which Slipher measured the red-shifts. Humason remeasured the red-shifts for four nebulae. In addition there were 22 other galaxies for which the red-shifts were obtained by Humason. Humason used a set of nebulae fainter than the ones whose spectral shifts were measured by Slipher and whose distances were estimated by Hubble. Hubble was not able to get estimates of the distances for the 22 nebulae whose red-shifts were measured by Humason.

Here are data from Hubble's article

The data on velocities (computed from the spectral-shifts) and distances used by Hubble are obvious rough estimates but they show a reasonably strong correlation coefficient of 0.790. The t-ratio for the regression of velocity on distance (with no constant term) is 10, strongly indicating that the relationship between galactic velocities and distance is not due to chance.

Despite all the limitations in their measurements Hubble and Humason had found confirmation of the de Sitter effect.

Hubble published in1929 his results in the Proceedings of the National Academy of Sciences in an article entitled "A Relation Between Distance and Radial Velocity Among Extragalactic Nebulae." Humason published his results in the same journal.

The de Sitter effect was now established and accepted by the physics community.

The final step was to note that an expanding universe implied that it arose from some small point in the past. Lemaître made this point in his article in the British scientific journal Nature. The title of his 1931 article was "The Beginning of the World from the Point of View of the Quantum Theory." This was the beginning of the Big Bang Theory although Lemaître did not use that term. He instead called it the Primordial Egg. Big Bang is an unfortunate term because it implies an explosion within the Universe whereas what was involved was an inflation of the spatial framework of the Universe.

The modern view of the Universe is of a structure of three spatial dimensions curved in a fourth dimension with time as an additional dimension that may or may not be curved. The radius of the four dimensional sphere of the Universe has been estimated to be 91 billion light-years.