General Relativity and the Universe

The stars in the night sky must have always fascinated human beings. We can only guess what the people of ancient times speculated about when they saw the stars return every night to the same spots in the sky. We know of Greek philosophers who proposed a heliocentric astronomical model with the Sun in the middle and the planets circulating around it as early as the 3rd century B.C., but it was Nicolaus Copernicus, who in the 16th century developed the first modern version of a model. It took Galileo Galilei’s genius in the beginning of the next century to really observe and understand the underlying facts, building one of the first telescopes for astronomy and hence laying the ground for modern astronomy. For the next three hundred years, astronomers collected evermore impressive tables of observations of the visible stars. In the Copernican system, the stars were assumed to be fixed to a distant sphere and nothing in the observations indicated anything to the contrary. In 1718, Edmund Halley discovered that stars actually could move in the sky, but it was believed that this happened in a static, fixed universe. Throughout the 18th and 19th century, the study of celestial bodies was placed on an ever-firmer footing with the famous laws of Kepler and Newton.

In November 1915, Albert Einstein (Nobel Prize in Physics 1921) presented his theory of gravity, which he nicknamed General Relativity (GR), an extension of his theory of special relativity. This was one of the greatest achievements in the history of science, a modern milestone. It was based on the Equivalence Principle, which states that the gravitational mass of a body is the same as its inertial mass. You cannot distinguish gravity from acceleration! Einstein had already checked that this could explain the precession of the perihelion of Mercury, a problem of Newtonian mechanics. The new insight was that gravity is really geometric in nature and that the curving of space and time, spacetime, makes bodies move as if they were affected by a force. The crucial physical parameters are the metric of spacetime, a matrix that allows us to compute infinitesimal distances (actually infinitesimal line elements - or proper times in the language of special relativity.) It became immediately clear that Einstein’s theory could be applied to cosmological situations, and Karl Schwarzschild very soon found the general solution for the metric around a massive body such as the Sun or a star.

In 1917, Einstein applied the GR equations to the entire Universe, making the implicit assumption that the Universe is homogenous; if we consider cosmological scales large enough such that local clusters of matter are evened out. He argued that this assumption fit well with his theory and he was not bothered by the fact that the observations at the time did not really substantiate his conjecture. Remarkably, the solutions of the equations indicated that the Universe could not be stable. This was contrary to all the thinking of the time and bothered Einstein. He soon found a solution, however. His theory of 1915 was not the most general one consistent with the Equivalence Principle. He could also introduce a cosmological constant, a constant energy density component of the Universe. With this Einstein could balance the Universe to make it static.

In the beginning of the 1920s, the Russian mathematician and physicist Alexander Friedmann studied the problem of the dynamics of the Universe using essentially the same assumptions as Einstein, and found in 1922 that Einstein’s steady state solution was really unstable. Any small perturbation would make the Universe non-static. At first Einstein did not believe Friedmann’s results and submitted his criticism to Zeitschrift für Physik, where Friedmann’s paper had been published. However, a year later Einstein found that he had made a mistake and submitted a new letter to the journal acknowledging this fact. Even so, Einstein did not like the concept of an expanding Universe and is said to have found the idea “abominable”. In 1924, Friedmann presented his full equations, but after he died in 1925 his work remained essentially neglected or unknown, even though it had been published in a prestigious journal. We have to remember that a true revolution was going on in physics during these years with the advent of the new quantum mechanics, and most physicists were busy with this process. In 1927, the Belgian priest and physicist Georges Lemaître working independently from Friedmann performed similar calculations based on GR and arrived at the same results. Unfortunately, Lemaître’s paper was published in a local Belgian journal and again the results did not spread far, even though Einstein knew of them and discussed them with Lemaître.

In the beginning of the 20th century it was generally believed that the entire Universe only consisted of our galaxy, the Milky Way. The many nebulae which had been found in the sky were thought to be merely gas clouds in distant parts of the Milky Way. In 1912, Vesto Slipher, while working at the Lowell Observatory, pioneered measurements of the shifts towards red of the light from the brightest of these spiral nebulae. The redshift of an object depends on its velocity radially away from us, and Slipher found that the nebulae seemed to move faster than the Milky Way escape velocity.

In the following years, the nature of the spiral nebulae was intensely debated. Could there be more than one galaxy? This question was finally settled in the 1920s with Edwin Hubble as a key figure. Using the new 100-inch telescope at Mt Wilson, Hubble was able to resolve individual stars in the Andromeda nebula and some other spiral nebulae, discovering that some of these stars were Cepheids, dimming and brightening with a regular period.

The Cepheids are pulsating giants with a characteristic relation between luminosity and the time interval between peaks in brightness, discovered by the American astronomer Henrietta Leavitt in 1912. This luminosity-period relation, calibrated with nearby Cepheids whose distances are known from parallax measurements, allows the determination of a Cepheid’s true luminosity from its time variation — and hence its distance (within ~10%) from the inverse square law.

Hubble used Leavitt’s relation to estimate the distance to the spiral nebulae, concluding that they were much too distant to be part of the Milky Way and hence must be galaxies of their own. Combining his own measurements and those of other astronomers he was able to plot the distances to 46 galaxies and found a rough proportionality of an object’s distance with its redshift. In 1929, he published what is today known as ‘Hubble’s law’: a galaxy’s distance is proportional to its radial recession velocity.

Even though Hubble’s data were quite rough and not as precise as the modern ones, the law became generally accepted, and Einstein had to admit that the Universe is indeed expanding. It is said, that he called the introduction of the cosmological constant his “greatest mistake”. From this time on, the importance of the cosmological constant faded, although it reappeared from time to time.

It should be noted for the historic records that Lemaître in his 1927 paper correctly derived the equations for an expanding Universe obtaining a relation similar to Hubble’s and found essentially the same proportionality constant (the “Hubble constant”) as Hubble did two years later. After Hubble’s result had spread, Arthur Eddington had Lemaître’s paper translated into English in 1931, without the sections about Hubble’s law. In a reply to Eddington, Lemaître also pointed out a logical consequence of an expanding Universe: The Universe must have existed for a finite time only, and must have emerged from an initial single quantum (in his words). In this sense, he paved the way for the concept of the Big Bang (a name coined much later by Fred Hoyle). It should also be noted that Carl Wirtz in 1924 and Knut Lundmark in 1925 had found that nebulae farther away recede faster than closer ones.

Hubble’s and others’ results from 1926 to 1934, even though not very precise, were encouraging indications of a homogeneous Universe and most scientists were quick to accept the notion. The concept of a homogeneous and isotropic Universe is called the Cosmological Principle. This goes back to Copernicus, who stated that the Earth is in no special, favoured place in the Universe. In modern language it is assumed that the Universe looks the same on cosmological scales to all observers, independent of their location and independent of in which direction they look in. The assumption of the Cosmological Principle was inherent in the work of Friedmann and Lemaître but virtually unknown in large parts of the scientific society. Thanks to the work of Howard Robertson in 1935-1936 and Arthur Walker in 1936 it became well known.

Robertson and Walker constructed the general metric of spacetime consistent with the Cosmological Principle and showed that it was not tied specifically to Einstein’s equations, as had been assumed by Friedmann and Lemaître. Since the 1930s, the evidence for the validity of the Cosmological Principle has grown stronger and stronger, and with the 1964 discovery of the CMB radiation by Arno Penzias and Robert Wilson (Nobel Prize in Physics 1978), the question was finally settled. The recent observations of the CMB show that the largest temperature anisotropies (on the order of 10^–3) arise due to the motion of the Milky Way through space. Subtracting this dipole component, the residual anisotropies are a hundred times smaller.

(http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/advanced.html)