Chapter 6

Stellar Equilibrium: Newton’s Theory, General

Relativity, Quantum Mechanics

E quindi uscimmo a riveder le stelle

Dante Alighieri

6.1 Introduction and Historical Outline

Einstein used to say that the left hand side of his field equations is carved into solid marble, while the right hand side is just only scribbled on perishable wood. By this he meant to emphasize the profound difference between the symmetric tensor Gμν , which is constructed out of geometrical quantities derived from first principles, and the stress-energy tensor Tμν , that encodes the matter content of space-time and looks like a black-box hiding our ignorance about the intimate structure of the latter. Einstein’s dream, pursued by the efforts of all his followers in contemporary theoretical research, is that of bringing the right hand side of the field equations to the left, namely of providing a unification of the metric tensor gμν with all the other fields that describe both Matter and the other interactions of Nature. Unified Theories like Supergravity and the microscopic Superstring Theory from which it derives, are presently the most favorite candidates to fulfill such a task.

Notwithstanding the undoubtable logical need of unifying gravity with the other interactions and constructing microscopic quantum theories that encode all fields, the major successes of General Relativity and most of its, by now well established, experimental confirmations are based on the marble-wood dualism that bothered its inventor. In this scheme one takes a drastically simplified description of Matter, by considering it to be a perfect fluid that has no viscosity and is completely described in terms of three local fields:

•the energy density scalar field ε(x) ≡ c2ρ(x) that expresses the amount of total energy1 stored in one infinitesimally small tri-volume around the space-time point x,

•the pressure scalar field p(x) that gives the pressure of the fluid at the same spacetime point x,

1By total energy we mean all kinds of energy, thermal, chemical, radiative and so on, including also the rest-mass energy.

Fig. 6.1 Sir Arthur Eddington, (1882–1944) was one of the most prominent Astrophycists of his time. During the solar eclipsis of 1919 he was the first to measure the deflection of light rays predicted by General Relativity and he was responsible for spreading the knowledge of Einstein Theory within the English speaking scientific community

enflate the bubble with the gravitational self-attraction of that large mass of fluid that tends to concentrate it into smaller volumes. Before the discovery of nuclear forces and nuclear reactions, what was the energy source powering the internal core of the Sun and of the other main sequence stars was not clear, yet as early as 1870 the American Physicist Jonathan Homer Lane2 was able to turn the integral-differential equation (6.1.1)–(6.1.2) into a manageable second order non-linear differential equation, the Lane-Emden equation [1], by using a class of equations of states (see (6.4.42)) which are still valid models for the mean behavior of stellar matter.

With the discovery of 1914 it became clear that there were states of matter, inside compact stars, that were very different from those so far known to Mankind and soon the discovery of the Quantum World provided the clue to interpret them. In his 1926 book on Internal Constitution of the Stars, Eddington (see Fig. 6.1) concluded that Prof. Adams has killed two birds with one stone: he has carried out a new test of Einstein General Theory of Relativity and he has confirmed our suspicion that matter 2000 times denser than platinum is not only possible, but is actually present in our Universe.

White Dwarfs, as stars of the type of Sirius B came to be known, were a puzzle since no classical mechanism could be imagined generating the pressure necessary to counterbalance the gravitational self-attraction of such dense objects. The same year 1926 that witnessed these astrophysical considerations saw also the birth of Fermi-Dirac statistics by means of the separate papers on quantum ideal gases published by Dirac and Fermi [3, 4]. Almost instantaneously, R.H. Fowler, in the very same 1926 year advanced the hypothesis that the electron degeneracy pressure could

2Jonathan Homer Lane (1819–1880) was an American physicist who worked most of his life at the US Patent Office. His studies on the Thermodynamics of the Sun were published in 1870 and were extended by Robert Emden in 1907. This latter (1862–1940) was of Swiss nationality but served as Professor of Physics and Metereology at the University of Munich in Germany.

Fig. 6.2 Subrahmanyan Chandrasekhar, (1910–1995). Born in Lahore (India) Subrahmanyan was the nephew of a Nobel Laureate and was to receive on his turn the Nobel Prize in 1983, sharing it with William Alfred Fowler, not to be confused with the British Physicist, Sir Ralph Howard Fowler of which Chandrasekhar was a student. R.H. Fowler was the first to imagine that white dwarfs could be sustained against gravitational collapse by the degeneracy pressure of an electron gas. Chandrasekhar developed this idea in 1930 and came to discover the mass limit which goes under his name which was also the motivation for his Nobel Laurea. Subrahmanyan obtained his doctorate from the University of Cambridge, where he came under the influence of Eddington, worked in Copenhagen in Bohr’s group and in 1937 was recruited by the University of Chicago where he served as professor until his death in 1995 at the age of 85. His classical studies on stellar structures were followed by exhaustive investigations on the mathematical theory of black-holes and in the very last years of his life of gravitational waves. Of very gentle character and deep culture, Subrahmanyan Chandrasekhar was affectionately named Chandra by all of his student, colleagues and collaborators

be the explanation for the missing mechanism sustaining white dwarfs against gravitational collapse.

It was the Indian born American physicist Subrahmanyan Chandrasekhar (see Fig. 6.2) who building on Fowler’s idea made the first accurate models of White Dwarfs in 1930 [9–11]. Working on this problem Chandra made the momentous discovery that white dwarfs had an upper mass limit that might be understood in terms of first principles and estimated in terms of fundamental physical constants.

Chandra’s argument could later be extended in a completely analogous way to neutron stars, whose existence was conjectured just two years later by Baade and Zwicky. They proposed that the formation of such even more compact astronomical objects was the mechanism lying behind supernova explosions. Neutron stars in the form of pulsars were actually discovered in 1964 by Hewish [8], who won the Nobel Prize for that.

Both White Dwarfs and Neutron Stars are a spectacular manifestation of Quantum Physics on large scales. Without quantum mechanics these objects could not be interpreted in any way. Yet also classical General Relativity imposes its own limit

Fig. 6.3 Robert Oppenheimer, (1904–1967) is the founder of the American School of Theoretical Physics. He is best known as Director of the Manhattan Project that constructed the Atomic Bomb and for his very strong opinions against the development of cold war with the Soviet Union which led him to oppose the development of the Hydrogen Bomb in the USA. Discriminated during the McCarthy period, Oppenheimer was rehabilitated under the Kennedy and Johnson’s presidencies. Kennedy honored him with the Enrico Fermi Award. Oppenheimer’s notable achievements in physics include the Born-Oppenheimer approximation, work on electron-positron theory, quantum tunneling and in particular the first shaping of the theory of gravitational collapse into black-holes

to the mass of possible stars. This is what became apparent through the general covariant updating of the equilibrium equation (6.1.1)–(6.1.2). Such an updating was constructed in 1939 by Robert Oppenheimer (see Fig. 6.3) and George Michael Volkoff in a paper [2] which built on previous results of Tolman [5, 6]. Working with the exact field equations of Einstein theory, Oppenheimer and his collaborator derived a much more complicated integral differential relation than (6.3.62). The consequence of such a general relativistic equation is that there exists a lower limit for the ratio between the radius of a star and its Schwarzschild radius. Below such a limit the star cannot exist and necessarily collapses further: to what? To a black-hole is the answer.

So the new principles of both General Relativity and Quantum Mechanics changed drastically the picture of stellar equilibrium which had been developed within Newton’s Theory. This came just at the time when new astronomical objects were discovered that could not have been interpreted without this new theoretical understanding.

In the course of the XXth century a completely new picture of the Universe emerged which is evolutive contrary to the static ones cheered by all Thinkers and Philosophers. Stars evolve and end up their life in very unusual states of matter like those of white dwarfs, neutron stars or black-holes. The Universe itself evolves as we plan to discuss in later chapters. The present one focuses on stellar equilib-