70 3 Rotating Black Holes and Thermodynamics

The proportionality factor κ(x) is a space-time function which can be proved to be constant on the horizon and there precisely equal to the expression introduced in (3.6.3). It is named surface gravity since it can be shown to be the limiting force that must be exerted at infinity to hold a unit test mass in place when approaching the

horizon. This	interpretation becomes obvious in the Schwarzschild limit (J						→	0).
		1
In this case we have κ =			and, by reinstalling the physical constants, we obtain
		2m
			κ	GM		(3.6.27)
				Rs2
which is indeed the Newton force on the horizon (r = Rs = 2 GMc2						).

By using quantum field theory in the background metric of a black-hole and carefully dealing with the creation of particle-antiparticle pairs near the horizon, Hawking found that all black-holes (including the Schwarzschild one) actually emit a faint thermal radiation whose temperature is the inverse of κ, evaluated at the horizon.

This intriguing semiclassical phenomenon gave the final evidence that the thermodynamical interpretation of the laws of black hole dynamics is quite sound and that a statistical interpretation of the Bekenstein Hawking entropy is compelling. The first example of such an interpretation was obtained by Strominger in 1996 in the context of string theory. We will not address such a topic in this book but we shall come back, in Chap. 9, to the structure of the black hole entropy, while discussing the classical black hole solutions of supergravity. There the entropy acquires a group theoretical interpretation that is also the main clue to its statistical interpretation in terms of string microstates.

References

1.Hawking, S.W.: Gravitational radiation from colliding black holes. Phys. Rev. Lett. 26, 1344– 1346 (1971)

2.Bekenstein, J.D.: Generalized second law of thermodynamics in black hole physics. Phys. Rev. D 9, 3292–3300 (1974)

3.Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199–220 (1975)

Chapter 4

Cosmology: A Historical Outline from Kant to WMAP and PLANCK

Vos calculs sont corrects, mais votre physique est abominable

Albert Einstein

4.1 Historical Introduction to Modern Cosmology

From most remote antiquity all humans have looked at the sky, have observed the stars and tried to obtain from them some answers to the most profound and intriguing questions that challenged their minds. Immanuel Kant (see Fig. 4.1), who was the first to understand the existence of other galaxies beyond the Milky Way, disposed that the inscription on his grave-stone should specify the most relevant targets of his life-long philosophical meditation, namely the star globe above our head and the moral law within it.

Essentially no human civilization, at any time of history and in whatsoever corner of the Earth was deprived of some kind of cosmology, namely of some general overview of the sky, of its contents, of its order and structure. Yet, until very recent times, all theories of the world have been very far from providing some realistic description of the existing cosmos, since all of them underestimated by several orders of magnitude the actual dimensions of the Universe and our distance from the closest stars. General Relativity came just at the time when the true ladder of cosmic distances started to be unveiled. Modern cosmology developed from these two seeds: a new geometrical theory of gravity, which is the leading force at very large scales, and the discovery that the Universe is indeed very large, its constituents being separated from each other by distances of a previously never suspected magnitude.

4.2 The Universe Is a Dynamical System

Einstein once said: The most incomprehensible thing about the Universe is that it is comprehensible.

Indeed his theory, General Relativity, proved to be the conceptual framework where, for the first time in human history, questions about the large scale structure of the visible universe could be formulated in an algorithmic way obtaining answers and predictions. To a certain extent such answers surprised and disappointed Einstein, whose fundamental philosophical attitude is revealed by his frequent ex-

P.G. Frè, Gravity, a Geometrical Course, DOI 10.1007/978-94-007-5443-0_4,	71
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