Starting from the early seventies, a precise analogy has been
found between classical thermodynamics and the general relativistic laws
governing black hole dynamics. In fact, as for ordinary thermodynamics,
it is possible to write the "four laws of black hole thermodynamics", where
black hole mass plays the role of energy, while the *superficial gravity*
and the area of the event horizon take the place of temperature and entropy.

Following the 1975 theoretical discovery of *Hawking radiation*
(consisting in the emission, by black holes, of a thermal spectrum of quantum
particles at a temperature proportional to the superficial gravity of the
event horizon), black hole thermodynamics has come to be regarded as much
more than a simple analogy. A more accurate statement of the Hawking effect,
actually, is the following: when one quantizes a classical field theory
on the background of a black hole space-time, one finds that the field's
*vacuum state,* as defined according to measurements performed at "early times",
appears to the "late" observers in the external region of the black hole
as a thermal bath of particles outgoing from the black hole's event horizon.

Quite surprisingly, particle production cannot be ascribed
to a direct interaction between the quantum field and the gravitational
one: rather, it appears as a *by-product of the different definition
of "particle"* by early and late observers. This fact has led us to
ponder what we called "Einstein's conflictual heredity": on the one hand,
the affirmation of Poincaré group as the *global symmetry group
* of space-time has been seminal to the great theoretical synthesis of the first
half of this century, begun with the full acknowledgement of Maxwell's
electro-magnetism as a special-relativistic theory, and beautifully climaxed
with quantum field theory. The concepts and interpretative paradigms of
these theories refer naturally to the privileged class of special-relativistic
inertial observers.

On the other hand, general relativity *equivalence principle*
does warrant Lorentz group as a simmetry group, *but only locally*.
This becomes a problem when one tries to generalise to curved space-time
geometries the concepts and paradigms inherited from special-relativistic
theories, especially when these are based on the *global* symmetries
of Minkowski space-time.

This is the case, in Maxwell's theory, of the concept of "radiation",
and, in quantum field theory, of the notion of "particle". It is interesting
to notice that a qualitative understanding of the difficulty can be gained
already in the simple case of a constant gravitational field, which can
be *simulated* in special relativity using accelerated references
frames (*apparent fields*). In classical electro-magnetism, one finds
that the apparently sound and well-established notion of *radiation*
of an accelerated particle *is not invariant* with respect to transformations
between inertial and accelerated reference frames.

For special-relativistic quantum field theory, a similar phenomenon
is found in *Unruh effect, *namely the measurement, by a uniformly
accelerated observer, of a thermal radiation in the vacuum state of the
quantum field: thus the usual *mathematical* definition of "particle"
(based on Fourier decomposition of field operators) cannot be translated
to accelerated reference frames. By means of a series of original "theoretical
experiments", based on a simple model of semiclassical *detector*
(a quantum point-like system moving on a given world-line in Minkowski
space-time), we have investigated the meaning that can be given *operatively*
to the concept of "particle"; we have found it in the discrete energy exchanges
between the *detectors* and the field: only in flat space-time does
this notion adhere to the usual mathematical definition. We have then extended
Unruh effect to a more general class of accelerated observers (i.e., constant
curvature world-lines), obtaining the *quasi-thermal* observed spectra.

More generally, quantum field theory on curved space-times
removes the last doubts as to the arbitrariness of a "particle" concept
in space-times other than Minkowski. It also provides a mathematical scheme
for a more convincing formulation and understanding of Hawking effect.
To further enlighten this question from our *operative* viewpoint,
we have examined the measurements that a *detector orbiting a black hole*
would report.

A weakness in black hole thermodynamics is still the absence
of a satisfactory *statistical interpretation* of their "entropy"
in terms of the dynamics of the matter fields on black hole geometries,
or, even better, of the *quantum gravitational dynamics* of the black
hole space-time itself. We end our work with a review of some of the most
interesting proposals towards this program.

It is amazing to think that most of the physics we have examined
is, in some sense, *physics of the void*: Hawking and Unruh effects
are measurements on the *vacuum state* of the quantum field (which
reveals unexpected dynamical properties); the absence of "particles" and
"radiation" (which, by common sense, we would use to define *void*)
proves elusive and badly defined; finally, black holes are solutions of
the *vacuum *Einstein equations: they are *pure geometry*, yet
they possess some attributes that we feel entitled to call *mass*,
*temperature*, and *entropy*!

- Corso di Laurea in Fisica, 1992/1996, Università di Parma, Facoltà di Scienze Matematiche, Fisiche e Naturali.
- Advisor: Prof.
**Massimo Pauri**; co-advisors: Prof.**Antonio Scotti**, Dott.**Roberto de Pietri**. Defended on**April 28, 1997**. - Cite:
*Mutamenti nella Nozione di Vuoto: Elettrodinamica dei Sistemi Accelerati, Radiazione di Unruh--Hawking e Termodinamica dei Buchi Neri*

M. Vallisneri

thesis, laurea in physics (University of Parma, Italy, 1997) 223 pp. [+]

*Illustration by Kola Krauze. The physics enthusiast Beatrice appears in several Galilean dialogues throughout the thesis.*

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© M. Vallisneri 2014 — last modified on 2012/10/20