On Gravitation and Quanta∗

We show that a gravitating mass M in thermal equilibrium behaves statistically like a system of some number N of harmonic oscillators whose zero-point energy ǫ 2 depends on N universally in such a way that Nǫ2 ∼ hc G , where G is the Newton constant, c is the velocity of light in vacuum, and h is the Planck constant. The large number N is also the number of weakly interacting gravitational atoms [2] which are the constituents of a black hole. The sum over all oscillators of the squares of zero-point energies is fixed and independent of the number of those oscillators. It is well known that the classical gravitating systems behave in the way foreign to statistical quantum mechanics. The negative specific heat of those systems and the phenomenon of a gravitational collapse are different facets of the same reality. The enigmatic Bekenstein entropy of black holes was not yet derived on the basis of microscopic theory. Our work may be considered a first step in direction of presenting such a basis [1-4]. The problem with all present approaches to this problem has been the silent assumption that the total entropy of gravitating systems (black holes) must be given by the Bekenstein formula Sbh = 4kπM 2, where the mass M is in Planck units. The postulate of gravitational constituents (gravitational atoms) and gravitational oscillators (quanta) leads to Bekenstein formula only after a part of mass-energy fluctuations is neglected [2]. The most unusual character of the gravitational mass-energy oscillators (quanta) is that they somehow manage, via the quadratic sum rule defining the Newton constant G, ∑ i ǫi 2 = b 5 G , to reduce their zero-point energy when the number N of gravitational atoms grows [2]. This also means that a cold large gravitational mass M ∼ μ √ N consists of N constituents. The formula M2 = μ 2 2πN , μ 2 = hc G , was derived long time ago by the present author. The physical meaning of the ‘phenomenological’ entropy of Bekenstein is that it is the measure of the number N of constituents making up a very ‘cold’ large body. The zero-point energy ǫi of gravitational quanta for a very large ‘cold’ mass is of an order of the Hawking thermal energy of quanta, ǫi ∼ μ 2 (4π)M . The more massive is a gravitating mass the softer are the gravitational quanta. The number N of constituent gravitational atoms of a given spin-zero quantum Schwarzschild black hole determine the energy of quasi-thermal quanta and the Bekenstein-Hawking entropy, kTbh ∼ μ √N , Sbh ∼ kN , but it is valid only in the particular limit when the interference terms are neglected [2]. Otherwise, as usual with oscillators, there are two sources of statistical fluctuations of mass-energy corresponding to the corpuscular and wave aspect of quanta [2]. We calculate the energy fluctuations This is a slightly extended version (with a footnote added on December 19, 1997) of an Abstract submitted in March 1997 to the Eight Marcel Grossmann Meeting, Jerusalem, June 22-27, 1997. E-mail address: [email protected]