ar X iv : g r - qc / 9 90 80 20 v 1 6 A ug 1 99 9 Gravitational Thermodynamics of Space - time Foam in One - loop Approximation

We show from one-loop quantum gravity and statistical thermodynamics that the thermodynamics of quantum foam in flat space-time and Schwarzschild space-time is exactly the same as that of Hawking-Unruh radiation in thermal equilibrium. This means we show unambiguously that Hawking-Unruh thermal radiation should contain thermal gravitons or the contribution of quantum space-time foam. As a by-product, we give also the quantum gravity correction in one-loop approximation to the classical black hole thermodynamics. PACS number(s): 04.60.Gw, 04.60.Dy, 05.30.Ch Typeset using REVTEX E-mail address: [email protected] E-mail address: [email protected] 1 The space-time foam-like structure (FLS) was firstly proposed by J. A. Wheeler about forty years ago[1]. He argued that the space-time may have a multiple connected non-trivial topological structure in Planck scale, though it seems smooth and simply connected in the large. The possible influence of FLS on field theory and the thermodynamical properties of FLS itself have been discussed by many authors[2-9]. In this paper, we would like to discuss the thermodynamical properties of FLS only from one-loop quantum gravity and statistical thermodynamics, so the result we get may be much more reliable. If time in Euclidean quantum gravity has an imaginary period of iβ = i(1/T ), (henceforth, we take h̄ = c = G = k = 1) then the partition function Z = ∑ n exp(−βEn) (1) of canonical ensemble can be rewritten as an Euclidean path integral Z = ∫ D(g, φ) exp(−Î(g, φ)), (2) where Î(g, φ) is the Euclideanized action of gravity, g, and matter field, φ, and En is the n-th energy eigenvalue of certain differential field operator on its eigenstate vector |g, φ〉n. For the pure quantum gravity case, we put φ = 0 and gab = g (0) ab + ḡab, (3) where ḡab is the metric fluctuation of the background metric g (0) ab , then we can expand the action in Taylor series about the background field g as Î(g) = Î(g) + Î2(ḡ) + [higher-order terms], (4) where Î2(ḡ) is the well known one-loop term of the Euclidean gravitational action. In oneloop approximation, the logarithm of partition function, Z, reads lnZ = −Î(g) + ln ∫ D(ḡ) exp(−Î2(ḡ)). (5) As Î(g) is equal to the Gibbons-Hawking’s surface term for vacuum Einstein gravity without cosmological term, the contributions to lnZ come from the surface term and Î2(ḡ) in one-loop approximation. 2 Now from quantum gravity and statistical thermodynamics we try to study the gravitational thermodynamics of the one-loop quantum gravity for flat space-time and Schwarzschild space-time background. Let us consider the gravitational field inside volume V and with an imaginary time period iβ. Hawking showed exactly that lnZ in one-loop approximation with flat space-time background reads[10] lnZ = 4πr 0T 3 135 = π 45 βV (6) for a system at temperature T = β, contained in a spherical box of radius r0, where the Casimir effect of the finite size of volume V is neglected. (Note that the factor 4π in Eq.(15.99) of Hawking’s original paper [10] should be corrected as 4π in Eq.(6). ) Hawking argued that Eq.(6) is just the contribution of the thermal gravitons to the partition function. However, in our opinion, if FLS is created from metric fluctuation, an equivalent interpretation of Eq.(6) as the contribution of FLS can also be given. Let Pn be the probability of FLS in volume V in the n-th energy eigenstate, then from