Quantum Gravity

6 WHY QUANTUM GRAVITY?number L and Faraday’s number F = eL. Withe, G, andc, one can constructthe same fundamental units as with ,G,andc (since the fine-structure constantis α = e 2 /c ≈ 1/137); therefore, Stoney’s units differ from Planck’s units byfactors of √ α. Quite generally one can argue that there are three fundamentaldimensional quantities (cf. Okun 1992).The Planck length is indeed very small. If one imagines an atom to be the sizeof the Moon’s orbit, l P would only be as small as about a tenth of the size of anucleus. Still, physicists have for some time entertained the idea that somethingdramatic happens at the Planck length, from the breakdown of the continuumto the emergence of non-trivial topology (‘space–time foam’); see, for example,Misner et al. (1973). We shall see in the course of this book how such ideas can bemade more precise in quantum gravity. Unified theories may contain an intrinsiclength scale from which l P may be deduced. In string theory, for example, this isthe string length l s . A generalized uncertainty relation shows that scales smallerthan l s have no operational significance; see Chapter 9. We also remark thatthe Einstein–Hilbert action (1.1) is of order only for TL ∼ l P t P ,wheretheintegration in the action is performed over a space–time region of extensionTL 3 .Figure 1.1 presents some of the important structures in our Universe in amass-versus-length diagram. A central role is played by the ‘fine-structure constantof gravity’,α g = Gm2 prc=(mprm P) 2≈ 5.91 × 10 −39 , (1.9)where m pr denotes the proton mass. Its smallness is responsible for the unimportanceof quantum-gravitational effects on laboratory and astrophysical scales,and for the separation between micro- and macrophysics. As can be seen fromthe diagram, important features occur for masses that contain simple powers ofα g (in terms of m pr ); cf. Rees (1995). For example, the Chandrasekhar mass M Cis given byM C ≈ α −3/2g m pr ≈ 1.8M ⊙ . (1.10)(A more precise value is M C ≈ 1.44M ⊙ .) It gives the upper limit for the massof a white dwarf and sets the scale for stellar masses. The minimum stellarlife times contain α −3/2g t P as the important factor. It is also interesting to notethat the geometric mean of the Planck length and the size of the observablepart of the Universe is about 0.1 mm—a scale of everyday’s life. It is an openquestion whether fundamental theories such as quantum gravity can provide anexplanation for such values, for example, for the ratio m pr /m P ,ornot.Tegmarket al. (2006) give a list of 31 dimensionless parameters in particle physics andcosmology that demand a fundamental explanation. We shall come back to thisin Chapter 10.As far as the relationship between quantum theory and the gravitational fieldis concerned, one can distinguish between different levels. The first, lowest level