Astroparticle Physics

192 9 The Early UniverseSchwarzschild radiusevent horizonTo see how the Planck scale arises, consider the Schwarzschildradius for a mass m (see Problem 5 in this chapter),R S = 2mGc 2 , (9.1)where for the moment factors of c and ¯h will be explicitlyinserted. The distance R S gives the event horizon of a blackhole. It represents the distance at which the effects of spacetimecurvature due to the mass m become significant.Now consider the Compton wavelength of a particle ofmass m,λ C = h mc . (9.2)Planck massPlanck energyPlanck lengthPlanck timeThis represents the distance at which quantum effects becomeimportant. The Planck scale is thus defined by the conditionλ C /2π = R S /2, i.e.,¯hmc = mGc 2 . (9.3)Solving for the Planck mass gives√¯hcm Pl =G ≈ 2.2 × 10−5 g , (9.4)or the mass of a water droplet about 1/3 mm in diameter.The rest-mass energy of m Pl is the Planck energy,√¯hcE Pl =5G ≈ 1.22 × 1019 GeV , (9.5)which is about 2 GJ or 650 kg TNT equivalent. Using thePlanck mass in the reduced Compton wavelength, ¯h/mc,gives the Planck length,√¯hGl Pl =c 3 ≈ 1.6 × 10−35 m . (9.6)The time that it takes light to travel l Pl is the Planck time,t Pl = l √Pl ¯hGc = c 5 ≈ 5.4 × 10−44 s . (9.7)The Planck mass, length, time, etc. are the unique quantitieswith the appropriate dimension that can be constructed fromthe fundamental constants linking quantum mechanics andrelativity: ¯h, c, andG. Since henceforth ¯h and c will be setequal to one, and one has