(ed.). Gravitational waves (IOP, 2001)(422s).

and the S-frame action (16.20) becomes∫S(g,φ)=−Kinematics: shrinking horizons 291d d+1 x ad e −φN [2dFH − 2d Ḣ − d(d + 1)H 2 + ˙φ 2 ]. (16.24)Modulo a total derivative, we can eliminate the first two terms, and the actiontakes the quadratic form∫S(g,φ)=− d d+1 x ad e −φN [ ˙φ 2 − 2dH ˙φ + d(d − 1)H 2 ]. (16.25)where, as expected, N plays the role of a Lagrange multiplier (no kinetic term inthe action).In the E-frame the variables are Ñ, ã, ˜φ, and the action (16.21), afterintegration by parts, takes the canonical form∫ [S( ˜g, ˜φ) =− d d+1 x ãd − 1 ]˙˜φ 2 + d(d − 1)H 2 . (16.26)Ñ 2A quick comparison with equation (16.25) finally leads to the field redefinition(no coordinate transformation!) connecting the Einstein and string frame:√2ã = ae −φ/(d−1) , Ñ = Ne −φ/(d−1) , ˜φ = φd − 1 . (16.27)In fact, the above transformation gives˜H = H −˙φ(16.28)d − 1and, when inserted into equation (16.26), exactly reproduces the S-frame action(16.25).Consider now a superinflationary, pre-big bang solution obtained in the S-frame, for instance the isotropic, d-dimensional vacuum solutiona = (−t) −1/√d , e φ = (−t) −(√ d+1) , t < 0, t → 0 − (16.29)(see appendix B, equations (16.160) and (16.161)), and look for the correspondingE-frame solution. The above solution is valid in the synchronous gauge, N = 1,and if we choose, for instance, the synchronous gauge also in the E-frame, we canfix Ñ by the condition:which defines the E-frame cosmic time, ˜t, as:Ñ dt ≡ Ne −φ/(d−1) dt = d˜t, (16.30)d˜t = e −φ/(d−1) dt. (16.31)